Domain Decomposition Techniques for Boundary Elements : Application to Fluid Flow (Advances in Boundary Elements)

Domain Decomposition Techniques for Boundary Elements Application to Fluid Flow WITPRESS WIT Press publishes leading b...

Author: V. Popov | H. Power | L. Skerget | V. Popov | H. Power | L. Skerget

59 downloads 794 Views 9MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Domain Decomposition Techniques for Boundary Elements Application to Fluid Flow

WITPRESS WIT Press publishes leading books in Science and Technology. Visit our website for the current list of titles. www.witpress.com

WITeLibrary Home of the Transactions of the Wessex Institute, the WIT electronic-library provides the international scientific community with immediate and permanent access to individual papers presented at WIT conferences. Visit the eLibrary at http://library.witpress.com

Advances in Boundary Elements Series Objectives The continuing interest in the application of the Boundary Element Method has generated a series of books and numerous scientific papers. In spite of all their advantages, the need exists for a serial publication in which the most recent advances in the method are demonstrated in a more complete form. Each volume in the Series comprises authored or edited books written by leading researchers in the field. The volumes are all self contained and cover a particular topic in sufficient detail for the analyst to understand the subject. Some books report on practical applications of the technique. The Series covers topics such as: Fluid Mechanics Heat Transfer Stress Analysis Fracture Mechanics Contact Mechanics Structural Dynamics Inelastic Problems Optimization and Sensitivity Plate Bending

Acoustics Cathodic Protection Problems High Performance Computing Sparse Methods Numerical Integration Industrial Applications Basic Principles Electrical and Electromagnetic Problems Mathematical & Computational Aspects

Series Editor C.A. Brebbia Wessex Institute of Technology, Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Email: [email protected] Telephone: 44 (0) 238 029 3223 Fax: 44 (0) 238 029 2853

Associate Editors

C. Alessandri Universita di Ferrara Italy

A. El-Zafrany Cranfield University UK

D.E. Beskos University of Patras Greece

P. Fedelinski Silesian Technical University Poland

M. Bonnet Ecole Polytechnique, Palaiseau France

J.I. Frankel University of Tennessee USA

M.B. Bush The University of Western Australia Australia

L. Gaul Universitat Stuttgart Germany

C.S. Chen The University of Southern Mississippi USA

G.S. Gipson Oklahoma State University USA

A.H.-D. Cheng The University of Southern Mississippii USA D.E. Cormack University of Toronto Canada T.G. Davies Glasgow University Scotland G. De Mey Ghent State University Belgium J. Dominguez University of Seville Spain Q.H. Du Tsinghua University China

M. Golberg Las Vegas USA S. Grilli University of Rhode Island USA K. Hayami National Institute of Informatics Japan D.B. Ingham The University of Leeds UK N. Kamiya Nagoya University Japan D.L. Karabalis University of Patras Greece

A.J. Kassab University of Central Florida USA

J.T. Katsikadelis National Technical University of Athens, Greece

J.T. Katsikadelis National Technical University of Athens, Greece

P. Skerget University of Maribor Slovenia

V. Leitao Inst. Superior Tecnico Portugal

V. Sladek Slovak Academy of Sciences Slovakia

W.J. Mansur COPPE Federal University of Rio de Janeiro Brazil

S. Syngellakis University of Southampton UK

R.A. Meric Research Institute for Basic Sciences Turkey K. Onishi Ibaraki University Japan

M. Tanaka Shinshu University Japan N. Tosaka Nihon University Japan

F. Paris University of Seville Spain

T. Tran-Cong University of Southern Queensland Australia

M. Predeleanu Universite Paris VI France

W.S. Venturini University of Sao Paulo Brazil

J.J. Rencis University of Arkansas USA

J.L. Wearing University of Sheffield UK

T.J. Rudolphi Iowa State University USA

Domain Decomposition Techniques for Boundary Elements Application to Fluid Flow

V. Popov Wessex Institute of Technology, UK H. Power University of Nottingham, UK L. Škerget University of Maribor, Slovenia

Editors: V. Popov Wessex Institute of Technology, UK H. Power University of Nottingham, UK L. Škerget University of Maribor, Slovenia Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: + 44 (0) 238 029 3223; Fax: + 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico WIT Press 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN-13: ISSN: LOC:

978-1-84564-100-9 1460-1419 2007922337

No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. © WIT Press 2007 Printed in Great Britain by Cambridge Printing All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publisher.

Contents

Preface ............................................................................................................. xiii Chapter 1 Diffusion–convection problems ..................................................................... A.E. Taigbenu 1 2

3 4 5

6 7

8

Introduction ................................................................................................ Boundary element formulations................................................................. 2.1 Formulation with the ED operator ................................................... 2.2 Formulation with the elliptic diffusion–convection (EDC) operator............................................................................................. 2.3 Formulation with the transient diffusion (TD) operator .................. 2.4 Formulation with the transient diffusion–convection (TDC) operator............................................................................................. Numerical implementation......................................................................... 3.1 Dual reciprocity method................................................................... 3.2 The Green element method .............................................................. Numerical stability for homogeneous diffusion–convection ..................... Numerical examples of diffusion–convection problems............................ 5.1 Example 1......................................................................................... 5.2 Example 2......................................................................................... 5.3 Example 3......................................................................................... 5.4 Example 4......................................................................................... 5.5 Example 5......................................................................................... Burgers’ equation ....................................................................................... Numerical formulations of Burgers’ equation ........................................... 7.1 ED formulation for Burgers’ equation ............................................. 7.2 TD formulation for Burgers’ equation ............................................. 7.3 TDC formulation for Burgers’ equation .......................................... Numerical examples of Burgers’ equation................................................. 8.1 Example 1 of Burgers’ equation ......................................................

1 1 3 3 4 5 6 8 10 11 12 16 16 17 17 19 19 20 21 21 23 24 25 25

8.2 Example 2 of Burgers’ equation ...................................................... 26 8.3 Example 3 of Burgers’ equation ...................................................... 27 Conclusion.................................................................................................. 29

9

Chapter 2 Viscous compressible fluid dynamics............................................................ 33 L. Škerget & M. Hriberšek 1 2 3 4 5 6 7

8

9 10

Introduction ............................................................................................... Conservation equations ............................................................................. Linear gradient type of constitutive models .............................................. Primitive variables formulation................................................................. Velocity–vorticity formulation.................................................................. 5.1 Velocity vector equation ................................................................... 5.2 Vorticity transport equation .............................................................. Pressure equation....................................................................................... Boundary-domain integral equations ........................................................ 7.1 Preliminary comments ...................................................................... 7.2 Integral representation of flow kinematics........................................ 7.3 Integral representation of flow kinetics ............................................ 7.4 Integral representation of pressure equation ..................................... 7.5 Integral representation of heat energy kinetics ................................. Discrete models ......................................................................................... 8.1 Single domain model ........................................................................ 8.2 Subdomain model ............................................................................. 8.3 Hybrid approaches ............................................................................ Test example: differentially-heated tall enclosure .................................... Conclusions ...............................................................................................

33 34 35 36 38 38 39 41 42 42 42 49 52 53 55 55 56 58 58 66

Chapter 3 Multi-domain DRM boundary element method for the numerical simulation of non-isothermal Newtonian and non-Newtonian flow problems.................................................................................................. 69 H. Power & W.F. Florez 1 Introduction ............................................................................................... 2 Thermal convection Newtonian flow problems ........................................ 2.1 Governing equations ......................................................................... 2.2 Integral representation formulae ....................................................... 2.3 Approximation of the domain integrals ............................................ 2.4 Particular solution ............................................................................. 2.5 Discretization and numerical solution............................................... 2.6 Numerical results .............................................................................. 3 Non-isothermal non-Newtonian Stokes flow with viscous dissipation.................................................................................................. 3.1 Governing equations .........................................................................

69 72 73 74 76 78 80 82 84 85

3.2 Multi-domain integral formulation ................................................... 87 3.3 Non-isothermal Couette flow with viscous dissipation .................... 88 4 Conclusion................................................................................................. 96 Chapter 4 Modelling flow and solute transport in fractured porous media using the DRM multidomain technique ....................................................... 99 A. Peratta & V. Popov 1

2 3

4

5 6

7

Introduction ................................................................................................ 1.1 General............................................................................................ 1.2 Fractured porous media .................................................................. 1.3 Overview ........................................................................................ 1.4 The continuum approach ................................................................ 1.5 The discrete fracture model ............................................................ 1.6 Historical development of porous media modelling....................... Governing equations .................................................................................. 2.1 Flow ................................................................................................ 2.2 Transport......................................................................................... Numerical method...................................................................................... 3.1 The boundary element method ....................................................... 3.2 The dual reciprocity method........................................................... 3.3 Miscellaneous ................................................................................. Numerical implementation......................................................................... 4.1 Introduction .................................................................................... 4.2 Space discretisation ........................................................................ 4.3 Blocks and 3D subdomains ............................................................ 4.4 2D subdomains ............................................................................... 4.5 1D subdomains ............................................................................... 4.6 Flow in the porous matrix............................................................... 4.7 Flow in fractures............................................................................. 4.8 Flow in pipes .................................................................................. 4.9 Transport in the porous matrix ....................................................... 4.10 Transport in fractures...................................................................... 4.11 Transport in pipes ........................................................................... Coupling strategy ....................................................................................... 5.1 Interconnected fracture network ..................................................... 5.2 Coupling the porous matrix with the fracture network................... Numerical results ....................................................................................... 6.1 Introduction .................................................................................... 6.2 Test case: 3D block without fractures ............................................ 6.3 Test case: fractures represented as 3D or 2D entities ..................... 6.4 Case study: application to 3d fractured crystalline rocks ............... Conclusion..................................................................................................

99 99 100 100 101 101 102 103 103 106 109 109 112 115 116 116 116 117 117 118 119 120 120 122 123 124 128 128 129 132 132 133 135 138 143

Chapter 5 Parallel domain decomposition boundary element method approach for large-scale transient and steady nonlinear heat conduction................. 147 E. Divo, A.J. Kassab & K. Erhart 1 2 3 4 5

Introduction ............................................................................................... Explicit domain decomposition................................................................. Iterative solution algorithm ....................................................................... Parallel implementation on a PC cluster ................................................... Applications in heat transfer...................................................................... 5.1 3D nonlinear heat conduction......................................................... 5.2 Transient heat conduction............................................................... 6 Numerical validation and examples .......................................................... 6.1 3D nonlinear heat conduction verification examples ..................... 6.2 Transient heat conduction verification examples ........................... 7 Conclusions ...............................................................................................

147 149 151 154 157 157 160 165 165 173 182

Chapter 6 Computational implementation for 3D problems........................................ 187 B. Natalini & V. Popov 1 2 3 4 5

6

7 8 9 10

Introduction ............................................................................................... Equations and DRM formulation .............................................................. The dual reciprocity method multidomain approach................................. Schematic view of a DRM code for Poisson’s problems .......................... General aspects of DRM-MD implementation for Poisson’s problems.... 5.1 Input data ........................................................................................ 5.2 Set geometrical information ........................................................... 5.3 Set boundary conditions and NCOL............................................... 5.4 Assemble system of equations........................................................ A 3D DRM-MD implementation for Poisson’s problems using discontinuous elements.............................................................................. 6.1 Input data ........................................................................................ 6.2 Set geometrical information ........................................................... 6.3 Set boundary conditions and NCOL............................................... 6.4 Assemble system of equations........................................................ A 3D DRM-MD implementation for Poisson’s problems using continuous elements .................................................................................. A 3D DRM-MD implementation for advection–diffusion problems........ A convenient arrangement......................................................................... Examples ................................................................................................... 10.1 The interpolation functions tested in this section .......................... 10.2 A 1D Poisson problem with constant source term......................... 10.3 A 1D steady state advection–diffusion problem with non-uniform velocity field.............................................................

188 189 192 195 196 196 197 198 198 198 200 201 202 202 204 206 208 210 210 212 217

10.4 The effect of the size of the problem ............................................. 10.5 Internal DRM nodes ...................................................................... 11 Transient advection–diffusion ................................................................... 11.1 Governing equation and DRM formulation................................... 11.2 Computational implementation ..................................................... 11.3 Comparison with analytical solution ............................................. 12 Conclusions ...............................................................................................

220 225 227 227 228 230 233

Chapter 7 Iterative schemes for the solution of systems of equations arising from the DRM in multidomains .................................................................... 237 M.I. Portapila & H. Power 1 Introduction ............................................................................................... 2 Preliminary remarks on the BEM.............................................................. 2.1 The BEM and the domain integrals ................................................ 3 The dual reciprocity approximation .......................................................... 3.1 Analysis of the convection–diffusion equation using DRM .......... 3.2 Overview of DRM in multidomains............................................... 4 Approximating functions fj ........................................................................ 5 Iterative solution methods for sparse linear systems................................. 5.1 Krylov subspace methods............................................................... 5.2 CG-type methods............................................................................ 5.3 Generalized minimum residual method.......................................... 5.4 LSQR, algorithm for sparse least-squares problems ...................... 6 Numerical analysis .................................................................................... 6.1 Computer implementation .............................................................. 6.2 Performance of Krylov iterative solvers......................................... 6.3 Direct methods................................................................................ 7 Conclusions ...............................................................................................

237 240 241 242 245 247 250 252 252 253 254 258 265 269 270 290 293

This page intentionally left blank

Preface The Finite Difference Method (FDM), the Finite Element Method (FEM) and the Finite Volume Method (FVM) currently exist as the principal numerical techniques used to solve fluid flow boundary value problems. Although there are analytical solutions for some simplified cases, mainly in one dimension, in many real-world applications, where an accurate prediction of the flow and transport processes is required, the above-mentioned numerical methods have provided significant contributions. On the other hand, the Boundary Element Method (BEM), whilst being recognised as a very efficient tool in various types of engineering applications, e.g. heat transfer, electrostatics, wave scattering and propagation, stress analysis, crack propagation etc., has not been extensively used in fluid flow problems. The BEM relies on transforming the governing partial differential equations (PDEs) through Green’s second identity into an equivalent system of integral equations, which are in turn evaluated over the boundaries of the domain. Necessarily, the BEM requires the fundamental solution for the governing PDE in closed form. In the case when this is not possible, or is very difficult to be achieved, the terms of the original PDE that are not considered when the fundamental solution is derived would appear in a domain integral. For the engineering applications noted above, there exist fundamental solutions in a closed form and therefore in absence of body loads the integrals are evaluated only over the boundaries of the problem domain. This provides an opportunity for the use of the BEM relative to domain methods as only the boundary of the problem domain needs to be discretized, thereby saving time in preparation of the model input data. The advantage of the BEM is most pronounced for model domains extending to infinity, e.g. wave radiation and scattering. Using the BEM only the near-field finite surface needs to be discretized since a fundamental solution that naturally obeys the radiation condition exists, consequently avoiding the need for discretization of an infinite domain. These features, combined with the second-order accuracy of the BEM, have allowed the BEM to remain a favourite engineering analysis/designing tool for certain types of problems. When the domain integrals cannot be avoided, there are several options available. The most commonly used technique in the past was to employ domain integration. Whilst being quite accurate this approach can also be time consuming for large problems, as the integration is not just over the boundaries but also over the domain of interest. An alternative is to convert the domain integrals into integrals over the boundary, by using, for example, the dual reciprocity (Nardini and Brebbia, 1983)

(DRM) or the multipole reciprocity (MRM) methods (Nowak and Brebbia, 1989; Nowak and Partridge, 1992). The dual reciprocity method appears to be the preferred approach nowadays when transforming the domain integrals to the boundary and its application will be explained in more detail in several chapters in this book. One of the earliest works on the application of boundary integral equation methods (BIEM) to the flow in porous media is due to Liggett (1977). Other early efforts include Butterfield and Tomlin [1972] and Lafe et al. [1981], who have used the so-called zoning technique to solve anisotropic problems. When the domain of the problem is non-homogeneous, as is often the case in groundwater problems, the BEM formulation requires the use of sub-domains. Liggett and Liu (1983) mentioned that the BIEM could be used for non-homogeneous porous media with a large number of sub-regions: ‘… the BIEM could be broken into sub-regions as small as elements. If such division proves necessary, however, the finite element method would be a better choice.’ That the BEM would loose its attractiveness if the number of sub-domains becomes large was certainly the view of the majority of the BEM users in the past. This view is easy to understand considering that the most important feature associated with the BEM is that the discretization requirements are restricted to the boundary of the problem. However, for certain applications, e.g., large problems, the BEM has a high CPU demand due to integration over the boundaries and solution of fully populated system matrix. Therefore, for very large problems, the main advantage of the BEM seems to be lessened and alternative BEM formulations are typically sought, e.g., the fast multipole BEM, BEM with domain subdivision, or other alternatives. Among the first efforts on the use of BEM with a large number of sub-regions, resulting in a mesh that looks like a FEM mesh, is the work of Taigbenu (1990) who called the numerical scheme the Green element method (GEM). The GEM uses the fundamental solution of the Laplace equation and accounts for the remaining terms of the governing equation using domain integration. The advantages of this approach relative to the conventional BEM are a consequence of the fewer number of integrations over elements evaluated per source node, the global coefficient matrix is sparse, and the GEM more readily accommodates flow and medium inhomogeneities. The disadvantages are related to the complications associated with the mesh generation, common for all domain methods, and since the GEM ensures only one degree of freedom per node, the flux at the internal node is expressed in terms of the primary dependent variable by a difference expression thereby compromising the second order accuracy commonly associated with the BEM. Taigbenu (1991) demonstrated that the accuracy can be improved with the use of higher interpolating function (quadratic) while still performing the boundary and domain integrals analytically. The method has been applied to 1D and 2D heterogeneous flow problems, Helmholtz and Boussinesq equations, diffusion and advection-diffusion problems using linear and quadratic interpolating functions (Taigbenu, 1995) and also to the nonlinear unsaturated flow using linear interpolating function (Taigbenu and Onyejekwe, 1995). Škerget and collaborators developed another BEM sub-domain approach, which is known as the Boundary-Domain Integral Method (BDIM). This approach has

been used to solve convection-diffusion problems (Žagar et al., 1994) using the fundamental solution of the convection-diffusion equation with constant coefficients, resulting in domain integrals with a convective term due to the perturbation velocity field. The formulation has been tested for high values of Pe (»106) and shows good accuracy. The BDIM has also been applied to transient non-linear convection-diffusion problems (Škerget and Rek, 1995) using the parabolic diffusion fundamental solution. The BDIM has the advantage over the GEM in that the fluxes are obtained directly from the integral equations, thereby preserving the inherent accuracy of the BEM. The advantage of the GEM with respect to BDIM is that for a same domain discretization the GEM would result in a smaller system matrix. Another BEM sub-domain approach is the Dual Reciprocity Method Multi-Domain (DRM-MD) approach, which has been introduced by Popov and Power. This formulation was a logical step after Popov and Power noticed substantial improvement in the performance of the DRM when used with sub-domains. The improvement relates to the accuracy and stability of the method. It also eliminates the problem of distribution of internal DRM nodes, usually used to improve the DRM approximation, since the distribution of the internal nodes in the DRM-MD is not any more an issue. The initial problem solved using this formulation was the flow of a mixture of gases through a porous media. The DRM-MD has also been applied to linear and non-linear advection-diffusion problems, driven cavity flow governed by the Navier-Stokes equations, flow of polymers inside mixers with complex geometries, flow through fractured porous media, two phase flow in porous media, etc. We must emphasize that the sub-domain techniques in the BEM are nowadays finding its place in the toolbox of numerical modellers, especially when dealing with complex 3D problems. We see their main application in conjunction with the classical BEM approach, which is based on a single domain, when part of the domain needs to be solved using a single domain approach, the classical BEM, and part needs to be solved using a domain approach, BEM subdomain technique. This has usually been done in the past by coupling the BEM with the FEM, however, it is much more efficient to use a combination of the BEM and a BEM sub-domain technique. The advantage arises from the simplicity of coupling the single domain and multi-domain solutions, and from the fact that only one formulation needs to be developed, rather than two separate formulations based on different techniques. There are still possibilities for improving the BEM sub-domain techniques. However, considering the increased interest and research in this approach we believe that BEM sub-domain techniques will become a logical choice in the future substituting the FEM whenever an efficient solution requires coupling of the BEM with a domain technique. The Editors 2007


CHAPTER 1 Diffusion–convection problems A.E. Taigbenu School of Civil and Environmental Engineering, University of the Witwatersrand, South Africa.

Abstract This chapter presents some solutions to the diffusion–convection equation that are based on the boundary element theory. Four formulations are discussed, but solutions from only three of them are presented. The formulations represent different replications of the differential equation along the lines of the singular integral theory. Their fundamental solutions come from different linear parts of the differential operator. The elliptic diffusion (ED) formulation that is based on the ED operator shows the most promise. Attempts at avoiding domain integrations through the dual reciprocity method are discussed, while full domain discretization through the Green element method for ease of evaluation of domain integrations and in solving heterogeneous and nonlinear transport is presented. The solutions from the three formulations to the nonlinear Burgers’ equation are also presented, with the ED formulation exhibiting superior performance.

1 Introduction There are many transport phenomena of theoretical and practical interest in a number of fields of science and engineering that are governed by the diffusion– convection equation. It can, under certain conditions, describe transport of mass, momentum, vorticity, and energy when mechanisms of diffusion or dispersion and convection or advection are of importance. Its solution continues to attract considerable interest in numerical circles because of its unique feature of being either a parabolic or a hyperbolic equation, depending on the values of the parameters of

2 Domain Decomposition Techniques for Boundary Elements the equation, and also because its solution process offers valuable computational experience which can be extended to the simulation of many other flows of engineering interest. Although there exist analytic solutions in one and two dimensions, their usefulness is limited to problems with simple regular geometries [1, 2]. Standard finite difference schemes produce, for advection-dominant cases, unacceptably large spurious oscillations or numerically diffused solutions [3–5]. From the reviews of finite element solutions provided by Anderson [6] and Zienkiewicz and Taylor [7], schemes that are based on Bobnov–Galerkin weighting functions showed similar spurious oscillations and diffused or smeared fronts because of their failure to correctly approximate the convective term. However, with upwind differencing of the convective term in what is now known as the Petrov–Galerkin weighting functions, more acceptable results have been reported [7–12]. Contemporary developments have been taking place in boundary element circles on accurate modeling of the transport equation. An earlier attempt by Brebbia and Skerget [13], which employed the fundamental solution to the temporal diffusion operator in two spatial dimensions, addressed only cases with small values of the Peclet number. Another approach, based on the fundamental solution to the 2D elliptic diffusion (ED) operator, treated the transient problem as a quasi-steady one and offered solutions which covered a wide spectrum of Peclet values [14–16]. All these formulations implemented the integral equations in the classical manner resulting in a fully populated global coefficient matrix. It has been recognized that the boundary-only character of the boundary element theory, observed for elliptic problems and considered one of the strengths of the boundary element method (BEM), is not retained when dealing with a parabolic–hyperbolic equation like that of diffusion–convection. The dual reciprocity method (DRM), proposed by Nardini and Brebbia [17], provided a way of transforming the domain integrations into boundary ones, though the method requires that some nodal points within the domain be provided. The DRM was extended to diffusion–convection problems by Wrobel et al. [18] and Aral and Tang [19]. A number of domain discretization formulations are available, ranging from those that subdivide the computational region into subdomains that result in block-banded coefficient matrices [20] to those which subdivide the domain into elements, thereby achieving a banded coefficient matrix in a manner similar to that encountered in finite element methodology (FEM). The latter formulation is referred to as the Green element method (GEM) [21, 22]. These formulations are discussed in this chapter with reference to diffusion–convection problems. A nonlinear form of the diffusion–convection equation is the Burgers’ equation. It provides a useful model for many diverse and seemingly unrelated phenomena such as shock flows, turbulence, wave propagation in combustion chambers, vehicular traffic movement, acoustic transmission, and many others. Boundary element formulations that have been applied to the Burgers’ equation are also discussed in this chapter.

Diffusion–Convection Problems

3

2 Boundary element formulations The mathematical statement of diffusion–convection transport in an incompressible medium can be written in general form as ∇ · (D∇c) = u · ∇c +

∂c +Q ∂t

(1)

where ∇ is the gradient operator, c is the basic dependent variable, D is a parameter that could depend on the medium (heterogeneous transport) and/or on the dependent variable (nonlinear transport), u is the velocity vector, t is the time dimension, and Q represents the source term which could have a functional dependence on c as in reactive transport. It should be noted that the basic dependent variable need not be a scalar but could be a vector as in the case of transport of momentum. In the spatial dimension, eqn (1) could apply either to one or two or three dimensions. The uniqueness of the solutions to eqn (1) depends on the prescribed boundary and initial conditions. The first type or the Dirichlet boundary condition specifies the value of the dependent variable on a part of the boundary, i.e. c(r, t) = C1 (t)

on 1 for t > t0

(2a)

while the second type of boundary condition specifies the normal flux on another part D∇c · n = qn (t) on 2 for t > t0 (2b) where n is the unit outward pointing normal vector on the boundary = 1 + 2 . The initial condition provides information on the distribution of c everywhere in the domain , i.e. c(r, t0 ) = C0 (t) on (2c) Various boundary element solution strategies arise from the choice of Green’s function or the fundamental solution that is applied. All the approaches restate eqn (1) as L(c) = b (3) where L stands for a linear differential part of eqn (1) and b can be referred to as a pseudo-forcing or source term that accounts for all heterogeneity and nonlinearity of the transport process. The solution to eqn (3) in an infinitely extensive spatial region is the free-space Green’s function or fundamental solution that is utilized in obtaining the integral representation of the transport equation. Different forms of eqn (3) have been used in boundary element methodology, and our discussion shall first be centered on the nontransient forms of L. 2.1 Formulation with the ED operator This formulation uses the nontransient portion of eqn (1) so that it can be rewritten as D∇ 2 c = b

(4)

4 Domain Decomposition Techniques for Boundary Elements where D is some averaged value of the parameter D and the pseudo-force term b accounts for the convection, transient, and source terms. b = u · ∇c +

∂c +Q ∂t

(5)

The corresponding integral equation arising from applying Green’s identity to eqn (4) is   ∂c cG∗ − G D λ(ζ )c(ζ ) + bGdA = 0 (6) ds + ∂n

The function G refers to the fundamental solution or the solution of the linear operator ∇ 2 G + δ(p, ζ ) = 0 (7) where δ is the Dirac delta function, and G∗ = ∂G/∂n is the normal derivative of the fundamental solution. For the 2D case, 1 lnr 2π η G∗ = − 2π r 2 G=−

(8a) (8b)

and for the 3D case, it is 1 4π r η ∗ G =− 4π r 3 G=

(9a) (9b)

where r = |r| is the magnitude of the distance vector r from the source point ζ to the field point p, and η = r · n is the normal distance from the source point to the boundary on which the field point is located. The parameter λ in eqn (6) has a value of unity if the source point is within the domain , and, when the source point is on the boundary, it is a fractional value that depends on the nodal angle at the source point. The approach that is based on the ED operator has been applied by Taigbenu and Liggett [16], Aral and Tang [19], Popov and Power [23], among others. The solutions from the computer model based on the elliptic diffusion operator are referred to as ED in this chapter. 2.2 Formulation with the elliptic diffusion–convection (EDC) operator This formulation is based on the nontransient ED and convection parts of eqn (1), i.e. D∇ 2 c − U · ∇c = b (10) where U is some reference uniform velocity field. In other words, the velocity vector ˆ u has been decomposed into a uniform velocity vector U and a perturbed one u,


5

ˆ The pseudo-force term b accounts for the perturbed convective i.e. u = U + u. velocity, transient, and source terms. b = uˆ · ∇c +

∂c +Q ∂t

(11)

The corresponding integral equation for this formulation is similar to eqn (6) and it is given as   ∂c λ(ζ )c(ζ ) + D  cG∗ − G bGdA (12) ds + c G Un ds + ∂n

where Un = U · n is the normal component of the uniform velocity U at the boundary . The fundamental solution for this linear operator is obtained from D∇ 2 G + U · ∇G + δ(p, ζ ) = 0

(13)

For the 2D case, 1 U·r K0 (µr) exp − 2π D 2D Un r 2 1 U·r G∗ = K0 (µr) exp − µηrK1 (µr) − 2π Dr 2 2D 2D G=

(14a) (14b)

and for the 3D case, 1 (U · r + Vr) K0 (µr) exp − 4π Dr 2D 2 ηVr + Un r U · r + Vr 1 ∗ G =− η+ exp − 4π Dr 3 2D 2D G=

(15a) (15b)

where µ = V /(2D) and V = |U|. Some of the investigators who have used the fundamental solution based on the EDC equation are Skerget et al. [24] and Ikeuchi and Onishi [25]. In this chapter, the computer model that is based on the elliptic diffusion–convection operator is referred to as EDC. 2.3 Formulation with the transient diffusion (TD) operator This formulation uses the temporal derivative term in conjunction with the diffusion term of eqn (1). It is based on a form of eqn (1) that is given by D∇ 2 c −

∂c =b ∂t

(16)

where the pseudo-forcing term b accounts for the convective and source terms b = u · ∇c + Q

(17)

6 Domain Decomposition Techniques for Boundary Elements The corresponding integral equation arising from applying Green’s identity to eqn (16) is ∂c dτ ds G (r, τ )c − G(r, τ ) ∂n

t λ(ζ )c(ζ )

(2)

+D 0

−

∗

t

G(r, t)c dA +

bG(r, τ ) dτ dA = 0

(1)

0

(18)

where the bracketed superscripts 1 and 2, respectively, denote the previous time level t1 and the current time level t2 . The fundamental solution is the solution in an infinite spatial region of D∇ 2 G +

∂G + δ(r, τ ) = 0 ∂τ

(19)

that is given by G=

H(τ ) r2 exp − [4π Dτ ]k/2 4Dτ η ∗ G =− G 2Dτ

(20a) (20b)

where k equals the number of spatial dimensions of the problem. The expression for the normal derivative of the fundamental solution can be inserted into eqn (18) to yield

t λ(ζ )c(ζ ) −

(2)

−

G (r, τ ) 0

η ∂c c+D dτ ds 2τ ∂n

t

G(r, t)c dA +


(1)

0

(21)

It should be noted that domain integration of the initial data is still required in this formulation. Some of the investigators who have used the fundamental solution of eqn (20a) are Taigbenu and Liggett [15], Zabaras and Mukherjee [26], and Young et al. [27, 28]. Young and his coworkers incorporated the method of characteristics into the formulation when solving the vorticity transport equation. The solutions from the computer model based on the transient diffusion operator are referred to as TD in this chapter. 2.4 Formulation with the transient diffusion–convection (TDC) operator This formulation uses the temporal derivative term in conjunction with the homogeneous diffusion–convection terms of eqn (1). Because the fundamental solution can only be derived with constant coefficients and a uniform velocity field, the


7

velocity in the convective term is split in the same manner as done earlier for the EDC formulation. The linear form of eqn (1) that is used is given by D∇ 2 c − U · ∇c −

∂c =b ∂t

(22)

where the pseudo-force term b accounts for the heterogeneous convective and source terms. b = uˆ · ∇c + Q (23) The corresponding integral equation arising from applying Green’s identity to eqn (22) is t λ(ζ )c(ζ )

(2)

∂c G (r, τ )c − G(r, τ ) dτ ds ∂n

+D

∗

0

t + Un

t

G(r, τ ) dτ ds −

0

G(r, t)c dA +


(1)

0

(24) The fundamental solution is the solution in an infinite spatial region of D∇ 2 G + U · ∇G +

∂G + δ(r, τ ) = 0 ∂τ

(25)

It is given by 2 H(τ ) r2 U·r τV G= + + exp − [4π Dτ ]k/2 4D 4Dτ 2D

η 1 G∗ = − + Un G 2D τ

(26a) (26b)

where k equals the number of spatial dimensions of the problem and, as in the nontransient version of this formulation, V = |U|. Inserting the expression for the normal derivative of the fundamental solution into eqn (24) simplifies the integral equation to t λ(ζ )c(ζ )(2) −

G (r, τ )

1 η ∂c − Un c + D dτ ds 2 τ ∂n

0

−

t G(r, t)c(1) dA +

bG (r, τ ) dτ dA = 0 0

(27)

It should be noted that domain integration of the initial data is also required in this formulation. Taigbenu [29] and Kakuda and Tosaka [30] used this fundamental

8 Domain Decomposition Techniques for Boundary Elements solution of eqn (26a) to solve diffusion–convection problems in one spatial dimension. In this chapter, the solutions from the computer model based on the transient diffusion–convection operator are referred to as TDC. The choice of formulation or linear differential operator adopted has largely depended on the personal inclinations of investigators. However, some general statements can be made concerning these formulations. The expressions of the fundamental solutions with the time variable tend to be more complicated than those without it. Their complicated expressions extend to the calculations of the integrations over elements so that more computational effort is generally required in evaluating the elements of the global coefficient matrix. A number of the integrations cannot be carried out analytically, and so have to be evaluated numerically. One disadvantage of the formulations that are based on nontransient fundamental solutions is that the temporal derivative has to be approximated by a finite difference expression. That provides some limit on the size of the time step that can be taken to ensure that the numerical scheme remains stable.

3 Numerical implementation The numerical implementation of the integral equations requires that the boundary and, sometimes, the domain be discretized into suitable elements and functional quantities be allowed to vary over those elements using some prescribed interpolation functions. The order of polynomial used as the interpolation function bears a close correlation to the computational effort involved in evaluating the element integrations. Zero and first order polynomials are commonly used as interpolation functions. For the boundary integrations, line segments or plane surfaces are commonly used in discretizing the boundary, while spline elements have occasionally been used. Over a typical element, functional quantities are interpolated according to the relationship c(r, t) ≈ Ni (r)ci (t) (28) where Ni is the interpolation function with respect to the nodal point i. A number of approaches have evolved in the numerical implementation of the integral equation. They have arisen largely because of the domain integrals that contain the initial data distribution and nonhomogeneous transport terms. The classical approach gives rise to a solution scheme in which the solution at any node is obtained from contributions from all the nodes in the computational region. For that reason, the coefficient matrix is always dense and fully populated (Fig. 1a). For large-scale regional transport problems, this could require an enormous amount of computing resources if the matrix has to be decomposed at each time step. In this age of vector and parallel computing, that is not too great a challenge to handle. In the classical approach, the evaluation of the domain integrations requires that the region be discretized into suitable polygonal elements so that some measure of accuracy is achieved in describing the distribution of the dependent variable over the elements. That makes BEM quite unattractive in the sense that its computational advantage as a boundary-only method is lost.


(a)

9

(b)

(c)

Figure 1: Typical forms of the coefficient matrix: (a) fully populated with single domain; (b) partial domain discretization; (c) full domain discretization. In the classical approach, the outcome of implementing the boundary and domain integrations gives rise to a matrix equation that takes, for the ED and EDC formulations that use nontransient fundamental solutions, the form Bij cj + Hij ϕj + Tij

dcj = Si dt

(29a)

and, for the TD and TDC formulations that use transient fundamental solutions, the form (2) (2) Bij cj + Hij ϕj = Si (29b) where ϕ = ∂c/∂n. It should be understood that the labels of the matrices in eqns (29a) and (29b) have been used in a generic sense, without implying that their elements are the same for the four different formulations. The vector Si is a known vector that, in eqn (29a), accounts for known sources while, in eqn (29b), accounts for known sources and data on c and ϕ at the previous time t1 . The matrix equation (29a) from the ED and EDC formulations can be solved only after the temporal derivative is discretized. This is done by finite differencing, i.e. dc c(2) − c(1) ≈ dt t

(30)

This simplification allows for the introduction of a difference weighting factor α that takes a value of between zero and unity. The weighting factor is used to weight the other terms in eqn (29a) so as to reflect the time within the time step t at which differencing is done. An α value of zero indicates that differencing is done at t1 ,

10 Domain Decomposition Techniques for Boundary Elements while a value of unity indicates that differencing is done at t2 . The introduction of eqn (30) into eqn (29a) yields Tij (2) (2) αBij + (31) c + αHij ϕj = Ri t j (1)

(1)

where Ri = [βBij + Tij / t] cj + βHij ϕj and β = α − 1. Equation (31) for the ED and EDC formulations, now in a similar form as eqn (29b) for the TD and TDC formulations, can be solved after effecting the known boundary conditions. The resultant matrix equation for all the formulations has the form (2)

Aij hj

= Pi

(32)

where h = {c, ϕ}T is a mixed vector of unknowns. In the classical approach, the global coefficient Aij is dense and fully populated. 3.1 Dual reciprocity method In the classical approach, domain integrations are implemented usually by Gauss quadrature methods. However, to still retain the boundary-only character of BEM, a number of initiatives have been pursued [17, 31]. The most notable one is the DRM that was first initiated by Nardini and Brebbia. It transforms these domain integrals into surface integrals by making use of radial basis functions. Though domain integrals are avoided using the DRM, it nonetheless still requires that some number of nodes be chosen within the computational region. The DRM is well documented in many works by Brebbia and his co-workers [17, 23, 32]. Here, it is only briefly described to ensure completeness of the chapter, and readers are directed to a more detailed discussion of the method in the references provided. We shall focus only on the domain integration term given as bGdA. The basic idea of the DRM is to express b in terms of radial interpolation functions, i.e. b(r, t) ≈

M

ωk (t)fk (r)

(33)

j=1

where fk (r) are the radial interpolation functions, M is the number of nodal points at which interpolation is carried out, and ωk are unknown coefficients. There are a number of such functions that can be used of which the one based on the distance vector between the field and the source node is most widely used, i.e. fk (r) = 1 + r. A new auxiliary function ωk is proposed such that its Laplacian equals the radial interpolation function, i.e. ∇ 2 ψk = fk . To this, Green’s identity is applied to yield an equation that is similar to eqn (6) ∂ψ ∗ ψk G − G fk GdA = −λ(ζ )ψk (ζ ) − ds (34) ∂n


11

Using all M radial functions fk in eqn (33), the expression for the domain integral now becomes   M

∂ψ ψk G ∗ − G bGdA = ωk −λ(ζ )ψk (ζ ) − ds (35) ∂n

k=1

Now the domain integration involves only boundary integrals. It has been achieved by applying the reciprocity principle used for the ED operator, hence, the name DRM. The unknown coefficients ωk are obtained by inversion of eqn (33) at the M nodal points, i.e. ω = F−1 b

(36)

3.2 The Green element method All boundary element implementing procedures so far applied to the diffusion– convection transport equation can be considered to fall within a spectrum of formulations with meshless formulations at one end and, on the other, full domain decomposition formulations to which the GEM belongs. Meshless formulations, which the DRM attempts to achieve, are bound to do well only for weakly domaindominant problems (weak nonhomogeneity and nonlinearities), but their numerical appeal lies in the retention of the original spirit of BEM. Domain decomposition formulations, on the other hand, hold the key in solving highly domain-dominant problems to which the diffusion–convection transport equation belongs. However, they tend to have less appeal because they conjure the FEM which the BEM sought to overcome in the first instance. Domain decomposition was originally proposed to allow the BEM to handle large field problems by decomposing the original domain into subdomains within which the integral equation is applied, while imposing the appropriate compatibility relations at the interfaces of adjacent subdomains. Whereas fully populated matrices are encountered in BEM matrices with single-domain formulations (Fig. 1a), those with domain decomposition result in block-banded matrices with the blocks of adjacent domains overlapping themselves (Fig. 1b). The solution of the global coefficient matrix from these domain discretization schemes cannot be effected because the system is generally overdeterminate. Several schemes are used to reduce the overdeterminate system to a closed system that can then be solved [20]. The GEM is not primarily motivated only by the achievement of a banded matrix (Fig. 1c), but by the ease and flexibility that the approach offers in evaluating the domain integrations. This latter numerical feature stems from the fact that when evaluating the domain integrations, the source and field nodes always share the same element, while the flexibility allows for heterogeneous and nonlinear terms to be correctly represented from element to element. It is this that has made it possible for the calculation of the convection–diffusion problems with heterogeneous parameters and reaction terms. The GEM essentially calculates the integral equation earlier obtained in an element or cell, and obtains the global coefficient matrix by aggregating the element

12 Domain Decomposition Techniques for Boundary Elements equations for all the elements. The aggregation is done such that compatibility conditions at inter-element boundaries are applied and ∂c/∂n is expressed in terms of c, so that the outcome is that the solution for c is calculated at all nodes, and ∂c/∂n at the external nodes. Details of the method can be found in references [21] and [22]. Popov and Power [23] have incorporated DRM into GEM so that domain integrations are avoided.

4 Numerical stability for homogeneous diffusion–convection BEM schemes for the diffusion–convection equation are generally time-marching in nature, and numerical stability analysis provides an insight into how errors arising from the discrete replication of the original differential equation and computer precision propagate in both amplitude and speed with time. The stability of the ED, TD, and TDC formulations are examined for homogeneous transport in one dimension, i.e. ∂c/∂n. The boundary element coefficients from the three formulations using linear interpolating functions in space and time take the form of eqn (29b) (2)

(2)

Bij cj + Hij ϕj

= Si

(37)

Since there is no source term, the known vector on the right-hand side of eqn (37) essentially accounts for known data at the previous time t1 . Essentially the three models are time-marching schemes which use the known data at the beginning of the time step to obtain their new values at the end of the time step on the basis of the numerical replication of the differential equation. For a well-posed initial-value problem, the solution at any time level is expected to be bounded, and this can only be achieved by a numerical scheme which does not allow the solution and accumulated computational errors to grow out of bounds. The stability characteristics of the three formulations are assessed by the method of Stone and Brian [33], which has been applied by Leendertse [34, 35], Gray and Pinder [8], Gray and Lynch [36], and Taigbenu [37, 38]. It represents the general solution of the linear differential equation by an infinite series of Fourier wave components, and the stability characteristics of the numerical scheme are assessed by comparing the numerical amplitude and speed of each wave component to their theoretical values. A scheme is said to be dissipative if it is unable to correctly reproduce the amplitude of the Fourier wave component, and it is dispersive if it is unable to correctly reproduce its speed. Dissipative schemes cause steep gradients in the solution to become smeared, while dispersive schemes cause the small wavelengths to be out of phase with the longer wavelengths which usually manifest as oscillations in the numerical solution. The proposed Fourier series solution is c(x, t) =

∞

m exp[î(σm x + βm t)]

(38)

m=−∞

in which, for each Fourier wave component, βm = 2π/Tm is the temporal frequency, Tm is the period, σm = 2π/Lm is the spatial frequency or wave number,


13

Lm is the wavelength, and î is the imaginary representation of a complex variable. Only one term of the series needs to be examined since the differential operator is linear. The dispersion relationship: β = σ (îDσ − U) is introduced into eqn (38) to have for one term of the series solution c = c(1) ∼ exp[îσ (x − Ut)] exp(−Dσ 2 t)

(39)

Equation (39) indicates that each Fourier wave component is a translatory wave with propagation speed of U, and amplitude modification of exp(−Dσ 2 t) after an elapse time t. In other words: c(2) = at c(1) , where at is the theoretical amplification of each Fourier wave after one time step. It has a magnitude of |at | = exp(−Dσ 2 t) and phase angle of θt = −σ U t. Denoting the magnitude and phase angle of the numerical amplification as |an | and θn , respectively, the amplification factor after one wavelength of propagation is Nt Nt an |an | Pe L 2 |an | T = = = at exp(−Dσ 2 t) exp(−4π 2 Cr x 2 )

(40)

where Nt = L/( xCr ) is the number of time steps for the wave to traverse one wavelength L, Cr = U t/ x is the element Courant number, and Pe = U x/D is the element Peclet number. The corresponding expression for the phase lag at the end of one wavelength of propagation is θn θn − 1 = 2π −1 (41) = 2π θt σ xCr To evaluate T and for the three numerical formulations, we consider a computational region that is discretized uniformly by linear elements each of which has a length of x such that a node of interest xi is sandwiched between two nodes xi−1 and xi+1 . The numerical scheme modifies c according to the relationship: c(2) = an c(1) , and similarly for ϕ, where an is the numerical amplification. Applying the element integral equation (37) to the two elements: [xi−1 , xi ] and [xi , xi+1 ] produces a quadratic equation in an that depends on the coefficients Bij and Hij from which the two roots are calculated, though one of them is inadmissible. For values of Cr = 0.2 and Cr = 1, Pe = 2 and Pe = 50, the propagation characteristics of the ED, TD, and TDC formulations are compared. The first sets of results, shown in Fig. 2a–d, give the variation of |an | with respect to L/ x and they indicate that all three formulations are stable. The next sets of plots compare the numerical amplification values to their corresponding theoretical values, i.e. the variation of T and . Ideally T should be unity and zero. These plots for Cr = 0.2 and Pe = 2, are in Fig. 3a and b, in Fig. 4a and b for Cr = 1 and Pe = 2, in Fig. 5a and b for Cr = 0.2 and Pe = 50, and in Fig. 6a and b for Cr = 1 and Pe = 50. The results show that at small values of Cr and Pe , the ED formulation with α = 0.5 (Crank–Nicholson scheme) and the TDC formulation exhibit the least phase lag, but the latter is more dissipative (Fig. 3a and b). (The harmonics are considered to be correctly propagated when the amplification factor is within 5%

14 Domain Decomposition Techniques for Boundary Elements 1.0 Numerical amplification, an

Numerical amplification, an

1.0

0.8

0.6 α=1

0.4

Pe=2, Cr=0.2 ED TD TDC

α=.75 α=.67 α=.5

0.2 1

10

0.8

α=.75 α=1

0.6

α=.5

0.4

α=.67

0.0 1

100

10

(a)

(b) 1.2

α=.5 α=.67

1.0

α=.75 α=1

0.8


0.7 0.6 1

10

Numerical amplification, an

1.1 Numerical amplification, an

100

L/∆x

L/∆x

0.9

Pe=2, Cr=1 ED TD TDC

0.2

1.0

α=.5 α=1 α=.75 α=.67

0.8 0.6 0.4


0.2 0.0 1

100

10

L/∆x

L/∆x

(c)

(d)

100

Figure 2: Numerical amplification for various Peclet and Courant number values.


2

α=1

1

α=.5

α=.75 α=.67

0

α=.5

-60 Phase lag,Θ

Amplification factor, T

3

-120

α=1

α=.75

α=.67

-180 -240

Pe=2, C r=0.2 ED TD TDC

-300

0

-360 1

10 L/∆x

(a)

100

1

10 L/∆x

100

(b)

Figure 3: Amplification factor and phase lag for Cr = 0.2 and Pe = 2. of unity and the phase angle shift is within 5%.) When the value of the Courant number is increased to unity for small values of the Peclet number (diffusiondominant transport), the propagation characteristics of the Fourier harmonics by the numerical schemes become slightly different (Fig. 4a and b). Only harmonics with

15

Diffusion–Convection Problems 60 Pe=2, Cr=1 ED TD TDC

2

α=.5

α=.67

1

0

1

10 L/∆x

-60

α=1

α=.75

α=.67

-120 -180


-240 -300

α=1

α=.75

α=.5

0 Phase lag,Θ


3

-360 1

100

10

100

L/∆x

(a)

(b)

Figure 4: Amplification factor and phase lag for Cr = 1 and Pe = 2. 0 Pe=50, C r=0.2 ED TD TDC

1.0

0.5

-60

α=.5

α=1 α=.67

Phase lag, Θ


1.5

α=.5 α=.75

-120 -180


-240

α=.75

-300 -360

0.0 1

10 L/∆x

(a)

100

α=1 α=.67

1

10 L/∆x (b)

100

Figure 5: Amplification factor and phase lag for Cr = 0.2 and Pe = 50.

wavelengths larger than four times the spatial element size are correctly propagated in speed by the ED (α = 0.5) and TDC formulations. The amplitudes of the wave components are damped more by the TD and TDC formulations than by the ED formulation. In contrast to the case when Cr = 0.2, all ED schemes amplify the amplitudes of the short wavelengths. What can be considered surprising is the result that the TD and TDC formulations do better in propagating the harmonics at larger values of the Courant number (larger time step) than at smaller values (smaller time step). When Cr = 0.2 and Pe = 50, the ED formulation with α = 0.5 does best in propagating all harmonics with wavelengths larger than four times the element size (Fig. 5a and b). At this large Peclet number (convection-dominant transport)

16 Domain Decomposition Techniques for Boundary Elements α=.5 1.0

α=.67 α=.75

0.5

α=1

0.0 1

0 α=.5


10 L/∆x (a)

100

-60 Phase lag,Θ


1.5

α=1

α=.75

α=.67

-120 -180 -240


-300 -360 1

10

100

L/∆x (b)

Figure 6: Amplification factor and phase lag for Cr = 1 and Pe = 50.

and small Courant number, the TD and TDC formulations damp the amplitudes of virtually all harmonics and only correctly propagate the speed of harmonics with large wavelengths 10 times the element size. For this large Peclet number, increasing the Courant number to unity does bring about improvement in the propagating characteristics of the TD and TDC formulations, but the converse is the case with the ED formulation (Fig. 6a and b). The above results are summarized as follows: (1) incorporating the Crank–Nicholson scheme into the ED model gives optimal stability characteristics at small values of the Courant number for all transport modes; (2) the stability characteristics of the TD and TDC are improved when a value of unity is used for the Courant number.

5 Numerical examples of diffusion–convection problems The ED, TD, and TDC formulations are applied to five examples of transient diffusion–advection transport. The first four examples are one-dimensional in space, while the fifth is two-dimensional. 5.1 Example 1 This first example is the classical 1D transport problem which finds use in evaluating the performance of many numerical schemes. The velocity field is uniform and there are no sources. The boundary and initial conditions are: c(0, t > 0) = 1, ∂c (x = ∞, t > 0)/∂x = 0, and c(x, t = 0) = 0; its well-known solution was provided by Ogata and Banks [39]. For Pe = 2 (diffusion-dominant transport) all the numerical schemes exhibit negligible spurious oscillations. The breakthrough concentration curves of the exact and numerical solutions at t = 0.5 are presented in Fig. 7a. The solutions of the TD and TDC formulations are presented when Cr = 1

Diffusion–Convection Problems 1.2

1.2

Pe=2, t=0.5 Exact ED TD & TDC

1.0 0.8

Pe=50, t=0.5 Exact ED TD TDC

1.0 0.8

c

0.6

c

17

0.6

0.4

0.4

0.2

0.2

0.0

0.0

-0.2 0.0

0.2

0.4

0.6

x (a)

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

x (b)

Figure 7: Breakthrough curves for diffusion–convection example 1.

because both formulations do better at this Courant value than at Cr = 0.2. The ED solution is obtained with Cr = 0.2. All three formulations are able to correctly reproduce the breakthrough curve which is not steep because of diffusion dominance. For the strong advection-dominant transport (Pe = 50), the TDC model does not exhibit any oscillations that are exhibited by the ED and TD models (Fig. 7b). All the models are able to capture the steep front of the breakthrough curve. 5.2 Example 2 This second example is that of an initial Gauss-hill concentration profile subjected to dispersion and uniform advection. The boundary conditions and initial Gauss-hill profile are: c(x = −∞, t) = 0, ∂c(x = ∞, t)∂x = 0, and c(x, 0) = exp[−(x − x0 )2 /2σ02 ]; the exact solution is c(x, t) = σ0 exp[−(x − x0 )2 /2σ 2 ]/σ , t where σ 2 = σ02 + 2Dt and x¯ = x0 + 0 U(s) ds. Our calculations use x0 = 0 and σ0 = 0.1 to obtain solutions at t = 0.5. The concentration breakthrough curves at t = 0.5 can be found in Fig. 8a and b. They are obtained for the diffusion-dominant transport (Pe = 2) and convection-dominant transport (Pe = 50). As in the previous example, a value of Cr = 0.2 is used in the ED model with the Crank–Nicholson scheme, while unit value of the Courant number is used in the TD and TDC models. None of numerical models exhibit any significant oscillations or wiggles upstream or downstream of the concentration hill, but the ED model does indicate superior performance in terms of being able to accurately model the concentration hill. 5.3 Example 3 The third example is that of a contaminated stream subjected to advection from a freshwater source and biological/chemical degradation. The initial and boundary

18 Domain Decomposition Techniques for Boundary Elements 0.8

1.0

Pe=50, t=0.5 Exact ED TD TDC

0.8

0.6

c

c

0.6

0.4

0.4

0.2

Pe=2, t=0.5 Exact ED TD & TDC

0.0 0.0

0.2

0.4

x

0.6

0.2 0.0

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

x (b)

(a)

Figure 8: Concentration hill of diffusion–convection example 2. 1.0

1.0

0.4

Exact ED TD

0.0 0.0

0.5

1.0

1.5

0.4 0.2

Pe=2

0.2

t=0.5

c

0.6

t=1

t=0.5

c

0.6

t=1

0.8

0.8

2.0

Pe=50 Exact ED TD

0.0 -0.2 0.0

0.5

1.0

x

x

(a)

(b)

1.5

2.0

Figure 9: Breakthrough curves of diffusion–convection example 3.

conditions are: c(x, t = 0) = 1, c(x = 0, t > 0) = 0, and ∂c (x = ∞, t > 0)/∂x = 0. The exact solution is given by van Genuchten and Alves [2]. Two cases of this example are examined with the following parameters: the first is U = 1, D = 0.025, and Pe = 2, and the second is U = 1, D = 0.0005, and Pe = 50. For both cases the source term is a reaction term expressed as Q = 0.2c. Only the ED and TD models are applied, and the numerical calculations use a uniform spatial element size of 0.025. The numerical and exact solutions, obtained at times 0.5 and 1.0, are presented in Fig. 9a for the first case, while those of the second case are presented in Fig. 9b. For both cases, the ED model reproduced the exact solution more closely than the TD model. The numerical solutions have trailing oscillations for the strong convection-dominant transport but their strength appears to decrease as simulation time increases.


1.2

U=5, D=1 Exact ED TD

1.2 1.0

U=20, D=1 Exact ED TD

1.0

t=0.01

t=0.01

0.8

t=0.03

19

t=0.03

0.6

c

c

0.8

0.4

0.6 0.4

0.2

t=0.1

t=0.1

0.0

0.2

-0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

x

x

(a)

(b)

0.8

1.0

Figure 10: Breakthrough curves of diffusion–convection example 4.

5.4 Example 4 This example is the case of transport of an initial concentration profile of unit value everywhere in a uniform flow field Uin the x direction. The boundary conditions are c(x = 0, t > 0) = 0 and c(x = 1, t > 0) = 0. The exact solution to this problem can be found in reference [19]. Twenty uniform rectangular elements are used to discretize the domain, while the time step employed in the numerical calculations is expressed in terms of the Courant number. These values are: 0.1, 0.25, and 0.5. The results of the ED model with Cr = 0.1 and that of the TD model with Cr = 0.5 are presented and in Fig. 10a for U = 5 and Fig. 10b for U = 20. The ED solution is slightly superior in approximating the exact solution than the TD model. 5.5 Example 5 This example is a transient transport problem in two dimensions for which Cleary [1] derived an exact solution. The pollutant, which is continuously injected into the flow medium from a line source, is dispersed in the longitudinal and transverse directions and advected by a uniform velocity field along the longitudinal direction. At the initial time, the concentration has zero value everywhere. The boundary conditions for this example are: c(0, y, t) = 1, 2 ≤ y ≤ 3;

c(0, y, t) = 0, 0 ≤ y < 2 and 3 < y ≤ 5 (42a)

∂c(x, 0, t) ∂c(x, 5, t) = = 0, x > 0; ∂y ∂y

∂c(∞, y, t) = 0, 0 ≤ y ≤ 5 (42b) ∂x

Finite element simulations of this example were carried out by Yeh [40], and the parameters employed in those simulations are also used here. Although the domain is semi-infinite, a finite domain of 15 × 2.5, which is discretized into 150 uniform


1.0 Pe=0.05, t=4 Exact FEM ED TD

0.8

0.8

0.6

c

c

0.6

Pe=1; t=4 Exact FEM ED TD

0.4

0.4

0.2

0.2

0.0

0.0

0

2

4

6

0

8

2

4

6

8

x (b)

x

(a) 1.1

Pe=50; t=4 Exact FEM ED TD

0.9

c

0.7 0.5 0.3 0.1 -0.1

0

2

4

6

8

x

(c)

Figure 11: Breakthrough curves of diffusion–convection example 5. rectangular elements each of 0.5 × 0.5, is used for the calculations. Following Yeh [40], three transport modes are examined: the case of diffusion-dominant transport (Pe = 0.05), equal dominance of diffusion and convection (Pe = 1), and the convection-dominant transport (Pe = 50). A uniform time step of 0.2 is employed in all the calculations, and the breakthrough concentration fronts at t = 4 along y = 2.5 are presented in Fig. 11a–c for the three cases. The finite element solutions are obtained by an orthogonal weighting scheme. For all three cases, the ED formulation reproduces the concentration front better than the TD formulation or the FEM. For lower values of the Peclet number, the solutions from the current formulation are about the same as the FEM solutions, while for the advection-dominant transport, the TD formulation is dissipative with a smeared front and the FEM is both dissipative and dispersive with the concentration shooting higher than the exact value at the upstream end of the front.

6 Burgers’ equation Burgers’ equation [41, 42] is a second-order simplification of the momentum transport equation or the Navier–Stokes equation, and it provides a useful model for


21

flow phenomena in which there exist the balancing effects of viscous and inertia or convective forces. Its unique features of a propagating wave front when inertia is dominant, and a stretched wave front when viscous forces are pronounced make the solution of Burgers’ equation quite a challenge. Exact solutions of the equation have been obtained mostly for infinite domains by the Cole–Hopf transformation [42, 43], and are summarized by Benton and Platzman [44]. A subdomain discretization of the integral equation in the Green element sense is used to solve the Burgers’ equation. Three formulations are applied; these are the ED, TD, and TDC formulations. The discretized integral equations are nonlinear, and they are linearized by either the Newton–Raphson or Picard algorithm. The efficiency and accuracy of the three formulations are evaluated with three numerical examples.

7 Numerical formulations of Burgers’ equation The mathematical statement that describes one-dimensional nonlinear fluid flow phenomenon of shock or wave propagation with viscous dissipation, widely known as the Burgers’ equation, is given by Lighthill [45] D

∂u ∂u ∂ 2u −u − =0 2 ∂x ∂x ∂t

on x0 ≤ x ≤ xL

(43)

in which D = 1/Re is the reciprocal of the flow Reynolds number which is a dimensionless parameter that gives the relative magnitude of inertia forces to viscous forces, and u(x, t) is the dependent variable which denotes the velocity field. Fletcher [46] discussed the important features of eqn (43) as a prototype of the force balance between the nonlinear acceleration or inertia or convective term u∂u/∂x and the diffusive term D∂ 2 u/∂x 2 . All three numerical formulations follow a unified approach of (1) constructing the Green’s function to an appropriate linear differential operator of the flow equation, (2) employing Green’s second identity in transforming the flow equation into an integral one, (3) discretizing the integral equation by suitable elements to obtain a system of discrete equations, and (4) linearizing the nonlinear discrete equations either by the Newton–Raphson or Picard algorithm. 7.1 ED formulation for Burgers’ equation The ED formulation derives its fundamental solution from the second-order diffusion term: d2 G/dx 2 + δ(x − ζ ) = 0, where G = −|x − ζ |/2. Applying Green’s identity to the differential equation yields the integral equation for a typical linear element [x1 , x2 ]

D λ(ζ )u(ζ , t) + [G

∗

u]xx21

− [Gϕ]xx21

x2 bGdx = 0

+ x1

(44)

22 Domain Decomposition Techniques for Boundary Elements in which b = u∂u/∂x + ∂u/∂t and ϕ ≡ ∂u/∂x. The line integral of eqn (44) is evaluated by approximating u and ϕ with linear interpolation functions Nj , i.e.

D λui + [G

∗

u]xx21

− [Gϕ]xx21

x2 +

duj G Nj Nn uj ϕn + Nj dx = 0 dt

(45)

x1

or in matrix form Bij uj + Hij (u)ϕj + Tij

duj = Si dt

(46)

where

1 Bij = D δij + G∗ (x2 − xi )δ2j − G∗ (x1 − xi )δ1j 2

(47a)

x2 Hij = −D(G(x2 − xi )δ2j − G(x1 − xi )δ1j ) +

GNj Nn un dx

(47b)

x1

x2 Tij =

GNj dx

(47c)

x1

Details of the expressions of these matrices can be found in reference [47]. It should be noted that the matrix Hij depends on the velocity field, hence, the nonlinear nature of the discrete equations. Using the same time discretization described earlier in Section 3, eqn (46) becomes similar to eqn (31), i.e. Tij (2) (2) αBij + uj + αHij ϕj − Ri = gi (48) t Denoting h = {u, ϕ}T as the mixed solution vector, the N–R algorithm for eqn (48) refines an estimate of the solution h(2,m) by the relationship h(2,m+1) = h(2,m) + h(2,m+1) , where m is the iteration number and h(2,m+1) is the increment that is obtained from: (2,m) (2,m+1) (2,m) Jij hj = −gi (49) (2,m)

in which the Jacobian Jij

(2,m)

Jij

takes the expression

 Tij dHik (2,m) ∂gi   + = α Bij + ϕk    ∂uj u =u(2,m) t duj j j = ∂gi    = αHij   ∂ϕj (2,m) ϕj =ϕ j

Equation (49) is solved at each time step until convergence is achieved.

(50)


23

7.2 TD formulation for Burgers’ equation The G(x, τ ) = H(τ ) exp[−x 2 /4Dτ ]/ √ TD formulation uses the fundamental solution 2 4π Dτ to the linear diffusion equation D∂ G/∂x 2 + ∂G/∂τ + δ(x, τ ) = 0. The integral equation presented earlier in eqn (21) is applied to a typical spatial element [x1 , x2 ] in time interval [t1 , t2 ] (2) λui

t

x2 ∗ x2 x2 + D G u x − [Gϕ]x1 dτ − G (x, τ ) u(1) dx 1

x1

0

t x2 G (x, τ ) uϕ dx dτ = 0

+

(51)

0 x1

A linear distribution is prescribed for u and ϕ within the spatial element [x1 , x2 ] and the temporal element [t1 , t2 ], while the convective term is approximated as u¯ ϕ, where u¯ is the average velocity within the element. With this approximation, eqn (51) in matrix form becomes (2)

(2)

Bij uj + Hij (¯u)ϕj

= Si

(52)

where 1 Bij = δij + D 2

t

τ [G∗ (x2 − xi , τ )δ2j − G∗ (x1 − xi , τ )δ1j ] dτ t

(53a)

0

t Hij = −D

τ [G(x2 − xi , τ )δ2j − G(x1 − xi , τ )δ1j ] dτ t

0

t x2 + u¯

τ G(x − xi , τ )Nj dx dτ t

(53b)

0 x1

 t

 τ ∗ 1 − [G (x2 − xi , τ )δ2j − G∗ (x1 − xi , τ )δ1j ] dτ Si = − D  t 0

x2 −

G(x





(1) − xi , t)Nj dx  uj

− D

x1

t

1−

τ [G(x2 − xi , τ )δ2j t

0

tx2

   τ (1) G(x − xi , τ ) Nj dx dτ ϕj 1−  t

− G(x1 − xi , τ )δ1j ] dτ − u¯ 0 x1

(53c)

24 Domain Decomposition Techniques for Boundary Elements The analytical expressions for these element matrices can be found in reference [22]. Equation (52) is nonlinear, and can be solved by applying a nonlinear solution scheme. Using the Picard algorithm, the nonlinear discrete eqn (52) is linearized by using known estimates of u = u(2,m) to calculate u¯ = u¯ (m) and, therefore, the right-hand side of eqn (52). The refined estimate of the solution h(2,m+1) = {u(2,m+1) , ϕ (2,m+1) }T , where m is the iteration, is obtained by solving eqn (52) after incorporating the known boundary conditions. The solution refinement process is continued at each time step until |h(2,m+1) − h(2,m) | at all nodes fall below a predetermined accuracy tolerance. 7.3 TDC formulation for Burgers’ equation This formulation is based on the 1D version of that presented in√Section 2.4. √ √It uses the fundamental solution G(x, τ ) = H(τ ) exp[−(x/ 4Dτ + u¯ τ /4D)2 ]/ 4π Dτ to the linear diffusion equation D∂ 2 G/∂x 2 + u¯ ∂G/∂x + ∂G/∂τ + δ(x, τ ) = 0. The 1D form of the integral equation (24) is

(2) λui +

t x2 t ∗ x x2 (1) 2 D [G u]x1 − [Gϕ]x1 dτ + G(x, t)u dx + [Gu]xx21 dτ = 0 x1

0

0

(54) Similar to the approach of Kakuda and Tosaka [30], u is approximated by linear interpolation functions in space and time, and ϕ by its value at the current time to simplify eqn (54) to

(2) λui

1 + 2

t

x2 x x2 x2 u¯ − Gu − D[Gϕ]x1 dτ − G (x, t) u(1) dx = 0 τ x1 x1

0

(55) In eqn (55), we have used the expression for G∗ = −(x/τ + u¯ )G/2D. In matrix form eqn (55) is (2)

(2)

Bij (¯u)uj + Hij (¯u)ϕj

= Si

(56)

where 1 Bij = δij + D 2

t

τ t

x − xi u¯ − τ

[G∗ (x2 − xi , τ )δ2j − G∗ (x1 − xi )δ1j ] dτ

0

(57a) t Hij = −D

[G(x2 − xi , τ )δ2j − G(x1 − xi , τ )δ1j ] dτ 0

(57b)


25

 t τ x − xi 1

1− u− [G∗ (x2 − xi , τ )δ2j − G∗ (x1 − xi , τ )δ1j ] dτ Si = − 2 t τ 0

x2 −



G(x − xi , t)Nj dx  uj

(1)

(57c)

x1

Equation (56) constitutes the system of discrete element equations of the TDC formulation for Burgers’ equation. The expressions for the element matrices can be found in references [30] and [22], and they depend on the value of the primary variable u. The Picard algorithm is employed to linearize the formulation in the same way as for the TD formulation.

8 Numerical examples of Burgers’ equation The capabilities of the three formulations are demonstrated with three examples. The first two examples have analytic solutions with which we validate the numerical solutions, while the finite element solutions of the third example are used for comparison. 8.1 Example 1 of Burgers’ equation The first example is an initial sinusoidal wave which is allowed to propagate and diffuse within a confined flow domain in the x-dimension. The problem has the following boundary conditions: u(0, t) = 0,

u(1, t) = 0

and

u(x, t) = sin πx

(58)

The exact solution to this problem, in the form of an infinite series, has been provided by Cole [43]. Using 100 linear elements and a time step of 0.01 in all the numerical calculations, the numerical and exact solutions are presented in Fig. 12a for Re = 102 , and in Fig. 12b for Re = 103 . In all the numerical calculations, the Crank–Nicholson scheme is used in the ED formulation. The results show that the ED formulation gives the most accurate solution, followed by the TD formulation. The computing times of the ED, TD, and TDC formulations are in the approximate ratio of 2:1:10, with the speed of the TD formulation being enhanced by the ease in the assembly of its global coefficient matrix. The numerical solutions of the velocity profiles, presented in Fig. 12a and b, reflect the general behavior expected of the propagation of the velocity which exhibits a sharp front in the vicinity of x = 1 at early times, and, as a result of pronounced viscous action in this region, a smeared front at later times. The solution of the ED formulation for Re = 103 in the neighborhood of x = 1 exhibits oscillations due to its inability to accurately represent the large spatial gradients and incorporate them into the numerical scheme in a stable fashion. However, these oscillations are eliminated by generating the same number of elements with the Chebyshev polynomials so that the nodes cluster in the vicinity of x = 1.


Re = 102 ED TD TDC

0.8

0.4

.8 t =0 .2 t=1

u

0.6

t=

0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

x (a) 1.2 Re = 10 3

1.0

ED TD TDC

u

0.8

t=0

.4 .8 t =0 .2 t= 1

0.6 0.4 0.2

ν = 0.001

0.0 0.0

0.2

0.4

0.6

0.8

1.0

x (b)

Figure 12: Numerical solutions for velocity of Burgers’ example 1. 8.2 Example 2 of Burgers’ equation The second example is an initial discontinuous wave form which is allowed to diffuse into a continuous wave form, while at the same time being propagated in time along the x-direction. The initial and boundary conditions are given by u(x, 0) =

1, 0,

u(−∞, t) = 1,

x≤0 x>0

(59)

u(∞, t) = 0

The exact solution, obtained by the Cole–Hoph transform, is given by Lighthill [45]. The computational domain is discretized into 100 elements and, for the flow with Re = 10, a time step of 0.025 is used, while, for Re = 102 , a time step of 0.01 is used. The plots of the numerical and exact solutions are presented in Fig. 13a and b for Re = 10 and Re = 102 . Although the flow domain for this example is infinitely extensive, the numerical simulations are carried out using finite domains: −1.5 ≤ x ≤ 2.5 for the Re = 10 and −0.7 ≤ x ≤ 1.3 for the case Re = 102 , so that


27

Re=10 Exact ED,TD & TDC

0.8 0.6

.0 t=1

.6 t=0

u

.2 t=0

0.4 0.2 0.0 -0.5

0.0

0.5

1.0

1.5

x

(a) 1.0

Re=102 Exact ED,TD&TDC

0.8

u 0.4

t=1.0

t=0.6

t=0.2

0.6

0.2 0.0 -0.2

0.0

0.2

0.4

0.6

0.8

x (b)

Figure 13: Numerical solutions for velocity of Burgers’ example 2. the exact solutions at the end nodes satisfy the specified boundary conditions at x = −∞ and x = ∞ throughout the simulation times. All the numerical solutions are identical, and that is the reason why only the solutions from one formulation are presented alongside the exact solutions. The L2 norm estimates of the error between the numerical and exact solutions are the same for the three formulations when Re = 10, and differ marginally when Re = 102 , as indicated in Fig. 14. That figure shows that the ED formulation is most superior in accuracy, followed by the TD formulation. 8.3 Example 3 of Burgers’ equation The third example, earlier employed by Varoglu and Finn [48] and Kakuda and Tosaka [30] to verify their numerical solutions, has the following initial and boundary conditions   x6 u(−100, t) = 1,

u(20, t) = 0


Error

1.0 0.8 0.6

Re=102 ED TD TDC

0.4 0.2 0.2

0.4

0.6

0.8

1.0

Time

Figure 14: Error plots of numerical schemes of Burgers’ example 2.

1.1

1.1

Re=10

Re=1 FEM ED TD TDC

0.9

0.7

u

0.5

t=

0.3

2

0.3

t=4

0.5

t=2

4 t=

u

0.7

FEM ED TD TDC

0.9

0.1

0.1

-0.1

-0.1 2.0

4.0

6.0

8.0

10.0

5

12.0

6

7

8

9

x

x

(b)

(a) 1.1 0.9

t=2

0.5

t=4

t=1

u

0.7

Re=102

0.3

FEM ED TD TDC

0.1 -0.1 5

6

x

7

8

(c)

Figure 15: Numerical solutions for velocity of Burgers’ example 3.

The solutions of the three formulations for the third example are compared with the finite element solutions of Varoglu and Finn [48] for values of Re = 1, 10, and 102 in Fig. 15a, b and c, respectively. There is good agreement between the finite element solutions and ED solutions for all three Reynolds number values. However, the TD and TDC formulations produce, for the case Re = 102 , a velocity front that advances faster than that of the FEM and ED formulations.


29

9 Conclusion This chapter has presented some of the computational approaches currently being implemented along the lines of the boundary element theory for the solution of transient diffusion–convection problems. The dual nature of the diffusion–convection equation as a parabolic–hyperbolic equation presents many interesting numerical challenges, especially for convection-dominant transport. Formulations that are based on nontransient and transient fundamental solutions have been applied. The nontransient fundamental solutions tend to be simpler in expression and easier to manipulate when evaluating the boundary and domain integrals. In particular, that based on the ED operator, referred to as the ED formulation in this chapter, has a lot of appeal. Efforts at achieving a boundary-only formulation for the diffusion– convection problem are continuing. The DRM has scored some successes in this direction. However, when applied in the classical manner of implementing BEM, it does not do well for the convection-dominant problem. Implementation of DRM in a Green element sense shows more promising results. Domain decomposition approaches in implementing the boundary element method arose largely because of the domain integrations that have to be evaluated. Starting from the division of the domain into subdomains that gave rise to blockbanded matrices, there are now full domain discretization methods to which the GEM belongs. The GEM has provided additional computational capability of handling heterogeneous and nonlinear transport. The nonlinear diffusion–convection equation, known as the Burgers’ equation, has also been solved with three boundary element formulations. They are timemarching schemes which update the solution at an earlier time to new values at the current time. The nonlinear nature of the problem requires iteration of the solution to be done at every time level by the three formulations. The ED formulation has an advantage over the other two formulations in terms of accuracy, but uses about twice the computing time of the TD formulation that utilizes the least amount of computing time.

References [1]

Cleary, R.W., Groundwater Pollution and Hydrology: Mathematical Models and Computer Programs, Report No. 78-WR-15, Water Resources Program, Princeton University, 1978. [2] van Genuchten, M. Th. & Alves, W.J., Analytical solutions of the onedimensional convective–dispersive solute transport equation, Technical Bulletin No. 1661, US Department of Agriculture, 1982. [3] Peaceman, D.W. & Rachford, H.H.Jr., Numerical calculation of multidimensional miscible displacement. Soc. Petrol. Eng. J., 2, pp. 327–339, 1962. [4] Price, H.S., Cavendish, J.C. & Varga, R.S., Numerical methods of higherorder accuracy for diffusion–convection equations. Soc. Petrol Eng. J., pp. 293–300, 1968.

30 Domain Decomposition Techniques for Boundary Elements [5]

[6]

[7] [8] [9]

[10]

[11]

[12] [13]

[14] [15] [16]

[17] [18]

[19]

[20] [21]

Spalding, D.B., A novel finite difference formulation for differential equations involving both first and second derivatives. Int. J. Num. Meth. Eng., 4, pp. 551–559, 1972. Anderson, M.P., Using models to simulate the movement of contaminant through groundwater flow systems. CRC Crit. Rev. Environ. Contr., 9(2), pp. 97–156, 1979. Zienkiewicz, O.C. & Taylor, R.L., Chapter 12, The Finite Element Method, McGraw-Hill: London, pp. 438–505, 1991. Gray, W.G. & Pinder, G.F., An analysis of the numerical solution of the transport equation. Water Resources Res., 12(3), pp. 547–555, 1976. Heinrich, J.C., Huyakorn, P.S., Zienkiewicz, O.C. & Mitchell, A.R., An ‘upward’ finite element scheme for two-dimensional convective transport equation. Int. J. Num. Meth. Eng., 11, pp. 131–143, 1977. Sun, N-Z. & Yeh, W.W-G., A proposed upstream weight numerical method for simulating pollutant transport in groundwater. Water Resources Res., 19(6), pp. 1489–1500, 1983. Brooks, A.N. & Hughes, T.J.R., Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis or incompressible Navier–Stokes equation. Comp. Math. Appl. Mech. Eng., 32, pp. 199–259, 1982. Dick, E., Accurate Petrov-Galerkin methods for transient convective diffusion problems. Int. J. Num. Meth. Eng., 19, pp. 1425–1433, 1983. Brebbia, C.A. & Skerget, P., Diffusion–advection problems using boundary elements. Proc. 5th Int. Conf. on Finite Elements in Water Resources, Burlington Vermont, USA, pp. 747–768, 1984. Taigbenu, A.E., A new boundary element formulation applied to unsteady aquifer problems, PhD Thesis, Cornell University, USA, 1985. Taigbenu, A.E. & Liggett, J.A., An integral solution for the diffusion– advection equation. Water Resources Res., 22(8), pp. 1237–1246, 1986. Liggett, J.A. & Taigbenu, A.E., Calculation of diffusion, advection–diffusion and Boussinesq flow by integral methods. Proc. VI Int. Conf. on the Finite Element Method in Water Resources, Lisbon, Portugal, pp. 723–733, 1986. Nardini, D. & Brebbia, C.A., A new approach to free vibration analysis using boundary elements. Applied Math. Modelling, 7, pp. 157–162, 1983. Wrobel, L.C., Brebbia, C.A. & Nardini, D., The dual reciprocity boundary element formulation for transient heat conduction. Proc. VI Int. Conf. on Finite Elements in Water Resources, pp. 801–812, 1986. Aral, M.M. & Tang, Y., A boundary-only procedure for transient transport problems with or without first-order chemical reaction. Applied Math. Modelling, 13, pp. 130–137, 1989. Banerjee, P.K. & Butterfield, R., Boundary Element Methods in Engineering Science, McGraw-Hill: London, UK, 1981. Taigbenu, A.E., The Green element method. Int. J. Num. Meth. Eng., 38, pp. 2241–2263, 1995.


31

[22] Taigbenu, A.E., The Green Element Method, Kluwer Academic Press: Boston, 1999. [23] Popov, V. & Power, H., The DRM-MD integral equation method: an efficient approach for the numerical solution of domain dominant problems. Int. J. Num. Meth. Eng., 44, pp. 327–353, 1999. [24] Skerget, L., Hribersek, M. & Kuhn, G., Computational fluid dynamics by boundary-domain integral method. Int. J. Num. Meth. Eng., 46, pp. 1291– 1311, 1999. [25] Ikeuchi, M. & Onishi, K., Boundary element solutions to steady convective diffusion equations. Applied Math. Modelling, 7, pp. 115–118, 1983. [26] Zabaras, N. & Mukherjee, S., An analysis of solidification problems by the boundary element method. Int. J. Num. Meth. Eng., 24, pp. 1879–1900, 1987. [27] Young, D.L., Yang, S.K. & Eldho, T.I., Solution of the Navier–Stokes equations in velocity–vorticity form using Eulerian–Lagrangian boundary element method. Int. J. Num. Meth. Fluids, 34, pp. 627–650, 2000. [28] Young, D.L., Wang, Y.F. & Eldho, T.I., Solution of the advection–diffusion equation using the Eulerian–Lagrangian boundary element method. Engineering Anal. Boundary Elements, 24, pp. 449–457, 2000. [29] Taigbenu, A.E., Three Green element formulations for Burgers’ equation. Hybrid Methods in Engineering, 2002 (in press). [30] Kakuda, K. & Tosaka, N., The generalized boundary element approach to Burgers’ equation. Int. J. Num. Meth. Eng., 29, pp. 245–261. 1990. [31] Ahmad, S. & Banerjee, P.K., A new method in vibration analysis by BEM using particular integrals. J. Eng. Mech., 113, pp. 682–695, 1986. [32] Partridge, P.W., Brebbia, C.A. & Wrobel, L.C., The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications: Southampton and Elsevier Applied Science: New York, 1992. [33] Stone, H.L. & Brian, P.L.T., Numerical solution of convective transport problems. A. I. CH. E. J., 9(5), pp. 681–688, 1963. [34] Leendertse, J.J., Aspects of a Computational Model for Long-Period WaterWave Propagation, Rep. RM-5294-PR, The Rand Corp.: Santa Monica, CA, 1967. [35] Leendertse, J.J., A Water-quality Simulation Model for Well-Mixed Estuaries and Coastal Seas. 1: Principles of Computation, Rep. RM-6230-RC, The Rand Corp.: Santa Monica, CA, 1970. [36] Gray, W.G. & Lynch, D.R., Time-stepping schemes for finite element tidal model computations, Surface Flow, ed. W.G. Gray, CML Pub. Ltd.: Southampton, pp. 1–14, 1984. [37] Taigbenu, A.E., Numerical stability characteristics of a Hermitian Green element model for the transport equation. Engineering Analysis with Boundary Elements, 22(2), pp. 161–165, 1998. [38] Taigbenu, A.E., Three Green element models for the diffusion–advection equation and their stability characteristics. Engineering Analysis with Boundary Elements, 23, pp. 577–589, 1999.


[40]

[41]

[42] [43] [44] [45]

[46]

[47]

[48]

Ogata, A. & Banks, R.B., A solution of the differential equation of longitudinal dispersion in porous media. U.S. Geol. Surv. Prof. Pap. 411-A., 1976. Yeh, G.T., Solution of groundwater flow equations using an orthogonal finite element scheme. Proc. of Int. AMSE Conf. on Modelling and Simulations, Tassin, France, pp. 219–351, 1983. Burgers, J.M.A., Mathematical model illustrating the theory of turbulence. Advances in Applied Mech., Vol. 1, eds. R. von Mises & T. von Karman, Academic Press: New York, pp. 171–199, 1948. Hopf, E., The partial differential equation ut + uux = µxx . Comm. Pure Appl. Math., 3, pp. 201–230, 1950. Cole, J.D., On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of App. Math., 23, pp. 225–236, 1951. Benton, E.R. & Platzman, G.W., A table of solutions of the one-dimensional Burgers’ equation. Quart. J. App. Math., pp. 195–212, 1972. Lighthill, M.J., Viscosity effects in sound waves of finite amplitude. Surveys in Mechanics, eds. G.K. Batchelor & R.M. Davis, Cambridge University Press: Cambridge, 1956. Fletcher, C.A.J., Burger’s equation: a model for all reasons. Num. Sol. of Partial Differential Equations, ed. J. Noye, North-Holland Pub. Comp.: Amsterdam, pp. 139–225, 1982. Taigbenu, A.E. & Onyejekwe, O.O., A mixed Green element formulation for transient Burgers’ equation. Int. J. Num. Meth. Fluids, 24, pp. 563–578, 1997. Varoglu, E. & Finn, W.D.L., Space-time finite elements incorporating characteristics for the Burgers’ equations. Int. J. Num. Meth. Eng., 16, pp. 171–184, 1980.

CHAPTER 2 Viscous compressible fluid dynamics L. Škerget & M. Hriberšek Faculty of Mechanical Engineering, University of Maribor, Smetanova, Maribor, Slovenia.

Abstract The development of boundary element methods for computation of fluid flow is predominantly focused on the flow of incompressible fluids. This assumption provides a good approximation for a wide variety of engineering applications, but to achieve a better representation of physical phenomena, a compressible fluid flow must be taken into account. The goal of the chapter is, first, to give an in depth explanation for the transformation of Navier–Stokes equations for compressible fluids into its velocity–vorticity formulation equivalent. In addition, the pressure equation for the velocity–vorticity formulation is derived. This is followed by integral transformations of the governing equations based on the use of a parabolic diffusion fundamental solution. Section 8 explains discrete models, including the subdomain type discretisation. As a test example, natural convection in a differentially heated tall enclosure is presented.

1 Introduction Most of the studies dealing with transport phenomena are based on presuming that the fluid is incompressible and viscous, where the mass density is a constant quantity, and the velocity does not depend on the mass density. Pressure in the incompressible fluid flow model is not a thermodynamic state variable, but simply a force in the linear momentum balance equation. Such an easy rheological model for the fluid is suitable for modelling of slow flows, or flows with small pressure and temperature gradients or no chemical reaction and where, therefore, the mass density differences may be neglected.

34 Domain Decomposition Techniques for Boundary Elements In this chapter we will deal with the real compressible viscous fluid with restriction to the subsonic flows where the difference in mass density significantly influences the vector velocity field. However, there are no shock waves and no sudden sharp changes in the values of the field functions. In the model of compressible fluid the pressure is a thermodynamic quantity p = p(ρ , T ).

2 Conservation equations The analytical description of the motion of a continuous viscous compressible media is based on the conservation of mass, momentum, and energy with associated rheological models and equations of state [1]. The present development is focused on the laminar flow of compressible isotropic fluid in solution region bounded by boundary . The field functions of interest are the velocity vector field vi (rj , t), the scalar pressure field p(rj , t), the temperature field T (rj , t) and the field of mass density ρ(rj , t). The mass, momentum, and energy conservation equations are given by ∂ρvj ∂ρ = 0, + ∂t ∂xj ∂σij ∂ρvi vj ∂ρvi = + ρgi , + ∂t ∂xj ∂xj ∂qj ∂ρT vj ∂ρT cp =− + S + , + ∂t ∂xj ∂xj

(1) (2) (3)

in the Cartesian frame xi , where ρ and cp denote variable fluid mass density and isobaric specific heat capacity per unit mass, t is time, gi is gravitational acceleration vector, σij represents the components of the total stress tensor, qi is specific heat diffusion flux, while S stands for the heat source term and is a Rayleigh viscous dissipation function which stands for the conversion of mechanical energy to heat and acts as an additional heat source. With the definition of Stokes material derivative of the variable (·), as given by D (·) /Dt = ∂ (·) /∂t + vk ∂ (·) /∂xk , the Navier–Stokes equations are given by ∂vj 1 Dρ = 0, + ρ Dt ∂xj

(4)

∂σij Dvi + ρgi , = Dt ∂xj

(5)

∂qj DT + S + , =− Dt ∂xj

(6)

ρ c

where c denotes specific heat per unit volume, c = cp ρ. If the incompressible fluid model is applied, than the total rate of mass density variation is identically zero Dρ = 0, (7) Dt

Viscous Compressible Fluid Dynamics

35

and the mass conservation is satisfied with a simple restriction condition for the solenoidal velocity vector ∂vj = 0. (8) ∂xj The set of eqns (4)–(6) represents an unclosed system of partial differential equations that has to be closed and solved in conjuction with appropriate rheological equations, equations of state and boundary, as well as initial conditions of the problem. The Cauchy total stress tensor σij can be decomposed into a pressure contribution p plus an extra deviatoric stress tensor field function τij σij = −pδij + τij ,

(9)

where δij is the Kronecker delta. The Rayleigh viscous dissipation term is given by = τij

∂vi . ∂xj

(10)

3 Linear gradient type of constitutive models In the general Reiner–Rivlins rheological model of viscous shear fluid the deviatoric stress tensor reads as τij = αδij + β ε˙ ij + γ ε˙ ik ε˙ kj . (11) By considering the equalities α = −2η˙εii /3 and β = 2η, the following form of the constitutive model for compressible viscous shear fluid may be written 2 τij = 2η˙εij − η Dδij , 3

(12)

where D = div v = ε˙ ii represents the divergence of the velocity field. Considering the eqn (10) we may write ∂vi ∂vi ∂vj ∂vi 2 =η − ηD2 . + (13) ∂xj ∂xj ∂xi ∂xj 3 In the case of intensive unsteady heat transfer, it is important to take into account a terminal velocity of a moving temperature front, namely qi = −k

∂T ∂qi −λ , ∂xi ∂t

(14)

where material constants k and λ are the heat conductivity and the heat relaxation time. For most heat transfer problems of practical importance, the simplification known as the Fourier law of heat diffusion is accurate enough, namely qi = −k

∂T . ∂xi

(15)

36 Domain Decomposition Techniques for Boundary Elements

4 Primitive variables formulation Combining constitutive models for stress tensor or momentum flux tensor and heat diffusion flux, eqns (12) and (15) in conservation eqns (5) and (6), the following system of nonlinear equations is developed ∂vj 1 Dρ =D=− , ∂xj ρ Dt Dvi ∂p ∂ 2 ρ 2η ε˙ ij − η Dδij − = + ρgi , Dt ∂xj 3 ∂xi DT ∂T ∂ c k + S + . = Dt ∂xj ∂xj

(16) (17) (18)

Because of analytical reasons in developing the velocity–vorticity formulation of governing equations, the operator div τ is worth writing in an extended form ∂ 2 η Lij + Lji − Dδij div τ = ∂xj 3 =2

∂Lij ∂Lji ∂η 2 ∂η D ε˙ ij + η +η − , ∂xj ∂xj ∂xj 3 ∂xi

(19)

where the term ∂Lji /∂xj is equal to the grad D, due to the continuity equation, div v = D. Thus the following relation is valid ∂Lji ∂vj ∂ ∂D = = . (20) ∂xj ∂xi ∂xj ∂xi ˙ ij , one can By considering the equality between tensor field functions ε˙ ij = Lij − derive an expression div τ = 2

∂L ∂η ∂η ∂D 2 ∂η D ˙ ij + η ij + η Lij − 2 − . ∂xj ∂xj ∂xj ∂xi 3 ∂xi

(21)

Finally, substituting the equalities ∂Lij ∂ 2 vi = ∂xj ∂xj ∂xj

and

2

∂η ∂η ˙ ij = −eijk ωk , ∂xj ∂xj

(22)

the first extended form of div τ may be formulated as follows 2 div τ = η v + grad η × ω + 2 grad v · grad η + η grad D − grad (ηD). (23) 3 By the derivation of the second extended form we also take into account the equalities div (grad v ) = grad D − rot ω = v,

(24)

rot (ηω) = η rot ω −ω × grad η,

(25)


37

obtaining the second extended form 4 + 2grad η × ω + 2grad v · grad η + grad (ηD) div τ = −rot(ηω) 3 − 2D grad η.

(26)

Considering the second extended form of the term div τ , eqn (26), the momentum equation (17) may be written in a form suitable to derive the velocity–vorticity formulation, e.g. in a vector form ρ

4 D v = −rot(ηω) + 2grad η × ω + 2grad v · grad η + grad (ηD) − 2D grad η Dt 3 − grad p + ρ g, (27)

or in Cartesian tensor formulation ρ

Dvi ∂ηωk ∂η ∂η ∂vi 4 ∂ηD ∂η + 2eijk ωk + 2 + − 2D = −eijk Dt ∂xj ∂xj ∂xj ∂xj 3 ∂xi ∂xi −

∂p + ρgi . ∂xi

(28)

Representing the material properties of the fluid, e.g. the dynamic viscosity η, heat conductivity k, the specific heat per unit volume c, and the mass density ρ, as a sum of a constant and a variable part η = ηo + ! η,

k = ko + ! k,

c = co +! c,

ρ = ρo + ρ !,

(29)

then the momentum and energy equations (27) and (18) can be written in analogy to the basic conservation equations formulated for the constant material properties 1 ρ 1 D v ×ω = −νo ∇ − ∇ p + g + f m , Dt ρo ρo ρo S Sm DT = ao T + + + , Dt co co co

(30) (31)

with the pseudo-body force term f m and the pseudo-heat source term S m introduced into the momentum equation (30) and into the energy equation (31), respectively, capturing the variable material property effects, and given by the expressions × (! η×ω v · ∇ η + 4 ∇(ηD) −ρ ηω) + 2∇ +2∇ − 2D∇η !a , f m = −∇ 3

(32)

or in tensor notation form as fim = −eijk

∂! η ωk ∂! η ∂η ∂vi 4 ∂ηD ∂η + 2eijk ωk + 2 + − 2D −ρ !ai , ∂xj ∂xj ∂xj ∂xj 3 ∂xi ∂xi

(33)

38 Domain Decomposition Techniques for Boundary Elements which for plane flow problems reduces to the relation fim = −eij

∂! ηω ∂! η ∂η ∂vi 4 ∂ηD ∂η + 2eij ω+2 + − 2D −ρ !ai , ∂xj ∂xj ∂xj ∂xj 3 ∂xi ∂xi

(34)

while the pseudo-heat source term is given by an expression DT ! ) −! S m = ∇( k ∇T c , Dt

(35)

in which the kinematic viscosity is νo = ηo /ρo , the heat diffusivity ao = ko /co , and the inertia acceleration vector is a = D v /Dt.

5 Velocity–vorticity formulation The divergence and the curl of a vector field function are the basic differential operators in vector analysis. Written for the velocity field vi (rj , t) they represent a local expansion rate D div v =

∂vj = D, ∂xj

1 Dρ ρ Dt

(36)

∂ωj = 0, ∂xj

(37)

D=−

and the local vorticity vector ωi (rj , t) rot ω = eijk

∂vk = ωi , ∂xj

representing a solenoidal vector by definition, the fluid motion computation procedure may be partitioned into its kinetics and kinematics.Vorticity definition equation (37) and the continuity equation (36) represent a differential description of the kinematic aspect of viscous compressible fluid motion. The kinematics deals with the relationship among the velocity field at any given instant of time, and the vorticity and mass density fields at the same instant. If the velocity and mass density fields are known in the solution domain, the corresponding vorticity field can be established through eqn (37). For the known vorticity and mass density field functions, the corresponding velocity vector can be determined by solving eqns (36) and (37), provided that appropriate boundary conditions for the velocity are prescribed. The kinetic aspect of the fluid motion is governed by the vorticity transport equation describing the redistribution of the vorticity in the fluid domain through the various transport phenomena. 5.1 Velocity vector equation By applying the curl operator to the vorticity definition equation (37) ×ω × (∇ × v ) = ∇( ∇ · v ) − ∇ =∇ v,

(38)


39

and by using the continuity equation (36), the following vector elliptic Poisson equation for the velocity vector is obtained ×ω = 0,

v+∇ − ∇D

(39)

∂ 2 vi ∂ωk ∂D + eijk − = 0. ∂xj ∂xj ∂xj ∂xi

(40)

or in tensor notation form

The eqn (40) represents the kinematics of a compressible fluid motion, expressing the compatibility and restriction conditions among velocity, vorticity, and mass density field functions. To accelerate the convergence and the stability of the coupled velocity, vorticity, and pressure computational iterative scheme, the false transient approach may be used for the elliptic velocity Poisson equation. By adding the artificial accumulation term, the eqn (40) can be written in its parabolic diffusion form ∂ 2 vi ∂ωk 1 ∂vi ∂D − − = 0, + eijk ∂xj ∂xj α ∂t ∂xj ∂xi

(41)

where α is the relaxation parameter controlling the diffusion and accumulation processes. It is obvious that the governing velocity equation is exactly satisfied only at the steady state of the artificial transient (t → ∞), when the false time derivative vanishes. For the two-dimensional plane motion the eqn (41) reduces to the relation ∂ 2 vi ∂ω 1 ∂vi ∂D − − = 0. + eij ∂xj ∂xj α ∂t ∂xj ∂xi

(42)

5.2 Vorticity transport equation Substituting the convection term by the following equality v = 1 ∇v 2 − v × ω, ( v · ∇) 2

(43)

the momentum equation (30) can be rewritten as ∂ v ×ω + ρ g + 1 f m , − ∇h +ω × v = −νo ∇ ∂t ρo ρo

(44)

where h = p/ρo + v 2 /2 is the total pressure. Finally, the vorticity transport equation is obtained by applying the curl differential operator to both the sides of eqn (44) ∂ω × ( ×∇ ×ω × ρ g + 1 ∇ × f m , =∇ v × ω) − νo ∇ +∇ ∂t ρo ρo

(45)

40 Domain Decomposition Techniques for Boundary Elements bearing in mind the following vector relations ×∇ ×ω ∇ · ω) ∇ = ∇( − ∇ 2 ω,

(46)

× ( · ω) ω · v ) + (ω v, ∇ v × ω) = v (∇ − ( v · ∇) − ω( ∇ · ∇)

(47)

and

where div ω = 0 and div v = D due to the vorticity definition and continuity equation, rendering Dω 1 1 m v − ωD + (ω · ∇) + ∇ = νo ∇ 2 ω × ρ g + ∇ ×f , Dt ρo ρo

(48)

or in Cartesian tensor notation form ∂f m Dωi ∂ 2 ωi ∂vi ∂ρgk 1 1 + ωj − ωi D + eijk + eijk k . = νo Dt ∂xj ∂xj ∂xj ρo ∂xj ρo ∂xj

(49)

On the other hand, considering the vector properties div ω = 0 and div v = D, another form of eqn (49) may be written, namely ∂f m ∂ωj vi ∂vj ωi ∂ 2 ωi ∂ρgk 1 1 ∂ωi = νo + + eijk + eijk k . + ∂xj ∂xj ∂t ∂xj ∂xj ρo ∂xj ρo ∂xj

(50)

For the two-dimensional plane flow, the vorticity vector ω(r j , t) has just one component perpendicular to the plane of the flow, e.g. ω = (0, 0, ω), and it can be treated as a scalar quantity ω. The vortex-twisting transport term is identically v = 0, reducing the vector vorticity equation to a scalar one for zero due to (ω · ∇) vorticity ω ∂ 2ω 1 ∂ρgi 1 ∂f m ∂ω ∂vj ω = νo − eij − eij i . + ∂t ∂xj ∂xj ∂xj ρo ∂xj ρo ∂xj

(51)

The vorticity transport statement equation (50) is a highly nonlinear partial dif j , t) in ferential equation due to the products of velocity v (rj , t) and vorticity ω(r convective and in stretching-twisting terms, and the velocity field function is kinematically dependent on vorticity. Because of this inherent nonlinearity, the kinetics of general viscous motion, and what is drastically true for high Reynolds number flows, represents greater numerical efforts than that considered by the kinematics. Due to the buoyancy force and variable material property terms acting as additional temperature and pressure dependent vorticity source terms, the vorticity transport equation is coupled to the energy and pressure equations, making the numerical solution procedure very severe. The dilatation and the vortical part of the flow, D and ω field functions, have to be under relaxed to achieve the convergence of the numerical solution procedure.


41

6 Pressure equation In compressible fluid dynamics the pressure is a temperature and mass density dependent thermodynamic quantity. Let us start with the momentum equation (30) for the pressure gradient grad p p = f p = −ηo ∇ ×ω ∇ − ρo a + ρ g + f m .

(52)

In an indicial tensor formulation this equation takes the form ∂p ∂ωk = fpi = −ηo eijk − ρo ai + ρgi + fim , ∂xi ∂xj

(53)

which simplifies in planar flow case to the following dependence ∂p ∂ω = fpi = −ηo eij − ρo ai + ρgi + fim , ∂xi ∂xj

(54)

where a = D v /Dt represents the acceleration vector. In the vector function f p the inertia, diffusion, and gravitational effects are incorporated as well as the effects of nonlinear material properties. To derive the pressure equation, depending on known field and material functions, the divergence of eqn (52) should be calculated, resulting in the elliptic Poisson pressure equation · f p = 0,

p − ∇ (55) or in a indicial tensor notation as ∂fpi ∂p − = 0. ∂xi ∂xi ∂xi

(56)

The Neumann boundary conditions for the pressure equation may be determined for the whole solution domain and the following relation is valid ∂p = fp · n ∂n

on .

(57)

By adding the false pressure transient term, the eqn (56) may be formulated as a false parabolic diffusion equation ∂p 1 ∂p ∂fpi − − = 0, ∂xi ∂xi α ∂t ∂xi

(58)

where α is a relaxation parameter. It is obvious that the elliptic pressure equation is exactly satisfied only at the false transient steady state, when the false pressure accumulation term vanishes.


7 Boundary-domain integral equations 7.1 Preliminary comments The unique property and advantage of the boundary element method originates from the application of the Green fundamental solutions as particular weighting functions [2, 3]. Since fundamental solutions only consider the linear transport phenomenon, an appropriate selection of a linear differential operator L [·] is of prime importance in establishing a stable and accurate singular integral representation corresponding to the original differential conservation equation. All differential conservation models of different flow field functions can be written in the following general form L [u] + b = 0,

(59)

where the operator L [·] can be either elliptic or parabolic, u(rj , t) is an arbitrary field function, and b(rj , t) is applied for nonlinear transport effects or pseudo-body forces. 7.2 Integral representation of flow kinematics Employing the linear parabolic diffusion differential operator ∂ 2 (·) ∂ (·) , − ∂xj ∂xj ∂t

L [·] = α

(60)

the following relationship may be obtained L [vi ] + bi = α

∂ 2 vi ∂vi − + bi = 0. ∂xj ∂xj ∂t

(61)

The singular boundary-domain integral representation for the velocity vector can be formulated by using the Green theorems for scalar functions or weighting residuals technique rendering the following vector integral formulation, e.g. written in the time incremental form for the time step t = tF − tF−1 c (ξ ) v (ξ , tF ) + α +

tF

tF−1

tF

tF

tF−1

v q dtd = α

tF−1

bu dtd +

v F−1 uF−1 d,

∂ v u dtd ∂n (62)

where u stands for the parabolic diffusion fundamental solution of the equation α

∂ 2 u ∂u + + δ(ξ , s)δ(tF , t) = 0, ∂xj ∂xj ∂t

(63)


43

given by the expression u =

1 2 e−r /4ατ , d/2 (4π ατ )

(64)

and q is its normal derivative, e.g. q = ∂u /∂n = q · n . The vector flux variable is defined as q = grad u or qi = ∂u /∂xi , where (ξ , tF ) and (s, t) are used for the source and reference field points, respectively, d is the dimension of the problem in τ = tF − t. Assuming a constant variation of all field functions within the individual time increment t, the time integrals in eqn (62) may be evaluated analytically, e.g. one can substitute tF tF U =α u dt, Q = α q dt, (65) tF−1

tF−1

and the integral representation, eqn (62), may be rewritten as ∂ v 1 c (ξ ) v (ξ , tF ) + v Q d = U d + bU d α ∂n + v F−1 uF−1 d.

(66)

Equating the pseudo-body force with the rotational and compressible fluid flow part ×ω b = α∇ − α ∇D,

(67)

renders an integral formulation

c (ξ ) v (ξ , tF ) +

v Q d =

∂ v × ω)U U d + (∇ d ∂n − (∇D)U d + v F−1 uF−1 d.

(68)

The pseudo-body force domain integral involves derivatives of the vorticity and dilatation fields, which can be eliminated by using the Gauss divergence theorem. Applying the vector equalities × ω)U × (ωU , (∇ =∇ ) + ω × ∇U

(69)

, (∇D)U = ∇(DU ) − D∇U

(70)

44 Domain Decomposition Techniques for Boundary Elements can be equated to flux vector Q = ∂U /∂xi , the domain where the grad U or ∇U i integrals can be written as sums × ω)U × (ωU d (∇ d = ω ×Q ∇ )d + d, =− ω × n U d + ω ×Q (71) d ∇(DU )d − (∇D)U d = DQ d. = D nU d − DQ (72)

The following form of the integral representation for the kinematics is obtained as follows ∂ v d c (ξ ) v (ξ , tF ) + v Q d = × n U d + ω ×Q U d − ω ∂n − D nU d + DQ d + v F−1 uF−1 d, (73)

or in tensor symbolic notation as ∂vi c (ξ ) vi (ξ , tF ) + vi Q d = ωj nk U d + eijk ωj Qk d U d − eijk ∂n − Dni U d + DQi d + vi,F−1 uF−1 d. (74)

The kinematics of plane motion is given by two scalar equations as follows ∂vi c (ξ ) vi (ξ , tF ) + vi Q d = U d + eij ωnj U d − eij ωQj d ∂n − Dni U d + DQi d + vi,F−1 uF−1 d, (75)

or in an extended form for the x and y components of the velocity vector ∂vx c (ξ ) vx (ξ , tF ) + vx Q d = ωQy d U d + ωny U d − ∂n − Dnx U d + DQx d + vx,F−1 uF−1 d, (76)

∂vy c (ξ ) vy (ξ , tF ) + vy Q d = U d − ωnx U d + ωQx d ∂n − Dny U d + DQy d + vy,F−1 uF−1 d. (77)


45

The eqn (74) represents the kinematics of viscous compressible fluid motion or the compatibility and restriction conditions among vorticity, velocity, and dilatation field functions in an integral form. The most important issue in numerical modelling of compressible fluid flow circumstances is to obtain a divergence-free final solution for the vorticity vector and to satisfy the mass conservation equation [4]. Thus, the kinematic integral representation should preserve the compatibility and restriction conditions for the velocity and vorticity field functions. By using additional compatibility and restriction requirements, e.g. ω = rot v and div v = D, a more convenient integral representation may be derived [5, 6]. Let us for a moment, due to derivation simplicity and clarity, focus our attention to the plane two-dimensional flow kinematics given by the eqn (75). By using the expressions for the velocity components’ normal derivatives, vorticity definition, and unit tangent and normal vector, e.g. ∂vi /∂n = ∂vi /xj nj , ω = eij ∂vj /∂xi = ∂vy /∂x − ∂vx /∂y, n = (nx , ny ), and t = (tx , ty ) = ( − ny , nx ), for i, j = 1, 2, and by applying the continuity equation (16), the following relation may be derived ∂vj ∂vi + eij ωnj − Dni = −eij . ∂n ∂t

(78)

Thus, the boundary integrals on the right-hand side of the eqn (75) can be rewritten, resulting in the following integral formulation

∂vj c (ξ ) vi (ξ , tF ) + vi Q d = −eij ωQj d U d − eij ∂t + DQi d + vi,F−1 uF−1 d.

(79)

The eqn (79) can be again reformulated as ∂vj U c (ξ ) vi (ξ , tF ) + vi Q d = +eij vj Qt d − eij d ∂t − eij ωQj d + DQi d + vi,F−1 uF−1 d,

(80)

where we called the tangential derivative of the fundamental solution Qt = ∂U /∂t, and by applying the Gauss theorem, the second boundary integral on the right-hand side of equation vanishes, e.g. as follows

∂vj U ∂vj U − ny + nx d ∂x ∂y ∂ ∂vj U ∂ ∂vj U = − + d ≡ 0, ∂y ∂x ∂x ∂y

∂vj U d = ∂t

(81)

46 Domain Decomposition Techniques for Boundary Elements resulting in the final integral representation for the two-dimensional plane kinematics vj Qt d − eij ωQj d c (ξ ) vi (ξ , tF ) + vi Q d = eij + DQi d + vi,F−1 uF−1 d. (82)

The boundary integral representation for the general flow situation can now be easily stated as, e.g. in the compact symbolic notation form for the cyclic combination of the indices, ijkij = 12312, c (ξ ) vi (ξ , tF ) + vi Q d = vk Qk ni − Qi nk d − vj (Qi nj − Qj ni )d + ωj Qk d − ωk Qj d + DQi d + vi,F−1 uF−1 d, (83)

or in the form of parabolic integral vector formulation d c (ξ ) v (ξ , tF ) + v Q d = Q × n × v d + ω ×Q + DQ d + v F−1 uF−1 d.

(84)

The vector integral statement, eqn (84), represents three scalar equations for individual x, y, z Cartesian coordinate directions, but only two of them are independent c (ξ ) vx (ξ , tF ) + vx Q d = vz (Qz nx − Qx nz )d − vy (Qx ny − Qy nx )d + ωy Qz d − ωz Qy d + DQx d + vx,F−1 uF−1 d, c (ξ ) vy (ξ , tF ) + vy Q d = vx (Qx ny − Qy nx )d − vz (Qy nz − Qz ny )d + ωz Qx d − ωx Qz d (85) + DQy d + vy,F−1 uF−1 d, c (ξ ) vz (ξ , tF ) + vz Q d = vy (Qy nz − Qz ny )d − vx Qz nx − Qx nz d + ωx Qy d − ωy Qx d + DQz d + vz,F−1 uF−1 d.


47

The boundary-domain integral representation, eqn (84), is completely equivalent to eqns (36) and (37), or eqn (40), together with velocity boundary conditions, expressing the kinematics, compatibility, and restriction conditions of a general viscous compressible fluid flow in the integral form. Boundary velocity conditions are included in the boundary integrals representing the contribution to the velocity field in the solution domain. The domain integral gives the contribution of the rotational and dilatation flow to the development of the velocity field. The last domain integral is due to the false initial conditions. A unique feature of the singular integral representation is that it enables the explicit determination of velocity components in the domain. This unique ability made it possible to confine the solution field to the vortical and dilatation region of the flow. Notice that for the irrotational and incompressible fluid flow the domain integrals vanish, and the kinematics of the potential fluid flow is given by the boundary integrals only. The compatibility between velocity, vorticity, and mass density field functions cannot exist for arbitrary specified vorticity and expansion rate distribution in a solution domain, since the vorticity field is kinematically restricted. Since the vorticity field in the interior of the fluid domain is determined uniquely from the solution of the vorticity transport equation (49), only the boundary vorticity values are subject to the kinematic restriction. In consequence, the vorticity boundary conditions are linked to the kinematic restriction given in the integral form within the domain integral. As was mentioned, the boundary vorticity values are expressed in the integral form within the domain integral, excluding a need for use of an appropriate approximative formula determining locally vorticity values on the boundary, which would bring some additional error in the numerical scheme employed [7]. Using this unique feature of global integral representation for boundary vorticity values, the vector equation (84) has to be written in its tangential form as follows

c (ξ ) n (ξ ) × v (ξ , tF ) + n (ξ ) × + n (ξ ) ×

v Q d = n (ξ ) ×

d + n (ξ ) × ω ×Q

+ n (ξ ) ×

× n × v d Q

d DQ

v F−1 uF−1 d,

(86)

in order to obtain an appropriate nonsingular implicit system of equations for unknown boundary vorticity or tangential velocity component values to the boundary. When the normal velocity component values to the boundary are unknown, the normal form of the mentioned equation has to be employed

× n × v d Q c (ξ ) n (ξ ) · v (ξ , tF ) + n (ξ ) · v Q d = n (ξ ) · d + n (ξ ) · DQ d + n (ξ ) · v F−1 uF−1 + n (ξ ) · ω ×Q d.

(87)

48 Domain Decomposition Techniques for Boundary Elements The eqns (86) and (87) basically represent the application of the boundary velocity conditions given for the normal and tangent velocity component to the boundary. For the closed cavity flow problem, where the boundary represents the nonpermeable and noslip solid wall on which v = 0, the eqn (84) reduces to c (ξ ) v (ξ , tF ) = ω × Q d DQ d + v F−1 uF−1 d. (88)

For such flow circumstances, the boundary integral vanishes, and the velocity field in the domain is simply given by a domain integral of the vorticity and the expansion rate fields. Finally, it should be stressed, that the parabolic kinematic integral representation given by eqn (84) is an equivalent to the parabolised version of the velocity equation (42). Notice, that for a large time increment value, the kernel function U reduces to an elliptic Laplace fundamental solution, e.g. t → ∞ and U → u , the domain integral due to the false initial conditions vanishes, and the integral representation (84) reduces to the integral representation for the elliptic velocity equation (41) as follows c (ξ ) v (ξ ) + v q d = ( q × n ) × v d + ω × q d + D q d,

(89) where u is now the elliptic Laplace fundamental solution and q = ∂u /∂n. 7.2.1 External flow kinematics Let us consider a body sunk in an infinite domain of viscous and compressible fluid flow. Velocity in the free stream v ∞ can be introduced in the elliptic kinematic formulation equation (89), splitting the domain by two separate boundaries, dividing the solid body from the fluid, and ∞ at a very large distance, where the free stream velocity applies. The contribution of the velocity field at the infinite distance, or the influence of the irrotational fluid motion to the development of the velocity field can be determined in the following way. The boundary ∞ should be a very large spherical surface of radius R, with the solid body in the middle. Integral statement equation (89) can now be written for the whole boundary denoted by + = + ∞ occupying the infinite solution domain c (ξ ) v (ξ ) + v q d = ( q × n ) × v d + ω × q d + D q d. +

+

(90) = n /4π R2 The contribution on ∞ can be evaluated using the value ∇u " # 1 1 1 lim − v∞ , − n · n v n × n × v ( ) ( ) ∞ ∞ d = − 2 4π R→∞ R2 ∞ R

(91)


49

and the boundary-domain integral equation of the external flow around the immersed body, which has the surface , becomes c (ξ ) v (ξ ) + v q d = ( q × n ) × v d + ω × q d + D q d + v ∞ .

(92) For the case of a nonrotating and impermeable solid surface (with v = 0), the above formulation equation (92) simplifies to c (ξ ) v (ξ ) = ω × q d + D q d + v ∞ , (93)

which may be recognised as the Biot–Savart’s law of induced velocity, which represents a relationship among the velocity, vorticity, and mass density field functions of an infinite domain. Since only the vorticity and dilatation distributions in a flow domain contribute to the velocity field, while the domain integrals in potential and incompressible flow parts vanish, e.g. (ω = 0), and (D = 0), the solution procedure or the discretisation has to be made only for the vortical and dilatation part of the whole domain. 7.3 Integral representation of flow kinetics Considering the kinetics in an integral representation, one has to take into account the parabolic diffusion convection character of the vorticity transport equation (50). With the use of the linear parabolic diffusion differential operator [8] L [·] = νo

∂ 2 (·) ∂ (·) − , ∂xj ∂xj ∂t

(94)

the vorticity equation can be formulated as a vector nonhomogeneous parabolic diffusion equation as follows L [ωi ] + bi = νo

∂ 2 ωi ∂ωi + bi = 0, − ∂xj ∂xj ∂t

(95)

with the integral representation written in a time increment form for a time step t = tF − tF−1 tF tF ∂ωi c (ξ ) ωi (ξ , tF ) + νo ωi q dtd = νo u dtd tF−1 tF−1 ∂n tF bi u dtd + ωi,F−1 uF−1 d, (96) + tF−1

where u stands for the parabolic diffusion fundamental solution, and α has to be replaced by νo . Assuming constant variation of all field functions within the

50 Domain Decomposition Techniques for Boundary Elements individual time increment, eqn (96) is written as follows ∂ωi 1 c (ξ ) ωi (ξ , tF ) + ωi Q d = U d + bi U d νo ∂n + ωi,F−1 uF−1 d.

(97)

The domain integral of the nonhomogeneous nonlinear contribution bi , represented as 1 ∂ bi = − ρo vj ωi − ρo ωj vi − ρeijk gk − eijk fkm , (98) ρo ∂xj includes the transport and source effects of the vorticity, and, therefore, we obtain the following parabolic integral representation for vorticity kinetics ∂ωi c (ξ ) ωi (ξ , tF ) + ωi Q d = U d ∂n 1 ∂ − ρo vj ωi − ρo ωj vi − ρeijk gk − eijk fkm U d ηo ∂xj + ωi,F−1 uF−1 d. (99)

The domain integral in eqn (99) incorporates space derivatives of field functions. The problem may be sorted out in various ways, i.e. using Gaussian theorem, the derivatives are transfered to the fundamental solution, and the following integral equation is obtained ∂ωi c (ξ ) ωi (ξ , tF ) + ωi Q d = U d ∂n 1 ρo vj ωi − ρo ωj vi − ρeijk gk − eijk fkm nj U d − ηo 1 + ρo vj ωi − ρo ωj vi − ρeijk gk − eijk fkm Qj d ηo + ωi,F−1 uF−1 d. (100)

The boundary-domain integral equation (100) expresses the space vorticty transport in an integral form. Vorticity diffusion is described by the first two boundary integrals, while the third boundary integral gives the terms representing the convective vorticity flux and transport across the surface due to twisting-stretching effect, which vanish for vn = 0 or v = 0 respectively, and the vorticity generation on the boundary due to buoyancy forces and variable material properties. The first domain integral gives the influence of the transport effects in the domain due to the convection, twisting-stretching, bouyancy forces, and variable material property values, while the second domain integral is due to the initial vorticity distribution effect on the development of the vorticity field in the next time interval.


51

7.3.1 Integral representation of plane flow kinetics Governing the two-dimensional plane vorticity equation (51) is a nonhomogeneous scalar parabolic diffusion–convection partial differential equation. Applying the linear parabolic diffusion differential operator equation (94), the plane vorticity equation can be treated as a scalar nonhomogeneous parabolic diffusion equation L [ω] + b = νo

∂ 2ω ∂ω + b = 0, − ∂xj ∂xj ∂t

(101)

where the nonhomogeneous term b stands for pseudo-body force or source term. Thus, the following boundary-domain integral statement corresponding to eqn (101) can be straightforwardly derived c (ξ ) ω (ξ , tF ) + νo +

tF

tF tF−1

bu dtd +

tF−1

tF

tF−1

ωq dtd = νo

∂ω u dtd ∂n

ωi,F−1 uF−1 d,

(102)

where u is now a parabolic diffusion two-dimensional plane fundamental solution. Equating the body force term b to the convection, buoyancy forces, and nonliner material properties b=−

1 ∂ ρo vj ω + ρeij gi + eij fim , ρo ∂xj

the following parabolic integral formulation can be written ∂ω c (ξ ) ω (ξ , tF ) + ωQ d = U d ∂n 1 ∂ − ρo vj ω + ρeij gi + eij fim U d + ωF−1 uF−1 d, ηo ∂xj

(103)

(104)

where a constant variation of all field functions within the individual time increment t = tF − tF−1 is assumed. Applying the Gaussian theorem to the first domain integral of eqn (104), the direction derivatives of vorticity and density are shifted to the Green function, resulting in a boundary-domain integral formulation without derivatives of field functions within the domain integral ∂ω c (ξ ) ω (ξ , tF ) + ωQ d = U d ∂n 1 − ρo vj ω + ρeij gi + eij fim nj U d ηo 1 + ωF−1 uF−1 d. (105) ρo vj ω + ρeij gi + eij fim Qj d + ηo

52 Domain Decomposition Techniques for Boundary Elements The eqn (105) represents plane vorticity transport in the integral form in a physically justified manner, showing a complete analogy with eqn (100) for space fluid flow, with an exception of the twisting-stretching transport term, which appears only in the three dimensional case. To stress the physics of the transport mechanism, the eqn (105) can be rewritten as ∂ω − ρo vn ω + ρgt + ftm U d ∂n 1 + ωF−1 uF−1 d, (106) ρo vj ω + ρeij gi + eij fim Qj d + ηo

c (ξ ) ω (ξ , tF ) +

1 ηo

ωQ d =

ηo

where vn , gt and ftm are the normal velocity, and the tangential gravity and nonlinear material source components, respectively, e.g. vn = v · n , gt = g · t = −eij gi nj , and ftm = f m · t = −eij fim nj . The boundary integrals describe the total vorticity flux on the boundary, due to molecular diffusion, convection, and vorticity generation by a tangential force and source terms. The first domain integral gives the influence of forced and natural convection, and nonlinear material behaviour, while the last domain integral is due to the initial vorticity distribution effect on the development of the vorticity field in a subsequent time interval. 7.4 Integral representation of pressure equation The pressure equation (57) is a parabolic equation, and, therefore, employing the linear parabolic diffusion differential operator L [·] = α

∂ 2 (·) ∂(·) − , ∂xj ∂xj ∂t

(107)

the following expression can be obtained ∂ 2p ∂p L p +b=α − + b = 0. ∂xj ∂xj ∂t

(108)

The corresponding singular parabolic integral representation is given by c (ξ ) v (ξ , tF ) + α +

tF

tF−1

tF

tF

tF−1

pq dtd = α

tF−1

bu dtd +

pF−1 uF−1 d,

∂p u dtd ∂n (109)

where u stands for the parabolic diffusion fundamental solution. Assuming a constant variation of all field functions within the individual time increment t, the time integrals in eqn (109) may be evaluated analytically, and the integral representation,


eqn (62), can be rewritten in the following form ∂p 1 U d + c (ξ ) v (ξ , tF ) + pQ d = bU d α ∂n + pF−1 uF−1 d.

53

(110)

By equating the pseudo-body force term with the expression b = −α

∂fpi , ∂xi

(111)

the integral equation is given by ∂fpi ∂p U d c (ξ ) p (ξ , tF ) + pQ d = U d − ∂n ∂xi + pF−1 uF−1 d.

(112)

Using the Gaussian theorem, the domain integral in eqn (112) may be rewritten as follows ∂fpi U d = fpi ni U d − fpi Qi d, (113) ∂xi and, because ∂p/∂n = f p · n , the final form of the pressure integral equation is obtained c (ξ ) p (ξ , tF ) + pQ d = fpi Qi d + pF−1 uF−1 d, (114)

where the vector f p is given by eqn (52). For given Neumann boundary conditions, the pressure field p(rj , t) is determined to within a constant, with the solution of the parabolic differential equation (57) or the corresponding integral equation (114) taking into account the known velocity and vorticity field functions, and material property values at a given time interval. Thus, grad p is uniquely determined by the current flow field functions, independent of the flow history. The integral statement has to be first applied on the boundary for implicit evaluation of pressure boundary values. The computation of domain pressure values follows in an explicit manner. 7.5 Integral representation of heat energy kinetics The integral representation of the nonlinear heat energy diffusion–convection transport equation is derived considering the linear parabolic diffusion differential operator ∂ 2 (·) ∂(·) L [·] = ao − , (115) ∂xj ∂xj ∂t

54 Domain Decomposition Techniques for Boundary Elements and, therefore, eqn (31) may be rewritten in the form L [T ] + b = ao

∂ 2T ∂T + b, − ∂xj ∂xj ∂t

(116)

with the following corresponding integral representation written for a time step t = tF − tF−1 c (ξ ) T (ξ , tF ) + ao +

tF

tF−1

T q dtd = ao

bu dtd +

tF−1

tF−1

tF

tF

∂T u dtd ∂n

Ti,F−1 uF−1 d,

(117)

where u is the parabolic diffusion fundamental solution. Assuming constant variation of all field functions within the individual time increment, the eqn (117) is written as follows

∂T 1 U d + bU d ao ∂n + TF−1 uF−1 d.

c (ξ ) T (ξ , tF ) +

T Q d =

(118)

The domain integral of the pseudo-body forces includes the effects of convection, Rayleigh dissipation function, the heat source term, and terms due to nonlinear material properties, namely 1 ∂ !∂T ∂c 1 ∂T b= k T vj − cvj T + + cT D −! c co ∂xj ∂xj co ∂xj ∂t +

1 ( + S). co

(119)

Therefore, the following integral representation can be evaluated ∂T ∂ !∂T 1 k − cvj T U d U d + ko ∂xj ∂xj ∂n 1 ∂c ∂T 1 + T vj U d + + cT D −! c ( + S) U d ko ∂xj ∂t ko + TF−1 uF−1 d. (120)

c (ξ ) T (ξ , tF ) +

T Q d =


55

By using the Gauss divergence theorem the field function derivatives in the first domain integral are transfered to the fundamental solution, and, therefore, the following integral representation of the heat energy kinetics is obtained ∂T 1 k − cvn T U d c (ξ ) T (ξ , tF ) + T Q d = ko ∂n 1 ∂c ∂T 1 ∂T ! − k T vj − cvj T Qj d + + cT D −! c U d ko ∂xj ko ∂xj ∂t 1 + TF−1 uF−1 d. (121) ( + S) U d + ko The boundary integrals describe the total heat flux on the boundary due to molecular diffusion and convection. The first domain integral gives the influence of the convection and the nonlinear diffusion flux, the second domain integral includes the nonlinear material effects, the third domain integral describes the Rayleigh dissipation and heat source term effects, while the last one is due to the initial temperature distribution effect on the development of the temperature field in subsequent time intervals.

8 Discrete models After the derivation of integral equations the next step consists of transforming these equations into their algebraic forms [9–11]. The transformation is done by the means of boundary and domain discretisation and by the use of interpolation polinomials for representation of field functions. Different discrete models were developed to study the stability and convergency of the developed numerical scheme. The basic discrete model consists of a single domain discretisation, which means that the domain is treated as a single entity. The outer boundary of the domain is divided into boundary elements and the domain interior is divided into internal cells. The second discrete model is connected with the idea of subdomain technique or a multi-domain model. The subdomain technique was introduced with the aim of reducing memory and computer time demands of BEM computations. Large subdomains found their application in various areas of BEM computations, especially in linear and weakly nonlinear physical problems, i.e. heat conduction, linear elasticity, etc. In the case of computation of highly nonlinear phenomena, such as the present nonisothermal viscous compressible fluid flow, the standard subdomain technique was not efficient enough. The application of the diffusion or the diffusion–convection fundamental solution and its local implementation in parts of the computational domain resulted in the concept of a macro-element. It can be viewed as a limit version of the classical subdomain technique for BEM. 8.1 Single domain model The single domain model has several advantages but also some severe limitations when considered in the BEM context. Among the advantages are its simple use,

56 Domain Decomposition Techniques for Boundary Elements as the computational mesh consists of an outer boundary element mesh (Fig. 1), whereas the interior of the domain is discretised by internal cells or by means of dual and multiple reciprocity approximations of domain integrals in the case of a weak nonlinearity only, e.g. low Peclet number value flows. The number of unknowns in the flow kinematics system matrix is defined by the number of nodes on the outer boundary of the computational domain, i.e. the system matrix is very small. Computation of field function values in the interior of the domain is done explicitly, after the boundary node values are resolved. This step is fast and can be done in full parallel mode. The disadvantages of the single domain models are the following: fully populated system matrix of both flow kinetics and flow kinematics, extremely large computing times for all the boundary as well as domain integrals, and relatively low numerical stability of the BEM algorithm when applied to highly nonlinear problems. The latter can be resolved by extending the system matrix of flow kinetics to all computational nodes, boundary as well as domain. Unfortunately, this results in a fully populated large system matrix, which is hard to store in core and solve when the computational mesh becomes more dense. 8.2 Subdomain model The subdomain discrete model divides the computational domain into large subdomains, each consisting of a large number of boundary elements and internal cells. An example of such a mesh is presented in Fig. 2. By this approach, the large system matrix of flow kinetics can be transformed into a block-type structure, allowing the use of storage reducing schemes and fast iterative solvers, [10]. The drawback of the subdomain technique in the BEM context is the need to use a mix of continuous and discontinuous boundary elements and internal cells in order to avoid problems with overdetermination of the final system of algebraic equations,

Figure 1: Discrete model: single subdomain, boundary element mesh (left), internal cell mesh (right).

Figure 2: Discrete model: large subdomain mesh, internal cell mesh the same as in Fig. 1.


57

resulting from the existence of star points. The other, more severe drawback, is the high sensibility of accuracy of flow kinematics computation, when a subdomain mesh is applied also for computation of flow kinematics. It was shown [10] that only the segmentation technique is applicable for flow kinematics in its present form. The main steps of the segmentation technique for flow kinematics can be described as: 1. Computation in original domain without interface boundaries. In this way, boundary values of unknown vorticities and velocities are computed without additional error due to additional interface boundaries. 2. After the boundary values for velocities and vorticities are computed, unknown values of velocities at points on the interface boundaries are evaluated, still in the original mesh. 3. With the additional known velocities from Step 2, computation moves to the subdomain mesh, where values of velocities inside each subdomain are computed independently of other subdomains. Another possibility is to decrease the size of a subdomain, that eventually merges with the dimension of one internal cell, resulting in a macro-element discretisation (Fig. 3). In [9], the macro-element discretisation was introduced and applied for both flow kinetics and flow kinematics, the latter in its velocity-vector form. With this approach, one macro-element or subdomain consists of one quadrilateral internal cell and four boundary elements (Fig. 4). The geometrical singularities are overcome by using discontinuous boundary elements and internal cells of different order. Constant linear discontinuous and quadratic discontinuous boundary elements were developed. Internal cells developed were constant, linear discontinuous, quadratic discontinuous, and quadratic continuous quadrilateral cells. Extensive testing proved that there is a need for a continuous function field at the corners of a cell, as only a combination of discontinuous quadratic boundary elements and

1 Figure 3: Discrete model: macro-element mesh.

r

r

r

r r

r

r r r

r

r

r

r r

r

r r

r

r

r

Figure 4: Layout of a macro-element.


subdomain i

subdomain j ωij = ωji

nji

nij

ij ji ω • ω A A A qω A U A

ij

q

∂ω 1 − vn ω ∂n ν

ωij

ij

= −q or

(122) ωji

∂ω 1 =− − vn ω ∂n ν

(123) ji (124)

ji

Figure 5: Interface boundary conditions for vorticity transport. quadratic continuous quadrilateral cells (Fig. 4) proved to ensure the stability of the BEM numerical algorithm. When using any kind of a subdomain technique, the interface between subdomains is discretised twice, once for each subdomain. This means we have two nodal points at the same geometrical point, resulting in an increase of the overall number of unknowns in the system matrix. The nodes are linked together by application of compatibility and continuity conditions, presented in Fig. 5 for the case of flow kinetics of vorticity transport. This also allows the use of heterogenous material properties in the computational domain, i.e. density, viscosity, or heat conductivity. 8.3 Hybrid approaches The computation of flow kinematics, resulting from the velocity Poisson equation, can be combined with the macro-element computation of flow kinetics, as shown recently, [11]. Here, the segmentation technique is used to lower the computational demands of flow kinematics, and macro-element discretisation is applied in flow kinetics, resulting in sparse system and integral matrices, suitable for fast solution by the use of preconditioned iterative methods. Another approach, where the macro-element technique is used, is the mixed boundary element discretisation, presented in [12]. Its main idea is the use of macroelements and a combination of continuous boundary elements for field functions and discontinuous boundary elements for fluxes. Although this approach does result in sparse systems of equations, its main drawback is the overdetermined system of equations.

9 Test example: differentially-heated tall enclosure The coupled momentum energy flow case in a square thermally driven cavity is frequently considered as an exercise for the incompressible flow by numerical


59

models with Boussinesq approximation, [9, 13, 14], in which a series of steady state reference numerical solutions for Rayleigh number values in the range 103 < Ra < 108 are presented. The study of natural convection in a non-Newtonian fluid saturated porous square cavity is presented in [15] for different Rayleigh and Darcy number values. When large temperature differences are considered, the use of a compressible numerical model solver, capable of solving the low Mach number value flows [6, 16, 17], is necessary in order to obtain physically relevant results. The natural convection inside a rectangular, differentially-heated enclosure is considered less frequently than the standard square case. The fluid motion is caused by buoyancy force due to the heated left wall while the right wall is cooled and the rest of the boundary is adiabatic. The problem has been studied in [18] specially at a near critical Rayleigh number value Ra = 3.1 × 105 inside a reactangular cavity of aspect ratio 8 : 1, when the field functions reach their quasi-steady states and oscilate around mean values, while all material properties are assumed to have constant values and the Boussinesq approximation is considered. The natural convection solutions in a differentially heated cavity of aspect ratio 4 : 1 under large temperature gradients are presented in [19]. For the constant physical properties, the transition to unsteadiness takes place with a critical Rayleigh number value 3.2 × 105 < Rac < 3.4 × 105 . If the material properties are nonlinear functions of temperature, the transition is quite different. As reported in [19], the solutions are steady at Ra < 2 × 105 , while by slightly increasing the Rayleigh number value, the steady state is obtained after a very long integration time, and for a Rayleigh number value Ra > 2.025 × 105 , the solution does not seem to be steady. When the Ra > 2.3 × 105 the unsteadiness is emphasised. First the incompressible Boussinesq fluid model is assumed in the BEM numerical simulation of natural convection study in a rectangular enclosure with aspect ratio 8 : 1, Fig. 6. In this case the Rayleigh number is defined as Ra = Pr

g β(Th − Tc ) W 3 , νo2

(125)

where Pr is the characteristic nondimensional Prandtl number, e.g. Pr = 0.71 for air, W is a characteristic width of the enclosure, Th and Tc are the hot and the cold wall temperatures, respectively, To is a reference temperature defined as To = (Th + Tc )/2 and νo (To ) is a corresponding reference kinematic viscosity. In our case we choose Th = 0.5 and Tc = −0.5 and the material properties are assumed to be constant values. The heat transfer through the wall is represented by a local ¯ respectively, defined as and an average Nusselt number, Nu(y) and Nu, α(y)W ∂T W Nu(y) = = (126) ko , ko ko (Th − Tc ) ∂x w 1 L ¯ = Nu Nu(y) dy. (127) L 0 In the current analysis the flow patterns for Rayleigh number values Ra = 103 , Ra = 104 , Ra = 105 , and Ra = 3.1 × 105 were studied. Only one nonuniform

60 Domain Decomposition Techniques for Boundary Elements y ∂T/∂n = 0 H

g Th

Tc

(To , ρo )

∂T/∂n = 0 0

x W

Figure 6: Presentation of the thermally driven cavity problem in a rectangular enclosure with aspect ratio 8 : 1: boundary and initial conditions; computational mesh (10 × 60 cells with ratio 4). disctretisation was considered consisting of a M = 10 × 60 mesh of three-node quadratic boundary elements and nine-node quadratic quadrilateral internal cells with the aspect ratio of 1 : 4 in the x− and y−directions. For all cases the existence of a steady state was assumed first and the time step value of t = 1016 was selected. The results for Ra = 103 −105 are presented in Figs 7 and 8. Next, for the Ra = 3.1 × 105 flow case the time dependent analysis was performed also by running the simulation from a steady-state solution for Ra = 105 , used as the initial condition. The dimensionless time step value of t = 0.002 was used and the simulation results after 100 time steps are presented in Fig. 8. The representative velocity and thermal boundary layers can easily be seen on the hot and cold walls. In the second study, large temperature differences are considered in the rectangular cavity of the same dimensions as for the Boussinesq case, which impose the


61

Figure 7: Velocity field and temperature isolines, incompressible flow: Ra = 103 , 104 , left to right.


Figure 8: Velocity field and temperature isolines, incompressible flow: Ra = 105 , 3.1 × 105 , left to right.


63

use of the compressible numerical model solver. For a compressible fluid motion the Rayleigh number is defined as Ra = Pr

g ρo2 (Th − Tc ) W 3 , To ηo2

(128)

where Pr is the characteristic nondimensional Prandtl number, e.g. Pr = 0.71 for air, W is the width of the cavity, Th and Tc are the hot and the cold wall temperatures, respectively, To is a reference temperature defined as To = (Th − Tc )/2, po is a reference pressure, and ρo (To , po ) is a corresponding reference mass density. In our case we choose To = 600 K and po = 101,325 Pa. The temperature difference can be represented by a nondimensional parameter , defined as =

T h − Tc . 2To

(129)

The heat transfer through the wall is represented by a local and an average Nusselt ¯ respectively, defined as number, Nu(y) and Nu, α(y)W W ∂T Nu(y) = = (130) k(T ) , ko ko (Th − Tc ) ∂x w 1 L ¯ = Nu Nu(y) dy, (131) L 0 where k(T ) is temperature-dependent heat conductivity and ko = k(To ). In the performed test cases the Pr number value is assumed to remain constant (0.71), the temperature dependence of the viscosity is given by the Sutherland’s model 3/2 ∗ T +S T η(T ) = , (132) η∗ T∗ T +S and the heat conductivity is expressed as k(T ) =

η(T )cp Pr

(133)

with T ∗ = 273 K, S = 110.5 K, η∗ = 1.68 × 10−5 kg/m/s, cp = κR/(κ − 1), κ = 1.4, and R = 287.0 J/kg K. The influence of temperature on cp is neglected. The parameters defining the problem are η(η∗ , S, T ∗ ), R, κ, k, To , po , , W , and g . The independent dimensionless parameters appearing in the problem are , κ, Pr, Ra, s/T ∗ , To /T ∗ , and ρo = po /RTo . The problem is completely defined by the Ra number value, the value of , a reference state po and To , and initial conditions, e.g. T (x, y) = To , p(x, y) = po , and v (x, y) = 0. The computations were performed for Ra = 103 , Ra = 104 , Ra = 105 , and Ra = 3.1 × 105 , with the temperature difference parameter = 0.6, a result of imposed Th = 960 K and Tc = 240 K. Figures 9 and 10 show velocity and temperature fields at the steady state. An evident difference against the incompressible flow computations is the asymmetry of the flow and temperature fields.


Figure 9: Velocity field and temperature isolines, compressible flow: Ra = 103 , 104 , left to right.


65

Figure 10: Velocity field and temperature isolines, compressible flow: Ra = 105 .

66 Domain Decomposition Techniques for Boundary Elements Table 1: Average Nusselt number values for different Rayleigh number values for incompressible and compressible fluid models. Ra Incompressible [18] [20] Compressible

103

104

105

3.1 × 105

1.056 – – 1.060

1.790 – – 1.807

3.377 – – 3.321

4.477 4.493 4.579 –

Table 1 shows the comparison of the computed Nusselt number values for different Rayleigh number values for incompressible and compressible fluid models.

10 Conclusions In the chapter the boundary element integral approach for the solution of compressible fluid motion in a thermally driven cavity is presented. The governing equations arise from the velocity–vorticity formulation of the Navier–Stokes equation for a compressible fluid, and consist of flow kinematics, vorticity transport, heat transport, and pressure equations. The flow kinematics and pressure equations are solved by using a false transient approach in order to increase the stability of the nonlinear iteration procedure. The variable material properties are accounted for in pseudo-body force terms, discretised by internal cells. The presented numerical scheme can further be improved by the use of subdomain-based elliptic difusion– convection fundamental solutions in deriving the integral representations of flow kinetics and energy transport equations. The results of the computed test example of buoyant flow of compressible fluid in a tall cavity confirm the applicability of a BEM-based numerical scheme also for a highly nonlinear transport phenomena such as compressible natural convection. The computed BEM results are in good agreement with the benchmark solutions presented in [17], especially taking into account the much coarser mesh applied for the BEM numerical model.

References [1]

Batchelor, G.K., An Introduction to Fluid Dynamics, Cambridge University Press: Cambridge, 1967. [2] Wrobel, L.C., The Boundary Element Method, Vol. 1, Applications in Thermo-Fluids and Acoustics, Wiley: Chichester, 2002. [3] Škerget, L., Hriberšek, M. & Žuniˇc, Z., Natural convection flows in complex cavities by BEM. Int. J. Num. Meth. Heat & Fluid Flow, 13(6), 2003. [4] Wu, J.C. & Thompson, J.F., Numerical solution of time dependent incompressible Navier-Stokes equations using an integro-differential formulation. Computers and Fluids, 1, pp. 197–215, 1973.


67

[5] Wu, J.C., Problems of general viscous fluid flow (Chapter 2). Developments in BEM, Vol. 2, Elsevier Appl. Sci. Publ.: London and New York, 1982. [6] Škerget, L. & Samec, N., BEM for the two-dimensional plane compressible fluid dynamics. Eng. Anal. Bound. Elem., 29, pp. 41–57, 2004. [7] Rizk, Y.M., An integral representation approach for time dependent viscous flow, PhD Thesis, Georgia Institute of Technology, 1980. [8] Škerget, L., Alujeviˇc, A., Brebbia, C.A. & Kuhn, G., Natural and forced convection simulation using the velocity–vorticity approach. Topics in Boundary Element Research, Vol. 5, pp. 49–86, Springer-Verlag: Berlin, 1989. [9] Škerget, L., Hriberšek, M. & Kuhn, G., Computational fluid dynamics by boundary domain integral method. Int. J. Num. Meth. Eng., 46, pp. 1291– 1311, 1999. [10] Hriberšek, M. & Škerget, L., Iterative methods in solving Navier–Stokes equations by the boundary element method. Int. J. Num. Meth. Eng., 39, pp. 115–139, 1996. [11] Hriberšek, M., Segmentation based boundary domain integral method for the numerical solution of Navier–Stokes equations. BEM 24, Split, WIT Press: Southampton, 2003. [12] Ramšak, M. & Škerget, L., Mixed boundary elements for laminar flows. Int. J. Num. Meth. Fluids, 31, pp. 861–877, 1999. [13] De Vahl Davis, G., Natural convection of air in a square cavity: a bench mark numerical solution. Int. J. Num. Meth. Fluids, 3, pp. 249–264, 1983. [14] Le Quere, P., Accurate solutions to the square thermally driven cavity at high Rayleigh number. Computers Fluids, 20(1), pp. 29–41, 1991. [15] Jecl, R. & Škerget, L., BEM for natural convection in non-Newtonian fluid saturated square porous cavity. Eng. Anal. Bound. Elem., 27, pp. 963–975, 2003. [16] Vierendeels, J., Merci, B. & Dick, E., Benchmark solutions for the natural convection heat transfer problem in a square cavity. Advances in fluid mechanicsIV, AFM 2002, eds. M. Rahman, R. Verhoeven & C.A. Brebbia, WIT Press: Southampton and Boston, pp. 45–54, 2002. [17] Vierendeels, J., Merci, B. & Dick, E., Numerical study of the natural convection heat transfer with large temperature differences. Int. J. Num. Meth. Heat & Fluid Flow, 11, pp. 329–341, 2001. [18] Ingber, M.S., A vorticity method for the solution of natural convection flows in enclosures. Int. J. Num. Meth. Heat & Fluid Flow, 13, pp. 655– 671, 2003. [19] Weisman, C., Calsyn, L., Dubois, C. & Le Quere, P., Sur la nature de la transition a l’instationare d’un ecoulement de convection naturelle en cavite differentiellement chauffee a grands ecarts de temperature. Comptes rendus de l’academie des sciences, Serie II b, Mecanique, pp. 343–350, 2001. [20] Xin, S. & Le Quere, P., An extended Chebyshev pseudo-spectral contribution to the CPNCFE benchmark. Computational Fluid and Solid Mechanics, Proc. of the First MIT Conf. on Computational Fluid and Solid Mechanics, Vol. 2, Elsevier: Amsterdam, pp. 1509–1513, 1991.


CHAPTER 3 Multi-domain DRM boundary element method for the numerical simulation of non-isothermal Newtonian and non-Newtonian flow problems H. Power1 & W.F. Florez2 1 The

University of Nottingham, School of Mechanical, Materials and Manufacturing Engineering, Nottingham, UK. 2 Energy and Thermodynamics Research Group, UPB University, Medellin, Colombia.

Abstract The multi-domain dual reciprocity method has been effectively applied to the solution of two-dimensional non-isothermal Newtonian and non-Newtonian flow problems, where the momentum and energy equations govern the motion of a viscous fluid. In the proposed boundary integral method the domain integrals are transformed into equivalent boundary integrals by the dual reciprocity approach applied in a subdomain basis. On each subregion or domain element the integral representation formulae for the velocity and temperature are applied and discretized using linear continuous boundary elements, and the equations from adjacent subregions are linked by additional continuity conditions. Some examples showing the accuracy, efficiency and flexibility of the proposed method are presented.

1 Introduction The main purpose of this work is to present the use of the multi-domain dual reciprocity method (MD-DRM) [1, 2] for the numerical simulation of non-isothermal Newtonian and non-Newtonian flow problems with viscous dissipation effects. The MD-DRM is a recently developed technique that allows an efficient and accurate boundary element numerical solution of highly nonlinear and convective problems. In this work, we will show how the MD-DRM technique can be used to solve this

70 Domain Decomposition Techniques for Boundary Elements type of problems, showing the efficiency and versatility of the proposed numerical approach. The application of the boundary element method (BEM) to Newtonian or nonNewtonian nonlinear problems requires finding a fundamental solution of the system of governing equations. However, since such a fundamental solution for a general model is not possible to be known, it is necessary to put the nonlinear terms into a pseudo-body force leading to domain integrals that can be evaluated by using the cell integration approach (cell–BEM) [3]. Although this method is effective and general, it makes the BEM lose its boundary-only nature resulting in a numerical scheme several orders of magnitude more time consuming than other domain techniques as the finite differences method (FDM) and finite element method (FEM). The computational cost of the cell integration approach mainly depends on the fact that the solution at each surface or internal point must involve the evaluation of the complete surface integrals yielding in this way to a fully populated matrix system. A handful of methods have been developed to take the domain integrals to the boundary. Among them, the dual reciprocity method (DRM) [4] and the particular solution method (PIT) [5] are the most popular ones. These two approaches have been particularly attractive in recent years because of the advances in multidimensional interpolation with radial basis functions (RBF) used in these techniques. Both the DRM and PIT approximate the nonlinear and non-homogeneous terms of a partial differential equation as a series of vector-valued interpolation functions, which are defined in terms of a set of surface and internal nodes. These interpolation functions lead to a particular solution for the problem which can be used together with Green’s identities to convert the domain integrals into boundary integrals. As in the cell integration approach, in these two boundary-only approaches the evaluation of the functions at each node requires the integration of the complete surface integrals yielding again a fully populated matrix system. A major problem encountered with the DRM and PIT is that the resulting algebraic system consists of a series of matrix multiplications of dense matrices. Although the computing time required by the DRM and PIT is lower than that with the cell integration scheme, it is still very high compared to domain approaches. Besides, in complex problems both the DRM and PIT have been limited to a small degree of nonlinearity. From these limitations, it appears that this type of boundaryonly formulations in their original single domain form cannot be extended to solve highly nonlinear problems. When dealing with the BEM solution of large problems it is usual to use the method of domain decomposition, in which the original domain is divided into subregions, and in each of them the full integral representation formula is applied. At the interfaces between the adjacent subregions, the corresponding continuity conditions are enforced. While the matrices which arise in the single domain DRM are fully populated, the subregion formulation leads to block-banded matrix systems with one block for each subregion and overlaps between blocks when sub-regions have a common interface. The implementation of the subregion BEM formulation in the limiting case of a very large number of subregions, including cell integration at each subregion was

Multi-Domain DRM Boundary Element Method

71

called by Taigbenu and collaborators as the Green element method (GEM) [6]. In this method, the resultant coefficient matrix is as sparse as that of the finite element method and, hence, its solution is as efficient as in the domain approach. Recently, Popov and Power [1] found that the DRM approximation of the internal potential of a nonlinear convection diffusion problem can be substantially improved by using a domain decomposition scheme. Popov and Power’s idea of using domain decomposition to improve the accuracy of the DRM approach was inspired by the work of Kansa and Carlson [7] on data approximation with RBFs. Kansa and Carlson observed that the best approximation is obtained when the original domain is split into matching subdomains. In a similar way, Popov and Power [1] noticed that the DRM approximation for nonlinear problems can also be improved when the original domain is divided into smaller subdomains. At each subregion the BEM formulation of the problem is separately applied. The corresponding domain integral resulting from the BEM formulation is transformed to surface integrals along the contour of the subregions by the DRM approach. The desired accuracy of the scheme is achieved by refining the size of the subdomains. For a large number of subdomains this technique was called a MD-DRM, and the discrete form of the integral equation formulation of the problem is given by a banded matrix system. Recently the MD-DRM has been successfully applied by Florez and Power [8] to the solution of isothermal non-Newtonian flow problems and to the Navier– Stokes equations for incompressible Newtonian fluids at moderately high Reynolds numbers [9]. In the present work it will be shown how the MD-DRM can be extended to more general non-isothermal problems including viscous dissipation terms. The problem of multiplication of several large fully populated matrices encountered in the DRM formulation in a single domain is reduced, in the present domain decomposition case, to the multiplication of very small matrices at each subdomain. Besides, if the subdomains are of similar shape all the evaluations and multiplications can be performed only in one subdomain, and the results for the others can be found by scaling due to the geometric characteristic of the DRM functions. Because of the nature of the integral equation formulation at each subdomain, the solution obtained with MD-DRM is a kind of Hermitian solution. In the present case of non-isothermal flow, the values of the velocity and shear stress fields as well as the temperature and its directional derivatives at every point defining the subdomains are found directly from the solution of the resulting matrix system. Although the solution of a fully populated system of equations is computationally very expensive, this type of solution for a sparse system can be carried out very efficiently by not taking into account the multiplications of the zero terms. In this manner the solution of the MD-DRM system could be obtained in a fraction of the solution time of an equivalent fully populated system resulting from any of the other standard BEM schemes. Besides, the required storage of the MD-DRM approach is substantially smaller than the one required by other BEM schemes. For the sparse system obtained with the MD-DRM approach, both the efficiency and the storage requirement are functions of the bandwidth of the matrix system. It is important to point out that the solution of any BEM approach could also be substantially improved with the use of indirect solvers which were not considered in this work.

72 Domain Decomposition Techniques for Boundary Elements In the following sections, we will introduce the MD-DRM solution of the coupled system of Navier–Stokes equations and the energy equation. Besides the efficiency and accuracy of the proposed numerical method, the MD-DRM has the advantage of the robust RBF interpolation used in the DRM. Due to their smoothness or noise minimization character, the radial functions guarantee high accuracy on the evaluation of the gradient of the field variables [10]. This property of the DRM interpolation permits a precise evaluation of the velocity and temperature derivatives directly from the solution of the matrix system without any additional burden.

2 Thermal convection Newtonian flow problems In natural convection phenomena the flow pattern and the velocity field of a viscous fluid is established by buoyancy forces caused by temperature differences within the fluid. For most incompressible Newtonian fluids the viscous dissipation and the temperature dependence of the viscosity can be neglected in the energy equation. In contrast, the density variation represents an important term that acts as a body force in the momentum equations and causes the fluid motion. Buoyancy is due to the combined presence of a fluid density gradient and a body force that is proportional to the density. The body force is usually gravitational, although it might be any other force such as centrifugal or a Coriolis force. There are several ways in which a mass density gradient may appear in a fluid but in the most common situation it is due to a temperature gradient. Although in the present work the fluid is going to be assumed incompressible, the effect of variable density in the buoyancy force will be accounted for using the Boussinesq approximation. Free convection flows may be classified according to whether the flow is bounded by a surface. In the absence of bounding surfaces, free boundary flows may occur in the form of plumes or buoyant jets [11]. In this work we focus on free convection flows of incompressible fluids bounded by a surface. In this type of problems a thermal boundary layer usually appears on the heated walls and in those zones the high convection terms in the momentum and energy equations dominate the solution. In the past, several BEM formulations based on cell integration have been successfully applied to the solution of the thermally driven cavity flow problem. Onishi et al. [12] proposed a cell–BEM formulation for natural convection problems in terms of the stream function, vorticity and temperature as the main variables. On the other hand Skerget et al. [13] used a velocity, vorticity and temperature formulation together with domain partition techniques and discontinuous boundary elements for the solution of both Newtonian and non-Newtonian flow problems. The penalty function formulation is another method that can be used to solve fluid flow problems and it was presented separately by Kuroki [14], Brebbia [15] and Kitawa [16]. In this formulation a penalty parameter is used to obtain a separate equation for the pressure and so the Navier–Stokes equations are reduced to the Navier equations of elastostatics with a body force including the convective and buoyancy terms.


73

The aim of this section is to show an efficient use of the MD-DRM technique for the solution of the coupled momentum and energy equations. The domain integrals that appear on each subdomain can be converted into a series of boundary integrals by dual reciprocity. 2.1 Governing equations The governing equations of the fluid flow (continuity, momentum and energy) assuming the Boussinesq approximation (density variations are relevant only in the buoyancy terms of the momentum equations), Newtonian fluid behaviour, negligible influence of pressure on temperature and constant fluid properties, may be written in tensor notation as: ∂ui =0 (1) ∂xi −

∂p ∂ui ∂ 2 ui +µ = ρuj − ρgi β(T − Tc ) ∂xi ∂xj ∂xj ∂xj k

for all x ∈

∂ 2T ∂T = ρCp uj ∂xj ∂xj ∂xj

(2) (3)

where ui is the flow velocity, ρ the fluid density, p the pressure, µ the viscosity, and is a bounded two-dimensional domain. In the energy equation (eqn (3)) T is the fluid temperature at any point, k is the thermal conductivity and Cp the thermal capacity. The lifting force ρgi β(T − Tc ) in eqn (2) represents the effect of temperature change on the density as a buoyancy term, where Tc is the reference temperature, i.e. the temperature at the cold wall in a convective flow problem, β is the coefficient of thermal expansion and gi is the magnitude of gravity acting in the i direction. Let us now consider the dimensionless parameters that govern free convective flow and heat transfer. These parameters can be obtained by non-dimensionalizing the governing equations. Defining, xi∗ =

xi L

ui∗ =

ui L α

θ=

T − Tc Th − T c

p∗ =

pL 2 µα

(4)

where α is the thermal diffusivity and L and Th are a reference length and the reference temperature of a hot wall, respectively. With these new variables the continuity, momentum and energy equations (eqns (1)–(3)) reduce to, ∂ui∗ =0 ∂xi∗ −

∂ 2 ui∗ 1 ∗ ∂ui∗ ∂p∗ u + = − Raθ ∂xi∗ Pr j ∂xj∗ ∂xj∗ ∂xj∗ ∂θ ∂θ = uj∗ ∗ ∗ ∗ ∂xj ∂xj ∂xj

(5) for all x ∈

(6) (7)

74 Domain Decomposition Techniques for Boundary Elements and the following dimensionless numbers have been defined: βgi (Th − Tc )L 3 ν ν Pr = α Ra = Gr Pr

Gr =

(8) (9) (10)

In the above equations, Gr is the Grashof number, Pr is the Prandtl number, ν is the kinematic viscosity and Ra is the Raleigh number. For simplicity, from now on, we adopt the notation u∗ = u, p∗ = p and xj∗ = xj . The Grashof number plays the same role in free convection that the Reynolds number plays in an isothermal fluid, and the former indicates the ratio of the buoyancy force to the viscous force acting on the fluid. In general, the flow field may satisfy mixed boundary condition of the form ui (ξ ) = Ui (ξ )

for all ξ ∈ SD

(11)

and ti (ξ ) = σij ( u, p)nj (ξ )

for all ξ ∈ SN

(12)

where S = SD + SN is the boundary surface with outward unit vector n , t is the surface force or traction and σij is the stress tensor defined as: ∂uj ∂ui (13) + σij = −pδij + µ ∂xj ∂xi And there is also a set of boundary conditions for the energy equations given by, T (ξ ) = Te

for all ξ ∈ SA

(14)

and q = −k

∂T = qe ∂n

for all ξ ∈ SB

(15)

where S = SA + SB is the boundary surface. Note that eqns (14) and (15) correspond to specified temperature and heat flux along different parts of the boundary S. 2.2 Integral representation formulae The integral representation formulae for the eqns (1) and (2) at a point x ∈ ∪ S are given by Ladyzhenskaya [17]: k ckj (x)uk (x) = Kkj (x, y)uj (y)dSy − Ui (x, y)tj (y)dSy + Uik (x, y)gi (y)d S

S

(16)


75

here g is the non-homogeneous term or pseudo-body force equal to the convective term, i.e. ∂ui − ρgi β(T − Tc ) (17) gi = ρuj ∂xj The coefficients cki have values between δki and 0, being equal to 21 δki for smooth boundaries and cki = δki for points inside the domain . The Green’s function Uik under the second integral in eqn (16) is the fundamental solution of the Stokes’ equation known as Stokeslet, with a corresponding pressure qk : 1 (xi − yi )(xk − yk ) 1 (18) ln δij + Uik (x, y) = − 4π r r2 qk (x, y) = −

1 (xk − yk ) 2π r2

(19)

and Kij represents the traction fundamental solution given by Kij (x, y) = σik (Ui (x, y), q j (x, y))nk (y) j

(20)

where $

j σik (Ui (x, y), q j (x, y))

j

j

∂Uk ∂Ui = δlk q (x, y) + µ + ∂xk ∂xi j

% (21)

and thus Kij (x, y) = −

1 (xi − yi )(xj − yj )(xk − yk ) nk (y) π r4

(22)

The parabolic partial differential equation (3), can be seen as a Poisson equation whose integral representation is obtained by applying Green’s second theorem for a scalar function [18]. However, due to the presence of the convective term in eqn (3), a domain integral is present in the integral representation and this latter integral can be transformed to the boundary using the DRM technique [4]. The integral representation of the energy transport equation at each domain element in which the domain will be divided, can be expressed as follows: G(x, y)h(y)dy (23) cT (x) = H(x, y)T (y)dSy − G(x, y)q(y)dSy + S

S

where ∂T ∂n ∂T h(x) = uj ∂xj q(x) =

(24) (25)

76 Domain Decomposition Techniques for Boundary Elements and G and H are the fundamental solution and its normal derivative along the integration path, respectively, i.e. G(x, y) =

1 log r(x, y) 2π

(26)

1 ∂ log r(x, y) 2π ∂ny

(27)

and H(x, y) =

where x and y are the position vectors of the source and field points, respectively [3]. 2.3 Approximation of the domain integrals As was explained in our previous articles [9, 19], the non-homogeneous terms that appear within the domain integrals in eqns (16) and (23) can be approximated using a series of particular solutions and interpolation functions. To express the domain integral in eqn (16) in terms of equivalent boundary integrals, the DRM approximation is introduced. The basic idea is to expand the g (y) term using radial interpolation functions at each subregion, i.e. gi (x) =

N+L+A

f m (x)αlm δil

(28)

m=1

The coefficients αlm are unknowns to be determined by collocation on a set of N nodes on the boundary and L internal nodes. It will be considered here that there are A = 3 augmentation global functions from the set {1, x1 , x2 }. It should be pointed out that the functions x1 and x2 alone give results that are not translatory symmetric. However, in the present work, the mentioned augmenting functions are calculated in a local system of coordinates whose origin is at the centroid of each subdomain. Thus, the global augmenting functions are different for each cell and scaled to avoid any numerical problems. An augmented spline consists of the RBF plus a series of additional global functions [20]. The RBF used in this work is the thin-plate spline: f m (x) = f (r(x, ym )) = r 2 log r

m = 1, . . . , N + L

(29)

where r = r(x, ym ) is the Euclidean distance between the field point x and the collocation point ym , other RBFs are discussed in the literature [20], such as Gaussian and multiquadratic functions. However, the thin-plate spline has features of smoothness and convergence that make it the recommended choice. Equation (28) when applied to the N + L collocation nodes will generate 2(N + L) linear equations with 2(N + L + A) unknowns and, therefore, 2A additional conditions are necessary which basically guarantee the maximum smoothness

77


of the interpolant [20]. These additional relationships are: N+L

j αl δil

=

j=1

N+L

j j x1 αl δil

=

j=1

N+L

j

j

x2 αl δil = 0

(30)

j=1

where x j represents the jth collocation node. Similarly, for the energy eqn (23) the convective term can be approximated as: h(x) = ρCp uj

N+L+A

∂T = f m (x)β m ∂xj

(31)

m=1

with the following additional equations for the global interpolation functions, N+L

βj =

j=1

N+L

j

x1 β j =

j=1

N+L

j

x2 β j = 0

(32)

j=1

After defining the auxiliary velocity field (Uˆ ilm (x), pˆ lm (x)) [21] solution of the following equations: µ

∂ 2 Uˆ ilm (x) ∂ pˆ lm (x) − = f m (x)δil ∂xj ∂xj ∂xi

(33)

∂ Uˆ ilm =0 ∂xi

(34)

and the auxiliary thermal field Tˆ solution of the following Poisson equation, k

∂ 2 Tˆ m (x) = f m (x) ∂xj ∂xj

(35)

we can apply Green’s identities to both the auxiliary velocity field and the auxiliary thermal field to obtain integral representation formulae in terms of only boundary integrals [4, 9, 19, 21], thus ckj (x)uk (x) − Kkj (x, y)uj (y)dSy + Uik (x, y)tj (y)dSy S

=

N+L+A

"

αlm ckj (x)Uˆ klm (x) −

S

S

m=1

Kkj (x, y)Uˆ jlm (y)dSy +

#

S

Uik (x, y)ˆtjlm (y)dSy (36)

and likewise for the energy equation, cT (x) − H(x, y)T (y)dSy + G(x, y)q(y)dSy =

S N+L+A

"

β m cTˆ m (x) −

m=1

where qˆ m = ∂ Tˆ m /∂n.

S

H(x, y)Tˆ m (y)dSy + S

#

G(x, y)ˆqm (y)dSy S

(37)

78 Domain Decomposition Techniques for Boundary Elements 2.4 Particular solution The particular solution for the flow field (Uˆ ilm (x), pˆ lm (x)) defined by the eqns (33)– (34) have been presented in earlier papers by Mingo [22] and Florez [19], and it can be obtained using the approach suggested by Power and Partridge [21]. In this chapter and based on our previous successful implementation of the MD-DRM technique using augmented thin-plate splines for the solution of the Navier–Stokes equations, we focus on the particular solution of the auxiliary Poisson equation, eqn (35). With the convective term interpolated using the functions f m (x) as in eqn (31), the particular solution Tˆ m (x) in eqn (35) can be found using cylindrical coordinates as suggested in [4] for the RBF, and the method of undetermined coefficients [20] for the polynomial interpolation functions. The results are summarized as follows [23]: Tˆ m =

r 4 log r r4 − 16 32

for f m (x) = r 2 log r

(38)

where r = x − ym is the Euclidean distance between the field node x and the collocation node ym . On the other hand, the particular solutions corresponding to the global functions f m (x) = 1, x1 , x2 are: x12 + x22 4 3 + x x2 x 1 2 Tˆ m = 1 8 x 3 + x2 x12 Tˆ m = 2 8 Tˆ m =

(39) (40) (41)

respectively. The corresponding normal derivatives are given by the following equations, qˆ m =

∂ Tˆ m r 2 xˆ i ni log r r 2 xˆ i ni = − ∂n 4 16

for f m (x) = r 2 log r

(42)

while for the global functions f m (x) = 1, x1 , x2 the corresponding fluxes are given by, x1 δ1i + x2 δ2i ni for f m (x) = 1 2 (3x12 + x22 )δ1i + 2x2 x1 δ2i qˆ m = ni for f m (x) = x1 8 (3x22 + x12 )δ2i + 2x1 x2 δ1i qˆ m = ni for f m (x) = x2 8 qˆ m =

(43) (44) (45)

where δij is the Kronecker delta. Having obtained the auxiliary non-homogeneous thermal field, we are now in a position to handle the convective term of the energy equation by the DRM approach.


79

Taking into account that in eqn (31) the velocity uj is considered as a known variable at each iteration of the coupled problem, we only need to approximate the temperature gradient ∂T /∂xj . Therefore, by expanding the temperature in terms of the interpolation functions f m (x), with unknown coefficients γ m , we obtain T=

N+L+A

f m (x)γ m

(46)

m=1

or in matrix notation T = Fγ

(47)

where the components of vector T are the temperature values at each node, the columns of matrix F contain the interpolation functions and vector γ contains the unknown coefficients. From eqn (47) it is easy to establish a mechanism that can relate the nodal values of T with the nodal values of its derivatives, ∂T /∂xj by taking the derivative of eqn (46) with respect to xj , thus N+L+A

∂f m (x) ∂T = γm ∂xj ∂xj

(48)

m=1

Substitution of eqn (47) into eqn (48) yields, ∂T ∂F −1 = F T ∂xj ∂xj

(49)

Now we can express the unknown coefficients β m in the approximation of the convective term that appears in eqn (31), as follows: β = ρCp F−1 uj

∂F −1 F T ∂xj

(50)

where vector β contains the unknown coefficients β m and uj is a diagonal matrix with nodal values of the velocity. The non-homogeneous term of the momentum equations given in eqn (17) requires not only the values of the temperatures but also the calculation of the velocity derivatives at the collocation nodes. In this work we have used a velocity interpolation scheme that satisfies the mass conservation principle and that was presented in [9]. In the mass conservative interpolation scheme the velocity derivatives are approximated in terms of the components of the velocity field, i.e. u1 , u2 in such a way that the mass balance (1) is satisfied in the domain. The main results can be summarized in the following equations (for more details see [9]): g = Fα

(51)

80 Domain Decomposition Techniques for Boundary Elements which is just eqn (28) written in matrix form. Likewise, the following approximation for the velocity components holds, u = Fλ

(52)

where u contains the velocities at every node, matrix F contains the interpolation functions from the mass conservative interpolation method and λ is the vector of unknown coefficients. Taking the derivatives on both sides of the latter equation we obtain, ∂F −1 ∂u = F u (53) ∂xj ∂xj With all the above approximations it is possible to discretize the integral equations eqns (36) and (37) and obtain a system of equations for the velocities and temperatures. 2.5 Discretization and numerical solution Boundary integral equations (36) and (37) were discretized using linear isoparametric linear boundary elements [18], hence, the momentum and energy integral equations can be rewritten as: c i ui −

n

Kik uk +

k=1

n

Uik tk =

N+L+A

α

cj Uˆ j −

j

j=1

k=1

n

Kik Uˆ kj +

k=1

n

& Uik ˆtkj

k=1

(54) and ci Ti −

n

k=1

Hik Tk +

n

k=1

Gik qk =

N+L+A

j=1

β

j

cj Tˆ j −

n

k=1

Hik Tˆ kj +

n

& Gik qˆ kj

k=1

(55) respectively, where Kik , Uik , Hik and Gik are the standard influence matrices resulting from the integration over the boundary elements, the index i represents the collocation nodes, k the nodes at the integration elements and j the DRM collocation points. For the numerical solution of the integral equations, we propose a multi-domain approach that divides the entire domain into smaller subdomains or domain elements where each of them is enclosed by four boundary elements. On each subdomain the eqns (54) and (55) hold. All the surface integrals are evaluated using Gauss– Legendre quadratures; however, special attention should be paid to the weakly singular integrals whose kernel is the Stokeslet Uik (x, y) when the source point tends to the field point x → y. This singularity can be dealt with using a special transformation of coordinates [24]. Besides the discretised integral equations, there are also additional continuity conditions at the interface between subdomains.


81

On the common interface j that connects the subdomains i and i + 1 the velocity matching conditions are of the form u i j = u i+1 j

(56)

and the dynamic equilibrium or traction matching conditions are: ti j = − ti+1 j

(57)

A similar procedure is applied to the integral equation for the temperature field; together with the temperature, Ti j = Ti+1 , and flux q i j = q i+1 , matching condij j tions at the interface between the subregions. The unknown coefficients vectors α j and β j are determined from eqns (28), (31), (50) and (51) by collocation on the boundary nodes of each subdomain. With the approximations explained in Section 2.4, the vector α j can be found in terms of nodal velocities only, therefore, eqn (54) represents a nonlinear system for the tractions and velocities. The resulting global nonlinear system of momentum equations for the entire domain is solved using the Newton–Raphson scheme combined with a line-search algorithm intended to reduce the residual at each iteration [25]. On the other hand, we used simple or direct iteration to cope with the coupling of the momentum and energy equations through the buoyancy term. In this way the velocity in the convective term from the energy equation is considered as a known variable from the previous solution of the momentum equations obtained by the Newton–Raphson method. Likewise, during each iteration of the iterative solution for the momentum equations, the non-homogeneous buoyancy force is considered as a known variable. These forces are determined from the previous temperature solution by direct iteration. For more details about the numerical algorithm see [26]. The use of linear boundary elements in the context of the multi-domain technique produces overdetermined systems of equations. Such systems can be solved in a least squares sense either directly or by the normal equations. The direct solution of the overdetermined set of algebraic equations is the simplest approach for finding a least squares solution by methods such as the QR factorization or the singular value decomposition [27]. Although these methods are robust for the case of dense matrices and even ill-posed problems, they both require high amounts of computer memory and time. Since in the multi-domain case the coefficient matrix is block sparse, an iterative solution such as that of Paige and Saunders [28] is at least in theory, much faster at the same solution accuracy. However, the velocity– traction formulation of the momentum equations yields extremely ill-conditioned, non-symmetric and non positive definite matrices that make the iterative solvers lose their advantages and speed unless the appropriate preconditioner is available. To avoid the numerical problems that may arise during the iterative solution of overdetermined systems of equations in the present investigation, a direct solver has been applied to the normal equations. It is also worth mentioning that the numerical ill-conditioning problems that exist for the momentum equations are not present in the solution of the scalar energy equations and so an iterative solver can be used for the latter case.

82 Domain Decomposition Techniques for Boundary Elements 2.6 Numerical results To test the developed numerical method, the problem of natural convection flow in a square cavity is considered. Figure 1 shows the geometry of the problem and boundary conditions. The left wall is maintained hot at a temperature Th = 310 K, the right wall is at a colder temperature Tc = 300 K, the four walls are stationary, and the top and bottom boundaries are adiabatic. As the fluid next to the left wall gets warmer it ascends due to its lower density, while the fluid next to the cold wall moves downwards. Both flows have to overcome the resistance due to viscosity and gravity, and as a consequence a big recirculation zone appears at the cavity centre. The natural convection within the cavity depends on two parameters: The Prandtl number Pr, and the Raleigh number Ra. The temperature difference between the heated walls is considered small enough to adopt the Boussinesq approximation which assumes that the fluid properties β, µ and Pr are constant and evaluated at the average temperature (Tc + Th )/2. Although the density was assumed constant, the effects of its variations are included in the buoyancy term in the momentum equations. Two different values of the Raleigh numbers Ra = 103 and Ra = 104 were considered in the simulation. For each of the cases studied, two different mesh densities were used 10 × 10 and 20 × 20 divisions. Both cases, uniform and non-uniform

q=0,U1=0,U2=0

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

g θ=1,U1=0,U2=0

L

θ=0,U1=0,U2=0

Y

X

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 q=0,U1=0,U2=0

L

Figure 1: Schematic of the thermally driven cavity problem.


83

grids, were studied in an attempt to capture the flow complexities near the walls. The distribution of nodes for non-uniform meshes is given by the expression [29]: 1 xi i i (58) − = sin 2π L imax 2π imax Even with the coarse meshes used in this work, the results obtained by the present MD-DRM technique are in excellent agreement with previous solutions [14, 15, 23, 30] calculated with much denser meshes and the use of cell integration. The maximum and minimum velocity values predicted by our MD-DRM almost coincides with the values reported in the mentioned references, these results together with the effect of mesh refinement are shown in Table 1. The variables that appear in this table are: ux max , ymax : maximum horizontal velocity at the vertical middle plane of the cavity and the position where the maximum occurs. vx max , xmax : maximum vertical velocity at the horizontal middle plane of the cavity and the position where the maximum occurs. From Table 1 note that the accuracy or the MD-DRM solution is remarkably good considering that the mesh used here is much coarser than the 120 × 120 nonuniform grid used by Vahl Davis in his finite difference solution. The convergence criteria used in this work was defined as unew − uold ≤ 10−6 unew for the momentum equations and Tnew − Told ≤ 10−6 Tnew for the temperature field. Table 1: Comparison of the results for different mesh densities and different Ra numbers. Ra

10 × 10 uniform

20 × 20 uniform

20 × 20 non-uniform

Vahl Davis [30]

103

ux max ymax uy max xmax

3.580 0.790 3.590 0.175

3.600 0.810 3.650 0.180

3.610 0.810 3.665 0.180

3.649 0.813 3.691 0.177

104

ux max ymax uy max xmax

14.400 0.800 17.100 0.100

16.100 0.820 19.200 0.110

16.100 0.820 19.120 0.110

16.219 0.823 19.549 0.123


3 Non-isothermal non-Newtonian Stokes flow with viscous dissipation For liquids with high viscosity and low thermal conductivity such as most polymers, viscous dissipation cannot be neglected because it has a strong influence on the temperature field which, in turn, affects the thermophysical properties of the fluid. This phenomenon is particularly relevant in the polymer industry (in extrusion and injection processes the fluid is forced to flow through channels where high power generation near the walls is observed [31]). Also in radiators design, where the working fluids are usually highly viscous oils, dissipation in laminar flow regime is important. The behaviour of non-Newtonian fluids is strongly dependent on the viscosity variations within the domain that are caused by the shear rate and temperature. Most non-Newtonian fluids such as polymers exhibit a viscosity that is a decreasing function of the shear rate. This characteristic is known as shear thinning [32]. The viscosity of an inelastic non-Newtonian fluid can be calculated on the one hand in terms of the shear rate through several mathematical models such as the power law model, the Carreau model and the hyperbolic tangent model [32, 33]. On the other hand, the viscosity of polymer melts changes drastically with temperature (a change of 1% in temperature can cause at least a 25% change in viscosity) and it can be calculated by either the Andrade law [32], or the WLF equation [32]. Since the viscosity of non-Newtonian fluids is dependent on shear rate and temperature, it follows that the mathematical models governing the flow motion are nonlinear even in the case when inertia effects can be neglected, i.e. at low Reynolds number. This type of flow process requires sophisticated numerical solution techniques. Only a few works have been reported in the literature on the BEM numerical solution of inelastic non-Newtonian flows. Phan-Thien [34] proposed a BEM solution of the non-homogeneous momentum equations based on the particular solution approach and RBF interpolation for the nonlinear terms. Davies [31] used the pseudo-body force technique and cell integration to model polymers and optimize mixing equipment. Davies also pointed out that the application of the DRM to the whole domain does not produce accurate results especially when there are high viscosity gradients. As mentioned before, the cell integration BEM combined with subdomain techniques such as the Green’s element method GEM [6] have been applied together with a velocity–vorticity formulation by Skerget and Samec [35], to model nonNewtonian flows in enclosures. As in other previous works by Skerget [36] for the Navier–Stokes equations, the integral formulation keeps the domain integrals without any further simplification. In this section, we will introduce the MD-DRM solution of the coupled system of Stokes equations and the energy equation for non-Newtonian fluids with viscous dissipation effects. Besides the efficiency and accuracy of the proposed numerical method, the MD-DRM has the advantage of the robust RBF interpolation used


85

in the DRM. Due to their smoothness or noise minimization character, the radial functions guarantee high accuracy on the evaluation of the gradient of the field variables [10]. This property of the DRM interpolation permits a precise evaluation of the velocity and temperature derivatives directly from the solution of the matrix system without any additional burden. 3.1 Governing equations The system of mass, momentum and energy conservation equations for the steadystate, non-isothermal flow of an incompressible fluid including buoyancy and viscous dissipation is given in tensor notation by ∂ui = 0, x ∈ ∂xi −

(59)

∂p ∂ + (ηεij ) + ρgi β(T − Tc ) = 0, x ∈ ∂xi ∂xj

(60)

∂ 2T ∂T + σij εij = ρCp uj , x∈ ∂xj ∂xj ∂xj

(61)

k

with boundary conditions ui = ui0 , x ∈ u

(62)

ti = σij nj = ti0 , x ∈ t

(63)

T = T0 , x ∈ T

(64)

q = −k

∂T = q0 , x ∈ q ∂n

(65)

where εij = 21 (∂ui /∂xj + ∂uj /∂xi ) is the strain rate tensor, σij the total stress tensor, ui is the velocity vector, ti is the traction vector, ni is the outward unit normal to the boundary = u + t = T + q of the volume , T is the fluid temperature at any point, p the pressure and q is the heat flux. The thermophysical properties are: ρ the fluid density, η the viscosity which is a function of temperature and the generalized strain rate, k is the thermal conductivity, Cp the thermal capacity and β is the coefficient of thermal expansion. The buoyancy force ρgi β(T − Tc ) in eqn (60) represents the effect of temperature change on density, where Tc is the reference temperature and gi is the magnitude of gravity acting in the i direction. The term σij εij in eqn (61) is the irreversible rate of internal energy increase per unit volume by viscous dissipation or in other words the degradation of mechanical to thermal energy. To convert this problem into a perturbation to a base Newtonian flow, the total stress tensor is decomposed into a Newtonian and a nonlinear component as follows, (e)

σij = −pδij + ηN εij + τij

(66)

86 Domain Decomposition Techniques for Boundary Elements where ηN is an arbitrary constant viscosity that can be chosen to be the zero shear (e) rate viscosity and τij represents the non-Newtonian effects in the stress tensor. For inelastic generalized Newtonian fluids the nonlinear terms are: (e)

τij (γ˙ ) = (η − ηN )εij

(67)

where the non-Newtonian viscosity η is a function of the generalized shear rate γ˙ given by ' (68) γ˙ = 2εij εji With the formulation introduced in eqn (66) the momentum equation (60) can be rewritten as (e)

−

∂τij ∂ 2 ui ∂p + ηN + ρgi β(T − Tc ) + =0 ∂xi ∂xj ∂xj ∂xj

(69)

The viscosity of most non-Newtonian fluids such as polymers is usually a decreasing function of the generalized shear rate γ˙ and this is known as shear-thinning behaviour. For a Newtonian fluid the viscosity η is a constant value µ, η(γ˙ ) = µ = constant

(70)

On the other hand, the most commonly used expression for the viscosity of a nonNewtonian fluid is the power law or Ostwald–de-Waele model [32], η(γ˙ ) = K γ˙ n−1

(71)

where K is called the consistency index and n ∈ [0, 1], the power law index. There are other more accurate semi-theoretical models to describe the behaviour of non-Newtonian fluids, some of them are: •

The Carreau model [32] η(γ˙ ) = η∞ +

•

η 0 − η∞ [1 + (λγ˙ )2 ](1−n)/2

(72)

This model has four adjustable parameters to fit the experimental data: λ is a characteristic time, η∞ is a constant viscosity at very high shear rates, η0 and n are the same as in the power law model. The hyperbolic tangent model [32] n γ˙ η(γ˙ ) = A − B tanh k

(73)

where the parameters A, B, k and n are obtained by data fitting techniques to experimental data.


87

These models account for the effect of the shear rate on viscosity. However, the actual viscosity is also dependent on temperature through the Andrade law [32, 37] of the form η = η(γ˙ )e

E R

1 1 T − T0

(74)

where E is the activation energy of the fluid; R the ideal gas constant and η(γ˙ ) is the viscosity evaluated at the reference temperature T0 . Besides, in the above system of equations the flow field and temperature field are completely coupled, i.e. two-way coupling, making the problem even more complex. 3.2 Multi-domain integral formulation In terms of the Stokes fundamental solution, the velocity field in this problem has the same integral representation formulae as the previous one, i.e. eqn (16), but with the vector density g in the domain integral given by (e)

gi = −ρgi β(T − Tc ) −

∂τij

∂xj

(75)

instead of the convective and bouyancy terms in (17) Similarly, the integral representation of the energy transport equation in terms of the Laplace’s fundamental solution, at each domain, is given by eqn (23) with the scalar density h in the domain integral given by h = uj

∂T − σij εij ∂xj

(76)

instead of the convective term given in (25). As before, the final set of equations is completed by assembling the integral equations for each domain element, using the traction equilibrium and velocity (+) (−) compatibility at the common interfaces between subregions, i.e. ui = ui and (+) (−) σij nj + σij nj = 0. Likewise, the temperature integral equations must also be assembled using the continuity of the temperature field and the heat flux balance (+) (−) condition, i.e. Ti = Ti and ((∂T /∂xj )(+) )nj + ((∂T /∂xj )(−) )nj = 0. To express the domain integral in equations in terms of equivalent boundary integrals, the DRM approximation is reemployed, as explained before. The evalu(e) ation of the extra stress tensor τij and other terms appearing in eqns (75) and (76), require the numerical approximation of derivatives of velocity and temperature, which are evaluated using the approach given in Section 2.4. Once these derivatives are known, they can be used to obtain the value of the non-Newtonian viscosity, the stress tensor and the convective terms at each point of each subdomain. For the numerical solution of the problem the surface n of each subregion or domain element can be discretized by means of isoparametric linear

88 Domain Decomposition Techniques for Boundary Elements boundary elements. Along each element the integrals are calculated in terms of the nodal values of the velocity and tractions and using linear interpolation functions. In the present case, the resulting system of integral equations are nonlinear and coupled through several temperature terms. Their corresponding matrix representation can be solved using a Newton–Raphson scheme combined with a line-search algorithm intended to reduce the error at each iteration [25]. To account for the coupling between the momentum and energy equations, we used direct iteration. The velocity in the convective term from the energy equation is considered as a known variable from the previous solution of the momentum equations obtained by the Newton–Raphson method. Likewise, at each iteration the temperature-dependent terms in the momentum equations can be known from the previous solution of the energy equation. For fluids with variable viscosity, there is an additional source of problems in the numerical iteration. In the multi-domain solution of non-Newtonian problems there will be subdomains where the viscosity remains nearly constant and other regions with higher gradients where the viscosity changes ostensibly. These differences between subregions can make the residual function of the iterative method have flat regions of local minima or valleys where the iteration process stagnates. Additionally, the DRM for the non-Newtonian case is based on a perturbation formulation given in eqn (69), so when the assumed constant viscosity ηN is very far from its actual value η, Newton’s method might not make good progress unless the initial guess for the iterations is very close to the true solution. To alleviate the problems mentioned above, a predictor-corrector approach was designed in which a sequence of linear problems can be solved to get a better initial guess for starting the nonlinear iterations. In the predictor stage each subdomain is assumed to have a constant viscosity equal to the average viscosity of the boundary nodes that define the subdomain. This average value is then corrected in the corrector stage of the process when the original nonlinear system of equations is solved by Newton’s method until convergence is reached. If Newton’s method stops at a local minimum, the predictor step can be performed once again but starting from that local minimum and so on. 3.3 Non-isothermal Couette flow with viscous dissipation In this example we consider the flow of an incompressible power law fluid between two axial cylinders as shown in Fig. 2. As the inner cylinder rotates, each cell of fluid rubs against the adjacent cells. This rubbing of adjacent layers of fluid produces heat; that is, mechanical energy is degraded into thermal energy. The magnitude of the viscous dissipation effect depends on the local velocity gradient. The surfaces of the inner and outer cylinders are maintained at the same temperature T = T0 . The geometry is defined by the radius R1 = 1 m and R2 = 5 m. In our numerical simulation of this problem, for the MD-DRM subdomain approach, a subdivision of 400 domain elements mesh was used as shown in Fig. 3.


89

R1 T0

T0

ω R2

Figure 2: Couette flow with viscous heat generation. The mathematical model for this example consists of the continuity equation, the momentum equations and the energy equation, i.e. ∂ui = 0, x ∈ ∂xi ∂p ∂ − + (ηεij ) = 0, x ∈ ∂xi ∂xj k

∂ 2T + σij εij = 0, x ∈ ∂xj ∂xj

(77) (78) (79)

There are three possible cases depending on whether the viscosity is dependent on temperature, on shear rate or on both of them. In all these cases the problem consists of a two-way coupling system of nonlinear equations. 3.3.1 Case I: Viscosity does not vary with temperature η = η(γ˙ ) For a power law fluid with a viscosity that is only a function of the generalized shear rate the rheological model is given by eqn (71) in terms of the two constants K = 1 Pa · sn (the consistency coefficient) and n (power law index). The numerical results for different values of the parameter n can be seen in Figs 4–6.


Figure 3: Uniform 20 × 20 domain elements mesh used for MD-DRM solution of the Couette problem.

Figure 4 presents our result for the different velocity profiles, Fig. 5 the viscosity profiles and Fig. 6 the temperature profiles. These results have been compared with the analytical solution obtained according to [33] by direct integration of the momentum and energy equations. In all cases the numerical error was less than 0.5%, as shown in the figures. 3.3.2 Case II: Viscosity is a function of temperature only η = η(T ) In this case we have considered that viscosity is constant (Newtonian fluid) and the temperature rise in the fluid results in variations of the viscosity which then affect the velocity profile. The equations of motion and energy are coupled and must be solved by an iterative method for each point of the flow system. The expression for the viscosity in this case is the following (see eqn (74)) E

η = η0 e R with η0 = 1 Pa · s.

1 1 T − T0

(80)


91

5

4.5

4 − Analytical o MD−DRM

n=1 3.5

3

u

n=0.8 2.5

2

1.5

1 n=0.6 0.5 n=0.4 0 0.5

1

1.5 r

2

2.5

Figure 4: Velocity profiles for the Couette flow problem with viscous dissipation.

12

10

− Analytical o MD−DRM

n=0.4

µ (Viscosity)

8

6

4

n=0.6 2 n=0.8 n=1 0 0.5

1

1.5 r

2

2.5

Figure 5: Viscosity profiles for the Couette flow problem with viscous dissipation.

92 Domain Decomposition Techniques for Boundary Elements 60 n=1.0 58

o MD−DRM − Theoretical

56

T

54

n=0.8

52

n=0.6

n=0.4

50

48 0.5

1

1.5 r

2

2.5

Figure 6: Temperature profiles for the Couette flow problem with viscous dissipation. In this problem the viscosity is constant and it does not change with temperature. In Figs 7–9, we show the comparison between our numerical solution for different values of the activity energy coefficient and the solution obtained with a completely different numerical scheme for the velocity, temperature and viscosity profiles, respectively. As before, the agreement between the two solutions is excellent. The numerical scheme used to validate our result was based on a finite difference three-stage Lobatto IIIa formula which is basically a collocation technique [38, 39]. 3.3.3 Case III: Viscosity is a function of both shear rate and temperature η = η(γ˙ , T ) This is the most complex case where the effects of shear rate and temperature affect the value of viscosity, i.e. E

η = K γ˙ n−1 e R

1 1 T − T0

(81)

As before, in this case we compare our results with those obtained with the finite difference method mentioned above [38, 39]. Figures 10–12 show the comparison between the velocity, temperature and viscosity profiles obtained with the two approaches. The numerical results were obtained for two different values of the power law index n of a non-Newtonian fluid at a constant activation energy E = 107 J/mol, and as can be seen in these figures the MD-DRM results are very close to the test solution with a numerical difference less than 1%.


93

6

E=109 5 − Shampine o MD−DRM

8

E=10 4

7

u

E=10 3

2

1

0 0.5

1

1.5 r

2

2.5

Figure 7: Velocity profiles for the Couette flow problem with temperature dependent viscosity. 59 7

E=10

58

o MD−DRM − Shampine 57

8

E=10

56

T

55

54

53 9

E=10 52

51

50 0.5

1

1.5 r

2

2.5

Figure 8: Temperature profiles for the Couette flow problem with temperature dependent viscosity.

94 Domain Decomposition Techniques for Boundary Elements 1

0.9 7

E=10 0.8

0.7

0.6

µ

0.5 8

o MD−DRM − Shampine

E=10 0.4

0.3

0.2 9

E=10

0.1

0 0.5

1

1.5 x

2

2.5

Figure 9: Viscosity profiles for the Couette flow problem with temperature dependent viscosity. 5

4.5 o MD−DRM − VBP4C

4

3.5

u

3

n=0.8 7 E=10

2.5

2

1.5

1

n=0.4 7 E=10

0.5

0 0.5

1

1.5 r

2

2.5

Figure 10: Velocity profiles for the Couette flow problem with viscous dissipation and temperature dependent viscosity.


95

56

o MD−DRM − BVP4C

n=0.8 7 E=10

55

54

T

53

52 n=0.4 7 E=10

51

50 0.5

1

1.5 r

2

2.5

Figure 11: Temperature profiles for the Couette flow problem with viscous dissipation and temperature dependent viscosity. 14

12 o MD−DRM − BVP4C 10 n=0.4 7 E=10

µ

8

6

4

n=0.8 7 E=10

2

0 0.5

1

1.5 r

2

2.5

Figure 12: Viscosity profiles for the Couette flow problem with viscous dissipation and temperature dependent viscosity.


4 Conclusion In this work a MD-DRM has been described and applied for the solution of nonisothermal Newtonian and non-Newtonian flow problems with viscous dissipation. Several other numerical examples are provided in our previous articles [26, 40], including cases of natural convection with viscous dissipation. The multi-domain technique is basically a domain partition method which divides the entire domain into smaller regions. In each subregion or domain cell the integral representation formulae for the flow and temperature field are applied, and between adjacent regions the corresponding matching or continuity conditions are imposed. The domain integrals in each domain element are treated by the DRM approximation in terms of the most efficient interpolation functions available in the mathematical literature. Despite the relatively rough meshes used and most simple boundary elements the results show convergence and high accuracy. Despite the fact that in the MD-DRM there are internal elements and a mesh, it must not be regarded as a domain method because all the domain integrals are converted into equivalent boundary integrals in each cell or element. Therefore the proposed multi-domain method preserves the boundary-only character of the BEM. The different test examples presented show the versatility and efficiency of the proposed numerical scheme for the solution of nonlinear flow problems. In the present article we have also extended the capabilities of the MD-DRM to nonisothermal Newtonian and non-Newtonian flow problems.

References [1]

[2]

[3] [4]

[5]

[6]

Popov, V. & Power, H., The DRM-MD integral equation method for the numerical solution of convection-diffusion equation. Boundary Element Research in Europe, Computational Mechanics Publications: Southampton, pp. 67–81, 1999. Florez, W.F. & Power, H., Comparison between continuous and discontinuous boundary elements in the multidomain dual reciprocity method for the solution of the Navier–Stokes equations. Engineering Analysis with Boundary Elements, 25, pp. 57–69, 2001. Brebbia, C.A., Telles, J. & Wrobel, L.C., Boundary Element Techniques, Springer-Verlag: Berlin and New York, 1984. Partridge, P.W., Brebbia, C.A. & Wrobel, L.C., The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications: Southampton, 1992. Ahmad, S. & Banerjee, P.K., A new method in vibration analysis by BEM using particular integrals. J. Eng. Mech. Div., ASCE, 113, pp. 682– 695, 1986. Taigbenu, A.E., The Green element method. Int. J. Numer. Methods Eng., 38, pp. 2241–2263, 1995.


[7]

[8]

[9]

[10] [11] [12]

[13] [14]

[15]

[16]

[17] [18] [19]

[20] [21] [22]

[23]

97

Kansa, E.J. & Carlson, R.E, Radial basis functions: a class of grip-free, scattered data approximations. Computational Fluid Dynamics, 3(4), pp. 479–496, 1995. Florez, W.F. & Power, H., Multi-domain dual reciprocity for the solution of inelastic non-Newtonian problems. Computational Mechanics, 27, pp. 396–411, 2001. Florez, W.F., Power, H. & Chejne, F., Conservative interpolation for the boundary integral solution of the Navier–Stokes equations. Computational Mechanics, 26(6), pp. 507–513, 2000. Hickernell, F., Radial basis function approximation as smoothing splines. Appl. Math. Comput., 102(1), pp. 1–24, 1999. Incropera, F.P. & DeWitt, D., Fundamentals of Heat and Mass Transfer, Wiley: New York, 1996. Onishi, K. & Kurok, T., An application of the boundary element method to incompressible laminar viscous flow. Engineering Analysis with Boundary Elements, 1, pp. 122–127, 1994. Skerget, L. & Samec, N., BEM for non-Newtonian flow. Engineering Analysis with Boundary Elements, 23(5–6), pp. 435–443, 1999. Kuroki, T. & Onishi, K., Thermal fluid flow with velocity evaluation using boundary elements and penalty function methods. Proceedings of the VII International BEM Conference, Computational Mechanics: Southampton, 1985. Brebbia, C.A., Tanaka, M. & Wrobel, L.C., A boundary element analysis of natural convection problems by penalty function formulation. Boundary Elements, Computational Mechanics: Southampton, 1988. Kitagawa, K., Brebbia, C.A. & Tanaka, M., A boundary element analysis for themal convection problems (Chapter 5). Topics in Boundary Element Research, Springer: Berlin, 1989. Ladyzhenskaya, O.A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach: New York, 1963. Brebbia, C.A. & Dominguez, J., Boundary Elements: An Introductory Course, Computational Mechanics Publications: Southampton, 1992. Florez, W.F. & Power, H., Multi-domain dual reciprocity BEM approach for the Navier–Stokes system of equations. Comm. Num. Meth. Engng., 16, pp. 671–681, 2000. Goldberg, M.A. & Chen, C.S., Discrete Projection Methods for Integral Equations. Computational Mechanics Publications: Southampton, 1997. Power, H. & Wrobel, L.C., Boundary Integral Methods in Fluid Mechanics, Computational Mechanics Publications: Southampton, 1995. Power, H. & Mingo, R., The DRM subdomain decomposition approach to solve the two-dimensional Navier–Stokes system of equations. Engineering Analysis with Boundary Elements, 24, pp. 107–119, 2000. Mingo, R. & Power, H., The DRM subdomain decomposition approach for two-dimensional thermal convection flow problems. Engineering Analysis with Boundary Elements, 24, pp. 121–127, 2000.

98 Domain Decomposition Techniques for Boundary Elements [24] Telles, J.C., A self-adaptive coordinate transformation for efficient numerical evaluation of general boundary element integrals. Int. J. Num. Meth. Eng., 24, pp. 959–973, 1987. [25] Dennis, J.E. & Schnabel, R.B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall: New Jersey, 1980. [26] Florez, W., Power, H. & Chejne, F., Numerical solution of thermal convection problems using the multidomain boundary element method. Numerical Methods for Partial Differential Equations, 18(4), pp. 469–489, 2002. [27] Press, W.H., Teulkolski, S.A., Flannery, B.P. & Vetterling, W.T., Numerical Recipes, Cambridge University Press: Cambridge, 1992. [28] Paige, C. & Saunders, M., LSQR: sparse linear equations and least squares problems. ACM Transactions on Mathematical Software, 8(2), pp. 195–209, 1982. [29] Caseiras, C.P., Development of finite volume methods for the solution of the incompressible Navier–Stokes equations, PhD Thesis, Zaragoza University, 1994. [30] Vahl, D., Natural convection of air in a square cavity: a benchmark numerical solution. Int. Journ. Num. Meth. Fluids, 3, pp. 249–264, 1983. [31] Davis, B.A., Investigation of non-Linear flows in polymer mixing using the boundary integral method, PhD Thesis, University of Wisconsin-Madison, Department of Mechanical Engineering, Madison, 1995. [32] Agassant, J.F.,Avenas, P., Sergent, J.Ph. & Carreau, P.J., Polymer Processing: Principles and Modeling, Hanser: New York, 1991. [33] Bird, R.B., Stewart, W. & Lightfoot, E.N., Transport Phenomena, John Wiley and Sons: New York, 1960. [34] Phan-Thien, N., Applications of boundary element methods in nonNewtonian fluid mechanics, Boundary Element Applications in Fluid Mechanics, Computational Mechanics Publications: Southampton, 1995. [35] Skerget, L. & Samec, L., BEM for non-Newtonian fluid flow. Engineering Analysis with Boundary Elements, 23, pp. 435–442, 1999. [36] Skerget, L. & Hribersek, M., Iterative methods in solving Naviers–Stokes equations by the boundary element method. Int. J. Num. Meth. Fluids, 39, pp. 115–139, 1996. [37] Osswald, T.A. & Menges, G., Materials Science of Polymers for Engineers, Hanser: Munich, 1995. [38] Shampine, L.F., Numerical Solution of Ordinary Differential Equations, Chapman Hall Mathematics: London, 1994. [39] The Mathworks Inc., MATLAB The Language of Technical Computing, Ver. 6. [40] Florez, W., Power, H. & Chejne, F., Multi-domain DRM boundary element method for non-isothermal non-Newtonian Stokes flow with viscous dissipation. International Journal of Numerical Methods for Heat and Fluid Flow, 13(6), pp. 736–768, 2003.

CHAPTER 4 Modelling flow and solute transport in fractured porous media using the DRM multidomain technique A. Peratta & V. Popov Wessex Institute of Technology, Southampton, UK.

Abstract The objective of this work is to develop a new numerical approach for the 3D modelling of flow and transient solute transport in fractured porous media which would provide an accurate and efficient treatment of 3D complex geometries and inhomogeneity distributions. For this reason, and in order to eliminate as much as possible the number of degrees of freedom, the fracture network, fractures and fractures intersections are solved with a coupled 2D–1D model while the porous matrix is solved independently with a 3D model. The interaction between both models is accounted for by a coupling iterative technique. In this way, it is possible to improve efficiency and reduce CPU usage by avoiding 3D mesh refinements of the fractures. The approach is based on the discrete fracture model in which the exact geometry and location of each fracture in the network must be provided as an input. The numerical solution is based on a multidimensional coupling strategy implemented using the boundary element method multidomain scheme for the flow and the boundary element dual reciprocity method multidomain scheme for the transport. Accurate results and high efficiency have been obtained and are reported in this chapter.

1 Introduction 1.1 General Modelling of flow and transport in fractured porous media (FPM) is of increasing interest in the engineering and environmental fields. The main objective of

100 Domain Decomposition Techniques for Boundary Elements the present work is to develop a new numerical technique based on the multidimensional coupling of the boundary element method (BEM) to simulate the processes of flow and transient solute transport in 3D FPM by making use of the discrete fracture network model (DFNM). This chapter is organised in the following way. The first section describes an introduction to FPM, the aim and motivation of the present work, some references to earlier publications on the subject, the background of the conceptual model and an introduction to the numerical approach. Section 2 presents the governing equations, and Section 3 is dedicated to a description of the BEM and its features used in this work. Section 4 deals with the numerical scheme used for the porous matrix, the fracture network and the fracture intersections, introducing the concept of ‘pipes’. Section 5 introduces some comments on the computational implementation of the computational code, and Section 6 shows the numerical results obtained with the method. The first part contains the validation of the method and the numerical performance of the scheme, while the second part shows results of the model applied to more realistic situations. 1.2 Fractured porous media In the continuum approach, a porous medium can be regarded as a multiphase material body in which a representative elementary volume (REV) can be determined for it, such that no matter where we place it within a domain occupied by the porous medium, it will always contain both a persistent solid phase and a void space. The size of the REV is such that the physical parameters that represent the distributions of the void space and of the solid matrix within it are statistically meaningful [1]. On the other hand, a fractured porous medium can be regarded as a composition of two well-distinguished entities, namely individual fractures, fracture networks and the solid medium, or porous matrix, existing between the fractures (named also the porous matrix). A thorough description of FPM and fracture networks can be found in reference [2]. Roughly speaking, individual fractures look like plane discontinuities when viewed from afar, or like two solid surfaces that surround an interstitial 3D space when viewed from close-up. Fractures are regarded as porous media with usually higher permeability than the adjacent porous matrix [3–5] and with apertures of many orders of magnitude smaller than the integration domain. 1.3 Overview At present there are many different approaches for modelling flow and solute transport in FPM. One possible way to classify most of the available models is in terms of the degree of detail and accuracy that each one of them can describe according to their length ranges and space scales. The two major divisions that appear [1] are the microscopic representation, in which the scale is such that it is possible to distinguish the void region inside a pore and the different phases within it, and the macroscopic representation, in which transport properties are usually averaged

Flow and Solute Transport in Fractured Porous Media

101

over REVs of porous material relatively large in comparison to the pore size, but at the same time small enough to describe local properties in the physical domain. 1.4 The continuum approach The macroscopic representation yields to the continuum approach. Hence, under the continuum assumptions and according to the scale of the problem, there are four major subdivisions, as defined by Bear and Berkowitz [6]: • •

•

•

The very near field zone: In which the region of interest is usually focused on a single well-defined fracture, embedded in the porous matrix. The near field flow: In which a small amount of fractures that may or may not intersect each other defines a small and bounded fracture network, which is deterministically known. This model allows statistical representation of the fractures with random shapes and orientations. Moreover, the fractures and the porous matrix represent well-defined separated regions in space. The far field model: In this case, transport may be regarded as taking place in two overlapping continua. One for the fluid in the fracture network and the other for the porous matrix. There are material fluxes exchanged between both media that occupy the same physical location. The very far field model: The fluid in the FPM as seen from afar may be regarded as a single continuum, with an equivalent permeability. The properties of the void space reflect both the presence of the fracture network and the porous matrix blocks. This kind of model is applicable in cases where the system under consideration allows sufficient mixing between fluid and contaminants in the fracture and in the porous blocks, bringing the two systems to a local equilibrium at every (macroscopic) point. Such a system can be described by a model analogous to that of a regular porous medium.

1.5 The discrete fracture model The discrete fracture network model adopted in this work can be regarded as a network of interconnected fractures embedded in a porous matrix. Each individual fracture is represented by an arbitrary surface, that can be plane or not, finite or infinite and of various shapes. In general, the main feature of the fractures is that they might cover a wide range of scales, from submillimetric fissures to long faults of hundreds of kilometres. Another distinctive feature is that fractures might intersect each other randomly, generating a complex interconnected network. The goal of this work is to model the fractures individually, in such a way that they are deterministically prescribed so that the exact location and geometry are known and predefined. When viewed from a close distance, each surface of the fracture network can be represented as a 3D object with one of its dimensions of several orders of magnitude smaller than the other two. This dimension is the aperture of the individual fracture, namely wf , which at the same time might vary from point to point within the fracture.

102 Domain Decomposition Techniques for Boundary Elements A convenient simplification of the model is to decompose each individual fracture of the fracture network into smaller fracture elements. Each element has associated a constant aperture that might be different from the others. In this way a piecewise varying field of aperture can be prescribed over every single surface of the network. Although each fracture has three dimensions, the model proposed in this work regards the aperture as a scalar field attached to a 2D surface built from a cluster of interconnected flat fracture elements, thus being represented by two local coordinates. The fracture network is itself made of porous media, usually of higher permeability than the adjacent porous matrix in which it is embedded. For the purpose of this work, it is enough to calculate the permeability of the fracture network by means of the Poiseuille law or by data collected from experimental measurements. 1.6 Historical development of porous media modelling In the field of deterministic fracture networks, Warren and Root [7] made a very important contribution with orthogonal structured fracture networks. Later Odeh [8] presented a generalisation of this work with arbitrary unstructured fracture network patterns. In the range of local scale modelling, it is worth mentioning the contribution of Snow [9] where he found exact expressions of permeabilities for infinite fractures. Following on, Kranz et al. [10] and Witherspoon et al. [11] worked out in the experimental field to determine permeabilities at local scale ranges. Brown and Scholz [12] and later Gentier [13] introduced a new method to systematically characterise natural fracture networks. This characterisation was subsequently used by Brown [14] and Moreno et al. [15] to determine the permeability of a single fracture by the integration of the 2D Reynolds equation. Barton and Larsen [16] compiled many years of research on hydromechanical joint properties in a coupled joint behaviour model. Lately, Mourzenko et al. [17] reformulated the calculation of the single fracture permeability by integrating the Stokes equations, and subsequently Koudina et al. [18] contributed with the calculation of the permeability of networks made of interconnected fractures regarded as polygons. On the fracture network scale, significant efforts were made to model transport properties and to match the numerical results with those obtained from the experiment. Bond networks were extensively used by Dienes [19], Long et al. [20] and Cacas et al. [21] with an elegant method to generate off lattice bond networks in three dimensions. Faust and Mercer [22] have demonstrated the circumstances in which the very far field approach can be used to describe flow and heat transport in porous media. During the eighties the trend was the application of concepts derived from the percolation theory for the study of fractures. Percolation deals with the effects of varying the richness of interconnections present in a random system. The basic idea of percolation is the existence of a sharp transition at which the long-range connectivity of the system disappears (or, going the other way, appears). This transition occurs abruptly when some generalised density in this system reaches a critical value called percolation threshold.


103

Researchers attempted to determine the percolation threshold, in this case interpreted as the density of fractures above which the connectivity of fractures is sufficient to enable flow through the network, or at least through part of the fractures. Dienes [19] and Charlaix et al. [23] provided key contributions in this subject bringing new light into network analysis. Also Robinson [24], Charlaix et al. [25], Wilke et al. [26] and Marsily [27] studied the issue of connectivity with percolation theory. Another fruitful point of view was that of Lévy [28] when she applied the technique of multiple scales to the theoretical determination of the permeability of fractured porous media. The general description about fracture networks in reference [1] together with reference [2] represent a convenient starting point that covers the necessary physical aspects for the numerical modelling. An innovative technique to deal with fractures as interfaces in porous blocks has been implemented and tested in two dimensions (the porous matrix is represented in two dimensions, while the fractures are represented as 1D curves) with the finite element method in reference [29]. This reference presents a strategy similar to the one employed here from the point of view of the theory involved in the numerical modelling and it represents a convenient starting point for the formulation and the derivation of the model. The present work is based on the same idea but employs a different numerical strategy since it solves 3D problems using the BEM.

2 Governing equations The mathematical formulation that describes the DFNM is divided into two parts, namely flow and transport. The former is concerned about the velocity field ( v) and the hydraulic head (h) in the FPM, whereas the latter is concerned about the concentration (c) and the flux of the pollutant ( q). Based on the hypothesis of low concentrations and incompressible medium, both parts of the model can be decoupled and solved in a sequential way. The flow problem is solved first and the velocity field becomes an input for the transport problem, driving the convective term. 2.1 Flow The flow model in porous media is based on the continuum approach, and it is generally described by: ∇ · (k∇h) + Qe = Sr

∂h , ∂t

(1)

which was derived [30] by assuming isothermal and homogeneous flow, hydraulic conductivity independent of pressure changes, and specific storativity and hydraulic conductivity unaffected by variations of the porosity, and where h = p/ρg + z is the hydraulic head, Sr = ρg(α + θβ) is the specific storativity, α = (1/(1 − θ))∂θ/∂p is the coefficient of rock matrix compressibility, β = (1/ρ)∂ρ/∂p is the coefficient of compressibility of the fluid, θ is the porosity of the medium, p is the pressure,

104 Domain Decomposition Techniques for Boundary Elements ρ is the density, g is the gravity acceleration, z is the coordinate along direction of gravity, and Qe refers to sources or sinks. Equation (1) represents the conservation of mass assuming the Darcy law for porous media, which relates the velocity (v) to the gradient of the hydraulic head according to v = −k∇h,

(2)

where k is the hydraulic conductivity. 2.1.1 Flow in the porous matrix When the characteristic diffusion time defined as τd = L 2 Sr /k, where L is the length scale of the problem, is several orders of magnitude smaller than the modelling time t, i.e. the time scale of practical interest, it is reasonable to neglect the timedependent term and to consider a steady-state solution for the flow. (In saturated groundwater flows, the compressibility factors are small and the conductivity high, such that τd t.) Under these circumstances, the governing equation in the porous matrix becomes: ∇ · (−km ∇hm ) = 0,

(3)

where subscript m stands for the porous matrix. 2.1.2 Flow in a single fracture Let us consider the tangential plane to the surface that represents the fracture, and a local system of coordinates (x , y , z ) such that z is coincident with the local normal to that plane (i.e. nˆ m,B in Fig. 1). The DFNM assumes that wf Lf ,

(4)

kf km ,

(5)

Porous Block A

nm

D

p2

p3

p1 Fracture Plane

np

nf

Γ nm

B

Pipe C

Figure 1: Intersection of two fractures.


105

where wf is the local aperture of the fracture and Lf is the extension of the fracture. Therefore, it is possible to represent the solution in the fracture as a superposition of the 2D solution on the tangential plane and a 1D profile g(z) along z h(x, y, z) = hf (x , y )g(z ),

0 ≤ z ≤ wf .

(6)

Using this last expression in (3) and integrating along z yields ∇2D · (−kf ∇2D hf ) =

1 vm,n,j , w¯ f

(7)

j=A,B

where vm,n,j = vm,j · nˆ m,j ,

j = A, B

(8)

is the normal velocity directed from the adjacent porous blocks to the fracture plane in nˆ m,j direction, and the following 2D operator has been employed: ∇2D · v := ∇ · v − ∇(v · zˆ ) · zˆ ,

and

∇2D h := ∇h − ∇h · zˆ ,

vm,j for j = A, B is the velocity in the jth porous block at the interface that limits with the fracture plane. For simplicity, in this work a fracture plane will be always confined between two blocks of PM, thus disregarding the case of fractures as surface boundaries of the whole integration domain that represents the FPM (Fig. 1) and w¯ f is the equivalent aperture of the fracture. wf g(z )dz . (9) w¯ f := 0

2.1.3 Fracture intersections In any arbitrary interconnected fracture network, a certain number (mf ) of fractures might intercept each other converging into a common intersection. The resulting pipe could have material properties significantly different from its adjacent environment (either fractures or porous matrix), in the same way that fractures represent a discontinuity of high conductivity for the adjacent blocks of porous matrix. Furthermore, since single fractures are represented by surfaces, it is natural to represent their intersections by 1D curves in the 3D space. In the fracture intersections or pipes (we will often refer to pipes as the theoretical representation of the real fracture intersections), integration of the continuity equation over the cross section Ap (Fig. 1) yields: ∂ 2 hp −kp 2 dA = vf,n d, (10) ∂η Ap where vf,n = −kf (∂hf /∂ nˆ f ) is the velocity along the fracture planes adjacent to the pipe in the direction of the outward normal unitary vector (ˆnf ) perpendicular to the longitudinal axis of the pipe, and η is the coordinate along the pipe (ˆnp ), see Fig. 1.

106 Domain Decomposition Techniques for Boundary Elements Furthermore, assuming that: (1) the main contribution of the flux in the righthand side integral of (10) comes from the adjacent fractures, thus neglecting the part of the integral involving the contact between the lateral surface of the pipe and the porous matrix and that; (2) the hydraulic head in a given cross section of the pipe is constant (depending only on the local coordinate along the pipe η); then eqn (10) can be expressed in the following way: f

∂ 2 hp = wf,i vf,n,i , ∂η2

m

−Ap kp

(11)

i=1

where i denotes each adjacent fracture element to the pipe, the subindex p stands for the pipe, and Ap is the area of the pipe cross section. 2.1.4 Flow in pipe connectors In the same way that fractures intersect each other creating pipes, an arbitrary number of pipes might intersect each other creating pipe connectors. These objects can be regarded as closed volumes of similar extension in all ' directions and comparable with the mean diameter of all the convergent pipes Ap . By analogy, it is consistent to represent pipe connectors by points called multiple pipe connectors (MPCs), disregarding their 3D structure by integration in volume. Thus, in an MPC, the following zero-dimensional version of the mass conservation is considered for the flow: mp

Ap,i vp,η,i = 0

(12)

i=1

together with the continuity of hydraulic heads at the end-points of the converging pipes, given by: hp,1 = hp,2 = · · · = hp,mp ,

(13)

where mp is the number of pipes joining in the point. 2.2 Transport This section presents the governing equations for transport in porous matrix, fractures, pipes, and MPCs. It is considered that all these entities offer the same kind of discontinuity for both flow and transport problems, so the derivation is similar to the one for the flow, this being the only difference and the leading operator of the partial differential equation. The formulation for the complete problem where all the entities interact together and the description of the coupling strategies will be presented in the next sections. In general, the transport process under consideration can be described by the time-dependent advection–diffusion reaction (ADR)


107

equation for the concentration of pollutant c: R

∂c + ∇ · q = kc + ρ, ∂t q = vc − D∇c,

(14) (15)

D = D + αv, M

(16)

where v = |v|, D is the dispersion coefficient, DM is the molecular diffusivity, R is the retardation factor, and α is the dispersivity. It is practical to define the p p-dimensional ADR operator (Lχ ) as: Lpχ := Rχ

∂ ∂ ∂2 − Dχ 2 − kR , + vχ,i ∂t ∂xi ∂xi

∀xi ∈ Rp ,

(17)

where subscript χ can be any of m, f, or p identifying porous matrix, fractures, or pipe element, respectively; Rχ is the retardation factor, the index p denotes the dimensionality of χ and i = 1, . . . , p identifies the corresponding space coordinate. Thus, the general formulation for the transport in any entity of the FPM can be summarised in the following expression: Lpχ [cχ ] = ρχ

(18)

where ρχ depends on χ and p, and represents the coupling term between entities. Table 1 summarises all the possibilities. It can be seen that the approach can be applied for FPM in 3D or 2D. When the problem is 3D the column m = 3 applies, and for 2D problems m = 2. 2.2.1 Transport in the porous matrix Assuming a small variation of solute concentration and low compressibility of the fluid, the fluid density variation can be neglected. In addition, considering a homogeneous isotropic media, the concentration in the porous matrix is described by eqn (18) with χ = m and: ρm = 0. (19) Table 1: Entities involved in the problem and their possible dimensionality. Entity

χ

p (3D case)

p (2D case)

Porous matrix Fracture Pipe MPC

m f p o

3 2 1 0

2 1 0 –

108 Domain Decomposition Techniques for Boundary Elements (For simplicity, the theory is presented for homogeneous properties, despite the fact that the domain decomposition technique mentioned below allows treatment of piecewise homogeneous media.) 2.2.2 Transport in fractures Integration of (15) along z in the same way as it was done in the flow problem yields: 1 ρf = qm,n,j , (20) w¯ f j=A,B

where qm,n,j = q · nˆ m,j is the normal solute flux coming from the porous matrix into the fracture. Thus, the influence of the two blocks of porous matrix (A and B) adjacent to the fracture (see Fig. 1) is considered as a source term inside the fracture. Finally, the formulation for a single fracture is represented by eqn (18) with χ = f and eqn (20) describing ρf . 2.2.3 Transport in pipes Integration of (15) over a given pipe cross section Ap yields 1 ρp (η, t) = q · nˆ r d, Ap

(21)

ˆ is where t represents time, η is the local coordinate along the pipe, nˆ r = nˆ p × d the unitary normal vector normal to the pipe at η, nˆ p is the local unitary vector along the pipe at point η, and is the boundary of the pipe cross section, see Fig. 1. If we assume that the solute flux between a pipe and its surrounding porous matrix, i.e. the contribution of the integral (21) between points p1 and p2 in Fig. 1, is negligible in comparison with the flux between a pipe and its adjacent fractures, the integral (21) between p2 and p3 , then ρp can be approximated by: ρp (η, t) =

mf 1 vf,n,k cf,k (η, t) − Df,k ∇2D cf,k (η, t) · nˆ f w¯ f,k . Ap

(22)

k=1

The 1D formulation for pipes is finally represented by eqn (18) with χ = p and the source term given by (22). 2.2.4 Transport in pipe connectors Similar to the flow problem, the 3D structure of an MPC is collapsed into a point where L0o is identically zero, and the source term of (18) becomes q p,x · nˆ p dSo (23) ρo = ∂Vo

thus leading to the continuity of concentration cp,1 = cp,2 = · · · = cp,mp

(24)


109

and the conservation of the normal solute flux mp

Ap qp,n,i = 0.

(25)

i=1

3 Numerical method This section provides an overview of the numerical methods involved in this work: the BEM in combination with the dual reciprocity method (DRM). There are also two additional explanations regarding the domain decomposition technique and the time integration scheme. 3.1 The boundary element method This subsection shows the main steps of the BEM applied to the Poisson equation, which is the leading equation in the present formulation. However, a complete introduction to the BEM can be found in references [31–33]. One of the main advantages of the method is that it only requires discretisation of the boundary into elements, avoiding the need to discretise in volume. The starting point of the BEM is the integral formulation of the governing differential equation. There are different ways to reach the same formulation, e.g. the Green’s third identity or weighted residuals, the latter being the one that offers a more flexible and general mathematical development, in the sense that it is easier to apply to general complex partial differential equations. Consider the Poisson equation valid in with an arbitrary source term ρ ∇ 2u = ρ (26) with the following conditions prescribed on = 1 + 2 (the boundary of ), u = u¯ ∂u = q¯ ∂n

on 1 , on 2 ,

(27) (28)

where n is the normal to the boundary . Then, the weighting of (26) with the weight function u∗ produces: ∂u 2 ∗ ∗ (∇ u − ρ)u d = (u − u¯ )q∗ d. (29) − q¯ u d − 2 ∂n 1 And integration by parts of the last expression yields ∂u ∗ ∂u∗ 2 ∗ ∗ u d + d u(∇ u )d = − q¯ u d − u 2 1 ∂n 2 ∂n ∂u∗ + u¯ ρu∗ d. d + 1 ∂n

(30)

110 Domain Decomposition Techniques for Boundary Elements The weight function u∗ is chosen to be the Green function of the Laplace operator, then it satisfies, for a concentrated source located at xi ∇ 2 u∗ + δ(x − xi ) = 0.

(31)

As a consequence, the integral expression (30) becomes: ci ui +

∂u∗ ud − ∂n

u∗

∂u d + ∂n

ρu∗ d = 0,

(32)

where ci =

1, if xi ∈ 1/2, if xi ∈ for smooth boundary.

(33)

Equation (32) represents the integral formulation of (26), and is the starting point of the BEM. The first three terms of (32) involve only boundary integrals. The boundary can be discretised into Ne2D elements. In the 3D case, the integration domain (used to represent the porous matrix blocks) is a volume, and its boundary is discretised by means of a collection of connected triangular or quadrilateral isoparametric elements. In the 2D case the integration domain () is a surface. The BEM for the 1D case will be used to represent pipes or the intersections that occur between the fractures of the fracture network. The integration domain () is a linear straight segment and their two geometrical endpoints become the boundary (). In ( general, the boundary is discretised into Ne boundary elements, such that e = N j=1 j . ci ui +

Ne

j=1

e

∂u∗ ud − ∂n

N

j

j=1

u j

∗ ∂u

∂n

ρu∗ d = 0.

d +

(34)

The treatment of the domain integral that appears in the last term of (34) will be deferred for the next section. Each boundary element contains a number Nfn of collocation nodes, where the potential or fluxes are evaluated. In this way, the values of the potential or its normal derivative at any point defined by the local coordinates (ξ ) on a given boundary element can be defined in terms of their values at the collocation nodes, and the interpolation functions ψk with k = 1, Nfn in the following way: u(ξ ) =

Nfn

ψk (ξ )uk ,

(35)

k=1

Nfn ∂u(ξ ) ∂u ψk (ξ ) . = ∂n ∂n k k=1

(36)


111

By applying the discretisation of the boundary and the collocation technique, the expression (32) can be rewritten in the following way: c i ui +

$ Nfn Ne

j=1 k=1

+

j

% $ % Nfn Ne

∂ukj ∂u∗ ∗ ψk dj ukj − u ψk dj ∂nj ∂n j j=1 k=1

ρu∗ d = 0.

(37)

The notation can be simplified by making use of the matrix notation, so the last expression can be written in the following way: H[u] − G[∂n u] = −

ρu∗ d,

(38)

where Hil = δil ci + Gil = j

j

∂ui∗ ψk (ξj )dj , ∂nj ξj

ui∗ (ξj )ψk (ξj )dj ,

(39a) (39b)

( e where index l = 1, . . . , Nfe and Nfe = N j=1 Nfn,j is the total number of collocation nodes. In fact, the index l is used to identify one of the adjacent freedom (collocation) nodes from a global point of view, and is a given as a function of the indicator of element (j), and the local collocation node of that element (k). The boundary element dj can be expressed in terms of the domain local coordinates (ξ ) through the Jacobian of the transformation |J| in the following way: dj = |J| dξ1 · · · dξh ,

(40)

where h is the dimension of . Finally, provided that the right-hand side term of eqn (38) can be written as a given vector in function of the source term, or a characteristic matrix in function of the unknown potentials and normal fluxes at the collocation nodes of the boundary, the application of the prescribed boundary conditions and the assembly of the linear set of equations, that (38) produces, yields a determined system of equations of dimension Nfe × Nfe of the form AX = b,

(41)

where the vector of unknowns (X) contains the potentials and normal fluxes that were not prescribed as boundary conditions, the matrix A involves the coefficients of H and G, and the right-hand side term involves the source term and the boundary conditions, multiplied with corresponding H and G coefficients.

112 Domain Decomposition Techniques for Boundary Elements 3.2 The dual reciprocity method This section provides an overview of the DRM including the generalities, the radial basis function considered, and the strategy for the reactive, convective, and timedependent terms. The previous section gave a general overview of the BEM for the Poisson equation, avoiding the treatment of domain integrals. In general, domain integrals arise from linear but inhomogeneous terms, nonlinear terms, or timedependent terms. In this case, the source term in (26) produced the domain integral in (32). The most often used techniques to deal with the domain integrals are: direct numerical approximation, elimination of non-homogeneous terms through exact or approximate particular solutions, and dual and multiple reciprocity methods. In principle, the domain integral would require some internal discretisation. Although internal discretisation has been extensively used up to the present, e.g. in the cell integration method [34], providing accurate results for a variety of partial differential equations, it has the main disadvantage of requiring an extra amount of data such as internal conectivities, hence, making the code more complex and more demanding in terms of computational resources. Here, the DRM is proposed in order to avoid this inconvenience. The DRM was first proposed in 1982 [35] and subsequently used by many other researchers [36]. A thorough introduction to the method can be found in reference [37]. The main idea is to translate the source term that appears in (38) to the boundary by means of a finite set of interpolating functions and applying again the same procedure that was used to derive the integral equation (32). 3.2.1 General approach The source term (ρ) in (26) can be written as a linear combination of the approximating functions fj Nr

ρ(x) = αj fj (x), (42) j=1

where Nr is the number of functions required for the approximation. The approximating functions are linked to the particular solution uˆ of the leading operator through ∇ 2 uˆ j = fj . (43) Thus, eqn (26) can be written in the following way: ∇ 2u =

Nr

αj (∇ 2 uˆ j ).

(44)

j=1

In the last expression it is possible to apply the weighting procedure with the fundamental solution in order to produce the integral equation

(∇ 2 u)u∗ d =

Nr

j=1

αj

(∇ 2 uˆ j )u∗ d.

(45)


113

Integration by parts on both sides of the last expression, and the subsequent discretisation of the boundary, as described in the last section, yields the following integral equation, for the source point located at the ith collocation node: $N % Nfn Nfn Nfn Nr fn

Hik uk − Gik [∂n uk ] = αj Hik uˆ kj − Gik [∂n uˆ kj ] , (46) k=1

j=1

k=1

k=1

k=1

where uˆ ij := uˆ j (xi ) is the jth particular solution evaluated at the ith collocation point, and ∂n uˆ kj := ∂ uˆ j (xk )/∂n[k] is derivative of the jth particular solution evaluated at the kth collocation node in the direction of the outward normal to the boundary element that contains the kth node. The vector of coefficients (αj ) in eqn (46) can be obtained by solving the linear system: Fα = ρ, (47) where ρ = ρ(xi ) is the vector of source terms evaluated at the collocation nodes (it is supposed to be known, as the source term is a prescribed scalar field), Fij = fj (xi ), and α is the vector of unknown coefficients. Note that in general the number of interpolating functions does not need to be equal to the number of freedom nodes. Therefore, in order to invert F it is necessary to introduce more equations or to remove information from the freedom nodes [38]. Finally, it is more practical to rewrite eqn (46) in the following matrix notation: Hu − G[∂n u] = Sρ

(48)

where the following DRM matrices have been defined ˆ := uˆ ij , U ˆ := Q

∂ uˆ j (xk ) , ∂n[k]

ˆ − GQ)F ˆ −1 . S := (HU

(49) (50) (51)

3.2.2 Radial basis functions In principle, any set of approximation functions f could be used in the DRM formulation. The only restriction is that the resulting matrix F must be non-singular. At the same time it is desirable to minimise high frequency oscillations without excessively smoothing the interpolation. The interpolating functions used in this work for the transport problem in the fractures, and in the porous matrix were the so called augmented thin plate splines (ATPS) [38], whereas the solution in the pipes is based on cubic splines. Table 2 summarises the sets of interpolating functions employed in each case. When an element of F matrix is represented by a radial basis function, then the following notation is equivalent: fij = fj (xi ) = f (rij ), where rij is the distance between the collocation nodes i and j.

114 Domain Decomposition Techniques for Boundary Elements Table 2: Radial basis function considered in each dimension. Entity Porous matrix Fracture Pipe MPC

Dimension

Interpolating functions

3D 2D 1D 0D

{r, 1, x, y, z} {r 2 log r, 1, x, y} Cubic splines –

3.2.3 The reaction term The reaction term (−kr u(x)) involves the evaluation of the unknown field u in the domain. Applying the linear combination (42) to u and inverting F it is straightforward to express the potential at any point inside the domain in terms of its values at the collocation nodes. In this way, the reaction term contributes with the vector ρ REACT,i = −kr ui ,

i = 1, . . . , Nfn .

(52)

3.2.4 The convective term The convective term introduces a first order derivative in space and is represented by: ∇(vu) = v · ∇u + u(∇ · v). (53) The value of ∇u at any point inside the domain can be expressed in terms of F by means of Nr ∂fj (x) ∂u(x) = αj . (54) ∂xp ∂xp j=1

The coefficients αj can be obtained by inversion of F as posed in eqn (47), enhancing the system if necessary to make it determined. In the case of potential flow, the second term on the right-hand side of (53) vanishes while the first one contributes to the discrete source vector ρ in (48) according to: ρ CONV,i =

Vikp Tkjp uj ,

(55)

kpj

where Vikp = δik vkp , Tkjp =

∂fkl −1 f . ∂xp lj

(56) (57)

Subscript p stands for the components of the Cartesian coordinates x, y, z in the 3D case or x , y in 2D, while subscripts i, j, k, and l represent any of the collocation nodes.


115

3.3 Miscellaneous 3.3.1 Time integration The integration in time is based on the finite difference method. A two-time level scheme has been implemented such that the time derivative is approximated in the following way ∂c cm+1 − cm ∂q qm+1 − qm ≈ , ≈ , (58) ∂t δt ∂t δt and the concentration c and its flux q are given by: c ≈ θc cm+1 + (1 − θc )cm

(59)

q ≈ θq q

(60)

m+1

+ (1 − θq )q . m

Superscripts m and m + 1 indicate previous and present time levels, respectively. The variable θφ can be adjusted from 0 to 1 yielding to different schemes (Crank– Nicholson, Euler implicit, Euler explicit, or any intermediate scheme). The time step δt is recalculated at every time level such that the maximum variation of concentration c/c remains bounded below a certain arbitrary threshold κ, according to: m−1 c m . (61) δt < κ ∂c/∂t 3.3.2 Domain decomposition In order to deal with different material properties, the integration domain is discretised into many subdomains. Each subdomain represents an isolated boundary element problem and produces its own system of equations. Then, every subdomain is connected to its neighbours through their interfaces in which the continuity of potentials and normal fluxes are prescribed. The matching conditions between two subdomains A and B that share a common boundary are: uA = uB , [ q · nˆ ]A = −[ q · nˆ ]B ,

(62)

where u denotes the potential (concentration for the transport problem, or hydraulic head for the flow problem), q is the flux of that quantity, and nˆ A is the outward normal unitary vector of the boundary of the subdomain A. Popov and Power implemented a scheme of this kind using DRM and called it the dual reciprocity method multidomain (DRM-MD) approach [36, 39]. In this way, the formulation can deal with piecewise homogeneous material properties. Moreover, by increasing the mesh refinement it is possible to solve problems with strong variations of the material properties or the solution fields, in spite of dealing with meshes more similar to the ones employed by the finite elements method (FEM) or finite volume method. The BEM applied to a single domain is characterised by producing a dense system of equations, which represents a real challenge for the solver, especially for large systems. On the other hand, domain decomposition offers the means to

116 Domain Decomposition Techniques for Boundary Elements produce sparse systems of equations. This is a desirable feature, since sparse systems can be solved in a more efficient way by iterative solvers, even though paying the price of increasing the number of degrees of freedom during the discretisation.

4 Numerical implementation 4.1 Introduction This section is focused on the numerical implementation of the DRM-MD BEM described in Section 3.1 applied to the flow and transport models described in Section 2. The objective is to solve a complete 3D DFNM of a fractured porous media. Section 4.2 describes the space discretisation for each entity appearing in the model, introducing, at the same time, the concept of block. Sections 4.6–4.8 for flow and 4.9–4.10 for transport describe the discrete matrix equations to be used in each subdomain of the porous matrix, in each fracture or pipe element, respectively. Once the sets of equations are produced for each subdomain of each entity, they are all linked together in Section 5, where it is described how to couple the whole problem with an assembly scheme and a fixed point iteration strategy. Each entity is associated with a different spatial dimension leading to a different formulation. However, all of them share a common feature: they are all based on the BE-DRM-MD with the fundamental solution of the Laplace equation. The governing equations of the flow and transport models can always be rewritten into Poisson-like equations, since any other term apart from the one involving the Laplace operator is considered as a generalised source term leading to a domain integral to be processed by the DRM. The following equations summarise the kernels used for each dimension. 1 x, x0 ∈ R3 (3D), 4π x − x0 1 1 ∗ u2D (x, x0 ) = ln x, x0 ∈ R2 (2D), x − x0 2π

∗ u3D (x, x0 ) =

∗ u1D (x, x0 ) =

x − x0 + K 2

x, x0 ∈ R1 (1D),

(63a) (63b) (63c)

where x0 is the source point, K is an arbitrary constant and . . . is the Euclidean norm in Rn . 4.2 Space discretisation In the 3D case, the integration domain is composed by a complex set of crossing fractures, represented by surfaces ∈ R2 and embedded in R3 . The 3D volume that is completely enclosed by surfaces with imposed boundary conditions and/or fractures will be called block (see Fig. 2). Thus the interstitial volumes in the fracture network are usually blocks.


117

BLOCK

BC

Fracture BC

Figure 2: Any 3D region enclosed by surfaces with known boundary conditions, and/or by surfaces representing fractures is called a block. 4

3

4

7

3

9

8

1

2

1

5

3

3

4

6

2

1

2

1

6

5

2

Figure 3: Discontinuous boundary elements for the discretisation of 3D subdomains. 4.3 Blocks and 3D subdomains A block can be discretised into one or many subdomains of arbitrary shape, in which case it will be defined as a single domain or multidomain block, respectively. On the other hand, the boundary of each subdomain of a given block is discretised into many boundary elements (at least four, when the subdomain is a tetrahedron). The following possibilities for the boundary elements of each 3D subdomain were considered in the implementation. In any case, the geometry can be described by quadratic or linear shape functions, and the interpolating functions follow the isoparametric approach. (Although this condition is not necessary at all, and it does not simplify the numerical implementation as in the case of finite elements.) The possibilities are sketched in Fig. 3. The computational implementation allows the mixing of quadrilaterals with triangular elements, provided that they all have the same degree of interpolating and shape functions (Table 3). This feature provides more flexibility when generating meshes for complex geometries. The expressions of the interpolating and shape functions associated to each element (i.e. Jacobians, definition of local coordinates) are the ones defined in [31] and [40]. For details about the geometry modeller and mesh generator see references [41–43]. 4.4 2D subdomains A general plane or curved surface representing a single fracture is represented by a number of 2D flat subdomains that can be triangular or quadrilateral and take part of a structured or unstructured mesh. The subdomains in the fracture described by

118 Domain Decomposition Techniques for Boundary Elements Table 3: Boundary elements for 3D subdomains where NDfn is the number of discontinuous freedom nodes (Dfn) and NGN is the number of geometrical nodes (GN). Geometry Quadrilateral Quadrilateral Triangular Triangular

η2 5

2

Node

1 1/4 0

Node

1 −1/2 −1

h1 h2 h1 h2

NGN

Quadratic Linear Quadratic Linear

9 4 6 3

9 4 6 3

η1

Discontinuous Freedom node

5

6

3 1

NDfn

η2

4

6

Degree

7

4

8

3 1

2

η1

Geometrical node DRM internal node

2D Triangular subdomains 3 4 5 6 DRM 3/4 1/4 0 0 1/3 1/4 3/4 3/4 1/4 1/3 2D Quadrilateral subdomains 2 3 4 5 6 7 1/2 1 1 1/2 −1/2 −1 −1 −1/2 1/2 1 1 1/2 2 3/4 0

8 −1 −1/2

DRM 0 0

Figure 4: Two-dimensional subdomains for fracture discretisation. linear shape functions and the isoparametric approach imposes linear interpolating functions. The boundary of each 2D subdomain is discretised into three (for triangles) or four (for quadrilaterals) discontinuous linear boundary elements with two freedom nodes per segment. At the same time it is possible to add internal DRM nodes in order to increase the accuracy of the DRM approximation. Figure 4 shows the location of the freedom nodes in the triangular and quadrilateral subdomains in terms of their natural local coordinates (η1 , η2 ). In any case, the implementation requires geometrical matching between the boundary elements of the 3D subdomains and the adjacent 2D subdomains, despite of the fact that the freedom nodes do not need to be coincident. 4.5 1D subdomains A pipe is discretised into one or many 1D subdomains with two internal freedom nodes in the case of flow and four freedom nodes in the case of transport, i.e. two


119

3 b

2

3

a3

b 2 a2

Pipe element

n

f3

n f2

4 b 1

1 a

1

3

n f1

bi

n fi

i a

2

n f mf

1

i i−th fracture element

h b mf a mf

m

f

Figure 5: Multiple intersection of fractures making a pipe element of fracture multiplicity mf .

internal plus the two geometrical endpoints. In any case the two internal freedom nodes are coincident with the discontinuous freedom nodes of the adjacent 2D subdomains as shown in Fig. 5. 4.6 Flow in the porous matrix The flow in a 3D subdomain of the porous matrix 3D is represented by eqn (3). The corresponding integral equation applied to each freedom node associated to 3D leads to the following system of equations: km H3D hm + ϒG˜vm,n = 0, k

(64)

120 Domain Decomposition Techniques for Boundary Elements where v˜m,n,i =

vm,n,i , k

i = 1, . . . , NDfn

(65)

is the scaled velocity array at every ith freedom node, vm,n,i = −km ∇hm · nˆ m,i is the Darcy velocity normal to the boundary evaluated at the ith freedom node. The scaling factor k can be taken as the mean value of the hydraulic conductivity, and it has been included in order to balance the order of magnitude of the coefficients taking part in the system of equations. ϒ :=

+1 if nˆ m,i is the outward normal of 3D , −1 if nˆ m,i is the inward normal of 3D .

(66)

The matrices H3D ∈ RNfn ×Nfn and G3D ∈ RNfn ×NDfn are calculated using (39a) and (39b) with (63a). Here, Nfn = NDRM + NDfn is the total number of freedom nodes adjacent to the 3D , NDRM is the number of internal (DRM) nodes within 3D , and NDfn is the total number of discontinuous freedom nodes of the subdomain, i.e. those freedom nodes attached to the boundary elements. 4.7 Flow in fractures The steady state flow problem in a single domain 2D belonging to any surface representing the fracture is governed by (7), a Poisson-like integral equation, that can be written in matrix notation in the following way:

w¯ f kf H2D hf + G2D v˜ f,n = −S2D v˜ m,n,k , k

(67)

k=A,B

where the matrices H2D ∈ RNfn ×Nfn and G2D ∈ RNfn ×NDfn are calculated using (39a) and (39b) with (63b). vf,n is the component of the velocity vector projected onto the fracture that is normal to the boundary of a 2D subdomain. The subscript k stands for the two blocks of porous matrices adjacent to the fracture. The scaled velocity v˜ ∈ R2×2 is derived from the normal velocity in a 2D fracture element according to vñ,f = −

w¯ f kf ∇2D hf · nˆ f . k

(68)

It has got dimensions of length and represents the variable to be stored in memory, from which the real physical velocity vn,f can be rebuilt, by multiplying with k/w¯ f . 4.8 Flow in pipes The multidomain discretisation of a 1D curve representing a pipe is a connected sequence of linear 1D subdomains or pipe elements, each one of them defined by


I

II

1

III

3

2

v t,p,1

4 vn,f,2

v n,f,2

-1.0

-0.5

121

0.0

0.5

1.0

Figure 6: Distribution of freedom nodes in a single pipe element.

2 geometrical nodes (endpoints 1 and 4 in Fig. 6) and two internal nodes (2 and 3). The integral formulation for a pipe element is based on eqn (11). The pipe element is coincident with the boundary element of the adjacent 2D fracture subdomain in both geometrical and functional senses. The geometrical matching means that it shares the same geometrical element and coordinates (a segment of a line) and the functional matching means that both entities share the same freedom nodes (nodes 2 and 3). In this work the fracture boundary elements are represented by 1D linear isoparametric discontinuous elements, with the following interpolating functions 1 ψa,b (ξ ) = ± ξ , (69) 2 where ξ ∈ [−1, 1] is the local coordinate along the pipe element. Consequently the non-dimensional version of (11) becomes a 1D Poisson-like equation with a linear source term: f

4kp Ap d 2 hp = − wf,i vf,n,i (ξ ), Lp2 dξ 2

(70)

vf,n,i (ξ ) = ψa (ξ )vf,n,2 + ψb (ξ )vf,n,3 .

(71)

m

i=1

where

So it is possible to integrate it twice to obtain an analytical expression for the hydraulic head along the pipe expressed in the following third-order polynomial

hp (ξ ) =

3

ck ξ k .

k=0

The coefficients ck can be calculated by imposing the usual matching conditions between the adjacent fractures and the pipe element (continuity of hydraulic head

122 Domain Decomposition Techniques for Boundary Elements and mass conservation). The result is summarised here: α3 − α2 , 6 α2 + α3 , = 4 α2 − α3 = + hp,3 − hp,2 , 24 hp,2 + hp,3 α 2 + α3 − , = 2 16 mf Lp2 =− vf,n,i,j wf,i , 4kp Ap

c3 =

(72)

c2

(73)

c1 c0 αj

(74) (75) (76)

i=1

where j = 2, 3 indicates any of the internal freedom nodes. 4.9 Transport in the porous matrix The BE-DRM formulation of the advection–diffusion equation for the porous matrix in the isotropic case is

∂cm Vp Tp c, +S ∂t 3

Dm Hcm + Gφ m = −Sρ + RS

(77)

p=1

where φ = −Dm ∇cm · nˆ m is the diffusive flux of concentration normal to the boundary of the subdomain in the outward direction. Substitution of the time derivative yields: n+1 n n+1 n M1 cm + M2 φ˜ m = Sρ + M3 cm + M4 φ˜ m

(78)

where M1 = θ

3

Dm θ R kr V p Tp − H3D − S3D S3D − θ S3D I3D , ¯ ¯ ¯ ¯ D D D δt D p=1

M2 = θ ϒG3D , M3 = (θ − 1)

(80) 3

Dm 1−θ V p Tp H3D + S3D ¯ ¯ D D p=1

R kr S + (1 − θ) S3D I3D , ¯ ¯ 3D D δt D M4 = (θ − 1) ϒ3D G3D . −

(79)

(81) (82)

¯ represents a mean value of the diffusion coefficient and is used only The quantity D to balance the orders of magnitude of the coefficients involved. The 3D normal


diffusive flux of concentration is defined as: Dm ∇cm · nˆ m , φ˜ m = − k¯

123

(83)

4.10 Transport in fractures The integral equation that represents (15) and (20) in the 2D BEM-DRM formulation can be written in the following way: Df H2D cf + G2D φ f = S2D ρ 2D

(84)

where the right-hand side term is ∂cf (qm,A + qm,B ) + V p Tp c f − ∂t w¯ f 2

ρ 2D = kR cf + Rf

(85)

p=1

and qm,γ ,i = vm,n,i cγ ,i − Dm

∂cm , ∂ nˆ m γ

i = 1, . . . , Nfn

(86)

is the normal flux of concentration coming from the two adjacent porous blocks γ = A, B. Therefore, after replacing the approximation for the time derivative, it results in the following system of equations for the 2D subdomain:

n ∗ M1 cf n+1 + M2 φ˜ f n+1 = M3 cf n + M4 φ˜ f + εf M5 q˜ m,i , (87) i=A,B

where εf is a relaxation factor, to be explained later, and the rest of the matrices are: M1 = θ

2

Rf Df w¯ f θ w¯ f θ Vf,p Tf,p − w¯ f H2D − S2D S2D − S2D kr I2D k¯ k¯ k¯ δt k¯ p=1

M2 = θ (1 + β) ϒ2D G2D

(88) (89)

Df w¯ f w¯ f Vf,p Tf,p H2D + (1 − θ ) S2D k¯ k¯ 2

M3 = (θ − 1)

p=1

1−θ Rf S2D kr I2D S2D + k¯ δt k¯ M4 = (θ − 1) (1 + β) ϒG2D , − w¯ f

M5 = −S2D ,

(90) (91) (92)

where φ˜ f = −

D f wf ∇2D cf · nˆ f k¯

(93)

is the scaled diffusive flux of concentration normal to the boundary of the fracture times the aperture of the fracture.

124 Domain Decomposition Techniques for Boundary Elements 4.11 Transport in pipes The partial differential equation that describes the 1D time-dependent homogeneous transport of pollutants with a variable source term ρ is Rp

∂cp (η, t) ∂cp (, t) ∂ 2 cp (η, t) = ρ(η, t), + vp − Dp ∂t ∂η ∂η2

η ∈ [η1 , η4 ],

(94)

where vp is the velocity along the pipe that might vary along η, Rp is the retardation factor of the pipe, and Dp is the dispersion coefficient in the pipe, assuming that it is piecewise constant. We are interested in external sources reaching the 1D domain from lateral advection and diffusion conditions. They can be expressed as ρ(x, t) = −vn,f cf (x, s, t) + qn,f (x, s, t)ds, (95)

where stands for the boundary of the pipe cross section. The spline method with four collocation nodes introduces four equations per each pipe element, see [44] and [45] for details on the stability analysis and calculation of the matrix elements. 4

m+1 M1ij cp,j +

j=1

=

m+1 M2ij φ˜ p,j +

j=1∧4 4

j=1

m ˆ ij cp,j + M1

m+1 M3ij φ˜ f,j

j=2∧3 m ˆ ij φ˜ p,j + M2

j=1∧4

m ˆ ij φ˜ f,j M3

Rp 1 θ Dp H1D − B5 − B8 − B3 M1 = δt k¯ Rp M2 = θ (−ϒG1D − B6) − B4 δt M3 = −θ B7 R ˆ = 1 (θ − 1) Dp H1D − B5 − B8 − p B3 M1 δt k¯ R ˆ = (θ − 1) (ϒG1D − B6) − p B4 M2 δt ˆ = (1 − θ ) B7, M3 where φ˜ p, j = −Dp

(96)

j=2∧3

∂cp ∂ nˆ p

(97) (98) (99) (100) (101) (102)

(103)

represents the diffusive flux of concentration along the pipe coordinate in the jth collocation node. The interpolation functions are cubic splines (CS). Various end


125

conditions can be used for CS. The common practice is to use ‘natural’ end conditions to close the system of equations by specifying the second-order derivative of the splines to be zero at the endpoints (nodes 1 and 4). In this scheme the first-order derivatives of concentration at the endpoints participate in the interpolation scheme, so the approximation also provides the physical fluxes at endpoints. The interval [x1 , x4 ] has been considered as subdivided into three regions I, II, and III, connected at the internal nodes 2 and 3 by the spline smoothness condition, see Fig. 6. In this way the concentration at any point of the pipe element between x1 and x2 can be expressed in terms of its values at the four collocation nodes cp,j , for j = 1, 2, 3, 4 and its tangential derivative at the endpoints ψp,1 and ψp,4 , according to cp (ξ , t) =

4

Sja (ξ )cp,j + S5a (ξ )

j=1

ψ(ξ , t) = =

δx δx ψ1 + S6a (ξ ) ψ4 2 2

(104)

2 ∂ cp (ξ , t) δx ∂ξ 4 2 â Sj (ξ )cp,j + Sˆ5a (ξ )ψ1 + Sˆ6a (ξ )ψ4 δx

(105)

j=1

where the superscript a, identifies one of the three regions   1, −1 ≤ ξ < −1/2 a = 2, −1/2 ≤ ξ ≤ 1/2   3, 1/2 < ξ ≤ 1 and ψ(ξ , t) := ∂cp /∂ξ is the derivative of cp along the dimensionless pipe coordinate ξ . The functions Sja (x) are cubic polynomials of ξ of the form Sia (ξ ) =

4

j=1

sija ξ (j−1)

and

dSia Sîa (ξ ) = dξ

i, . . . , 6,

where the superscript between parenthesis indicates power. They are derived from the solution of the CS interpolation problem with imposed first-order derivatives at the endpoints 1 and 4 of the interval. These derivatives might be regarded as additional degrees of freedom. The constraints for the spline problem are the continuity of second-order derivatives in nodes 2 and 3. To close the system, zero second-order derivatives at nodes 1 and 4 are prescribed. The development of the spline method can be found in standard textbooks of numerical analysis, see for example [46]. The matrix elements sija are shown in reference [44]. Figure 7 shows the shape of the base functions for the CS method. Next, the integrals appearing on the right-hand side of (94) are explained.

126 Domain Decomposition Techniques for Boundary Elements s1

s2

-0.5

0

0.5

1

s3

x -0.5 0

s4

0.5

1

x

s5

-0.5

0

0.5

1

x

-0.5 0

0.5

-0.5

0.5

1

x

s6

-0.5

0

0.5

1

x

0

1

x

Figure 7: Interpolation functions for the spline interpolation scheme applied to the pipe element. s1 , s2 , s3 , and s4 , affect the potentials at collocation nodes 1 to 4, whereas s5 and s6 affect the derivatives of the potential at the endpoints of the pipe element.

4.11.1 The time-dependent term for the CS scheme The volume integral in (94), that involves the time-dependent term, is approximated in the following way

x4

∗ u1D (x, x0 )Rp

x1

4

∂cp ∂cp,j ∂ B3ij Rp B4ik Rp ψ3k−2 , dx ≈ + ∂t ∂t ∂t j=1

(106)

k=1,2

where B3 ∈ R4×4 and B4 ∈ R4×2 are defined as: δx B3ij = 2 δx B4ik = 2

1

−1

1

−1

∗ u1D (ξ , ξi )Sja (ξ )dξ , ∗ a u1D (ξ , ξi )Sk+4 (ξ )

i, j = 1, 2, 3, 4,

δx dξ , 2

i = 1, 2, 3, 4; k = 1, 2.

(107) (108)

For definition of B3 and B4 see [44]. 4.11.2 The convective term for the CS scheme The volume integral involving the convective term is approximated in the following way

x4

x1

∗ u1D (x, x0 )vp

∂cp B5ij vp cp,j + B6ik vp ψ3k−2 , dx ≈ ∂x 4

2

j=1

k=1

(109)


where B5 ∈ R4×4 and B6 ∈ R4×2 are defined as 1 ∗ B5ij = u1D (ξ , ξi )Sˆ ja (ξ )dξ , i, j = 1, 2, 3, 4, B6ik =

(110)

−1 1 −1

∗ a u1D (ξ , ξi )Sˆ k+4 (ξ )

δx dξ , 2

127

j = 1, 2, 3, 4; k = 1, 2.

(111)

Here δx := x4 − x1 is the length of the pipe element. The matrix elements of B5 and B6 can be found in [44]. 4.11.3 The source term for the pipe element The approximation of any additional source term, say b(x) is similar to the timedependent term, and involves only matrices B3 and B4

x4 x1

∗ u1D (x, xi )bj dx ≈

4

B3ij bj +

j=1

B4ik

k=1,2

∂bj . ∂x

(112)

In the case of a prescribed time-dependent source term, the same approach as the one used in (58) and (60) can be applied for b(x, t), leading to bj ≈ (1 − θb )bjm+1 + θb bjm .

(113)

4.11.4 The exchange term for the pipe element The exchange term is the normal flux of concentration exchanged with the fractures. This exchange term represents a source term for the pipe element and requires domain integration. Under the pipe element approximation, its contribution can be expressed in the following way:

x4

x1

∗ u1D (x, xi )

mf

qf,n,k wf,k k=1

Ap

dx ≈

B7i,l Zfk,l+1 +

B8i,l cp,l ,

(114)

l=2,3

l=1,2 k=1,...,mf

where ∂cf,n,l+1 . Zfkl := wf,k,l+1 −Df,k ∂n

(115)

The matrix B7 affects the normal derivative of concentration coming from the fractures, and can be calculated according to: B7i,l =

+1

−1

∗ u1D (ξ , ξi )

1

= 1/2 − ξ ,

2

= 1/2 + ξ.

l (ξ )δxdξ ,

(116)

128 Domain Decomposition Techniques for Boundary Elements For i = 1, . . . , 4 and l = 1, 2. On the other hand, matrix B8 affects the convective part of the concentration flux coming from the adjacent fractures, and can be calculated according to: B8i,l =

+1 −1

∗ u1D (ξ , ξi )

mf

1 vp,2

+

2 vp,3

δx dξ. 2Ap

(117)

wf,n δx A8 48Ap

(118)

l wf,k

k=1

Then, the matrix elements of B7 and B8 are: t δx 2 4 2 13 20 B7 = 4 96Ap 20 13 2

B8 =

and

where ) A8 := δx vf,n,2

$

3 −1

35 16 − 19 16

123 16 − 19 16

11 −1

$

%t + vf,n,3

−1

− 19 16

− 19 16

11

123 16

35 16

%t * −1

.

3 (119)

5 Coupling strategy This section describes the strategy employed to solve the complete 3D DFNM. In principle, the calculation at every time step involves the simultaneous solution of different sets of equations introduced by: • • • •

the blocks of porous matrix (3D problem), the fractures (2D problem), the pipes network (1D problem), the MPCs (0D problem)

For simplicity, the coupling is divided into two parts. The first one refers to the way in which an interconnected fracture network can be solved by gathering the 2D equations of single fractures, the 1D equations from pipes, and the 0D equations from MPCs, whereas the second part explains the way in which the interconnected fracture network is coupled with all the 3D blocks of the porous matrix. In general, the coupling strategy is similar for both flow and transport problems. (The flow problem has slightly fewer degrees of freedom than the transport problem, as the flow in the pipes is calculated with the analytical solution of the Poisson equation.) So, in order to simplify the description, this section will refer to potentials (u) and fluxes ( q), which should be understood as concentrations and concentration fluxes in the case of transport, and hydraulic heads and velocities in the case of flow, respectively. 5.1 Interconnected fracture network The interconnected fracture network is the coupled set of fractures, pipes, and MPCs, namely the Fracture Network System (FNS). The equations coming from


129

those entities are assembled together into a unique system of equations, of the form AFNS x = b,

(120)

where AFNS is the linear system of equations involving the fracture equations, the pipe equations, and the MPC equations, x is the vector of unknowns involving normal fluxes and potentials at the freedom nodes, and vector b is the right-hand side vector that involves boundary conditions, solutions at any previous time step, and generalised source terms. In a multidomain scheme, the fractures contribute with a number of N S2D

Nfn,i i=1

equations of type (67) for flow, or (87) for transport to the FNS, where NSpD is the number of p-dimensional subdomains in the problem (p = 0, 1, 2, 3) and Nfn,i is the number of total freedom nodes in each subdomain (keeping into account the internal DRM nodes). In the case of the transport problem, the pipes, with four collocation nodes will contribute with a number of 4 × NS1D equations of type (96), and each MPC with more than two concurrent pipe elements (1D subdomains) will provide the remaining conservation equations leading to the assembly of a square FNS of equations. The conservation of fluxes and continuity of potentials (62) in the interface between two p-subdomains is automatically fulfilled in the assembly of the multidomain scheme, but when the degree of concurrence of entities is greater than two, then additional mass conservation equations must be considered. Thus, the total number of degrees of freedom in the FNS is ND,NFS =

N S2D

Nfn,i + 4 × NS1D + 2 × NMPC>2 ,

(121)

i=1

where NMPC>2 is the number of MPCs with pipe multiplicity higher than 2 (mp > 2). 5.2 Coupling the porous matrix with the fracture network The coupling between the porous matrix and the fracture network is similar to a fixed point iterative scheme. In general, the complete DFNM can be decomposed into an arbitrary number of blocks of porous media (NBL ). Each one of them is associated with a closed system of equations of dimension NS3D,BL

ND,BL =

Nfn,i

i=1

where NS3D,BL is the number of 3D subdomains in the block, and Nfn,i is the number of total freedom nodes in each subdomain (taking into account the internal DRM nodes). Therefore, the total number of degrees of freedom is NDOF = ND,NFS +

NBL

BL=1

ND,BL .

(122)

130 Domain Decomposition Techniques for Boundary Elements The collection of blocks must be solved simultaneously with the FNS. The iterative method consists of the following steps: 1. Guess an initial value for the exchange flux between the fracture and its adjacent ∗ ). For example, its value at every freedom node can be taken 3D blocks (qm,n to be equal to the previous time step. ∗ m qm,n = qm,n . ∗ . 2. Update the right-hand side term of the FNS with the most recent value of qm,n The exchange flux between blocks and FNS becomes a source term for the latter to be adjusted at every iteration step. ∗ and the potentials u∗ at every freedom 3. Solve the FNS to obtain the fluxes qf,n f node of the fracture network. 4. The potentials in the fracture uf∗ that come out from the solution of the FNS provide some temporary Dirichlet-type boundary conditions to its adjacent porous blocks, so this step consists of the transfer of the solution of the fracture network to the boundaries of the blocks that are in close contact. 5. For each one of the NBL 3D blocks in the model: • Update the right-hand side term with the new boundary conditions imposed by the FNS. • Choose the solver of the block according to the previous performance. • Solve the system of equations to get the normal fluxes at the freedom nodes ∗∗ . of their boundary elements in contact with fractures qm,n 6. Update the normal flux by swapping ∗ ∗∗ qm,n = qm,n .

7. Test the convergence criterion according to the maximum relative variation of uf∗ between two successive iteration steps, e.g. uf∗

uf∗∗ − uf∗ < ε, = MAX ufm

(123)

where ε establishes a convergence threshold and return to step 2 if it is not achieved. 8. Update the potentials and fluxes of the new time level (in case of the transport) at every freedom node φ m+1 = φ ∗∗ and return to the initial step. In the case of steady state flow, stop the calculation and postprocess the results. The strategy of decoupling the 3D problem into independent blocks of porous media bounded by fractures and solving them in a sequential way has the big advantage


131

of reducing the size of the linear systems to be solved, and the main assumption is that the flow and transport are mainly taking place through the FNS. Therefore, one can solve the FNS first separately, and then account for the influence of the matrix blocks using the iterative scheme. The iterative scheme is sketched in Fig. 8, where the vertical axis represents the normal flux, and the horizontal axis the potential in the fracture. The process starts by imposing zero normal flux (point A) as initial guess, then solving first the FNS and then the blocks, the iteration progress in the sequence A → B → C · · · approaching to the solution (point S) established in the intersection of the two curves representing the FNS and the equations introduced by the adjacent blocks. The local slope and curvature of the curves that represent the 3D block systems of equations and the FNS at the solution (point S) determine the convergence of the method. They both depend on the geometrical and physical properties of the fractures and the adjacent porous blocks (diffusivity, hydraulic conductivity, etc.). For example, if the diffusivity of the porous block is high, then a small variation in the concentration inside the fracture will induce high fluctuations of the exchange flux. Thus the local slope of the porous matrix system of equations will become higher, and the method will diverge. This situation is sketched in Fig. 9a, where the iteration starts at point A and after point F the concentrations inside the fracture become negative. Moreover, during the iteration, it might happen that the intermediate exchange flux becomes so high that it creates large oscillations, overshooting, negative concentrations in the fracture, or even divergence in the iterative scheme. To circumvent this problem, a relaxation factor affecting the normal flux was introduced, so that the corrections of the source term in the FNS become moderated and progress in a smooth way with the iterations. In this way, the iterative scheme in the qm,n –cf plane becomes similar to the one sketched in Fig. 9b and the leading

ufs

A

Fracture System

B

{ uf } Block A

qm

Fracture

uf

F

E qms

S Block B

C D

Porous Matrix System

{ qm }

Figure 8: Iterative scheme between fracture and adjacent blocks.

132 Domain Decomposition Techniques for Boundary Elements u

D q

u

fs E

A

A

B

{u } f

C q

ms C

fs

{u } f E ms

F D

{q (a)

m

}

{q (b)

m

}

B

Figure 9: (a) Divergency of the iterations and (b) constant relaxation factor applied to the normal flux. equations in the fractures, eqns (67) and (85), become

w¯ f kf v˜ m,n,k H2D hf + G2D v˜ f,n = −S2D k

(124)

k=1,2

∂cf (qm,A + qm,B ) + = kR cf + Rf V p Tp c f − ∂t w¯ f 2

ρ 2D

(125)

p=1

respectively. It is worth mentioning that although the presence of the relaxation factor slows down the convergence of the method by moderating the correction terms in the iterative scheme, it is always safer and preferable since it makes the scheme more stable. Values of relaxation factors ≈ 0.2 are used, with which less than 30 iterations are usually required in order to reach variation of concentrations in the fracture of cf∗ /cf∗ ≈ 0.05. Figure 16 presents a real convergence history in the qm,n –cf plane of the first five time steps between a fracture and its adjacent porous blocks in one of the runs of the computational code implemented for a test example.

6 Numerical results 6.1 Introduction This section presents the results obtained using the developed formulation. They are intended to provide a reference for the code validation and its modelling capabilities. The code validation presented in the first two examples consists of a series of numerical tests applied. Its objective is to evaluate the accuracy and consistency of the computational code in simple reference cases. The third test example will present the modelling results for flow and transport applied to the conceptual model


133

of a real rather complex situation which was defined according to the requirements defined in reference [47]. 6.2 Test case: 3D block without fractures The aim of this test is to compare the numerical solution obtained with the 3D block formulation against the available analytical expressions. A 3D unitary cube composed of a single multidomain block was used as a test example in which all lateral concentration fluxes were zero in order to recover 1D time-dependent profiles for which analytical solutions are available, see reference [48]. The description of the geometry involved 81 geometrical nodes. The block was discretised into 268 tetrahedral unstructured subdomains with linear discontinuous elements. This type of discretisation leads to a single system of equations with 3216 degrees of freedom, for both flow and transport problems, and, therefore, no iterative scheme is required. Figure 10a shows the dimensions and the system of coordinates, and Table 4 shows the applied boundary conditions. At initial time, the concentration and the concentration flux are 0. The retardation factor is Rm = 1, the diffusion coefficient is

(a)

(b)

Figure 10: (a) Diffusive flux of concentration at t = 31.7 year. (b) Concentration at t = 31.7 year. Table 4: Boundary conditions for the 3D test example. Boundary yz at x = 0 yz at x = 1 xy at z = 0 and z = 1 zx at y = 0 and y = 1

Flow hm hm vm vm

=1 =0 =0 =0

Transport cm = 1 qm,n = 0 qm,n = 0 qm,n = 0

134 Domain Decomposition Techniques for Boundary Elements Dm = 10−10 m2 /s and the hydraulic conductivity is km = 10−10 m/s, so the global Péclet number becomes 1. Figure 10a shows the vector field of diffusive flux at time t = 31.7 year, calculated according to eqn (54). Figure 10b shows the solution for the concentration at the same time during the transient. The time step was kept constant δt = 3.17 year. Figure 11a shows a comparison between the exact solution (hT = x) and the numerical solution hN for the hydraulic head along the x axis. The error in Fig. 11b is the absolute value of the difference hT − hN . Figure 12 shows a comparison between the numerical and the analytical solutions for the

1e-09

Analytical Numerical

0.8

Absolute Error Hyd Head

Hydraulic Head

1

0.6 0.4 0.2

1e-10 1e-11 1e-12 1e-13 1e-14 1e-15 1e-16

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

x (b)

x (a)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1.6 Numerical Analytical

Numerical Analytical

1.4 qx [1010 m/s]

C

Figure 11: Relative error of the flow problem in steady state.

1.2 1 0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

0

0.2

0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0

0.2

0.4

0.6 x/L

0.4

0.6

0.8

1

0.6

0.8

1

x/L

| qN - qT |/ qT,max

|CN - CT| / CT,max

x/L

0.8

1

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0

0.2

0.4 x/L

Figure 12: Comparison between analytical and numerical results for a 3D block at t = 31.7 year.


135

transport problem at t = 31.7 year. It can be observed that the numerical results of the concentrations are more accurate than the numerical results of the diffusive flux. 6.3 Test case: fractures represented as 3D or 2D entities In this example, the developed formulation is being tested and compared against a classical 3D BEM-DRM-MD method. The test problem consists of a cube of porous media of dimensions 1 m × 1 m × 1 m traversed by a fracture of aperture wf = 4 cm which divides the porous matrix into two blocks, namely A and B. The fracture is parallel to the plane xy, and it is located at z = 0.5 m. Table 5 shows the boundary conditions, which were chosen such that the advective transport is along the x axis. The solution is symmetric in the z direction having the fracture as a plane of symmetry and is constant along the y axis. Figure 13 shows the geometry of the problem and the boundary conditions for the transport. The porous media has hydraulic conductivity km = 10−10 m/s and diffusion coefficient Dm = 10−10 m2 /s, whereas the fracture zone has kf = 10−7 m/s and Df = 10−7 m2 /s. This numerical example consists of solving the problem with the developed formulation (method A) explained in this chapter in which the fracture is regarded as a plane and points C (H) collapse to points D (G), see Fig. 13. The results are compared with the numerical solution obtained with the classical 3D version of the BEM-DRM-MD approach (method B) in which the fracture is considered as a 3D entity of porous media. In method A, each block of porous media has been discretised into a structured array of 6 × 6 × 3 hexahedral subdomains in the x, y, and z directions, respectively. Each face of a hexahedron represents a quadrilateral linear isoparametric boundary element of the subdomain. The fracture has been discretised into a structured array of 6 × 6 quadrilateral subdomains in the x and y directions. This mesh matches geometrically with the 3D boundary elements of the hexahedral subdomains of the

Table 5: Boundary conditions for example 2. Boundary

Flow

Transport

Plane yz at x = 0, blocks A and B

hm = 1

qm,n = 0

Plane yz at x = 1 blocks A and B

hm = 0

qm,n = 0

Planes xy at z = 0 and z = 1 Planes xz at y = 0 and y = 1 at blocks A, B and fracture

vm,n = 0 vf,n = 0

qm,n = 0 qf,n = 0

Fracture x = 0 Fracture x = 1

hf = 1 hf = 0

cf = 1 qf,n = 0


∂c ∂n ˆ

L =0

F

∂c ∂n ˆ

E

Porous Matrix

=0 G

D

c=1 ∂c ∂n ˆ

Fracture H

C

Porous Matrix

=0 A

=0

∂c ∂n ˆ

= 0 wf

∂c ∂n ˆ

=0

B

∂c ∂n ˆ

(a)

∂c ∂n ˆ

=0

(b)

Figure 13: Conditions of the test example 2: (a) geometry and mesh used for method B; (b) cross section defined by points A, B, E, and F showing the boundary conditions applied for the transport problem. Table 6: Systems of equations involved in method A. System 1 2 3

Entity

NS

NDOF

NNZ (flow/transport)

Block A Block B Fracture

108 108 36

2592 2592 324

110592/110592 110592/110592 4524/5076

adjacent porous blocks. Each side of the quadrilateral represents a 2D discontinuous linear isoparametric boundary element. In this way, method A leads to three different linear systems of equations of the form Ax = b (The method leads to three linear systems of equations for the flow problem and another three linear systems of equations for the transport.), one for each block and one for the fracture, their corresponding dimensions are shown in Table 6, in which NS is the number of subdomains, NDOF is the number of degrees of freedom, and NNZ is the number of non-zero coefficients in the system of equations. In the method B, the same discretisation as before has been used for the porous matrix; however, the fracture has been discretised with a structured array of 6 × 6 × 1 hexahedral subdomains with linear isoparametric discontinuous elements. In this way, the whole problem can be expressed by two unique systems of equations one for flow and another for transport. The number of DOFs involved in method B is 6048 (i.e. number of rows of the system) and the number of non-zero entries is NNZ = 267,264. The time step for the transport problem in both methods was δt = 5.79 days. The computational code has been implemented in Fortran 90 for a Pentium III PC with 1 GHz CPU clock and 2 GB RAM. Table 7 shows the wall-clock CPU time used to solve the problems in both methods. The preprocessing time is the


137

Table 7: CPU time usage in seconds for methods A and B. Calculation stage

Method A

Method B

34.15 73.52 532

39.74 2023 4868

Preprocessing Flow (steady state) Transport (20 time steps)

Concentration across the fracture 0.8

Method A Method B

Method A Method B

0.7 0.6 0.5 C/C0

C/C0

Concentration along the fracture 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

t = 115.7 days

0.4 0.3

t = 115.7 days

0.2 t = 11.57 days

t = 11.57 days

0.1 0

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

x/L

z/L

(a)

(b)

0.8

1

Figure 14: Comparison of numerical solutions for the transport between methods A and B: (a) concentration along the fracture and (b) concentration across the fracture (z = 0 m to z = 1 m, at x = y = 0.4 m). time used to numerically integrate the BEM matrices H and G, and to build the system of equations. In this test it was used as a direct LU solver based on Gauss elimination for sparse systems [49]. The convenience of the method A from the point of view of computational costs can be observed. Method A produces three systems of equations, each one of them solved with a direct solver, but the coupling between them involves the iterative technique described in Section 5.2. However, the CPU time used for both methods employing the same solver shows that the same problem that method B solves in approximately 1 h 55 min, can be more efficiently solved in approximately 10.7 min by method A. Figure 14a shows the concentration profile obtained with methods A and B, at two different times (115.7 days and 11.57 days) along with a line contained in the fracture plane that extends from x = 0 to x = 1 m at z = 0.5 m. Figure 14b shows the concentration profile across the fracture on a line that extends from z = 0 to z = 1 at constant x = 0.4 m and y = 0.4 m. It can be observed that method A is consistent with the reference method B. However, method A overestimates the concentrations in the fracture by a factor smaller than 10%. This overestimation observed in Fig. 14a and b is due to the simplification of the fracture into a surface, by neglecting its aperture. The hypothesis of flat potential profiles across the aperture of the fracture represents a good approximation, as can be seen in Fig. 14b. Density plots of concentration for both methods at t = 115.7 days are shown in Fig. 15a and b. Figure 16a shows the


(a)

(b)

Figure 15: Density plot of concentration for: (a) method A and (b) method B at t = 115.7 days. qn 0.05

0.1

0.15

cf

0.2

5 A

10 15

5

6

5 e5

4 3

9

7 8

1

2

20 25 30

B

10 e5 15 e5

20 e5

25 e5

(a)

(b)

Figure 16: Iterations between a fracture subdomain and its adjacent blocks of porous media.

convergence history of the first five time steps. The horizontal axis corresponds to the concentration cf at one of the freedom nodes of a fracture element – freedom node 3 in Fig. 16b. The vertical axis is the normal exchange flux of concentration between the fracture element and block A of the porous matrix. Each time step required less than 30 fixed point iterations to achieve convergency with ε < 0.04 in (123). 6.4 Case study: application to 3d fractured crystalline rocks The proposed formulation, implemented in a computer code which was validated against several test examples, has been applied to large-scale problems. One of the conceptual models of the LowRiskDT project [47] has been numerically modelled, and part of the reported results are presented in this section in order


139

to show the capabilities of the developed formulation. The aim of the project is to provide risk assessments and to estimate the feasibility of conversion of abandoned underground mines into waste repositories for chemical pollutants coming from industry. The conceptual model involves a room and a tunnel both filled with clay and initially containing the chemical pollutant, in this case, Dichlorvos at 10,000 ppm. The room and its tunnel are located in a fractured crystalline matrix at 400 m deep from ground level. The modelling domain covers an extension of 600 m wide by 700 m long and 800 m deep. The numerical modelling involves near and far field models. The problem is considered as initially saturated. This hypothesis, not only simplifies the calculation but also, and most importantly, it represents the less favourable situation from the risk assessment point of view. Fracture zones in the crystalline rock are represented as fracture planes. And fracture intersections are modelled with pipe elements. Figure 17 shows the geometry and the dimensions of the problem. The room is 100 m long, 50 m wide and 50 m high. The tunnel is 150 m long and has a cross section of 5 m × 5 m. The properties of the material involved are given in Table 8. The geometrical description of the problem

(a)

(b)

Figure 17: (a) Plan view and (b) lateral view of the case study. Table 8: Assumed properties for the lowrisk example.

Material

Hydraulic conductivity (m/s)

Diffusion coefficient (m2 /s)

Aperture (m)

Clay Rock Fracture EDZ

10−10 10−10 10−7 10−7

10−10 10−10 10−7 10−7

– – 1 1 and 3

140 Domain Decomposition Techniques for Boundary Elements is given in Fig. 17 The boundary conditions for the flow problem are summarised in Table 9. In the transport problem, all the surfaces of the boundaries have zero normal derivative of concentration: ∂cm /∂ nˆ m = 0, ∂cf /∂ nˆ f = 0, and ∂cp /∂ nˆ p = 0. The excavation disturbed zone (EDZ) is a layer of high hydraulic conductivity and diffusivity that surrounds the room and the tunnel. It is caused by the excavation process, and takes into account the fissures and fractures that occur in the crystalline rock due to the excavation process. In the present model the EDZ was considered as a fracture, thus represented as a closed surface that wraps the room and the tunnel, and it is connected at the same time with some fractures coming from the boundaries. Here, a thickness of 3 m near the room, and 1 m around the tunnel was adopted as the parameters of the analogous fracture. The influence of the EDZ on the flow and transport processes has been reported in [50]. The far field is composed of 320 single-domain blocks, whereas the near field is represented by 17 multidomain blocks, composed of tetrahedral linear subdomains. All the fracture planes were discretised with triangular linear discontinuous elements. The complete model involved 337 3D blocks, 11,428 3D subdomains, 10,636 2D subdomains (for fractures), 1,788 pipe elements, and a total number of Nfn = 188,352 collocation nodes was required, leading to a total number of 266,895 degrees of freedom for the flow problem and 269,187 for the transport problem. In this way the calculation involved 337 closed systems of equations for each block plus one for the FNS of dimension 74,452 × 74,452. All the underlying linear systems of equations were solved with an iterative sparse solver, employing the conjugate gradient normal residues (CGNR) method from [51]. The computational implementation used the SPARSKIT library [52]. In addition, a previous preprocessing was required in order to remove the zeros of the diagonal and to improve the convergence of the method.

Table 9: Boundary conditions for the case study. Boundary

Flow

Transport

hm = 0.05y hf = 0.05y hp = 0.05y

qm,n = 0 qf,n = 0 qp,n = 0

Plane yz at x = 700

hm = 0 hf = 0 hp = 0

qm,n = 0 qf,n = 0 qp,n = 0

Planes xy at z = 0 xy at z = 600 xz at y = 800

vm,n = 0 vf,n = 0 vp,n = 0

qm,n = 0 qf,n = 0 qp,n = 0

xz at y = 0

hm = 0 hf = 0 hp = 0

qm,n = 0 qf,n = 0 qp,n = 0

Plane yz at x = 0,


141

For this purpose, the MC64 preconditioner [53] was used in order to calculate the corresponding row and column scale factors, and to perform the corresponding column pivoting. This preprocessor transforms the original matrix in the following way: Aij → Pij Aij exp (ui + vj ), where P is the permutation matrix and u and v are the exponential row and column scale factors, respectively. In Fig. 18a the mesh used for blocks, fractures, and pipes in the near and far field zones is shown. For simplicity the figure shows a cut. In Fig. 18b the numerical results for the flow problem is shown. They are represented by a density plot of hydraulic head and a vector plot for the velocity field. The flow and transport calculation for 20 time steps took 47 h 17 min. Figure 19 shows the time evolution of the concentration in the near and far fields. The shape of the plume reflects the typical behaviour of a moderate Pèclet number solution, the local grid Pèclet number is smaller than unity in the whole domain. The pollutant is conveyed to the surface in a slow convective motion thanks to the vertical component of the velocity field. Figure 20 shows the variation of concentration with time in a point inside the room and in the exploration well at 400 m depth. The transient time of the transport process is of the order of magnitude of 200,000 years. The concentrations found in the observation point are four orders of magnitude less than the initial concentration at the room. The maximum concentration of Dichlorvos found in the observation well will take place at approximately t = 200,000 years, and its value will be approximately c = 3 ppm.

(a)

(b)

Figure 18: Test example 3: (a) Mesh and characteristic lengths. (b) Numerical solution for the flow problem and density plot of hydraulic head together with the corresponding velocity vector field.


(a)

(b)

(c)

(d)

Figure 19: Density plots of concentration at different times: (a) t = 2 × 1012 s; (b) t = 6 × 1012 s; (c) t = 12 × 1012 s; (d) t = 20 × 1012 s. Concentration across the fracture 1.2

Well Room

1

C/C0

0.8

4E-4

Left

0.6

3E-4 Right

0.4

2E-4

0.2

1E-4

0 -0.2 -200

5E-4

0 0

200

400

600

800

1000 1200

Time [ x 1000 years]

Figure 20: Variation of the concentration in the middle of the room, and in the observation point O1 of Fig. 17b.


143

7 Conclusion A new 3D boundary element formulation based on the DRM and domain decomposition has been developed in order to solve in an efficient way the problem of flow and solute transport in FPM. The method presents a new way to couple all the different entities, taking advantage of the underlying physical properties of the model. Moreover, for large-scale problems involving complex 3D fracture networks where fracture apertures are much smaller than their extensions L, a standard BE-DRMMD method would be unacceptable from the point of view of the computational cost. Early results such as the comparison presented in the second example in this chapter have demonstrated the convenience of this technique in terms of accuracy and CPU time. From the practical point of view, the degree of mesh refinement and domain decomposition can be tuned from a single domain to the extreme FEMlike meshes, thus allowing a high flexibility when modelling complex situations. The BEM employed is based on the kernel of the Laplace operator. In the scheme used, any term apart from the Laplace operator is dealt with as if it was part of a generalised source term. This feature constitutes one of the biggest advantages of the formulation: its flexibility. The future developments of the computational code might include models with more complex transport phenomena. Under these circumstances, the changes are only done in the non-homogeneous part of the Laplace equation, without having to reformulate the kernel and the fundamental equations or the DRM matrices. Therefore, as long as the influence of the Laplace operator is relatively significant in the physical process (i.e. Pèclet numbers not very high) and the radial basis functions employed are able to represent the main features of the generalised source term, then the numerical scheme will be robust enough to solve the model in an accurate way. The present work has recently contributed with concrete results to the LowRiskDT project in the estimation of large-scale pollution impacts, and the developed computational code offers the possibility to investigate a large number of interesting features and a variety of physical situations.

Acknowledgements This research was sponsored by the LowRiskDT project (Contract number EVG1CT-2000-00020) part of the FP5, Energy, Environment and Sustainable Development European Commission Programme.

References [1]

Bear, C.-F.T.J. & de Marsily, G. (eds.), Flow and Contaminant Transport in Fractured Rock, Academic Press Inc.: San Diego, 1993. [2] Adler, P.M. & Thovert, J.-F. (eds.), Theory and Applications of Transport in Porous Media. Fracture and Fracture Networks, Vol. 15, Kluwer Academic Publishers: Dordrecht, 1999. [3] Wilson, C. & Witherspoon, P.M., An Investigation of Laminar Flow in Fractured Porous Rocks, Report 70-6, Soil Mech. Bitum. Mater. Res. Lab., Dep. of Civil Eng., Univ. of Calif., Berkeley, 1970.

144 Domain Decomposition Techniques for Boundary Elements [4] [5]

[6]

[7] [8] [9] [10]

[11]

[12] [13]

[14] [15]

[16]

[17]

[18] [19]

Nelson, R.A. & Handin, J., Experimental study of fracture permeability in porous rock. AAPG Bulletin, 61, pp. 227–236, 1977. Gale, J.E., Quinn, O., Wilson, C., Forster, C., Witherspoon, P.A. & Jacobson, L., Hydrogeologic characteristics of fractured rocks for waste isolation – The Stripa Experience. Symp. Sci. Basis for Nucl. Waste Mgmt., Boston, MA, 27–30 November 1979, pp. 507–518, 1980. Bear, J. & Berkowitz, B., Groundwater flow and pollution in fractured rock aquifers. Developments in Hydraulic Engineering, Volume 4, Elsevier Science Publishing Co.: New York, pp. 175–238, 1987. Warren, J.E. & Root, P.J., The behavior of naturally fractured reservoirs. Society of Petroleum Engineers Journal, 3, pp. 245–255, 1963. Odeh,A.S., Unsteady-state behavior of naturally fractured reservoirs. Society of Petroleum Engineers Journal, 5(I), pp. 60–66, 1965. Snow, D.T., Anisotropic permeability of fractured media. Water Resources Research, 5(6), pp. 1273–1289, 1969. Kranz, R.L., Frankel, A.D., Engelder, T. & Sholz, C.H., The permeability of whole and jointed Barre granite. Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 16, pp. 225–234, 1979. Whiterspoon, P.A., Wang, J., Iwai, K. & Gale, J.E., Validity of cubic law for fluid flow in a deformable rock. Water Resources Research, 16, pp. 1016– 1024, 1980. Brown, S.R. & Scholz, C.H., Broad-band study of the topography of natural rock surfaces. J. Geophys. Res., 90, pp. 12575–12582, 1985. Gentier, S., Morphology and hydromechanical behavior of a natural fracture in granite under normal stress. Doctoral thesis, Université d’Orléans, Orléans, France, 1986. Brown, S.R., Fluid flow through rock joints: the effect of surface roughness. J. Geophys. Res., 92(B2), pp. 1337–1347, 1987. Moreno, L., Tsang, Y.W., Tsang, C.F., Hale, F.V. & Neretnieks, I., Flow and tracer transport in a single fracture: a stochastic model and its relation to some field observations. Water Resources Research, 24(12), pp. 2033–2048, 1988. Barton, C.C. & Larsen, E., Fractal geometry of two-dimensional fracture networks at Yucca Mountain, southwest Nevada. Fundamentals of Rock Joints: International Symposium on Fundamentals of Rock Joints, Bjorkliden, Lapland, Sweden, September 1985, ed. O. Stephansson, Centek Publishers: Lulea, Sweden, Proceedings, pp. 77–84, 1985. Mourzenko, V.V., Thovert, J.F. & Adler, P.M., Permeability of a single fracture; validity of the Reynolds equation. J. Phys. II France, 5, pp. 465–482, 1995. Koudina, N., Gonzalez Garcia, R. & Thovert, J.-F., Permeability of threedimensional fracture networks. Phys. Rev. E., pp. 4466–4479, 1998. Dienes, J.K., Percolation Theory and the Permeability of Fractured Oil Shales, Los Alamos Nat. Lab. Rept, LA 8553 PR, 1981.


[20]

[21]

[22]

[23]

[24] [25] [26]

[27] [28] [29]

[30] [31] [32]

[33] [34]

[35]

145

Long, J., Remer, J., Wilson, C. & Witherspoon, P., Porous media equivalent for networks of discontinuous fractures. Water Resources Research, 18, pp. 645–658, 1982. Cacas, M.C., Ledoux, E., de Marsily, G., Tillie, G., Barbreau, A., Durand, E., Feuga, B. & Peaudecerf, P., Modeling fractures flow with a stochastic discrete fracture network: 1. The flow model. Water Resources Research, 26, pp. 479– 489, 1990. Faust, C.R. & Mercer, J.W., Geothermal reservoir simulation: 1. Mathematical models for liquid- and vapor-dominated hydrothermal systems. Water Resources Research, 15(1), pp. 23–30, 1979. Charlaix, E., Guyon, E. & Roux, S., Permeability of a random array of fractures of widely varying apertures. Transport in Porous Media, 2(1), pp. 31–43, 1987. Robinson, P.C., Numerical calculations of critical densities for lines and planes. J. Phys. A: Math. Gen., 17, pp. 2823–2830, 1984. Charlaix, E., Guyon, E. & Rivier, N., Criterion for percolation threshold in a random array of plates. Sol. St. Commun., 50(11), pp. 999–1002, 1984. Wilke, S., Guyon, E. & de Marsily, G., Water penetration through fractured rocks: test of a tridimensional percolation description. Mathematical Geology, 17(1), pp. 17–24, 1985. de Marsily, G., Quantitative Hydrogeology: Groundwater Hydrology for Engineers, 1st edn, Elsevier, 1986. Lévy, T., Ecoulement d’un fluide dans un milieu poreux fissuré. C.R. Acad. Sci. Paris, 206, pp. 1413–1417, 1988. Alboin, C., Jaffre, J., Roberts, J.E. & Serres, C., Modeling fractures as interfaces for flow and transport in porous media. Flow and Transport in Porous Media: Mathematical and Numerical Treatment, eds. Z. Chen & R.E. Ewing, Vol. 295, Springer-Verlag: Heidelberg, pp. 13–24, 2002. Ligget, J.A. & Liu, P.L.-F., The Boundary Integral Equation Method for Porous Media Flow, George Allen and Unwin: London, 1983. Telles, J.C., Brebbia, C.A. & Wrobel, L.C., Boundary Elements Techniques, Springer-Verlag: Berlin and New York, 1984. Brebbia, C.A. & Dominguez, J., Boundary Elements An Introductory Course, 2nd edn, Computational Mechanics Publications: Southampton, McGraw-Hill: New York, 1992. Banerjee, P.K., The Boundary Element Methods in Engineering, McGrawHill: London and New York, 1994. Zagar, I. & Skerget, L., Integral formulations of a diffusive-convective transport equation. Boundary Element Applications in Fluid Dynamics, eds. H. Power & C.A. Brebbia, Computational Mechanics Publications: Southampton and Boston, pp. 153–176, 1995. Nardini, D. & Brebbia, C.A., A New Approach to Free Vibration Analysis Using Boundary Elements, Computational Mechanics Publications: Southampton, Springer-Verlag: Berlin and New York, 1982.


[37]

[38]

[39]

[40]

[41] [42] [43] [44]

[45]

[46] [47] [48]

[49]

[50]

[51] [52]

[53]

Popov, V. & Power, H., The DRM-MD integral equation method: An efficient approach for the numerical solution of domain dominant problems. Int. J. Numer. Meth. Engng., 44, pp. 327–353, 1999. Brebbia, C.A., Partridge, P.W. & Wrobel, L.C., The Dual Reciprocity Boundary Elements Method, Computational Mechanics Publications: Southampton and Boston, 1992. Golberg, M.A. & Chen, C.S., The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations. Boundary Elements Communications, (5), pp. 205–228, 1994. Popov, V. & Power, H., DRM-MD approach for the numerical solution of gas flow in porous media, with application to landfill. Engineering Analysis with Boundary Elements, 23, pp. 175–188, 1999. do Rego Silva, J.J., Acoustic and Elastic Wave Scattering Using Boundary Elements,Vol. 18, Computational Mechanics Publications: Southampton and Boston, 1994. CIMNE, GID, The Personal Pre/Postprocessor Manual, International Center for Numerical Methods in Engineering, Barcelona, Spain. GID resources, http://gid.cimne.upc.es TCL-TK Developer Xchange, http://www.tcl.tk Peratta, A. & Popov, V., Numerical stability of the BEM for advection diffusion problems. International Journal of Numerical Methods for Partial Differential Equations, 20(5), pp. 675–702, 2004. Taignenu, A.E., Numerical stability characteristics of a Hermitian Green element model for the transport equation. Engineering Analysis with Boundary Elements, 22(2), pp. 161–165, 1998. Scheid, F., Theory and Problems of Numerical Analysis, McGraw-Hill: Venice, Italy, 1990. LowRisk Disposal Technology (LowRiskDT), European Community Project Documentation, Contract No EVG1-CT-2000-00020. van Genuchten, M.Th. & Alves, W.J., Analytical Solutions of the 1D Convective-Dispersive Solute Transport Equation, Technical Report 1661, US Department of Agriculture, 1982. Duff, I.S., ME28: A Sparse Unsymmetric Linear Equation Solver for Complex Equations, Technical report, Computer Science and Systems Division, April, 1980. Peratta, A. & Popov, V., Modelling flow and solute transport in fractured porous media. Boundary Elements XXV, eds. D. Poljak, C.-A. Brebbia & R. Roje, Vol. 25, WIT Press: Southampton, pp. 63–72, 2003. Saad, Y., Iterative Methods for Sparse Linear Systems, Manchester University Press: Massachussets, 1992. Saad, Y., SPARSKIT: A Basic Tool Kit for Sparse Matrix Computations, Technical Report RIACS-90-20, Research Institute for Advanced Computer Science, NASA Ames Research Center, Moffett Field, CA, 1990. HSL libraries, http://www.cse.clrc.ac.uk/nag/hsl/

CHAPTER 5 Parallel domain decomposition boundary element method approach for large-scale transient and steady nonlinear heat conduction E. Divo1 , A.J. Kassab2 & K. Erhart2 1 Engineering

Technology Department, University of Central Florida, Orlando, FL, USA. 2 Mechanical, Materials, and Aerospace Engineering Department, University of Central Florida, Orlando, FL, USA.

Abstract It is well known that the boundary element method (BEM) requires only a surface mesh to solve linear, nonlinear, and transient heat conduction problems, but the resulting matrix is fully populated. This poses serious challenges for largescale 3D problems due to storage requirements and solution times for large sets of non-symmetric equations. Therefore, it is often necessary to employ a domain decomposition approach. This is done herein using an artificial subsectioning technique, along with a region-by-region iteration algorithm particularly tailored for parallel computation to address these issues. A coarse surface grid solution coupled with an efficient physically based procedure provides an effective initial guess for the fine surface grid model. The process converges very efficiently and offers substantial savings in memory. The iterative domain decomposition technique is ideally suited for parallel computation and its MPI implementation on clusters is discussed. Results from 3D BEM heat conduction models including models of upwards of 85,000 nodes arising form an intricate film-cooled vane are presented. We demonstrate that the BEM can be readily applied to solve large-scale linear, nonlinear, and transient heat conduction problems.

1 Introduction Numerical solutions of engineering problems often require large complex systems of equations to be set up and solved. For any system of equations the amount of

148 Domain Decomposition Techniques for Boundary Elements computer memory required for storage is proportional to the square of the number of unknowns, which for large problems can exceed machine limitations. For this reason almost all computational software use some type of problem decomposition. For methods that result in sparse matrices the storage alone can be decomposed to save memory, but for techniques such as boundary element methods (BEM) fully populated matrices generally result and another approach is needed. The BEM requires only a surface mesh to solve a large class of field equations, and further, the nodal unknowns appearing in the BEM equations are the surface values of the field variable and its normal derivative. Thus, the BEM lends itself ideally not only to the analysis of field problems, but also to modeling coupled field problems such as those arising in conjugate heat transfer (CHT). However, in implementing the BEM for intricate 3D structures, the number of surface unknowns required to resolve the temperature field can readily number in the tens to hundreds of thousands. Since the ensuing matrix equation is fully populated, this poses a serious problem both from the storage requirements as well as the need to solve a large set of non-symmetric equations. The BEM community has generally approached this problem by: (1) artificial subsectioning of the 3D model into a multi-region model, an idea originated for piecewise non-homogeneous media [1–3], in conjunction with block-solvers reminiscent of FEM frontal solvers [4, 5] or iterative methods [6–9], and (2) fastmultipole methods adapted to BEM coupled to a GMRES non-symmetric iterative solver [10, 11]. The first approach is readily adapted to existing BEM codes, while the multipole approach, although very efficient, requires rewriting of existing BEM codes. Recently, a technique using wavelet decomposition has been proposed to compress the BEM matrix once it is formed and stored in order to accelerate the solution phase without major alteration of traditional BEM codes [12]. In this chapter, a domain decomposition or artificial multi-region subsectioning technique is developed along with a region-by-region iteration algorithm tailored for parallel computation [13]. Once the domain decomposition details are presented, two applications will be discussed and detailed. The first is a BEM algorithm to solve large-scale, 3D, steady, nonlinear heat conduction problems, which will allow for multiple regions of different nonlinear conductivities. A non-symmetric update of the interfacial fluxes to ensure equality of fluxes at the subdomain interfaces is formulated. The second application is a BEM algorithm to solve large-scale, transient heat conduction problems. The transient heat conduction equation is transformed into a modified Helmholtz equation using a Laplace transform. The time domain solution is retrieved with a Stehfest numerical inversion routine. The domain decomposition technique described herein employs an iteration scheme, which is used to ensure the continuity of both temperature and heat flux at the region interfaces. In order to provide a sufficiently accurate initial guess for the iterative process, a physically based initial guess for the temperatures at the domain interfaces is derived, and a coarse grid solution obtained with constant elements is employed. The results of the constant grid model serve as an initial guess for finer discretizations obtained with linear and quadratic boundary element models. The process converges very efficiently, offers substantial savings

Parallel Domain Decomposition BEM Approach

149

in memory, and does not require complex data-structure preparation required by block-solvers or the multipole approaches. Moreover, the process is shown to converge for steady linear and nonlinear problems as well as for transient problems. The nonlinear problems are treated using the classical Kirchhoff transform. Results from a series of numerical examples are presented. The numerical examples range from validations of the approach against exact solutions in simple geometries to increasingly intricate BEM heat conduction models including a plenum-cooled and a film-cooled turbine blade. The transient cases are then verified in a similar fashion including an intermediate verification for the BEM solution of the modified Helmholtz equation.

2 Explicit domain decomposition In a standard BEM solution process, outlined below for both cases discussed above, if N is the number of boundary nodes used to discretize the problem, the number of floating-point operations (FLOPS) required to generate the algebraic system is proportional to N 2 . Direct memory allocation is also proportional to N 2 . Enforcing imposed boundary conditions, yields [H]{T } = [G]{q} ⇒ [A]{x} = {b}

(1)

where {x} contains nodal unknowns T or q, whichever is not specified in the boundary conditions. The solution of the algebraic system for the boundary unknowns can be performed using a direct solution method, such as LU decomposition, requiring FLOPS proportional to N 3 or an iterative method such as bi-conjugate gradient or general minimization of residuals that, in general, require FLOPS proportional to N 2 to achieve convergence. In 3D problems of any appreciable size, the solution becomes computationally prohibitive and leads to enormous memory demands. A domain decomposition solution process is adopted instead, where the domain is decomposed by artificially subsectioning the single domain under interest into K subdomains. Each of these is independently discretized and solved by standard BEM, while enforcing continuity of temperature and heat flux at the interfaces. It is worth mentioning that the discretization of neighboring subdomains in this method of decomposition does not have to be coincident, i.e. at the connecting interface, boundary elements and nodes from the two adjoining subdomains are not required to be structured following a sequence or particular position. The only requirement at the connecting interface is that it forms a closed boundary with the same path on both sides. The information between neighboring subdomains separated by an interface can be effectively passed through an interpolation, for instance by compactly supported radial basis functions. The process is illustrated in 2D in Fig. 1, with a decomposition of four (K = 4) subdomains. The conduction problem is solved independently over each subdomain, where initially a guessed boundary condition is imposed over the interfaces in order to create a well-posed problem for each subdomain. The details of an efficient initial guess technique will be addressed later. The problem in subdomain 1

150 Domain Decomposition Techniques for Boundary Elements G4

G1

G3 W G2

G41

G11

G42

G43

G I31 + G I12 G II32 + G II13 G III33 + W1 G21

W2 G22

G44

W3 G23

G34

G III14 W4 G24

Figure 1: BEM single region discretization and four domain BEM decompositions. is transformed into ∇ 2 T1 (x, y) = 0 ⇒ [H1 ]{T1 } = [G1 ]{q1 }

(2)

The composition of this algebraic system requires (n2 ) FLOPS where n is the number of boundary nodes in the subdomain as well as (n2 ) for direct memory allocation. This new proportionality number n is roughly equivalent to n ≈ 2N/(K + 1), as long as the discretization along the interfaces has the same level of resolution as the discretization along the boundaries. Direct memory allocation requirement for later algebraic manipulation is now reduced to n2 , as the influence coefficient matrices can easily be stored in ROM memory for later use after the boundary value problems on the remaining subdomains have been effectively solved. For the example shown here, where the number of subdomains is K = 4, the new proportionality value n is approximately equal to n ≈ 2N/5. This simple multi-region example reduces the memory requirements to about n2 /N 2 = (4/25) = 16% of the standard BEM approach. The algebraic system for subdomain 1 is rearranged, with the aid of given and guessed boundary conditions, as: [H1 ]{T1 } = [G1 ]{q1 } ⇒ [A1 ]{x1 } = {b1 }

(3)

The solution of the new algebraic system of subdomain 1 now requires a number FLOPS proportional to n3 /N 3 = (8/125) = 6.4% of the standard BEM approach if a direct algebraic solution method is employed, or a number of floating point operations proportional to n2 /N 2 = (4/25) = 16% of the standard BEM approach if an indirect algebraic solution method is employed. For both, FLOPS count and


151

direct memory requirement, this reduction is dramatic. However, as the first set of solutions for the subdomains were obtained using guessed boundary conditions along the interfaces, the global solution needs to follow an iteration process and satisfy a convergence criteria. Globally, the FLOPS count for the formation of the algebraic setup for all K subdomains must be multiplied by K, therefore, the total operation count for the coefficient matrices computation is given by: Kn2 /N 2 ≈ 4K/(K + 1)2 . For this particular case with K = 4, this corresponds to Kn2 /N 2 = 16/25 = 64% of the standard BEM approach. Moreover, the more significant reduction is revealed in the RAM memory requirements as only the memory needs for one of the subdomains must be allocated at a time, as the others can temporarily be stored into ROM, and when a parallel strategy is adopted the matrices for each subdomain are stored by its assigned processor. Therefore, for this case of K = 4, the memory requirements are reduced to only n2 /N 2 = 4/25 = 16% of the standard single region case. In order to reduce the computational efforts needed with respect to the algebraic solution of the system of eqn (1), a direct approach LU factorization is employed for all subdomains. The LU factors of the coefficient matrices for all subdomains are constant. As they are independent of the right-hand side vector, they can be computed only once at the first iteration step and stored on disc for later use during the iteration process. Therefore, at each iteration only a forward and a backward substitution will be required for the algebraic solution. This feature allows a significant reduction in the operational count through the iteration process, as only a number of floating point operations proportional to n as opposed to n3 is required at each iteration step. The access to ROM memory at each iteration step must also be added to this computation time, which is usually larger than access to RAM. Typically, however, the overall convergence of the problem requires few iterations, and this ROM access is not a significant addition. Additionally, iterative solvers such as GMRES may offer a more efficient alternative.

3 Iterative solution algorithm The initial guess is crucial to the success of any iteration scheme. In order to provide an adequate initial guess for the 3D case, the problem is first solved using a coarse grid constant element model, obtained by collapsing the nodes of the discontinuous bilinear element to the centroid, and supply that model with a physically based initial guess for interface temperatures. This converged solution then serves as the initial guess to a finer grid solution obtained using iso-parametric bilinear elements, and this solution, in turn, may be used to provide the starting point to a super-parametric biquadratic model, see Fig. 6a–c where these three elements are illustrated. While the constant element solution can be used as an initial guess for the later runs, an initial guess is still required in the solution of the constant element case. An efficient initial guess can be made using a 1D heat conduction argument for every node on the external surfaces to every node at the interface of each subdomain. An area over distance argument is then used to weight the contribution of an external temperature node to an interface node, see Fig. 2.

152 Domain Decomposition Techniques for Boundary Elements Aj Tj rij

Ω1

Ti

Ai

Ω2

Figure 2: Initial guess at the interface node i illustrated in 2D for a two-region subdomain decomposition.

Ti

qj

Tj

qj

Ti

rij k

Tj

rij k

(a)

qj

T•j

1 (b)

hj

Figure 3: Electric circuit analogy to 1D heat conduction from node i to node j: (a) heat flux node i and (b) convective node j.

Relating any interface node i to any exterior node j, one can estimate: (Ne Ti =

Aj Tj j=1 rij (Ne Aj j=1 rij

(4)

→ where rij = |− rij | is the magnitude of the position vector from interfacial node i to surface node j. There are Ne exterior nodes which are imposed with boundary conditions such that NT exterior nodes are imposed with temperatures, Nq exterior nodes are subjected to heat flux conditions, and Nh exterior boundary nodes are subjected to convective boundary conditions. Using a 1D conduction argument for the flux and convective nodes (see Fig. 3), the following initial guess for any interfacial node can be readily obtained in terms of a simple algebraic expression: (NT Ti =

(Nq ( h Bij Hij T∞j − j=1 Bij Rij qj + N j=1 Hij +1 (NT (Nh Bij Hij Si − j=1 Bij + j=1 Hij +1

j=1 Bij Tj

(5)


153

and the coefficients are explicitly Aj Bij = , rij

rij Rij = , k

hj Hij = rij , k

and

Si =

N

Aj j=1

rij

(6)

with N = NT + Nq + Nh , the thermal conductivity of the medium is k, the film coefficient at the jth convective surface is hj . The area of element j denoted by Aj is readily computed as: + Aj =

d(x, y, z) =

j

+1 +1

−1

−1

|Jj (η, ζ )|dηdζ

(7)

For a nonlinear problem, the conductivity of the medium is taken at a mean reference temperature. Once the initial temperatures are imposed as boundary conditions at the interfaces, a resulting set of normal heat fluxes along the interfaces will be computed. These are then non-symmetrically averaged in an effort to match the heat flux from neighboring subdomains. Considering a two-domain substructure, the averaging at the interface is explicitly given as, I I q = q − 1 1

I + qI q 2 1

2

and

I I = q − q 2 2

I + qI q 1 2

2

(8)

I = −qI after averaging. Compactly to ensure the flux continuity condition q 2 1 supported radial basis interpolation can be employed in the flux averaging process in order to account for unstructured grids along the interface from neighboring subdomains. Using these fluxes the BEM equations are again solved, leading to mismatched temperatures along the interfaces for neighboring subdomains. These temperatures are interpolated, if necessary, from one side of the interface to the other side using compactly supported radial basis functions to account for the possibility of interface mismatch between the adjoining substructure grids. Once this is accomplished, the temperature is averaged out at each interface. Illustrating this for a two-domain substructure, the interface temperatures for regions 1 and 2 are:

TI 1

=

TI 2 =

TI 1 + TI 2 2 TI 1

+ TI 2 2

+

I R q 1

2

+

I R q 2

(9)

2

in general to account for a case where a physical interface exists and a thermal contact resistance is present between the connecting subdomains, where R is the thermal contact resistance imposing a jump on the interface temperature values. These matched temperatures along the interfaces are used as the next set of boundary conditions.

154 Domain Decomposition Techniques for Boundary Elements It is important to note that when dealing with the nonlinear problem the interfacial temperature update is performed in terms of temperatures T and not in terms of the Kirchhoff transform variable U. That is, given the current values of the transform variable from either side of the subdomain interface at the current iteration, these are both inverted to provide the actual temperatures and it is these temperatures that are averaged. This is an important point, as the Kirchhoff transform amplifies the jump in temperature at the interface leading to the divergence of the iterative process reported previously in the literature [14–16]. Also, if a convective boundary condition is imposed at the exposed surface of a subdomain, a sublevel iteration is carried out for that subdomain. However, as the solution for such a subdomain is part of the overall iterative process, the sublevel iterations are not carried out to convergence, rather a few sublevel iterations are carried out. For such cases, the number of sublevel iterations is set to a default number of 5, with an option for the user to increase that number as needed. The overall iteration is continued until a convergence criterion is satisfied. A measure of convergence may be defined as the L2 norm of mismatched temperatures along all interfaces as: , K NI - 1

I − TI 2 T (10) L2 = . u K × NI k=1 i=1

This norm measures the standard deviation of BEM computed interface temperatures T I and averaged-out updated interface temperatures TuI . The iteration routine can be stopped once this standard deviation reaches a small fraction of Tmax , where Tmax is the maximum temperature span of the global field. It is noted that an iteration is referred to as the process by which an iterative sweep is carried out to update both the interfacial fluxes and temperatures such that the above norm may be computed. Here it is important to note that for the steady problems a value of = 5 × 10−4 is sufficient for accurate solutions. However, due to the amplification effects of the Stehfest transform the transient cases require values as small as = 10−15 .

4 Parallel implementation on a PC cluster The domain decomposition BEM formulation detailed above is ideally suited to parallel computing. A small Windows XP-based cluster consisting of 10 Intel-based P3 and P4 CPUs (1.7 GHz∼2 GHz) equipped with RAMBUS memory ranging from 768 MB to 1,024 MB was built to implement and test these algorithms. This small cluster is interconnected through a local workgroup in a 100 base-T Ethernet network with full duplex switches. An additional cluster comprised of 12 Intel P4 CPUs (2 GHz) each equipped with 1 GB SDRAM was built separately and made accessible through a separate workgroup in a 100 base-T Ethernet network. A parallel version of the code is implemented under MPICH libraries which conform with MPI and MPI2 standards [17–19] and using the COMPAQ Visual FORTRAN compiler. The parallel code collapses to serial computation if a single processor is assigned to the cluster.


155

Upon launching the code under MPI, the processors are identified and given a rank. A small BEM problem is solved on each processor to identify its relative performance. A static load balancing routine is then performed to optimally assign domains to each processor by minimizing an objective function that contains information with regard to subdomain sizes and relative computational capabilities. Specifically, the following objective function is minimized: S=

NPRO

[LOAD(N) − FRA(N)]2

(11)

N=1

where there are N = 1, 2, . . . , NPRO processors available to the cluster. The fraction of the overall inverse time it took the Nth computer to solve the test problem is defined as [1/t(N)] FRA(N) = (NPRO N=1 [1/t(N)]

(12)

where t(N) is( the running time it took the Nth processor in the solution of the test problem and NPRO N=1 FRA(N) = 1. The faster the Nth processor, the larger FRA(N) assigned to the Nth processor. The LOAD(N) vector is a function measuring the load to the Nth processor in solving the actual problem and that is defined as LOAD(N) =

NR

k=1

NE(k) ILOAD(N, k) × NEtotal

a (13)

Here, there are k = 1, 2, . . . , NR regions or subdomains in the actual problem to be solved, region k has NE(k) boundary elements used to discretize that region, and there are NEtotal number of elements in the model. The power a = 3 when a direct solver is used for each subdomain problem and a = 2 when GMRES is used to solve each subdomain problem. The terms ILOAD(N, k) come from a matrix [ILOAD] assigning loads of certain regions to certain processors. [ILOAD] is a Boolean matrix depending on whether a region is assigned or not to a processor. For example, the [ILOAD] matrix shown in eqn (14) shows a current configuration for an 8-region (NR = 8) and 5-processor (NPRO = 5) problem . In the displayed load configuration, processor 1 is assigned regions 2 and 8, processor 2 is assigned regions 3, 5, 6 while the remaining processors are each assigned a region.

[ILOAD] =

1 2 .. . NPRO

1 0 0 1 0 0

2 1 0 0 0 0

3 0 1 0 0 0

··· 0 0 0 0 1

0 1 0 0 0

0 1 0 0 0

0 0 0 1 0

NR 1 0 0 0 0

(14)

This optimization is performed using a discrete genetic algorithm. A key step in the domain decomposition is to keep each subdomain discretization to a number of

156 Domain Decomposition Techniques for Boundary Elements elements that allows the problem to be stored in available RAM memory, avoiding disk paging (accessing ROM). A typical cluster identification and benchmarking is illustrated for the 10-node cluster in Fig. 4. The resulting load balancing for a 20-region model performed with the discrete genetic algorithm process described above is shown in Fig. 5.

Cluster Identification

Augustus Tiberius

Gaius

Claudius Nero(1)

Nero(2) Vespasian Titus

Trajan(1) Trajan(2)

Cluster Benchmarking

Augustus Tiberius

Gaius

Claudius Nero(1)

Nero(2) Vespasian Titus

Trajan(1) Trajan(2)

Figure 4: Illustration of cluster identification and benchmarking of 8-PC (10 processors) cluster. Two PCs are dual processors. Load Balancing

Augustus Tiberius Regions

Regions

7 (4776)

17 (6624)

9 (4776)

Gaius

Claudius

Nero(1)

Nero(2) Vespasian

Regions

Regions

Regions

Regions

Regions

Regions

Regions

3 (5040)

4 (5040)

6 (3600)

8 (3600)

18 (2400) 20 (2400)

Titus

11 (5256)

1 (4176)

10 (3600) 14 (5256)

13 (4176)

2 (3552)

16 (3936)

15 (4560)

Trajan(1) Trajan(2) Regions

5 (4776) 12 (3744) 19 (3936)

Figure 5: Load balancing for a film-cooled blade 20 region and 85,224 degrees of freedom problem using the 8-PC (10 processors) cluster.


157

5 Applications in heat transfer The following sections detail the BEM formulations for the two cases introduced above, steady 3D nonlinear heat conduction, and transient 2D linear conduction. The discussion begins with the derivations of each technique and is followed by verification examples and comparisons. 5.1 3D nonlinear heat conduction The initial discussion shall focus primarily on nonlinear 3D heat transfer, where the governing equation under consideration is the steady-state nonlinear heat conduction equation, or: ∇ · [k(T )∇T ] = 0

(15)

where, T denotes the temperature, and k is the thermal conductivity of the material. If the thermal conductivity is taken as constant, then the above reduces to the Laplace equation for the temperature as: ∇ 2T = 0

(16)

When the thermal conductivity variation with temperature is an important concern, the nonlinearity in the steady-state heat conduction equation can readily be removed by introducing the classical Kirchhoff transform, U(T ) [20], such that: U(T ) =

1 ko

T

k(T )dT

(17)

To

where To is the reference temperature and ko is the reference thermal conductivity. The transform and its inverse are readily evaluated, either analytically or numerically. As the transform U is nothing but the area under the k vs. T curve, it is a monotonically increasing function of T , and the back-transform T (U) is unique. The heat conduction equation then transforms to a Laplace equation for the transformed parameter U(T ), as: ∇ 2U = 0

(18)

The boundary conditions are transformed linearly as long as they are of the first or second kind, such that: T |rs = Ts → U|rs = U(Ts ) = Us ∂U ∂T = qs −k = qs → −ko ∂n rs ∂n rs

(19)

158 Domain Decomposition Techniques for Boundary Elements Here rs denotes a point on the surface. In the case of boundary conditions of the third kind the problem is nonlinear in the boundary condition as: ∂T ∂U −k = hs [T |rs − T∞ ] → −ko = hs [T (U|rs ) − T∞ ] (20) ∂n rs ∂n rs In this case, iteration is required. This is accomplished by rewriting the convective boundary condition as: ∂U −ko = hs [U|rs − T∞ ] + hs [T (U|rs ) − U|rs ] (21) ∂n rs and first solving the problem with the linearized boundary condition: ∂U −ko = hs [U|rs − T∞ ] ∂n rs

(22)

to provide an initial guess for iteration. The heat conduction equation thus reduces to the Laplace equation in any case. The notation will be hereon changed back to T as the dependent variable with the understanding that when dealing with a nonlinear problem: T → U. The Laplace equation is readily solved by first converting it into a boundary integral equation (BIE), [20, 21], as: + + C(ξ ) T (ξ ) + q(x)G(x, ξ ) dS(x) = T (x)H(x, ξ ) dS(x) (23) S

S

where S(x) is the surface bounding the domain of interest, ξ is the source point, x is the field point, q(x) = −k∂T (x)/∂n is the heat flux, G(x, ξ ) is the fundamental solution, and H(x, ξ ) = −k∂G(x, ξ )/∂n. The fundamental solution is the response of the adjoint governing differential operator at any field point x due to a Dirac delta function acting at the source point ξ , and is given by G(x, ξ ) = 1/4πk r(x, ξ ) in 3D where /r(x, ξ ) is the Euclidean distance from the source point ξ . The free term C(ξ ) = S(x) H(x, ξ )dS(x) can be shown analytically to be the internal angle subtended at source point divided by 4π when ξ is on the boundary and one when ξ is at the interior. In the standard BEM, polynomials are employed to discretize the boundary geometry and distribution of the temperature and heat flux on the boundary. The discretized BIE is usually collocated at the boundary points, leading to the algebraic analog of eqn (23), or [H]{T } = [G]{q}

(24)

These equations are readily solved upon imposition of boundary conditions. Subparametric constant, isoparametric bilinear, and superparametric biquadratic, discontinuous boundary elements are used as the basic elements in the 3D BEM codes developed to implement these algorithms. These are illustrated in Fig. 6a–c. Such elements avoid the so-called star-point issue and allow for discontinuous fluxes.

159

Parallel Domain Decomposition BEM Approach ζ

Geometry Node

z

n

3

T and q Node

1

4

3

2

4

1

∆S

h

-1

1

y 1

x

(a) n

z

ζ

Geometry Node

n

n

T and q Node

3

4

4

1

4

3

3

4

3 n

0.75

2

∆S

2 -1

-0.75 0.75

1 1

h

1

-0.75

y

1 1

x

2 2

-1

(b) n

z

n

n n

3

n n

n

Geometry Node

ζ

T and q Node

1

4

n

3

g

d

c

0.75

2

4

-1

∆S ∆

h -0.75 0.75

1

f

1

h

-0.75

y

a 1

x

2

-1

e -1

b 2

(c)

Figure 6: Constant, bilinear, and biquadratic discontinuous elements used for coarse and refined BEM solutions: (a) discontinuous subparametric constant element used for coarse solution, (b) discontinuous isoparametric bilinear element used for first level of the refined solution, and (c) discontinuous superparametric biquadratic element used for second level of the refined solution.

160 Domain Decomposition Techniques for Boundary Elements Moreover, the biquadratic elements used here are superparametric, with bilinear model of the geometry and biquadratic model of the temperature and heat flux. This type of element provides compatibility of geometric models with grids generated by structured finite volume grid generators. 5.2 Transient heat conduction The BEM has been traditionally used to solve transient heat conduction problems through three different approaches: (1) using the convolution scheme where a timedependent Green’s function is introduced to build a transient boundary integral equation model, (2) using the dual reciprocity method (DRM) to expand the spatial portion of the governing equation using radial basis functions and a finite difference scheme to march in time, and (3) using the Laplace transformation of the governing equation to eliminate the time derivative leading to a modified Helmholtz equation that can be solved using a steady BEM approach and further inverting the BEM solution back into real space–time using a numerical Laplace inversion scheme [20, 22–31]. The first approach will require the generation and storage of BEM influence coefficient matrices at every time step of the convolution scheme making the technique unfeasible for medium or large problems, particularly in 3D applications, as the computational and storage requirements become unrealistically high. The second approach poses a different issue as the global interpolation functions for the dual reciprocity technique, such as the widely used radial basis functions lack convergence and error estimation approximations and can at times lead to unwanted behavior and significantly increase the conditioning number of the resulting algebraic system. The third approach, originated in a BEM application by Rizzo and Shippy [23], does not require time marching or any type of interpolation, but it requires fine-tuning of the BEM solution of the modified Helmholtz equation and a numerical Laplace inversion of the results. Real variable-based numerical Laplace inversion techniques such as the Stehfest transformation [28–29] provide very accurate results for non-oscillative type of functions, such as the ones expected to result from transient heat conduction applications, as all poles of the transformed solution are real and distributed along the negative real axis. One type of parallelization has been discussed by Davies and Crann [30] where individual solutions, as required for the numerical Laplace inversion, are solved simultaneously in multiple processors. This type of parallelization does reduce the computational time, but does not aid in storage requirements as the entire domain must be handled by each processor, thus leaving room for efficiency improvements. The following sections will formulate a Laplace-transformed BEM algorithm structured with the iterative parallel domain decomposition scheme detailed above, which acts to significantly reduce the computational and storage requirements of large-scale problems. Accurate results are reported for different cases in regular and irregular geometries imposed with different types of boundary conditions. The iteration algorithm is proven to efficiently converge for all the reported situations under different levels of domain decomposition and offers much promise for the efficient solution of 3D transient heat conduction problems using the BEM.


161

5.2.1 Governing equation and the Laplace transform Transient heat conduction is governed by the well-known diffusion equation, which for a 2D rectangular coordinate system is given by: ∇ · [k ∇T (x, y, t)] = ρc

∂T (x, y, t) ∂t

(25)

The purpose of applying the Laplace transform to the transient heat conduction equation is to eliminate the time-dependency of the equation. Taking the Laplace transform of the diffusion equation yields the following: ∇ · [k ∇ T¯ (x, y, s)] = ρcs T¯ (x, y, s) − ρc T (x, y, 0)

(26)

where, T¯ (x, y, s) is the Laplace-transformed temperature and new dependent variable. The above expression can be further reduced by requiring the initial condition: T (x, y, 0) = 0

(27)

which is true for any case of uniform initial condition with a proper superposition. Equation (26) is also no longer a function of time, but now contains the Laplace transform parameter, s. This parameter can now simply be treated as a constant in all remaining derivations. The dependency of the temperature field on s can now be eliminated, and the initial condition can be applied resulting in the following expression: ∇ · [k ∇ T¯ (x, y)] − ρcs T¯ (x, y) = 0 (28) The thermal conductivity can be independent of temperature and the above expression will have the form of the modified Helmholtz equation: ∇ 2 ψ − m2 ψ = 0

(29)

The solution to this equation is well known, since many other physical problems are governed by the modified Helmholtz equation, such as acoustic propagation (where ψ, would be the acoustic potential) [31]. The BEM has been effectively implemented for the solution of such acoustic propagation problems. Therefore, the Laplace transformation of the diffusion equation yields a suitable form for the desired solution method. Finally, the boundary conditions must be transformed in order to refer the entire problem in the proper Laplace transform space. The boundary conditions are transformed using the following relations: T (x, y) T¯ (x, y, s) = (30) s q(x, y) q¯ (x, y, s) = (31) s

where, , represents the boundary (control surface).

162 Domain Decomposition Techniques for Boundary Elements 5.2.2 BEM for the modified Helmholtz equation The development of a BEM solution begins by reducing the governing equation to a boundary-only integral equation. The current form of the Laplace-transformed transient heat conduction problem can be expressed in integral form by pre-multiplying the equation by a generalized function G(x, y, ξ ) and integrating over the domain of interest of the problem at hand ( : control volume) to obtain:

[G(x, y, ξ )∇ · (k ∇ T¯ (x, y))]d − ρcs

G(x, y, ξ ) T¯ (x, y) d = 0

(32)

Integration by parts is applied twice to the above equation (Green’s second identity) and the equation below is obtained: +

∂ T¯ G k d − ∂n

+

T¯ k

∂G d + ∂n

T¯ ∇ · (k ∇G) d − ρcs

G T¯ d = 0

(33)

where the dependencies have been omitted for clarity. The following two expressions can be defined as: ∂ T¯ (x, y) ∂n ∂G H(x, y, ξ ) = −k (x, y, ξ ) ∂n q¯ (x, y) = −k

(34) (35)

Here, n is the outward-drawn normal. Introducing the above expressions into the integral equation yields:

T¯ [∇ · (k∇G) − ρcsG]d =

+

+

H T¯ d

G q¯ d −

(36)

The Dirac delta function can now be used to perturb the adjoint operator over the function G as: ∇ · [k∇G(x, y, ξ )] − ρcsG(x, y, ξ ) = −ρcδ(x, y, ξ )

(37)

This equation is then solved in free space to determine the fundamental solution, G as: 0 s 1 Ko r (38) G(x, y, ξ ) = 2π α α where α is defined as α = 'k/ρc, Ko is a modified Bessel function of the second kind of order zero, and r = (x − xi )2 + (y − yi )2 . Now that the function G has been determined the function H can be evaluated from its definition as: 0 0 s s ∂Ko −ρc ∂Ko r nx + r ny (39) H(x, y, ξ ) = 2π ∂x α ∂y α


163

where nx and ny are the x- and y-components of the outward-drawn normal, n. Noting that the derivative of Ko (z) is −K1 (z), the above expression is simplified to the following: 0 0 −ρc s s H(x, y, ξ ) = (40) K1 r · [(x − xi ) nx + (y − yi ) ny ] 2πr α α Finally, the sifting property of the Dirac delta function is used to eliminate the only domain integral in the integral equation and obtain a boundary-only integral equation as: + + ¯ ¯ ρc C(ξ ) T (ξ ) = H(x, y, ξ ) T (x, y) d − G(x, y, ξ ) q¯ (x, y) d (41)

The first step of the BEM implementation is to divide the boundary into N-discrete elements as: N

= j (42) j=1

The boundary integral equation is discretized along the boundary as shown below: N N

1 y, ξ ) T¯ (x, y) d − ρcC(ξ )T¯ (ξ ) = H(x, G(x, y, ξ ) q¯ (x, y) d j=1

j

j=1

j

(43) where C(ξ ) is a geometry-dependent term calculated as the internal angle at point ξ divided over 2π. Therefore, C(ξ ) = 1 if ξ ∈ and C(ξ ) = 21 when ξ ∈ and is a smooth boundary. The functions C(ξ ) and T¯ (ξ ) can now be evaluated at a specific collocation point ξi , in order to obtain the expression below: ρcCi T¯ i =

N

1ij T¯ j − H

j=1

N

Gij q¯ j

(44)

j=1

1ij = where Gij = j G(x, y, ξi )d and H j H(x, y, ξi )d are known as the influence coefficients, and these are evaluated numerically. Adaptive Gauss–Kronrod quadratures are used for this purpose. The equation above can now be simplified and the following expression results: N

j=1

Hij T¯ j =

N

Gij q¯ j

(45)

j=1

1ij − (ρc/2) δij , such that δij = 0 if i = j, and δij = 1 if i = j. BoundHere, Hij = H ary conditions can be further applied to reduce the system of equations above to the standard algebraic form [A]{x} = {b}. Once the system is solved by standard linear algebra methods, the solution must be inverted numerically from the Laplace space to the real (transient space). The numerical inversion routine of choice is described in the following section.

164 Domain Decomposition Techniques for Boundary Elements 5.2.3 Numerical inversion of the Laplace transformed solution The final step of the overall numerical solution is the inversion of the Laplace transformed BEM solution. While many techniques exist for such an inversion, the Stehfest transform has the advantages of being quite stable, very accurate, and simple to implement. The Stehfest transform works by computing a sample of solutions at a specified number of times and predicting the solution based on this sample [32]. Due to the non-oscillative behavior of the transient heat conduction equation, the Stehfest transform works exceptionally well. The accuracy of the Stehfest transform as coded for this algorithm was verified by testing several equations with known analytical inversions, and satisfactory results were obtained. A function, f (t), which has a Laplace transform, f¯ (s), can be inverted using the relation: f (t) ∼ lim In (t) n→∞

(46)

where the sample In (t) is defined on the basis of a delta sequence δn (t, s) as: ∞ In (t) = δn (t, s) f¯ (s)ds (47) 0

A delta sequence [33] is defined such that as n → ∞ then, the delta sequence tends to the Dirac delta function δ(t), i.e.: lim δn (t, s) → δ(t)

n→∞

(48)

The Stehfest inversion is considered the best attempt at an improvement using extrapolation methods on the result of an asymptotic expansion for the difference In (t) − f (t) resulting from a specific delta sequence first proposed by Garver in 1966 [32]. The Stehfest inversion of the Laplace transform f¯ (s) of a function of time f (t) is given by: N 2 f (t) = ln Kn f¯ (sn ) (49) t n=1

where the sequence of s-values is provided explicitly by: sn = n

ln2 t

(50)

and the series coefficients are: Kn = (−1)n+N/2

min (n,N/2) k=(n+1)/2

k N/2 (2k)! (N/2 − k)!k!(k − 1)!(n − k)!(2k − n)!

(51)

The coefficients Kn are computed once and stored. Double precision arithmetic is mandatory to obtain accurate solutions. This method has been shown to provide accurate inversion for heat conduction problems in the BEM literature and is adopted in this study as the method to invert Laplace-transformed BEM solutions. Typically, the upper limit in the series is taken as N = 12 ∼ 14, as cited by


165

Stehfest [28]. However for these types of BEM solution inversions, Moridis and Reddell [34] reported little gains in accuracy for N = 6 ∼ 10, and demonstrated accurate results using N = 6. Davies and Crann [35] also report accurate results using N = 8, for BEM problems with periodic boundary conditions. For this work N = 12 has been used following the original results of Stehfest, and for maximum accuracy. It is also notable that due to amplification effects of the large factorial coefficients, Kn , on both round off and truncation errors, BEM solutions must be carried to very high levels of precision. For this reason very accurate integration, linear solver, and iteration routines are necessary in the BEM solution. This requirement acts to further increase the computational power and time needed for accurate transient results. The advantage of this inversion method still remains, however, due to its consistent requirements for any time solution. The computation is independent of the given time value, which is a major advantage over time-marching schemes that require much longer run times for large time solutions compared to small time solutions.

6 Numerical validation and examples 6.1 3D nonlinear heat conduction verification examples Steady nonlinear heat conduction problems are now considered. All geometric lengths as well as temperatures are dimensionless. Thermophysical properties are scaled to the reference length and temperature. A validation example comparing a numerical and exact solution for a nonlinear problem in a rectangular slab is first presented. Here, the nonlinear conductivity is taken to vary as k(T ) = 1 + T , and use the following expression for temperature: 0 T (x, y, z) = −1 +

1+

3 (2 − 2x 2 + y2 + z2 ) 19

(52)

This expression satisfies the heat conduction equation, eqn (15), for the chosen thermal conductivity variation. The above is used to impose boundary conditions on a rectangular slab of length 4 and height and width of 1. The slab is discretized into four equal subregions with a total of 600 elements, and this corresponds to 600 DOF for the constant element model, 2,400 DOF for the bilinear model, and 4,800 DOF for the biquadratic model. Using the exact solution, first-kind boundary conditions were imposed at the two end-caps at y = 0 and y = 4 while secondkind boundary conditions were imposed on the barrel surfaces. The BEM problem converged in a total time of 73 s including seven iterations for the constant element model to provide the initial guess for the bilinear case, one converged sweep of temperature and flux updates in the bilinear grid, and subsequently one converged sweep of temperature and flux updates in the biquadratic grid. The total time to solution reported here includes load balancing, generation of H and G matrices, and iteration to solution. The additional advantage of this approach is that it allows to state that the solution is grid independent as a grid refinement is performed on the

166 Domain Decomposition Techniques for Boundary Elements Z X

T:

Z Y

X

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Te:

Y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a)

(b) Z X

DevT: -0.0125 -0.0075 -0.0025 0.0025

0.0075

Y

0.0125

(c)

Figure 7: Comparison of BEM domain decomposition solution vs. exact solution for a nonlinear conduction problem in a slab: (a) BEM-computed solution, (b) exact solution, and (c) absolute deviation less than 0.0125 over Tmax = 1. way toward the final solution. Results are plotted in Fig. 7a–c for the BEM computed and exact solutions as well as the absolute which is less than 0.0125 over Tmax = 1. Next, a cylinder of radius 1 and length 10 is considered. The cylinder is decomposed into 10 equal subdomains corresponding to a discretization of 2,080 elements and 2,080 DOF for the constant element discretization, 8,320 DOF for the bilinear discretization, and 16,640 DOF for the biquadratic discretization. Here, two cases are considered: (1) a rod with nonlinear conductivity taken as k(T ) = 1.93[1 + 9.07 × 10−4 (T − 720)], and (2) a composite rod with endcaps comprising 10% of the geometry with a low nonlinear conductivity taken as k(T ) = 7.51[1 + 4.49 × 10−4 (T − 1420)] while the remainder of the rod has the same conductivity as in case (1) or k(T ) = 19.33[1 + 4.53 × 10−4 (T − 1420)] over 80% of the interior. Convective boundary conditions were imposed everywhere on the cylinder walls, with the ends cooled by convection with T∞ = 0 and h = 10 while the perimeter is heated by convection with h = 1 and T∞ varying from 1000 to 4000. The timings and total iterations for convergence of the solutions are provided in Table 1. The results are illustrated in Fig. 8a and b in the form of surface temperature contour plots.


167

Table 1: Number of iterations and timings for the rod problem. 6-P4 cluster ∼ 2,080 elements Constant elements (2,080 DOF) Bilinear elements (8,320 DOF) Biquadratic elements (16,640 DOF) Total time to solution

Case 1 (Homogeneous)

Case 2 (Composite)

5 iterations 1 iteration 1 iteration 284 s

9 iterations 1 iteration 1 iteration 292 s

X Z Y

T: 200 400 600 800 1000 1200 14001600 1800

(a) X Z Y

T: 400 600 800 1000 1200 1400

(b)

Figure 8: Results from the BEM solution in a rod: (a) homogeneous rod with nonlinear conductivity, temperature span over the rod is T = 200 − 1800, and (b) composite rod with nonlinear conductivity, temperature span over the rod is T = 400 − 1400. The next problem considers a more intricate geometry and subsequently a larger scale problem. Here, a plenum-cooled turbine blade is decomposed into the six subdomains illustrated in Fig. 9. There are a total of 5,014 elements or 5,014 DOF for the constant element discretization, 20,056 DOF for the bilinear element discretization, and 40,112 DOF for the biquadratic element discretization.


Figure 9: Domain decomposition of the plenum-cooled blade. Table 2: Number of iterations and timings for the blade problem. 6 × P4 cluster ∼ 5,014 elements

Case 1 (Linear)

Case 2 (Nonlinear)

Constant elements (5,014 DOF) Bilinear elements (20,056 DOF) Total time to solution

3 iterations 1 iteration 765 s

9 iterations 1 iteration 781 s

Again, two cases are solved: (1) assumes a constant conductivity k = 1 and (2) takes the nonlinear conductivity to vary as k(T ) = 1.93[1 + 9.07 × 10−4 (T − 720)]. A mixed set of boundary conditions are imposed on the surface of the blade. The plena are imposed with convective conditions with T∞ varying linearly from 300 to 500 along the plenum depth and h = 5. The end surfaces in the span-wise direction are insulated, while the remaining surfaces are imposed with convective conditions with T∞ = 1000 and h = 10. The timings and number of iterations for convergence of the solutions are provided in Table 2. The results are illustrated in Fig. 10a and b as temperature contour plots. Only constant and bilinear analyses were performed in this case. Next, a conduction problem in an intricate geometry of a film-cooled blade is considered. The domain decomposition and discretization for this blade are shown in Fig. 11a–c. Here the discretization is comprised of 21,306 elements distributed over 20 subdomains. This corresponds to 21,306 DOF for the constant elements discretization and 85,224 DOF for the bilinear discretization. Two cases are considered: (1) linear with a constant conductivity of k = 1.34, and (2) nonlinear with k(T ) = 1.09[1 + 4.29 × 10−4 (T − 1620)]. The endwall surfaces in the span-wise direction are taken as adiabatic, while temperature boundary conditions imposed on the blade surfaces were obtained from a conjugate analysis


169

Y

Z

T:

600

650

700

750

800

850

900

X

950 1000

(a) Y

Z

T:

700

750

800

850

900

X

950 1000

(b)

Figure 10: Results from BEM solution in a plenum-cooled blade: (a) plot of the temperature over the plenum-cooled blade, temperature span over the blade is T = 600 − 1000, (b) nonlinear conductivity case, T = 700 − 1000. carried out on the blade coupling the 3D BEM code for heat conduction to the Glenn-HT finite volume code for the flow analysis [1, 2]. The temperatures varied from 1600–3100 over the surfaces exposed to film cooling, plenum air, and hot gas flowing over the external surfaces. The timings and number of iterations for convergence of the solutions are reported in Table 3. The solutions are shown in Fig. 12a and b as temperature contour plots. Here, only a constant element and a bilinear analysis were carried out. Finally, a large-scale conduction problem in a U-tube is considered. The domain decomposition for this case is shown in Fig. 13. Here the discretization is comprised


(a)

(b)

(c)

Figure 11: Domain decomposition of film-cooled blade, 21,306 element model and 20 subregions: (a) overall domain decomposition of film-cooled blade, (b) close-up of leading edge, and (c) close-up of mesh and domain decomposition.

Table 3: Number of iterations and timings for film-cooled blade problem. 10 × PC cluster ∼ 21,306 elements Constant elements (21,306 DOF) Bilinear elements (85,224 DOF) Total time to solution

Case 1 (Linear)

Case 2 (Nonlinear)

10 iterations 1 iteration 3,222 s



171

Y

Z

X

T: 1600 1725 1850 1975 2100 2225 2350 2475 2600 2725 2850 2975 3100

(a)

Y

Z

X

T: 1600 1725 1850 1975 2100 2225 2350 2475 2600 2725 2850 2975 3100

(b)

Figure 12: Plot of converged solutions for the film-cooled blade: (a) trailing edge linear model and (b) trailing edge for the nonlinear case.


(a)

(b)

Figure 13: Domain decomposition of U-tube, 44,640 element model and 31 subregions: (a) overall domain decomposition of U-tube and (b) close-up of mesh. Table 4: Number of iterations and timings for U-tube problem. 12 × PC cluster ∼ 44,640 elements Constant elements (44,640 DOF) Bilinear elements (178,560 DOF) Total time to solution

Case 1 (Linear)

Case 2 (Nonlinear)



of 44,640 elements distributed over 31 subdomains. This corresponds to 44,640 DOF for the constant elements discretization and 178,560 DOF for the bilinear discretization. Again, two cases are considered modeling stainless steel as the composing material: (1) uniform conductivity taken as k = 14.9, and (2) nonlinear conductivity taken as k(T ) = 14.9[1 + 4.7 × 10−4 (T − 500)]. The perimetric surface of the U-tube is kept insulated while heat is added through one of the endcaps at a rate of 1000 and heat is removed through the other endcap by convection at 0 degrees with a heat transfer coefficient of 100. The timings and number of iterations for convergence of the solutions are reported in Table 4. The solutions are shown in Fig. 14a and b as temperature contour plots. Here, only a constant element and a bilinear analysis were carried out.


T:

0

40 80 120 160 200 240 280 320 360 400

T:

0

173

40 80 120 160 200 240 280 320 360 400

(a)

(b)

Figure 14: Plot of converged solutions for the U-tube: (a) Uniform thermal conductivity solution and (b) variable thermal conductivity solution. 6.2 Transient heat conduction verification examples The following section details the verification procedures used to test the transient domain decomposition Laplace transform BEM heat conduction solver described above. An initial test was performed to verify the accuracy of the BEM solution of the modified Helmholtz equation. A L = 4 by l = 1 rectangular region imposed with T¯ = 0 on the bottom and left walls and with q¯ = −1 on the top and right wall is considered. The selected parameters for the modified Helmholtz equation are: s = 0.005, ρ = 1000, c = 2, and k = 1. The analytical solution is given by the following series expansion: T¯ (x, y) =

∞

En sinh (δn x) sin (λn y) +

n=0

∞

Fn sin (γn x) sinh (νn y)

(53)

n=0

where 2 lλn δn cosh (δn L) 2 π λn = (2n + 1) and δn = λ2n + ρcs 2l 2 Fn = Lγn νn cosh (νn l) 2 π (2n + 1) and νn = γn2 + ρcs γn = 2L En =

(54)


T:

0

0.1

0.2

0.3

0.4

0 .5

0 .6

0.7

(a)

(b)

(c)

(d)

Figure 15: Contour plots of the modified Helmholtz solution: (a) exact temperature distribution, (b) one-region model, (c) two-region model and (d) fourregion model.

The analytical solution followed by the BEM solutions for a varying number of regions are displayed in Fig. 15a–d in the form of contour plots. A total of 100 quadratic discontinuous boundary elements were employed in the single region case and 10 additional boundary elements were added at each interface for the multiregion cases. The accuracy of the BEM routine is clearly shown, as the iso-lines are virtually identical to the exact solution, including the case of multiple regions cases. The stability of the BEM solver is also readily recognized in the plot of residual vs. iteration for the multiple region cases shown in Fig. 16. Here, it is shown that the residual decays rapidly for both multi-region cases. The convergence criterion used was = 10−8 and achieved in less than 10 iterations for the four-region case.


175

1.00E-01 Two Region

1.00E-03

Norm

Four Region

1.00E-05 1.00E-07 1.00E-09 1.00E-11 0

5

10

15 Iteration

20

25

Figure 16: Plot of the L2 norm development for the two- and four-region BEM solutions of the Helmholtz equation using interface temperature differences.

It should be noted that the two-region model below corresponds to a storage reduction of 64% when compared to a single region, while the four-region case obtains an 84% reduction in RAM memory requirements. As mentioned in the multi-region discussion above, the number of FLOPS is also significantly dropped, with a 56% and 74% reduction in FLOPS for the two- and four-region cases, respectively. Next, a transient test was performed to verify the accuracy of the BEM solution combined with the numerical inversion of the Laplace transform. Again, a L = 4 by l = 1 rectangular region imposed with T = 0 on the bottom and left walls and with q = −1 on the top and right walls is considered. The domain is initially at a temperature T (x, y, 0) = 0. The selected parameters for the diffusion equation are: ρ = 1000, cp = 2, and k = 1. A total of 100 quadratic discontinuous boundary elements were employed in the single-region case and 10 additional boundary elements were added at each interface for the multi-region cases. An analytical solution was derived for the problem at hand yielding the following form:

T (x, y, t) = Tss (x, y) +

∞ ∞

Cnm sin (βn x) sin (λm y)e−αδnm t 2

(55)

n=0 m=0

where

Tss (x, y) =

∞

n=0

En sinh (δn x) sin (δn y) +

∞

n=0

Fn sin (γn x) sinh (γn y)

(56)

176 Domain Decomposition Techniques for Boundary Elements with En =

2 l δn2 cosh (δn L)

π (2n + 1) 2l 2 Fn = Lγn2 cosh (γn l) π γn = (2n + 1) 2L δn =

(57)

and Cnm =

4 Ll

L

l

To (x, y) sin (βn x) sin (λm y)dydx 0

(58)

0

with To (x, y) = T (x, y, 0) − Tss (x, y) π βn = (2n + 1) 2L π λm = (2m + 1) 2l 2 δnm = λ2m + βn2

(59)

The analytical solutions followed by the BEM solutions for a varying number of regions are displayed in Fig. 17a–d in the form of contour plots. This time, the

(a)

Figure 17: Contour plots of the transient temperature solution: (a) exact solution at t = 25 s, t = 100 s, t = 200 s, t = 500 s; (b) one-region BEM at t = 25 s, t = 100 s, t = 200 s, t = 500 s; (c) two-region BEM at t = 25 s, t = 100 s, t = 200 s, t = 500 s; and (d) four-region BEM at t = 25 s, t = 100 s, t = 200 s, t = 500 s.


177

(b)

(c)

(d)

Figure 17: Continued

temperature contours are shown at different times (t = 25 s, t = 100 s, t = 200 s, t = 500 s) and excellent agreement is found between the exact and BEM solutions in both single and multi-region cases. In addition, a plot of the temperature evolution from t = 0 s to t = 1000 s is shown in Fig. 18 for the point (x, y) = (3.8, 0.8), revealing virtually perfect agreement between exact and BEM solutions.


Temperature Evolution 1.2 1

Tem p(C)

0.8 0.6 Exact 1 Region

0.4

2 Region

0.2

4 Region

0 0

200

400

600

800

1000

time (sec)

Deviations 4.0E-04 1 Region

3.5E-04

2 Region Temp(C)

3.0E-04

4 Region

2.5E-04 2.0E-04 1.5E-04 1.0E-04 5.0E-05 0.0E+00 0

100

200

300

400

500

time(sec)

Figure 18: Exact and BEM temperature evolution and deviations at point (x, y) = (3.8, 0.8).

Figure 19 depicts the residuals of the iterative BEM solution, for the region case for three of the twelve total inverse transform steps at t = 100 s. Although the norm for the first case, s1, takes the longest to decay, it is the least critical as the transform coefficient is small for the first steps. The convergence criterion for the multi-region BEM iteration process for the transient case was reduced to = 10−15

179

Parallel Domain Decomposition BEM Approach Transient Iterative Norm

Norm

1.00E-01 1.00E-03

s1

1.00E-05

s6

1.00E-07

s12

1.00E-09 1.00E-11 1.00E-13 1.00E-15 1.00E-17 0

5

10

15

20

25

30

35

40

45

Iteration

Figure 19: Interface iterative norms for three different s-values (steps) used in the Stehfest transform: s1 , s6 and s12 in eqn (49).

y

x hx = 1000*e -0.05x

T• = 2500K

Figure 20: BEM discretization and BCs for the turbine guide vane problem.

to account for the error magnification caused by the large factorials found in the Stehfest transform algorithm. The final example presented here shows a transient heat transfer analysis performed on a typical guide vane under the estimated turbine conditions given below. The vane geometry is shown in Fig. 20, along with the applied convective boundary condition equations, noting that all temperatures are measured in degrees centigrade above ambient. The free-stream temperature, T∞ , has been assumed constant at 2500 degrees, and the heat transfer coefficient is given an exponentially decaying value with the maximum at the vane tip. As this is only a showcase of the abilities of the conduction solver all values have been estimated and are not perfectly representative of turbine conditions. The results of the analysis show the smooth, rapid propagation of heat through the vane over time in Fig. 21a–d. The same vane shape is then investigated with the addition of cooling passages using a coolant flow at 1800 degrees, see Fig. 22, two-passage case. The second


(a)

(b)

(c)

T:

0

500

1000

1500

2000

2500

(d)

Figure 21: Temperature fields at various times for a solid turbine guide vane: (a) t = 0.2 s and t = 0.4 s, (b) t = 0.6 s and t = 0.8 s, (c) t = 1.0 s, and (d) temperature scale for all time values.

T• = 1800K h = 250

y

x hx = 1000*e -0.05x

T• = 2500K

Figure 22: BEM discretization and BCs for the turbine guide vane problem with cooling passages.

case contains a single, centrally located cooling passage. Figure 23a–d show the slow down in heat transfer caused by the addition of the single cooling passage. The final case shows how multiple passages may also be employed to produce similar effects, see Fig. 24a–d. It is noted that both cases increase the heat up time as desired, but neither significantly reduces the steady temperature values. The hottest point on the vane is the front tip and the temperature of the tip is tracked and plotted over time for each of the three cases above in Fig. 25.


181

(a)

(b)

(c)

T:

0

500

1000

1500

2000

2500

(d)

Figure 23: Temperature fields for the turbine guide vane with cooling passage: (a) t = 0.2 s and t = 0.4 s, (b) t = 0.6 s and t = 0.8 s, (c) t = 1.0 s and (d) temperature scale for all time values.

(a)

(b)

(c)

T:

0

500

1000

1500

2000

2500

(d)

Figure 24: Temperature fields for the turbine guide vane with two cooling passages: (a) t = 0.2 s and t = 0.4 s, (b) t = 0.6 s and t = 0.8 s, (c) t = 1.0 s and (d) temperature scale for all time values.


Vane Tip Temperature

Temp (above amibient)

2500 2000 1500

Solid

1000

1 Passage 2 Passage

500 0 0

0.25

0.5 time (s)

0.75

1

Figure 25: Vane tip temperature over time for various vane configurations.

7 Conclusions The BEM is often an efficient choice for the solution of various engineering field problems as it acts to decrease the dimensionality of the problem. However, the solution of large problems is still prohibitive as the BEM coefficient matrices are typically fully populated and difficult to subdivide or compress. This chapter has presented an efficient iterative domain decomposition method to reduce the storage requirement and allow the solution of such large-scale problems. The decomposition approach lends itself ideally to parallel message passing type computing due to the independence of each of the BEM subregion solutions. With this approach, large-scale problems can be readily solved on small PC clusters. The domain decomposition approach is setup in general and can be applied to any type of BEM problem. This method of domain decomposition was then applied to steady, 3D, linear, and nonlinear heat conduction as well as to transient, 2D, linear heat conduction problems. These formulations are general and can be directly applied to other field problems such as elasticity, thermoelasticity, and acoustics. Results have been presented and excellent agreement with analytical solutions was found.

Acknowledgment This research is supported by a grant NAG3-2691 from NASA-Glenn.


183

References [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10] [11]

[12]

[13]

Kassab, A., Divo, E., Heidmann, J., Steinthorsson, E. & Rodriguez, F., BEM/FVM conjugate heat transfer analysis of a three-dimensional film cooled turbine blade. Int. J. Num. Methods Heat and Fluid Flow, in press. Heidmann, J.D., Kassab, A.J., Divo, E.A., Rodriguez, F. & Steinthorsson, E., Conjugate heat transfer effects on a realistic film-cooled turbine vane. ASME Paper GT2003-38553, 2003. Rizzo, F. & Shippy, D.J.,A formulation and solution procedure for the general non-homogeneous elastic inclusion problem. Int. J. of Solids and Structures, 4, pp. 1161–1179, 1968. Bialecki, R.A., Merkel, M., Mews, H. & Kuhn, G., In-and out-of-core BEM equation solver with parallel and nonlinear options. Int. J. Num. Methods in Engineering, 39, pp. 4215–4242, 1996. Kane, J.H., Kashava-Kumar, B.L. & Saigal, S., An arbitrary condensing, non condensing strategy for large scale, multi-zone boundary element analysis. Comp. Methods App. Mech. Eng., 79, pp. 219–244, 1990. Baltz, B. & Ingber, M.S., A parallel implementation of the boundary element method for heat conduction analysis in heterogeneous media. Engineering Analysis with Boundary Elements, 19, pp. 3–11, 1997. Kamiya, N., Iwase, H. & Kita, E., Parallel implementation of boundary element method with domain decomposition. Engineering Analysis with Boundary Elements, 18, pp. 209–216, 1996. Davies, A.J. & Mushtaq, J., The domain decomposition boundary element method, on a network of transputers. BETECHXI: Proc. of 11th Conf. Boundary Element Technology, Honolulu, Hawaii, ed. R.C. Ertekin, Computational Mechanics Publications: Southampton, pp. 397–406, 1996. Mai-Duy, N., Nguyen-Hong, P. & Tran-Cong, T., A fast convergent iterative boundary element method on PVM cluster. Engineering Analysis with Boundary Elements, 22, pp. 307–316, 1998. Greengard, L. & Strain, J., A fast algorithm for the evaluation of heat potentials. Comm. Pure Appl. Math., 43, pp. 949–963, 1990. Hackbush, W. & Nowak, Z.P., On the fast multiplication in the boundary element method by panel clustering. Numerische Mathematik, 54, pp. 463– 491, 1989. Bucher, H. & Wrobel, L.C., A novel approach to applying wavelet transforms in boundary element method. BETEQII: Advances in Boundary Element Techniques, II, Piscatawy, New Jersey, Hogaar Press: Switzerland, pp. 3–13, 2000. Rodriguez, F., Divo, E. & Kassab, A.J., A strategy for BEM modeling of large-scale three-dimensional heat transfer problems. Recent Advances in Theoretical and Applied Mechanics, Vol. XXI: Proc. of SECTAM XXI, Orlando, Florida, eds. A.J. Kassab, D.W. Nicholson and I. Ionescu, Orlando, Florida, Rivercross Press: Orlando, pp. 645–654, 2002.

184 Domain Decomposition Techniques for Boundary Elements [14] Azevedo, J.P.S. & Wrobel, L.C., Non-linear heat conduction in composite bodies: a boundary element formulation. Int. J. Numerical Methods in Engineering, 26, pp. 19–38, 1988. [15] Bialecki, R. & Nahlik, R., Solving nonlinear steady-state potential problems in non-homogeneous bodies using the boundary element method. Numerical Heat Transfer, Part B, 16, pp. 79–96, 1989. [16] Bialecki, R. & Kuhn, G., Boundary element solution of heat conduction problems in multi-zone bodies of non-linear materials. Int. J. for Numerical Methods in Engineering, 36, pp. 799–809, 1993. [17] Gropp, W., Lusk, E. & Thakur, R., Using MPI: Portable Parallel Programming with the Message-Passing Interface, The MIT Press: Cambridge, MA, 1999. [18] Gropp, W., Lusk, E. & Thakur, R., Using MPI-2: Advanced Features of the Message-Passing Interface, The MIT Press: Cambridge, MA, 1999. [19] Sterling, T.E., Beowulf Cluster Computing with Windows, The MIT Press: Cambridge, 2001. [20] Brebbia, C.A., Telles, J.C.F. & Wrobel, L.C., Boundary Element Techniques, Springer-Verlag: Berlin, 1984. [21] Wrobel, L.C., The Boundary Element Method – Applications in Thermofluids and Acoustics, Vol. 1, John Wiley and Sons: New York, 2002. [22] Kassab, A.J. & Wrobel, L.C., Boundary element methods in heat conduction (Chapter 5), Recent Advances in Numerical Heat Transfer, Vol. 2, eds. W.J. Mincowycz and E.M. Sparrow, Taylor and Francis: New York, pp. 143–188, 2000. [23] Rizzo, F.J. & Shippy, D.J., A method of solution for certain problems of transient heat conduction. AIAA Journal, 8(11), pp. 2004–2009, 1970. [24] Divo, E. & Kassab, A.J., a boundary integral equation for steady heat conduction in anisotropic and heterogeneous media. Num. Heat Transfer, Part B: Fundamentals, 32(1), pp. 37–61, 1997. [25] Divo, E. & Kassab, A.J., A generalized bie for transient heat conduction in heterogeneous media. J. Thermophysics and Heat Transfer, 12(3), pp. 364–373, 1998. [26] Divo, E., Kassab, A.J. & Rodriguez, F., A parallelized iterative domain decomposition approach for 3D boundary elements in non-linear heat conduction. Numerical Heat Transfer, Part B: Fundamentals, 44(5), pp. 417– 437, 2003. [27] Cheng, A.H.-D. & Ou, K., An efficient Laplace transform solution for multiaquifer systems. Water Resources Research, 25(4), pp. 742–748, 1989. [28] Stehfest, H., Numerical inversion of Laplace transforms. Communications of the ACM, 13, pp. 47–49 and p. 624, 1970. [29] Stehfest, H., Remarks on algorithm 368: numerical inversion of Laplace transforms. Communications of the ACM, 13, pp. 624, 1970. [30] Davies, A.J. & Crann, D., Parallel Laplace transform methods for boundary element solutions of diffusion-type problems. Journal of Boundary Elements, BETEQ 2001, No. 2, pp. 231–238, 2002.


[31]

[32]

[33] [34]

[35]

185

Divo, E., Kassab, A.J. & Ingber, M.S., Shape optimization of acoustic scattering bodies. Engineering Analysis with Boundary Elements, 27(7), pp. 695– 704, 2003. Davies, B. & Martin, B., Numerical inversion of the Laplace transform: a survey and comparison of methods. Journal of Computational Physics, 33, pp. 1–32, 1979. Greenberg, M., Applications of Green’s Functions in Engineering and Science, Prentice-Hall: Englewood Cliffs, New Jersey, 1971. Moridis, G.J. & Reddell, D.L., The Laplace transform boundary element (LTBE) method for the solution of diffusion-type equations. Boundary Elements XIII, pp. 83–97, WIT Press: Southampton, 1991. Davies, A.J. & Crann, D., The Laplace transform boundary element methods for diffusion problems with periodic boundary conditions. Boundary Elements XXVI, pp. 393–402, WIT Press: Southampton, 2004.


CHAPTER 6 Computational implementation for 3D problems B. Natalini1 & V. Popov2 1 Facultad de Ingeniería, Universidad 2Wessex Institute of Technology, UK.

Nacional del Nordeste, Argentina.

Abstract The dual reciprocity method (DRM) is a boundary element technique to approach domain-dominant problems without losing the boundary-only nature of the boundary element method. The DRM converts domain integrals into boundary integrals by means of approximation functions. The DRM is general and the number of applications solved using the procedure has been increasing in the literature since the early 1990s. However, the DRM faces a serious drawback when applied to large problems: the resulting system of equations is dense and frequently ill-conditioned. A way to overcome this inconvenient feature is by using domain subdivision in the limiting case when the resulting internal mesh looks like a finite element grid. This technique is known as the dual reciprocity method multidomain (DRM-MD) approach. The DRM-MD approach produces a sparse and well-conditioned system of equations. It has been successfully applied to a variety of problems in 2D domains and has showed good performance. The extension of the procedure to 3D cases is not straightforward since factors such as continuity of the elements, DRM approximation function, scaling, number of internal DRM nodes, etc. that largely affect the performance of the code need to be selected. In this chapter, several aspects regarding the computational implementation of the DRM-MD codes are presented and reviewed. A general assembly procedure is proposed, which can be used in both 2D and 3D problems and can be easily adapted to approach problems with different governing equations. Results of numerical examples using several different schemes for 3D problems are shown to provide an insight on 3D DRM-MD implementation.


1 Introduction The dual reciprocity method multidomain (DRM-MD) approach, a boundary element procedure introduced by Popov and Power [1] to approach domain-dominant problems, uses domain subdivision in the limit case in which the mesh looks like a finite element method (FEM) mesh, similar to the Green element method (GEM) [2] and the boundary-domain integral method (BDIM) [3]. The procedure applies the Green’s integral representation formula in each subdomain and imposes the full matching conditions on the interfaces of adjacent subdomains. Unlike the GEM and the BDIM, the DRM-MD computes the volume integral through the DRM. Some of the characteristic features of the procedure are: • • • • •

As it does not compute volume integrals, it preserves the boundary-only nature of the boundary element method (BEM). The matrix of coefficients of the system of equations produced by the DRM-MD is block–banded. The local F matrices (see Section 2) are well conditioned independently of the size of the problem. The number and position of the DRM nodes is not critical since small subdomains require few or no internal DRM nodes. When using continuous elements, the resulting system of equations is overdetermined.

The DRM-MD has been applied to a variety of problems that are governed by linear and nonlinear PDEs. Among the 2D cases that have been solved using this formulation can be mentioned the flow of a mixture of gases through a porous media [4–6], linear and nonlinear advection–diffusion problems [1, 7], driven cavity flow of a Newtonian fluid [8], pressure-driven flow and the Couette flow of nonNewtonian fluids [9], the flow of polymers inside mixers with complex geometries [10], and flow and transport in fractured porous media [11]. The procedure has been also applied to 3D cases: Poisson’s and steady-state advection–diffusion [12], flow and transport of pollutant in saturated porous media [13, 14], and flow in unsaturated porous media [13]. The DRM-MD has shown to be an accurate and versatile tool for the analysis of engineering problems. However, its implementation can be done under different schemes which largely affect the performance of the method. Portapila and Power [15, 16] carried out research on the influence of solvers on the efficiency of the procedure. Other aspects such as the type of DRM approximation function, representation of partial derivatives, scaling, etc., have been discussed by Popov and Power [1, 5, 17] and Natalini and Popov [13, 18]. Little can be found in the literature about the implementation of DRM-MD codes. In this chapter, a general scheme, which is suitable for different 2D and 3D problems, for the implementation of this numerical procedure is presented and discussed. For 3D cases, the proposed scheme is exemplified for Poisson’s and advection–diffusion problems in domains that have been discretized with meshes made up of tetrahedrons. Details are given when every side of the

Computational Implementation for 3D Problems

189

tetrahedron is either a continuous or a discontinuous quadratic triangular boundary element. It shall be seen that the proposed scheme can be easily adapted to different problems using different kinds of elements.

2 Equations and DRM formulation Let us consider the following Poisson’s equation, with the constitutive coefficient equal to one. ∇ 2 u(x) = b(x) (1) where u(x) is a scalar field (potential function), b(x) is the non-homogeneous term and x is a position vector in the domain with components xi . Given a point x belonging to a domain , which is enclosed by a contour , the Green integral representation formula for (1) gives the value of u at x in terms of integral equations involving the fundamental solution of the Laplace equation. λ(x)u(x) +

∗

q (x, y)u(y)dy −

∗

u (x, y)q(y)dy = −

u∗ (x, y)b(y)dy (2)

Here, u∗ (x, y) is the fundamental solution of the Laplace equation given by u∗ (x, y) =

1 4π r

(3)

for 3D problems, where r is the distance from the point of application of the concentrated unit source to any other point under consideration, i.e. r = |x − y|, q( y) = ∂u( y)/∂n and q∗ (x, y) = ∂u∗ (x, y)/∂n and n is the unit vector normal to the boundary of the subdomain. Notice that in eqn (2) all the integrals are over the boundary of the domain except for the one corresponding to the term b(y), which represents the sum of the non-homogeneous terms. The constant λ(x) has values between 1 and 0, being equal to 1/2 for smooth boundaries. It is also important to note that the above equation holds for points inside the domain, in which case λ(x) is equal to 1. For eqn (2) to be solved, appropriate boundary conditions must be applied. To express the domain integral in (2) in terms of equivalent boundary integrals, the DRM approximation is introduced. The basic idea is to expand the b(y) term using approximation functions, i.e. b(y) ∼ = b˜ =

J+I

αk f (y, zk )

(4)

k=1

The functions f ( y, zk ) are approximation functions that depend only on the geometry of the problem, and the constants αk are unknown coefficients. The approximation is done at (J + I) nodes, with J boundary nodes around the boundary of the domain and I nodes inside the domain.

190 Domain Decomposition Techniques for Boundary Elements With the DRM approximation for the non-homogeneous term b(y), the domain integral in eqn (2) becomes:

u (x, y)b(y)dy ∼ = ∗

J+I

αk

k=1

u∗ (x, y)f (y, zk )dy

(5)

Using (4) the value of the coefficients αk can be expressed in terms of the nodal values of the function b α = F−1 b

(6)

where F is a matrix with coefficients Fij = f (yi , zj )

(7)

A new auxiliary non-homogeneous Laplacian field, uˆ , for each collocation point zk , can be defined in the following way: ∂ 2 uˆ (y, zk ) = f (y, zk ) ∂xl ∂xl

(8)

Applying the Green’s formula to the non-homogeneous Laplacian field in (8) at a point x, yields −

u∗ (x, y)f (y, zk )dy = λ(x)û(x, zk ) + q∗ (x, y)û(y, zk )dy − u∗ (x, y)ˆq(y, zk )dy

(9)

where qˆ = ∂ uˆ /∂n. Substituting the last equation into (5), the domain integral can be recast in terms of a series of surface integrals, and using the resulting expression in (2), one finally arrives at a boundary-only integral representation formula λ(x)u(x) +

q∗ (x, y)u(y)dy −

∼ =

u∗ (x, y)q(y)dy

# J+I "

k ∗ k ∗ k αk λ(x)û(x, z ) + q (x, y)û(y, z )dy − u (x, y)ˆq(y, z )dy k=1

(10) For the numerical solution of the problem, the contour is discretized in j elements and the density of the integrals in the above equation is defined in terms of nodal


191

values by means of interpolation functions. In this way the eqn (10) for the ith node can be written as: $ % n

∗ ∗ λi ui + qij uj dj − uij qj dj j=1

=

n+m

k=1

j

j

 λi uˆ ik +

$ n

j=1

j

qij∗ uˆ jk dj −

j

% uij∗ qˆ ik dj  αk

(11)

After the application of the collocation technique to all boundary nodes, the previous ˆ and Q ˆ which depend equation can be written in terms of four matrices, H, G, U only on the geometry of the problem. ˆ − GQ)α ˆ Hu − Gq = (HU

(12)

In (12) the vector α is unknown but it can be expressed as a function of the nonhomogeneous term using (6), yielding ˆ − GQ)F ˆ −1 b Hu − Gq = (HU

(13)

In this chapter, the implementation of DRM-MD codes to solve two kinds of equations is discussed. The first one is the Poisson equation with a constant source term ∇ 2 u(x) = k (14) The resulting system of equations in every subdomain comes straightforwardly from (13) as ˆ − GQ)F ˆ −1 k Hu − Gq = (HU

(15)

ˆ − GQ)F ˆ −1 yields Calling S the matrix (HU Hu − Gq = Sk

(16)

The second one is the advection–diffusion equation, + ku D∇ 2 u = V · ∇u

(17)

where D is the coefficient of dispersion, V is the vector of flow velocity and k is the reaction rate. Here, D and k will be considered constant in the whole domain for the sake of simplicity. The system of equations resulting from the DRM formulation, in this case becomes S ∂u ∂u ∂u Hu − Gq = Vx + Vy + Vz + ku (18) D ∂x ∂y ∂z whereV x ,V y andV z are diagonal matrices containing the flow velocity components.

192 Domain Decomposition Techniques for Boundary Elements Replacing the partial derivatives by ∂u ∂F −1 = F u ∂xi ∂xi then Hu − Gq =

(19)

S ∂F ∂F ∂F V x F−1 u + Vy F−1 u + V z F−1 u + ku D ∂x ∂y ∂z

(20)

Denoting the matrix (S/D)[V x (∂F/∂x)F−1 + V y (∂F/∂y)F−1 + V z (∂F/∂z)F−1 ] as T and reordering (20) produces S H − T − k u − Gq = 0 (21) D

3 The dual reciprocity method multidomain approach The domain discretization in the BEM is commonly used when there are only a few parts of the domain that have different properties. In that case the method of domain decomposition is often used, in which the original domain is divided into subregions, and on each of them the full integral representation formula is applied. A case of a domain, which is divided into four subdomains, is shown in Fig. 1. Though in Fig. 1 a 2D domain is considered for reasons of simplicity, the conclusions can be extended to 3D cases as well. As the nodes at the interfaces between two subdomains introduce more unknowns than equations, every system has more unknowns than equations. However, when the matching conditions for the two subdomains are applied, the system of equations becomes closed.

C 2

3 E q 12

F q 12

q E23 F

B

q E43

E

q E14

F q21

D 4

1 A

Figure 1: Subdivision of a domain into subdomains.


193

Matching conditions for a potential problem establish that at every node at the interface: 1. the value of the potential is the same for both subdomains u1 (xk ) = u2 (xi )

(22)

2. the physical flux is the same for both subdomains 1 (uk , qk ) = −2 (ui , qi )

(23)

where the form of the function depends on the physical problem under consideration. While the BEM matrices that arise in the single domain formulation are fully populated, the subregion formulation leads to block-banded matrix systems with one block for each subregion and overlaps between blocks when subregions have a common interface. Equation (24) represents the structure of the system of equations in matrix form that corresponds to domain subdivision shown in Fig. 1.



A1 0  AMD x =  0 0

Ai12 Ai21 0 0

0 A2 0 0

Ai14 0 0 Ai41

0 Ai23 Ai32 0

0 0 A3 0

0 0 Ai34 Ai43

     xi1    x  12          0   x2  b1           x i  0 b  14 2 =  i b3   x23  0            b   4 x3  A4     i   x       34  x4

(24)

Aj represents the influence coefficients obtained by integration over the external boundary that bounds the subdomain j and xj represents the unknown potentials and derivatives at the nodes on this part of the boundary. For example, for the subdomain 1 the external part of the boundary is given by the curve from A to B. Aikl represents the influence coefficients obtained by integration over the interface i represent the unknown potentials and derivatives of the subdomains k and l and xkl at the nodes on the interface. When considering nodes on the interface several different situations may occur, of which only the two most characteristic will be explained in this text. The first one will be analysed using the node F on the interface between subdomains 1 and 2. Node F is a discontinuous node. In this node there are four unknowns, two potentials and two normal derivatives. Two equations can be written collocating from the F node, one for the subdomain 1 and one for the subdomain 2. Using eqns (22) and (23) the contribution of this node towards a closed system of equations is achieved. The situation with node E, which is a continuous node, is more complicated as this node is shared between four subdomains. In each subdomain there will

194 Domain Decomposition Techniques for Boundary Elements be three unknowns, two derivatives and one potential, which would overall make twelve unknowns. However, as the potential is unique in this node, using eqn (22) three of the unknowns are eliminated reducing the number of unknowns to nine. Further, by using eqn (23) the number of unknowns will be reduced to five, i.e. four normal derivatives, e.g. the ones shown in Fig. 1, and the potential. With four equations that can be written collocating from the node into each of the subdomains, the contribution of this node towards the closed system of equations has not been yet achieved and unless the medium is homogeneous and the line/s B–E–D or/and A–E–C are smooth in the node E, this node would need to be converted to a discontinuous node in order that a closed system of equations is achieved. Node E when discontinuous will have four freedom nodes instead, moved for a small distance from the location of node E on the lines A–E–C, in the direction of the A and C nodes, and on the line B–E–D, in the direction of the B and D nodes. In each of the new freedom nodes a situation equivalent to the situation in node F will appear. When the medium is homogeneous and the intersection lines are smooth in E, it can be shown that q12 = q43 and q23 = q14 , reducing the number of unknowns to three, making the final system of the equations overdetermined. Therefore, when continuous, node E may have three degrees of freedom, contributing towards an overdetermined system, or four degrees of freedom contributing towards a closed system of algebraic equations, depending on whether both or just one of the B–E– D and A–E–C lines are smooth in the node E. Node E can be continuous if all of the subdomains 1 to 4 have the same properties, or two by two of the neighbouring domains have the same properties, i.e. 1–2 and 3–4, or, 1–4 and 2–3. In any other combination, node E must be discontinuous, which produces eight degrees of freedom, contributing towards a closed system of algebraic equations. A similar analysis could be applied to nodes that are shared between three or more than four subdomains in 2D or 3D. Next, let us define the degree of overdetermination of a continuous node, Ov , as Ov = Neq − Nun

(25)

where Neq is the number of equations introduced by the node and Nun is the number of unknowns at the node. Provided all the subdomains around the node have the same properties, the value of Neq and Nun can be calculated as: Neq = Nsub + Nco

(26)

Nun = 1 + Nin

(27)

where Nsub is the number of subdomains around the node, Nco is the number of independent conditions of collinearity (in 2D problems) or coplanarity (in 3D problems) feasible to be formulated about the interfaces joining the node and Nin is the number of interfaces joining the node.


195

4 Schematic view of a DRM code for Poisson’s problems A typical computer code that uses a DRM algorithm to solve eqn (14) follows the sequence of Fig. 2. This sequence is common to any code independently of particular features such as the type of element or the DRM approximation function. Since a thorough description of implementation of DRM codes is given by Partridge et al. [19], only the general aspect shall be described here with emphasis on those aspects that are important to DRM-MD implementation. In the ‘Input data’ process, information on the geometry of the problem and the mesh is introduced together with the boundary conditions and values of the constant k. The information on the geometry of the problem and the mesh comprises geometrical nodes coordinates, connectivities and some further information to identify whether the node is an internal DRM node or any particular feature such as being in a corner of the domain. The ‘Calculate H and G’ process performs the integrals of the left-hand side of eqn (2). The process will produce only one pair of matrices H and G. To arrive at MAIN

Input data

Calculate H and G

Calculate S

Assemble system of equations

Solve system of equations

Distribute results

Output results

END

Figure 2: Flow chart of a DRM code for Poisson’s problems.

196 Domain Decomposition Techniques for Boundary Elements this point, the code requires some auxiliary processes to order the information, for instance: each row of the matrix corresponds to a source point and every column to a collocation point, which entails that nodes have been numbered with some criteria, which depends on the programmer preferences. Another example: when dealing with continuous elements there will be nodes in the interface between adjacent elements. In this case, in order to assemble the matrices H and G the code must be able to recognize the elements around every node. The way in which the information is ordered depends on the particular features of the code and the preferences of the developer who wrote it. ˆ − GQ)F ˆ −1 . There is only one S The ‘Calculate S’ process computes S = (HU matrix for the problem and for large cases, F grows ill-conditioned and F−1 cannot be calculated, which is one of the main drawbacks of the DRM. The ‘Assemble system of equations’ imposes the boundary conditions on eqn (16) to reduce it to the form Ax = y

(28)

Unlike AMD in eqn (24), the resulting matrix of coefficients A is fully populated. The ‘Solve system of equations’ solves eqn (28) using a suitable available solver and gives back a vector of results, in which the formerly unknown potential and normal derivatives are contained. The results are taken from the vector of results and distributed into the vectors u and q by ‘Distribute results’. Finally, results are presented as output in a given format.

5 General aspects of DRM-MD implementation for Poisson’s problems Figure 3 displays the sequence that is proposed to implement DRM-MD codes for Poisson’s problems. This scheme is general for 2D and 3D codes no matter what kind of elements are used. Some of the processes of Figs 2 and 3 have features that are inherent to every procedure but are still conceptually the same. For instance, the ‘Input data’ process is different in every case since DRM-MD uses subdomains, which is not the case in the DRM procedure. This is especially true for the last three processes of the main program, and to some extent it is also so for the ‘Calculate H and G’ and ‘Calculate S’ processes. Other processes, such as ‘Set boundary conditions and NCOL’, which do not have an equivalent in the DRM, are conceptually different. It shall be seen later that the scheme proposed in Fig. 3 (and its corresponding code) can be used for a wide range of equations with minor changes. Let us now make some remarks about the first four processes. 5.1 Input data It is conceptually the same as it is homologous to DRM. The difference is that the connectivities correspond to domain elements.


197

MAIN Set geometrical information Input data Order mesh information Set geometrical information Calculate H and G Set boundary conditions and NCOL


Calculate S Return


Distribute results

Output results

END

Figure 3: Proposed scheme for DRM-MD codes for Poisson’s problems. 5.2 Set geometrical information This process comprises three subprocesses in which all information that depends only on the mesh and the geometry of the domain is processed. It is independent of the boundary conditions and the governing equation considered. 5.2.1 Order mesh information Subsequent processes make use of a set of arrays containing all information that can be obtained from the geometrical data. For instance, given a subdomain which has subdomains around it, or given the normal vector components at a given node, etc., such sets of arrays are created here. 5.2.2 Calculate H and G It is conceptually similar to its homologous DRM, but, as the full integral representation formula is applied on each subdomain, there will be as many small H and G matrices as there are subdomains.

198 Domain Decomposition Techniques for Boundary Elements 5.2.3 Calculate S This is similar to its homologous DRM. There will be as many small S matrices as subdomains. Now, the problem of having an ill-conditioned matrix F has been eliminated no matter how large the problem under analysis is. 5.3 Set boundary conditions and NCOL The assembly process requires specifying the boundary conditions in a format that is different from the format in which the boundary conditions are input. That task is performed here. In eqn (24) some coefficients multiply unknown values of u and q, and some others multiply values that have been specified as boundary conditions. In this process, those coefficients multiplying known values are identified, and a column in the matrix of coefficients AMD of eqn (24) is assigned to every coefficient multiplying an unknown. This information, i.e. the number of the column that every coefficient has been assigned in the matrix AMD , is stored in an array called NCOL. Note that this process depends on the geometry, mesh and boundary conditions independently of the governing equation. 5.4 Assemble system of equations This is the only process that is linked to the governing equation. The key concept in the assembly is that every node of freedom generates coefficients that multiply the variables involved. These coefficients depend on the governing equation. The assembly is done by going through the mesh subdomain by subdomain, and at every subdomain going through the nodes of freedom. At every node the values of the coefficients are calculated and assigned to the matrix AMD or incorporated into the right-hand side according to the value of the array NCOL.

6 A 3D DRM-MD implementation for Poisson’s problems using discontinuous elements The codes that are described in this and subsequent sections are intended to be used in solving general problems of arbitrary geometry. Consequently, the domain is subdivided using unstructured tetrahedrons with quadratic shape functions to represent the geometry of every subdomain, which is very flexible when used to discretize complicated geometries. Ten nodes that are called geometrical nodes describe every tetrahedron. Every tetrahedron is made up of four triangular faces, each of which has a mesh point at the corner and at the mid-side position (Fig. 4). At this point it is convenient to introduce a few conventions for a better description. Numeration can be described as belonging to a given side of a given tetrahedron. Let us call this description as an element point of view (EPV). The subdomain


199

2 9

5 6

8

4

1 10

7 3

Figure 4: Tetrahedral subdomain. The geometrical nodes are numbered from a SPV. 3

5

6

1 4

2

Figure 5: Triangular boundary element. The geometrical nodes are numbered from an EPV.

point of view (SPV) is a description that sees the node as belonging to a given subdomain of the mesh, while the mesh point of view (MPV) considers the node as a part of the mesh. For instance, a given node can be at the same time the node 6 (of a given tetrahedron side) from an EPV, node 8 (of a given subdomain) from a SPV and node 2,648 (of the mesh) from a MPV. The elements used in the code described in this section are quadratic, discontinuous, triangular boundary elements. As the triangular elements are quadratic, each of them is defined by 6 geometrical nodes (Fig. 5). The relative position between nodes from SPV and EPV, which is displayed in Figs 4 and 5, is always the same. Table 1 shows the correspondence between the numeration of nodes from the EPV and the SPV. The BEM implementation for triangular elements used in the codes presented in this chapter follows the treatment proposed by Do Rêgo Silva [20]. As he described the implementation in detail, only general aspects will be mentioned here. Over the discontinuous triangular elements, 6 nodes of freedom, on which the eqn (11) is applied, are distributed according to Fig. 6. At every node of freedom there are two variables, the values of which can be unknown or can have been specified among the boundary conditions: the potential and the normal derivative. Inside every tetrahedron, internal DRM nodes can be added. For the sake of simplicity,

200 Domain Decomposition Techniques for Boundary Elements Table 1: Correspondence between the numeration of a node from the EPV and the SPV. Node from an EPV 1 2 3 4 5 6

Node from an SPV Side I

Side II

Side III

Side IV

1 3 2 7 6 5

4 2 3 9 6 10

4 3 1 10 7 8

4 1 2 8 5 9

3 5

6 1

4

2

Figure 6: Nodes of freedom on a discontinuous triangular boundary element. The nodes are numbered from an EPV. the code that is described here has no internal DRM nodes; therefore the total number of freedom nodes is equal to 24 (SPV). The flow chart of a DRM-MD code to solve problems that are governed by the Poisson’s equation (14) has been given in Fig. 3. Let us now study some features of every subroutine for this particular case. 6.1 Input data The input data file needs to be generated by a preprocessor due to the large amount of information involved even in the simplest 3D meshes. Here GID™ was successfully used with minor customizations. The input file has the following information: total number of subdomains, total number of geometrical nodes, length-scale factor, list of geometrical node coordinates, list of connectivities, list of geometrical nodes that are in the boundary, boundary conditions and list of values of the source term, k. The geometrical node coordinates are given and read from a MPV. The connectivities of each subdomain is a sequence of ten numbers that identify the ten geometrical nodes defining the subdomain. The list of geometrical nodes that are in


201

the boundary is given and read from a MPV. The boundary conditions are defined on the sides of subdomains. They are given as a list of four numbers that identify: subdomain (MPV), side of the subdomain (SPV), type of boundary condition (whether it is a potential or a normal derivative) and value of the boundary condition. Finally, k values are assigned to subdomains. 6.2 Set geometrical information 6.2.1 Order mesh information Table 2 shows the different auxiliary arrays that are set up in this process. Although the code has a subroutine for this task, some of the arrays of Table 2 have been already set up during the input process because they were needed to read some parts of the input file. The arrays are listed in the order in which the code sets them up. The arrays of Table 2 depend on the mesh; setting them up requires loops to go through the mesh extracting the information needed in every particular case. Apart from these, there are other auxiliary arrays that contain information from the SPV and the EPV. For instance, there is the array elv ij , which gives the geometrical node number from a SPV, of the node j of the side i of any tetrahedron. As they always have the same values independent of the mesh, they will not be explicitly mentioned in this work. During this process, the freedom nodes are generated and labelled. Table 2: Auxiliary arrays that depend only on the mesh and geometry of the problem. Array

Description

nshc1ij nshc2ij xgn1ij ygn1ij zgn1ij

Number of the cell sharing the side j of the cell Side (SPV) of the cell nshc1ij that shares the side j of the cell i

xnfij ynfij znfij nei

Coordinates (MPV) of the freedom node j (SPV) of the cell i

Coordinates (MPV) of the geometrical node j (SPV) of the cell i

Array indicating what rows of theAMD matrix correspond to the equations of the subdomain i: they occupy rows ne(I) + 1 to ne(I) + 24

nfij

Number of the freedom node (MPV) of the freedom node j (SPV) of the cell i

normverXij normverYij normverZij

Components of the outward (SPV) normal vector at the freedom node j (SPV) of the cell i

202 Domain Decomposition Techniques for Boundary Elements 6.2.2 Calculate H and G In this process, the full integral representation formula is applied on each subdomain producing H24 × 24 and G24 × 24 matrices. The triangular elements used here belong to a family of quadrilateral and triangular elements proposed by Do Rêgo Silva [20]. In order to have the capability of using both quadrilateral and triangular elements, the code applies a transformation of coordinates that maps the triangular elements into quadrilateral ones. When the integrals are regular, the standard Gaussian quadrature is used with a fixed 40 × 40 Gauss point number. The singular integrals are approached using polar coordinates, Gaussian quadrature and the self-adaptative coordinate transformation proposed by Telles [21]. Telles’ transformation is also used for regular integrals because it prevents the failure of the code when the source point is too near the collocation point. The Gauss point number for singular integrals was 32 for the radius and 40 for the angle. The diagonal elements of H are calculated through the rigid body motion hypothesis. In the code, H and G matrices are stored in two hijk and gijk arrays, where the subindex i stands for subdomain number. 6.2.3 Calculate S A S24 × 24 is obtained for every subdomain. As in the case of the DRM, the DRM-MD is highly dependant on the approximation function appearing in eqn (4). Natalini and Popov [12] studied the performance of the method in 3D codes using 10 different radial basis functions (RBFs). The preferred choice of the authors is the Duchon’s augmented thin plate splines (ATPS). However, any other approximation function can be used without modifying a single sentence of the process other than this one. As in the case of the BEM matrices, in the code, the S matrices are stored in a sijk array. 6.3 Set boundary conditions and NCOL In the same way as the ‘Order mesh information’ process, this process sets up a group of auxiliary arrays (see Table 3) by going throughout the mesh extracting the information needed in every particular case. Note that there is a significant conceptual difference between both processes: the former process analyses only geometrical information while this one requires, in addition, information on the boundary conditions. 6.4 Assemble system of equations Here the AMD matrix and the right-hand side vector of eqn (24) are assembled. As both the matrix and the vector are made up of the coefficients arising when applying eqn (13) to every subdomain, the process does an outer loop on subdomains and two inner loops on collocation and source nodes (SPV), respectively. For each pair of source and collocation nodes there are a coefficient multiplying the


203

Table 3: Auxiliary arrays that depend on the mesh, geometry and boundary conditions of the problem. Array

Description

kfnpi

Position of the freedom node i (MPV). When it is equal to 1, the node is on the boundary; when it is equal to 0, the node is in between two cells; when it is equal to 2, the node is an internal (SPV) DRM node Concentration at the freedom node j (SPV) of the cell i. Only the known values (if any) are initially assigned. Normal derivative at the freedom node j (SPV) of the cell i. Only the known values (if any) are initially assigned. Column number of the AMD matrix assigned to the coefficients of the cell i. The subindex j indicates the column number (SPV) of the A matrix. The subindex k indicates that, in the system (24), the coefficient multiplies the potential at the node j (SPV) if k = 1, or the normal derivative at the node j if k = 2. If the coefficient multiplies a value that is specified as a boundary condition, ncolijk is equal to −1. In order to apply the matching condition (23), a precedence order must be defined between adjacent subdomains. The criterion used here is that the subdomain labelled with the lower number is the master and the other one the slave. chsignij stores this information in a format that is convenient for the assembly: it indicates whether the subdomain i is master in regard to the subdomain adjacent to its node j (SPV). The matrix of coefficient, AMD , is stored in compressed sparse row format. iai is an integer array containing the pointers to the beginning of each row in the vector containing the non-zero coefficients of AMD .

cij derij ncolijk

chsignij

iai

potential at the collocation node and another one multiplying the normal derivative at the collocation node. The code calculates the value of these coefficients and asks whether they have been specified as boundary condition (this information is stored in NCOL). Then, according to the answer, the value of the coefficients are stored in some position of the AMD matrix or incorporated into the right-hand side vector. In the former case, the position is given by the NE and NCOL arrays. Figure 7 shows the flow chart of the process. Once this point is reached, there is not too much to say about the remaining part of the code; the system of equations must be solved and the result distributed and given as output. The performance of iterative solvers in 2D DRM-MD codes has been exhaustively tried by Portapila and Power [15, 16]. Based on their work, different solvers were tried with the discontinuous element code (DEC) that has been presented, and those that performed the best were a direct solver using a Gaussian elimination algorithm and an iterative solver using a conjugate gradientnormal residual (CGNR) algorithm [22].

204 Domain Decomposition Techniques for Boundary Elements Assemble system of equations

2

I=1, NC Loop on subdomains

3

J=1, 24 Loop on collocation points

1 Incorporate s(I,K,J)*k(I) into the right-hand-side vector

4

K=1, 24 Loop on source points

4 aux(1) = h(I,K,J) kn(1) = c(I,J) kn(2) = der(I,J)

3 2

chsgn(I,J) ≠ 0

Yes

aux(2) = g(I,K,J)

Return

No aux(2) = -g(I,K,J)

L=1, 2 Loop on variables

ncol(I,J,L) ≠ -1 Is the value of kn(L) unknown?

Yes Assign the value of aux(L) to a coefficient of AMD according to NE and NCOL

No Incorporate aux(L)* kn(L) into the right-hand-side vector according to NE

1

Figure 7: Flow chart of the ‘assembling system of equations’ process of the code for Poisson’s problems using discontinuous elements.

7 A 3D DRM-MD implementation for Poisson’s problems using continuous elements In this section, a code using tetrahedrons, whose sides are quadratic, continuous, triangular boundary elements, is described. In this case the freedom nodes are coincident with the geometrical nodes, as a result of which each tetrahedron


205

produces ten equations, provided there are no internal DRM nodes. This is a noticeable advantage with regard to the discontinuous elements code. Among the relative disadvantages are that the resulting system of equations is overdetermined, as was seen in Section 3. Also continuous elements cannot be used in between subdomains with different properties. In this case discontinuous elements must be used. Another important factor must be considered when using continuous elements. Equations (25)–(27) give the degree of overdetermination, Ov , in a continuous node. Ov must always be greater than or equal to zero. According to eqns (25)–(27) this is possible only when at least one independent condition of coplanarity (in 3D problems) of the interfaces joining the node exists. When the mesh is structured, that condition can be fulfilled, but in a general case the mesh is unstructured, therefore, the condition is not fulfilled, unless the node is in the boundary. Consequently, the eqn (16) was manipulated in such a way that the variables at nodes were potential and partial derivatives instead of potential and normal derivatives. The change of variables has been done by applying eqn (16) to a tetrahedral subdomain and expressing it in index notation. hij uj − gik qk = sij kj

1 ≤ i ≤ 10 with 1 ≤ j ≤ 10 1 ≤ k ≤ 24

(29)

Note that the subindex k goes from 1 to 24 because there are 24 normal derivatives in a single tetrahedron. At the same time, every normal derivative is the scalar product of the gradient vector and the normal vector: ∂u ∂u ∂u qk = (30) nx + ny + nz ∂x k k ∂y k k ∂z k k By replacing (30) into (29) ∂u ∂u ∂u hij uj − gik nxk − gik nyk − gik nzk = sij kj ∂x k ∂y k ∂z k

(31)

In the previous equation, there are twenty-four partial derivatives with respect to x, y and z, respectively. However, some of them are repeated, since there are only ten freedom nodes where the partial derivatives are computed. Thus, the products between the elements of the G matrix and the components of the normal vectors can be assembled in three 10 × 10 matrices that we will denote as Gx , Gy and Gz . The resulting system of equations is now hij uj − gijx

∂uj ∂uj y ∂uj − gij − gijz = sij kj ∂x ∂y ∂z

(32)

A further condition arises in order to apply (32): there must be at least four subdomains around every node to have a closed system of equations. This condition is not always fulfilled in apex nodes that are located in the boundary of the domain (MPV), where, instead, at least one condition of coplanarity can always be found. In those nodes the problem is defined in terms of potential and normal derivatives.

206 Domain Decomposition Techniques for Boundary Elements The continuous element code (CEC) is conceptually identical to the one that has been described in the previous section. The structure is the same and the same set of auxiliary arrays is used, though some of them are set up in a different order than in the previous case. The main differences between the codes are that in CEC: 1. The boundary conditions are specified as potential and partial derivatives, except for apex nodes, for which they are specified as potential and normal derivatives. 2. The size of the H, G and S matrices are 10 × 10, 10 × 24 and 10 × 10, respectively. 3. ‘Setting NCOL’ and the ‘Assembling system of equations’ processes are a bit more complex because every collocation node must be processed in different ways according to whether they are inside the domain (from a MPV) or in the boundary; and if they are in the boundary, a distinction must be made between apex and vertex nodes. 4. The overdetermined system of the CEC is solved here in a lest-square sense by means of a LSQR algorithm [23], which is algebraically equivalent to applying the symmetric conjugate gradient method to the normalized equations.

8 A 3D DRM-MD implementation for advection–diffusion problems The codes that have been described in the previous sections can be easily adapted to solve steady-state advection–diffusion problems, whose governing equation is (21). The chief modifications are that the definition of the coefficient multiplying potential (aux(1) in Fig. 8) and the matching condition for derivatives are changed. For the DEC, these mean changing two sentences and deleting one in the assembling subroutine, as can be seen in Fig. 8. Figure 9 shows the flow chart of the codes; it is common to continuous and discontinuous element codes and it uses the formulation (19) for first order partial derivatives, which leads to the resulting system of equations (21). Most of the structure is similar to the codes that solve the Poisson’s equation. Apart from the assembly process, the following processes need to be slightly modified: • • •

Input data: it has to incorporate the list of coefficients of dispersion D, and porosities n. Set geometrical information: it has one subprocess more, in which the partial derivatives of the f function are calculated. Distribute results: because the matching condition for derivatives is different from that in the Poisson case.

In addition, as it can be seen in Fig. 9, two new processes have to be incorporated in the main program: ‘Read velocities’ and ‘Calculate T’. The former process reads the velocities from a file and the latter calculates T = (S/D)[V x (∂F/∂x)F−1 + V y (∂F/∂y)F−1 + V z (∂F/∂z)F−1 ]. As in the case of the


207


1 2


2 3


4


3 4 Return

aux(1) = h(I,K,J)- t(I,K,J)-k(I)* s(I,K,J)/D(I) kn(1) = c(I,J) kn(2) = der(I,J)

chsgn(I,J) ≠ 0

Yes

aux(2) = D(infront)* n(infront) g(I,K,J) D(I)* n(I)

No aux(2) = -g(I,K,J)





1

Figure 8: Flow chart of the ‘Assembling system of equations’process of the code for steady-state advection–diffusion equation problems using discontinuous elements.

208 Domain Decomposition Techniques for Boundary Elements MAIN

Input data

Set geometrical information

Set boundary conditions and NCOL


Order mesh information

Calculate H and G

Calculate S Read velocities

Calculate T

Calculate ∂F F −1 , ∂F F −1 ∂x

∂y

and ∂F F −1 ∂z

Assemble system of equations Return Solve system of equations

Distribute results

Output results

END

Figure 9: Flow chart of a DRM-MD code to solve the steady-state advection– diffusion equation. Dashed line indicates processes that have been modified and bold line indicates new processes with regard to the Poisson’s code. BEM matrices, there is one T matrix for every subdomain, the values of which are stored in an array tijk .

9 A convenient arrangement When working with the codes that have been previously described, it is convenient to split them out in to two codes. Given a problem, the first code calculates the BEM and DRM matrices and the auxiliary arrays in order to save them in a file.


209

The second code reads the file produced by the first code and produces the solution of the problem. Let us suppose that there is a problem under study and it is intended to obtain the solution for a variety of cases with different parameters and values of boundary conditions. Provided that mesh, scale factor and kind of boundary conditions are not modified, this arrangement requires running the first code just once. Considering that the first code is several orders of magnitude more time-consuming than the second one, this scheme is highly convenient. Figure 10 illustrates this arrangement

PREPROCESS

Input data


Set boundary conditions and NCOL

MAIN

Input data

Read matrices and auxiliary arrays

Set boundary conditions

Write matrices and auxiliary arrays

Read velocities

END

Calculate T



Distribute results

Output results

END

Figure 10: Flow chart of the DRM-MD code with pre-processing to solve steadystate advection–diffusion. Processes in bold line are new with regard to the flow chart of Fig. 9.

210 Domain Decomposition Techniques for Boundary Elements for the steady-state advection–diffusion code. All the subroutines, but the three which are in bold line, are the same as those of Fig. 8. It is possible to use the PREPROCESS subroutine of the advection–diffusion case in codes to solve the Laplace and Poisson’s equations. However, it is preferable to use a particular PREPROCESS subroutine for each kind of problem since some problems require the calculation of matrices which are not used in other ones, and in large problems this means a noticeable difference in CPU time.

10 Examples Numerical examples on the 3D Poisson and the steady-state advection–diffusion equations tested under a variety of schemes, with different RBFs, continuity of elements, scaling and number of internal nodes, are presented in this section. 10.1 The interpolation functions tested in this section 1. 1 + R: Until the 90s, this was the most popular choice for the function f ( y, zk ) in the DRM approach [19]. 2. R + P0 : It has been demonstrated that f = R is conditionally positive definite (CPD) of order one [24, 25]. To assure a safe reconstruction a polynomial of degree zero must be added: b( y) ∼ =

J+I

αk R( y, zk ) + a

(33)

k=1

and αk coefficients must satisfy the condition J+I

αk = 0

(34)

k=1

3. ATPS(R + P1 ): Golberg [26] proposed the use of ATPS, which in R3 is represented with b( y) =

J+I

αk R( y, zk ) + ayx + byy + cyz + d

(35)

k=1

where the αk coefficients must satisfy the condition: N

k=1

αk =

N

k=1

αk yxk =

N

k=1

αk yyk =

N

αk yzk = 0

(36)

k=1

4. R2 log R + P1 : This is the ATPS in R2 . In this case the implementation is given by b( y) ∼ =

J+I

k=1

αk R2 ( y, zk ) log (R( y, zk )) + ayx + byy + cyz + d

(37)


211

where the αk coefficient must satisfy the condition N

αk =

k=1

N

αk yxk =

k=1

N

αk yyk =

k=1

N

αk yzk = 0

(38)

k=1

5. R2 log R: It can be found in any survey on RBFs that this function is a conditionally positive definite of order two, i.e. it needs a polynomial of order one to assure a safe reconstruction. This function is tested in order to check whether the solution is less accurate without the polynomial. 6. Wendland-C0: Wendland [27] proposed several positive definite piecewise polynomial CS-RBFs that have minimal degree for prescribed smoothness and space dimension. From the point of view of DRM-MD, what is of interest is that CS-RBFs are unconditionally definite positive. The fact that these functions are compactly supported is of secondary importance because the multidomain approach always produces sparse system of equations. In all the examples tried in this work the support has been set in such a way that βR ≤ 1. Let us call Wendland-C0 the function defined by $

R( y, zk ) f ( y, zk ) = 1 − β

%2 when 0 ≤

R ≤1 β

(39)

where β is a scaling parameter to control the support. 7. Wendland-C2: let us call Wendland-C2 the function defined by $

R( y, zk ) f ( y, zk ) = 1 − β

%4 ) $

% * R( y, zk ) 4 +1 β

when 0 ≤

R ≤1 β

(40)

8. Wendland-C4: let us call Wendland-C4 the function defined by  $ %6  $ %2 k) k) k) R( y, z R( y, z R( y, z 35 f ( y, zk ) = 1 − + 18 + 3 β β β when 0 ≤

R ≤1 β

(41)

9. Wendland-C6: Let us call Wendland-C6 the function defined by %8  $ %3 $ k R( y, zk ) R( y, z )  k f ( y, z ) = 1 − 32 β β  %2 $ R R( y, zk ) R( y, zk ) + 1 when 0 ≤ ≤ 1 (42) + 25 +8 β β β

212 Domain Decomposition Techniques for Boundary Elements 10. Bhumann: Bhumann [28] proposed a new family of CS-RBFs. Here the following one has been tested 1 6

R =0 (43) β  $ %4 $ % $ %4 J+I k) k) k)

R( y, z 7 R( y, z R( y, z f ( y, zk ) = − αk 2 ln β β 2 β k=1  $ $ %3 %2 16 R( y, zk ) R( y, zk ) 1 R + −2 + when 0 < ≤ 1 3 β β 6 β f ( y, zk ) =

when

(44) 10.2 A 1D Poisson problem with constant source term Equation (14) with k = 40 was solved with the DEC using all the RBFs introduced in Section 10.1. The 1D analytical solution of (14) is k u (x) = x 2 + 2

$

5 % Ul − U0 − kL 2 2 x + U0 L

(45)

where U0 = u(x = 0) and U1 = u(x = L) are the boundary conditions. In order to produce equivalent 1D results from the 3D codes, prismatic domains of length L = 1 in the x direction and width W = 0.2 in the y and z directions were used (Fig. 11), and the following boundary conditions were applied: u(0, y) = U0 = 10;

u(L, y) = U1 = 4

(46)

W=0.2 W=0.2

L=1.0

Figure 11: Geometry of the domain and the boundary conditions used in the numerical example.


and

∂u ∂u ∂u ∂u = = = =0 ∂n y=W /2 ∂n y=−W /2 ∂n z=W /2 ∂n z=−W /2

213

(47)

Figure 12 shows the analytical solution for this example, while Fig. 13 shows the discretization of the considered domain. Figures 14–18 present error distributions for this case using a mesh of 173 subdomains/tetrahedrons. The error distributions are grouped in different figures according to the order of magnitude of the error O(Error). There are several results for every CS-RBF because they have been tried with different supports. The support was set locally in every subdomain as: β = max R · s

(48)

12 POTENTIALS

10 8 6

k=40

4 2 0 0

0.2

0.4

0.6

0.8

1x

Figure 12: Analytical solution of the Poisson case.

Figure 13: Mesh with 173 subdomains used in the examples.

214 Domain Decomposition Techniques for Boundary Elements Error [%]

4 2 0 0

0.2

0.4

0.6

1 X

0.8

-2 -4 R^2 log R W-C0 (1.05) W-C2 (10) W-C4 (1.05) W-C6 (1.05) B (1.05)

-6 -8 -10 -12 -14

Figure 14: Error distribution along the x-axis using the DEC for the Poisson case and different RBFs (cases in which O(Error) = 1). Error [%]

0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1 X

-0.1 -0.2 -0.3 1+R

-0.4 -0.5

W-C0 (10) W-C2 (10000)

-0.6 -0.7

Figure 15: Error distribution along the x-axis using the DEC for the Poisson case and different RBFs (cases in which O(Error) = 10−1 ).

215

Computational Implementation for 3D Problems Error [%]

0.08

0.06

W-C0 (100) W-C2 (100)

0.04

W-C4 (10) W-C6 (10)

0.02

B (10)

0 0

0.2

0.4

0.6

1 X

0.8

-0.02

-0.04

-0.06

-0.08

Figure 16: Error distribution along the x-axis using the DEC for the Poisson case and different RBFs (cases in which O(Error) = 10−2 ). Error [%]

0.01

0.008 R + P0 ATPS R^2 log R + P1 W-C0 (1000) W-C4 (100) W-C6 (100) W-C6 (1000)

0.006

0.004

0.002 X

0 0

0.2

0.4

0.6

0.8

1

-0.002

-0.004

-0.006



Error [%]

0.0003 0.0002 0.0001 X

0 0

0.2

0.4

0.6

0.8

1

-0.0001 -0.0002

W-C0 (1000000) W-C0 (10000000)

-0.0003

W-C2 (1000) W-C4 (1000)

-0.0004

B (100)

-0.0005


where max R is the maximum R(y, zk ) of the subdomain and s is a scale factor. Every CS-RBF was tested with values of s equal to 1.05, 10, 102 , 103 , . . . , 108 . For every CS-RBF, the code fails to converge to the correct solution when s is below or above a certain level. For instance, the code fails when using the Wendland-C2 function and values for s below 10 or above 104 . The results when the code does not converge are not displayed. In the legends of Figs 14–27, the number appearing in brackets to the right of the label identifying the CS-RBF used is the value of s. Figure 18 shows that the most accurate results are given by CS-RBFs. However, the accuracy is strongly influenced by the support β, which depends on s. It is well known that when using CS-RBFs the accuracy increases with the size of the support, which is confirmed by the results in the figures. In conventional DRM, enlarging the support means including more DRM nodes in the approximation, which implies worsening the condition number and the sparseness of the matrix F. This is not the case in the multidomain approach, since the number of nodes involved is the same regardless of the size of the support, provided β ≥ max R. And yet the results show that the accuracy increases with the size of the support up to a point at which the code starts losing accuracy progressively until it fails. This creates a serious drawback to the use of CS-RBFs. The problem is analogous to using multiquadrics in 2D problems, where it is one of the most accurate approximation functions, but its performance depends on a parameter known as the shape parameter, for which there is no standard procedure to obtain the optimum value. As for the other


217

functions, all of them being conditionally positive definite, it can be seen that the results agreed with what the theory predicts: a proper implementation requires a polynomial term. Once this is done, the results of ATPS, R + P0 and R2 log R + P1 gave almost identical results with good accuracy (Fig. 17). The reason could be that the polynomial part of the RBF becomes dominant and fits well with the analytical solution. Next, the approximation functions will be tested for the advection–diffusion problem. 10.3 A 1D steady state advection–diffusion problem with non-uniform velocity field Equation (17) was solved with the DEC using all the RBFs introduced in Section 10.1. The boundary conditions, geometry of the domain and mesh were identical to the Poisson case. Figure 19 shows the analytical solution for the concentration for the different values of the reaction rate k. Figures 20–23 show the error distributions when k = 20. It can be seen that the errors are higher in the case of the advection–diffusion equation than in the case of the Poisson’s equation, and this is mainly due to the advective term. In the case of the advection–diffusion equation Wendland-C0 did not outperform ATPS, and although some other CS-RBFs did, their accuracy was of the same order of magnitude as the accuracy of the ATPS. The problem of defining the optimum value of s becomes more obvious, since the optimum values of s are not the same for the Poisson and advection–diffusion cases, indicating that they vary from case to case. 1 + R, R + P0 and R2 log R + P1 were unable to represent accurately the solution 12 CONCENTRATIONS

10

k=40 k=20 k=5 k=0

8 6 4 2 0 0

0.2

0.4

0.6

0.8

Figure 19: Analytical solution of the tested cases.

1x


Error [%]

50

X

R^2 log R

0 0

W-C0 (1000000)

0.2

0.4

0.6

0.8

1

W-C2 (1.05) W-C4 (1000)

-50

-100

-150

-200

Figure 20: Error distribution along the x-axis using the DEC for the advection– diffusion (k = 20) case and different RBFs (cases in which O(Error) = 102 ). 30

Error [%] 1+R R+Po R^2logR+P1 W-C0 (1000) W-C2 (100) W-C2 (1000) W-C4 (100) W-C6 (100) W-C6 (1000)

20

10

0 0

0.2

0.4

0.6

0.8

1 X

-10

-20

-30

Figure 21: Error distribution along the x-axis using the DEC for the advection– diffusion (k = 20) case and different RBFs (cases in which O(Error) = 10).


219

Error [%]

9 ATPS W-C0 (100000) W-C2 (10) W-C4 (1.05) W-C4 (10) W-C6 (1.05) W-C6 (10) B (1.05) B (10) B (100)

7

5

3

1

-1

0

0.2

0.4

0.6

0.8

1 X

-3

-5

Figure 22: Error distribution along the x-axis using the DEC for the advection– diffusion (k = 20) case and different RBFs (cases in which O(Error) = 1). Error [%]

2.5

2 W-C2 (10) W-C4 (10)

1.5

W-C6 (10)

1

B (10) B (100)

0.5

0 0

0.2

0.4

0.6

0.8

1 X

-0.5

-1

-1.5


220 Domain Decomposition Techniques for Boundary Elements of (17), mainly because they failed to approximate accurately the partial derivatives of the advective term, which for this case is the dominant term in (17). Figures 24–27 show the same but for k = 5, i.e. a case with a much smaller convective term. Apart from the fact that the error decreases by two orders of magnitude, it can be seen that the trends regarding the performance of the RBFs are the same as in the previous case, except that for k = 5, and R2 log R + P1 was almost as accurate as ATPS. Considering the two examples, of the globally supported RBFs only ATPS performed satisfactorily in all the cases. In terms of accuracy CS-RBFs outperformed ATPS in both cases. However, the performance of CS-RBFs depends on the size of the support, which introduces an additional variable in the problem. As a standard procedure to determine the optimum value of the size of the support is not yet available, and as it was shown that the optimal size of support for the CS-RBFs varies from case to case, ATPS appears to be the best choice of the tested functions for the 3D DRM-MD code at the moment, as it produced one of the most accurate and certainly the most consistent results and does not introduce any additional parameters. 10.4 The effect of the size of the problem Another important issue in these kinds of codes – the size of the problem – was analysed. A given problem can be converted into a similar one but with a different 30

Error [%]

20 10 R^2 log R

0 0

W-C0 (1000000)

0.2

0.4

0.6

0.8

1 X

-10

W-C2 (1.05) W-C4 (1000) W-C6 (1000)

-20 -30 -40 -50 -60 -70



221

Error [%]

4.5

3.5

1+R R+Po W-C0 (1000) W-C0 (100000) W-C2 (1000) W-C6 (100)

2.5

1.5

0.5

-0.5

0

0.2

0.4

0.6

1 X

0.8

-1.5

-2.5

Figure 25: Error distribution along the x-axis using the DEC for the advection– diffusion (k = 5) case and different RBFs (cases in which O(Error) = 1). 1.4

Error [%]

1.2 W-C4 (1.05)

1

W-C4 (100)

0.8

W-C6 (1.05) B (1.05)

0.6 0.4 0.2

X

0 0

0.2

0.4

0.6

0.8

1

-0.2 -0.4 -0.6

Figure 26: Error distribution along the x-axis using the DEC for the advection–diffusion (k = 5) case and different RBFs (cases in which O(Error) = 10−1 ).


Error [%]

0.1

0 0

0.2

0.4

0.6

0.8

1 X

-0.1 ATPS R^2 log R + P1 -0.2

W-C2 (10) W-C2 (100) W-C4 (10)

-0.3

W-C6 (10) B (10) B (100)

-0.4

Figure 27: Error distribution along the x-axis using the DEC for the advection–diffusion (k = 5) case and different RBFs (cases in which O(Error) ≤ 10−1 ).

size if a scale factor is properly applied on both the boundary conditions and the parameters of the governing equations. Once a result corresponding to this new problem is obtained, using the same scale factor again the solution of the original problem can be retrieved. From a mathematical point of view the results should be identical no matter whether scaling has been applied or not. In the codes presented here the scale affects the time needed for the iterative solver to converge. The reason is that in BEM formulations some of the matrices have a length dimension, for instance, in eqn (21) the term (H − T − (S/D)k) has no dimensions, but the matrix G has a dimension of length, consequently, the size at which the problem is solved affects the condition number of the resulting system of equations. Figure 28 shows the variation of the condition number with the inverse of the scale factor for the advection–diffusion case. The inverse of the scale factor indicates how many times the length of the problem has been enlarged. Both the DEC and the CEC have been tried using ATPS as the DRM approximation function. It can be seen that the condition number has a minimum for an inverse of


223

Condition number

4000 3500

k=20

3000

DEC

k=5

2500

k=0

2000 1500 1000 500 1/scale factor 0 0

5

10

15

20

Condition number

200000 180000 160000

CEC

140000

k=20

120000

k=5

100000

k=0

80000 60000 40000 1/scale factor

20000 0 0

10

20

30

Figure 28: Variation of the condition number with the scale factor for both the DEC and the CEC.

the scale factor around two for the DEC, and around sixteen for the CEC. Note that the DEC is better conditioned than the CEC. Figure 29 shows the variation of the maximum error with the inverse of the scale factor. For the DEC the best accuracy is in the range where the condition number is minimum. For the CEC the situation is different depending on the influence of the advective term. When the advective term is dominant, the accuracy is not noticeably affected by the scaling, but when the advective term is small or null, the error reaches a minimum for a scale factor that is not the best from the point of view of the condition number. At the same time these optimal scales vary with the value of k.

224 Domain Decomposition Techniques for Boundary Elements Maximum error [%]

10 9

DEC

8 7 6 5 4

k=20

3

k=5

2

k=0

1

1/scale factor

0 0

5

10

15

20

Maximum error [%]

2.5

k=20

2

k=5 k=0

1.5

1

CEC

0.5 1/scale factor 0 0

10

20

30

Figure 29: Variation of the maximum error with the scale factor for both the DEC and the CEC. The values corresponding to the CEC with k = 20 are not displayed because they stay uniformly around 14%.

It is evident that it is convenient to scale the problem using these codes. A single change in the unit system in which a given problem is being specified or a refinement of the mesh can cause the code to fail if scaling is not implemented. Tables 4 and 5 compare the time that the solver needs to give the solution of the problem and the maximum error for both situations: with and without scaling. The problem solved was the advection–diffusion case with a refined mesh of 1456 subdomains. Therefore, a better accuracy was expected with regard to the previous cases, in which a mesh of 173 subdomains was used. The scaling criterion was that

225


Table 4: DEC – time needed by the solver (the iterative one) to obtain a solution and maximum error with and without scaling. Refined mesh of 1,456 subdomains. k = 40

Solver time (s) Maximum error (%)

k = 20

k=5

k=0

Not Not Not scaled Scaled scaled Scaled scaled Scaled

Not scaled

Scaled

2,366

746

2,666

823

2,928

895

3,005

923

6.3

6.5

0.94

0.92

0.039

0.36

0.0078

0.52

Table 5: CEC – time needed by the solver to obtain a solution and maximum error with and without scaling. Refined mesh of 1,456 subdomains. k = 40

k = 20

k=5

k=0

Not Not Not Not scaled Scaled scaled Scaled scaled Scaled scaled Scaled Solver time (s) Maximum error (%)

1,451

103

1,503

145

1,621

158

1,661

169

33.2

23.0

2.0

1.9

0.13

0.78

0.066

1.0

the average size of the subdomains had to be kept the same as in the 173 subdomains case when the inverse of the scale factor was two for the DEC, and fifteen for the CEC. As the scaling criterion is up to a certain point a matter of preference, it can be implemented with different degrees of sophistication. The results displayed in Tables 4 and 5 show how the rate of convergence of the solver improves with scaling. It reduces roughly by three times the CPU time for the DEC and by ten times for the CEC. The accuracy worsens when k is equal to five and zero. However, for these values of k the codes achieved good accuracy so the error was low. For the CEC and k = 40 the accuracy improves substantially with scaling. 10.5 Internal DRM nodes The advection–diffusion DEC and the CEC, were tested using internal DRM nodes (IDRMN). Two situations were considered: (a) when one IDRMN was added in the mesh in the middle of every subdomain, and (b) when five IDRMNs were added, preserving as much as possible an equal distance between the nodes in order to

226 Domain Decomposition Techniques for Boundary Elements avoid an ill-conditioned system of equations. The distribution of the IDRMNs is done automatically by the code, once the number of the IDRMNs is defined in the input data. It was expected that the accuracy would improve as more IDRMNs were added. Tables 6 and 7 show that this was true for both codes using a mesh of 173 subdomains. Table 8 shows that this is still the trend for the DEC when using a refined mesh of 1,456 subdomains. Surprisingly, Table 9 shows that not always using IDRMNs a better accuracy for the case of CEC is achieved. An exhaustive study on using IDRMNs is beyond the scope of this thesis. However, it can be seen that though Table 9 corresponds to results obtained using the mesh with 1,456 subdomains, the accuracy is not better than the one obtained using 173 subdomains for the case of k equal to 5 or 0, which means that the code does not show convergence any more with the refinement of the mesh. The accuracy of this kind of code can be improved

Table 6: DEC – maximum error with 1 and 5 internal DRM nodes. Mesh of 173 subdomains. Direct solver. k = 20 No. of IDRMNs None Maximum 4.42 error (%)

1 4.19

k=5 5 None 3.16 0.18

1 0.18

k=0 5 None 0.16 0.13

1 0.10

5 0.06

Table 7: CEC – time needed by the solver to obtain a solution and maximum error with 1 and 5 internal DRM nodes. Mesh of 173 subdomains. k = 20 No. of IDRMNs Solver time (s) Maximum error (%)

None 10 14.4

1 19 12.0

k=5 5 47 9.2

None 12 0.63

1 20 0.43

k=0 5 55 0.33

None 13 0.99

1 24 0.65

5 63 0.34

Table 8: DEC – time needed by the solver (the iterative one) to obtain a solution and maximum error with 1 and 5 internal DRM nodes. Refined mesh of 1,456 subdomains. k = 40 No. of IDRMNs Solver time (s) Maximum error (%)

1 802 6.08

5 1,115 4.41

k = 20 1 885 0.90

5 1,220 0.67

k=5 1 959 0.24

5 1,304 0.050

k=0 1 986 0.36

5 1,350 0.07

227


Table 9: CEC – time needed by the solver to obtain a solution and maximum error with 1 and 5 internal DRM nodes. Refined mesh of 1,456 subdomains. k = 40 No. of IDRMNs Solver time (s) Maximum error (%)

1 201 26.3

5 637 30.1

k = 20 1 231 1.61

5 707 1.37

k=5 1 306 0.51

5 972 0.67

k=0 1 325 0.65

5 1,027 0.71

by refining the mesh, but this procedure has a limit because the condition number worsens with the refinement of the mesh. Further research on using IDRMNs should consider what happens when this limit is reached. Adding 5 IDRMNs seems to be a suitable tool to significantly reduce the error when using the DEC in cases in which the advective term is dominant, even though the solver time increases by about 50%.

11 Transient advection–diffusion 11.1 Governing equation and DRM formulation In this section, the DRM-MD is applied to a case of transport by advection, dispersion and diffusion, considering adsorption but excluding reaction, in a saturated, isotropic, homogeneous media, whose governing equation is R

∂c + s = D∇ 2 − V · ∇c ∂t

(49)

where c is the concentration of a given substance, R is the retardation factor, D is the coefficient of mechanical dispersion and s stands for a source or a sink. The theoretical considerations that lead to eqn (49) have been extensively treated by Bear and Verruijt [29] among others. In order to apply the DRM formulation on (49) let us express it as a Poisson’s equation

∂c 1 2 ∇c = V · ∇c − s+R (50) D ∂t Then, replacing the right-hand side into (13) Hc − Gq =

∂c ∂c ∂c ∂c S Vx + Vy + Vz − s+R D ∂x ∂y ∂z ∂t

(51)

By using (19), (51) can be expressed as

∂F −1 ∂F −1 ∂F −1 ∂c S Vx F c + Vy F c + Vz F c − (52) s+R Hc − Gq = D ∂x ∂y ∂z ∂t

228 Domain Decomposition Techniques for Boundary Elements If the linear finite difference time approximation is now introduced: c = (1 − θc )cm + θc cm+1 qc = 1 − θqc qc m + θqc qc m+1 ∂c 1 m+1 = (c − cm ) ∂t t

(53) (54) (55)

where the super-script indicates the number of time iterations and t the timestep. Substituting (53)–(55) into (52) yields H (1 − θc )cm + θc cm+1 − G (1 − θqc )qc m + θqc qc m+1 S ∂F ∂F = V x F−1 (1 − θc ) cm + θc cm+1 + V y F−1 (1 − θc ) cm + θc cm+1 D ∂x ∂y

R m+1 ∂F +V z F−1 (1 − θc )cm + θc cm+1 − (56) s+ c − cm ∂z t Reordering produces

RS S ∂F ∂F ∂F V x F−1 +V y F−1 +V z F−1 − cm+1 − θqc Gqc m+1 D ∂x ∂y ∂z D t ∂F ∂F ∂F RS m S = (1 − θc ) V x F−1 +V y F−1 + V z F−1 − (1 − θc )H − c D ∂x ∂y ∂z D t S s (57) + 1 − θqc G qc m − D

θc H − θc

Calling T the matrix (S/D)[V x (∂F/∂x)F−1 + V y (∂F/∂y)F−1 + V z (∂F/∂z)F−1 ] yields

RS cm+1 − θqc Gqc m+1 D t RS S = (1 − θc )T − (1 − θc )H − cm + (1 − θqc )G qc m − s D t D

θc H − θcT −

(58)

Equation (58), once the boundary and initial conditions are applied, is the system of equations produced by the DRM formulation on the transient advection–diffusion eqn (49). 11.2 Computational implementation Figure 30 shows the flow chart of a DEC with preprocessing to solve eqn (49) in 3D domains. The code can be easily obtained by introducing minor changes


229

MAIN

1

Input data

Distribute results

Read matrices and auxiliary arrays

CP=C Update concentrations of the previous timestep

Set boundary conditions Read velocities

IPRN1 = IPRN No

Calculate T

Yes Output results of this timestep

2 IPRN1=0 END

2

ITER = 1, NT Loop on timesteps IPRN1= IPRN1+1 Yes ITER = 1 No

Set initial conditions -

DERP = DER Update derivatives of the previous timestep Assemble system of equations Solve system of equations

1

Figure 30: Flow chart of a DRM-MD code to solve the transient advection–diffusion equation. Dashed line indicates processes that have been modified and bold line indicates new processes with regard to the steady-state advection–diffusion code.

in the code presented in Section 8. Differences with regard to the code in Section 8 are: •

Main program: There is a loop to perform the time steps and a new process called ‘Setic’.

230 Domain Decomposition Techniques for Boundary Elements •

•

•

Input data: It has to incorporate values of the retardation factor, size of the time steps, t, number of time steps, NT, number of time steps at which results are output, IPRN and initial conditions. Setic: This process sets two arrays up, cpij and derpij , which contains the initial concentration and normal derivative, respectively, at the node of freedom ‘j’ of the subdomain ‘i’. According to the notation of eqn (58), cpij has the values of cm and derpij those of qc m . Assembly: It assembles the system of eqn (58).

Figure 31 shows the flow chart of the assembly. 11.3 Comparison with analytical solution A 1D case of transport was solved using the DEC and results were compared with the analytical solution. Several combinations of the parameters of eqn (49) were tested, of which only one case will be displayed in this section. The quality of the agreement was similar in all cases provided suitable time steps were used. The 1D case is a column of a porous material of D = 1 in which fluid is flowing at a uniform constant velocity equal to 1. The length of the column is 1 and the boundary conditions are concentration 1 at the inlet of the domain and zero normal derivative at the outlet. The reaction constant is equal to 1, no sources are present and the initial condition is zero concentration inside the domain. The analytical solution is given in reference [30]. Again, in order to emulate the 1D case, a prismatic domain of 1 × 0.2 × 0.2 was subdivided in 173 subdomains and a constant, uniform, parallel-to-the-longitudinal axis velocity equal to 1 was imposed. Apart from the boundary conditions corresponding to the 1D case, zero normal derivatives were specified on the lateral sides of the domain. The example was solved using the DEC with ATPS as DRM approximation function, the same scaling criterion as the one used in Section 10.4 and no DRM nodes. θqc and θc were set equal to 1. Figures 32–34 show results at times 0.004, 0.02 and 0.06. Two time steps of 0.002 each were used to arrive at t = 0.004, then a further 8 time steps of 0.002 each were used to arrive at t = 0.02 and a further 10 time steps of 0.004 each were used to arrive at t = 0.06. Results are compared with the ones of the analytical model. For every situation the results are presented in natural and semi-logarithmic axes. When results are compared in natural axes, the agreement between the analytical and numerical values is good in all cases. The plot in semi-log axes in Fig. 32 shows that results have different orders of magnitude for values of the x coordinate higher than 0.22, where concentrations are very small. However, the shape of the analytical profile in that figure is highly suspicious, especially considering that the analytical model gives negative values for x higher than 0.6, which is meaningless. The order of magnitude of the difference between the analytical and numerical results decreases as the time (and concentrations) increases.


231


1 2


3


4


Incorporate into the right-hand-side vector the term [ (1 - c ) *(t(I,K,J)- h(I,K,J))- R(I)* s(I,K,J)/(D(I)* ∆t )]*cp(I,J) + (1 - qc ) * g(I,K,J)* derp(I,J)- s(I,K,J)*sour(I)/D(I) according to NE

4 3

aux(1) =

c *h(I,K,J)- c * t(I,K,J)-R(I)* s(I,K,J)/(D(I)* ∆t ) kn(1) = c(I,J) kn(2) = der(I,J)

chsgn(I,J) ≠ 0

Return

Yes aux(2) =

qc

*

D(infront) * n(infront ) *g(I,K,J) D ( I ) * n( I )

No aux(2) = -

2

qc

g(I,K,J)





1

Figure 31: Flow chart of the ‘Assemble system of equations’ process of the code to solve the transient advection–diffusion equation using discontinuous elements. Two factors have a decisive influence on the accuracy of the code: mesh and time step. Coarse meshes can prevent the code from converging to a correct solution and a exceedingly fine mesh can make the problem too cumbersome. If the time step is not chosen properly it can introduce errors. For instance, Fig. 35 compares results

232 Domain Decomposition Techniques for Boundary Elements Concentration

Concentration

1.2

1.0E+00 1.0E-01 0.0

1.0

0.2

0.4

0.6

0.8

1.0

1.2

1.0E-02 1.0E-03

0.8

1.0E-04


0.6

1.0E-05 1.0E-06 1.0E-07

0.4

1.0E-08


1.0E-09

0.2 X

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

-0.2

1.0E-10 1.0E-11 1.0E-12 X

1.0E-13

Figure 32: Profile of concentration at t = 0.004. Concentration

Concentration

1.0E+00

1.2

0.0

1.0

0.2

0.4

0.6

0.8

1.0

1.2

1.0E-01


0.8

1.0E-02

0.6

1.0E-03

0.4


1.0E-04

0.2 1.0E-05 X

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

X

1.0E-06

Figure 33: Profile of concentration at t = 0.02. Concentration

1.2

Concentration

1.0E+00 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.0 Analytical Numerical

0.8 0.6

1.0E-01

0.4


0.2 X

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

X

1.0E-02

Figure 34: Profile of concentration at t = 0.06.

Computational Implementation for 3D Problems 1.2

Concentration

233

Concentration

1.2 1.0

1.0 Analytical Numerical

0.8


0.8

0.6

0.6

0.4

0.4 0.2

0.2 X

0.0 0.0

0.2

t = 0.002

0.4

0.6

0.8

1.0

1 x10 timesteps timestep

1.2

X

0.0 0.0

0.2

t = 0.02

0.4

0.6

0.8

1.0

1.2

1 x1 timestep timestep

Figure 35: Profile of concentration at t = 0.02 using different time steps. The velocities in this case are equal to 15. using 10 time steps of 0.002 and 1 time step of 0.02 when using the velocities equal to 15.

12 Conclusions In this work, several aspects regarding the implementation of the DRM-MD codes have been presented and reviewed. Implementation of the DRM-MD code has two critical aspects: the domain mesh and the assembly of the system of equations. Producing and handling a domain mesh can have been a problem in the past, but nowadays, with the variety of pre- and postprocessors that are available on the market, it cannot be considered to be a serious objection. For the examples presented in this work, GIDTM , a pre and postprocessor with a very convenient price, was used with minor customisations. As for the assembly of the system of equations, a general scheme for the implementation of DRM-MD codes was proposed in which the assembly is split up into two parts. The first part is common to all codes regardless of the equation that is being solved. The subroutine based on this part processes the data related to the geometry of the domain, mesh used and the boundary conditions; it defines the size of the system of equations and the position of every coefficient in the system matrix. The second part, which differs for different types of problems, processes the boundary conditions and produces the coefficients of the system of equations and the right-hand side vector. In this way, if one code has been written to solve a given problem, adapting the code for different kinds of problems is straightforward, as the first part of the process, which has been referred to as ‘Set boundary conditions and NCOL’, is independent of the equations that are being solved and concentrates on the most difficult and annoying part of the assembly. This has been explained in Sections 6–9 and 11, where details of the application of this scheme to Poisson’s and both steady-state and transient advection–diffusion equations were given.

234 Domain Decomposition Techniques for Boundary Elements Examples of 3D cases were presented in which the sensitivity of the implementation of the DRM-MD to the choice of RBFs, continuity of the elements, scaling and number of internal nodes were tested on a set of DRM-MD codes to solve the Poisson and the steady-state advection–diffusion equations in 3D domains. Ten different RBFs were tested with the code using discontinuous elements. The highest accuracy was achieved when using the Wendland-C2, Wendland-C4, Wendland-C6 and Bhumann CS-RBFs. However, when using these functions, a further variable is introduced: the size of the support. As there is no rigorous guideline on how to choose the size of the support, which largely affects the accuracy and the convergence of the code, this is an obstacle at the moment for using the CS-RBFs in the DRM-MD codes. On the other hand, the ATPS showed satisfactory accuracy and convergence, and since the implementation of the ATPS is straightforward, this is the preferred choice of the authors of this work. The tests on the influence of the geometrical size of the problem were made using ATPS and both discontinuous and continuous elements. The size of the subdomains, which depends on the size of the problem and the degree of refinement of the mesh, affects the condition number of the resulting system of equations and, consequently, the time needed by the iterative solvers to converge. By using simple scaling of the equations, the solver time was reduced by three and by ten times for the DEC and the CEC, respectively. For the examples tested, internal DRM nodes in every subdomain improved the accuracy of the DEC, especially when using five IDRMNs in a problem with a dominant advective term. The same cannot be said about the CEC, which for some cases produced lower accuracy with the increase in the number of IDRMNs. Regarding the use of continuous and discontinuous elements, it is difficult to say which strategy is the best. The DEC offers higher accuracy, especially for highly advective transport, but it is much slower than the CEC. At the present stage, the DEC has the advantage that it can solve non-homogeneous domains. The most convenient strategy is to have a choice to use both types of elements in a single mesh, which would preserve the versatility of dealing with non-homogeneous domains, while offering scope for CPU and memory usage reduction. In Section 11, a linear finite difference approximation has been used to evaluate the time derivative of the governing equation. The code showed itself to be sensitive to the size of time steps. A small time step value improves the accuracy, but beyond a lower limit, which depends of the problem under consideration, an exceedingly small time step causes the failure of the code. On the other hand, an unsuitably big time step can produce significant errors in the results.

References [1]

Popov, V. & Power, H., The DRM-MD integral equation method: an efficient approach for the numerical solution of domain dominant problems. International Journal for Numerical Methods in Engineering, 44, pp. 327–353, 1999.


235

[2] Taigbenu, A.E., The Green element method. International Journal for Numerical methods in Engineering, 38, pp. 2241–2263, 1995. [3] Žagar, I., Škerget, L. & Alujević, A., Diffusion-convection problems using boundary-domain integral formulation for non-uniform flows. Boundary Element Methods in Fluid Dynamics II, eds. H. Power, C.A. Brebbia & D.B. Ingham, Computational Mechanics Publications: Southampton and Boston, pp. 75–82, 1994. [4] Popov, V. & Power, H., DRM-MD approach for the numerical solution of gas flow in porous media with application to landfill. Engineering Analysis with Boundary Elements, 23, pp. 175–188, 1999. [5] Popov, V., Power, H. & Baldasano, J.M., BEM solution of design of trenches in a multi-layered landfill. Journal of Environmental Engineering, 124(1), pp. 59–66, 1998. [6] Popov,V. & Power, H., Numerical analysis of the efficiency of landfill venting trenches. Journal of Environmental Engineering, 126(1), pp. 32–38, 2000. [7] Peratta, A. & Popov, V., Modelling flow and solute transport in fractured porous media. Proc. of Boundary Elements XXV, eds. C.A. Brebbia, D. Poljak & V. Roje, WIT Press: Southampton and Boston, pp. 63–72, 2003. [8] Florez, W.F. & Power, H., DRM multidomain mass conservative interpolation approach for the BEM solution of the two-dimensional Navier–Stokes equations. Computers & Mathematics with Applications, 43(3–5), pp. 457– 472, 2002. [9] Florez, W.F. & Power, H., Multi-domain mass conservative dual reciprocity method for the solution of the non-Newtonian Stokes equations. Applied Mathematical Modelling, 26(3), pp. 397–419, 2002. [10] Florez, W.F., Nonlinear Flow Using Dual Reciprocity, WIT Press: Southampton and Boston, 2001. [11] Samardzioska, T. & Popov, V., Numerical comparison of the equivalent continuum, non-homogeneous and dual porosity models for flow and transport in fractured porous media. Advances in Water Resources, 28, pp. 235–255, 2005. [12] Natalini, B. & Popov, V., Tests of radial basis functions in the 3D DRM-MD. Communications for Numerical Methods in Engineering, in print, available online 11 July 2005. [13] Natalini, B., The boundary element dual reciprocity method-multidomain approach for solving 3D potential problems, PhD thesis, Wessex Institute of Technology, University of Wales, 2005. [14] Peratta, A. & Popov, V., A new scheme for numerical modelling of flow and transport processes in 3D fractured porous media. Advances in Water Resources, in print, available online 15 July 2005. [15] Portapila, M.I. & Power, H., Iterative schemes for the solution of system of equations arising from the DRM in multi domain approach, and a comparative analysis of the performance of two different radial basis functions used in the interpolation. Engineering Analysis with Boundary Elements, 29(2), pp. 107–125, 2005.


[17] [18]

[19]

[20]

[21]

[22] [23]

[24] [25]

[26] [27]

[28] [29] [30]

Portapila, M.I. & Power, H., Efficiency and accuracy of an iterative solution for the DRM-MD. Boundary Elements Communications, 12(1), pp. 26–37, 2001. Popov, V. & Power, H., A domain decomposition on the dual reciprocity approach. Boundary Elements Communications, 7(1), pp. 1–5, 1996. Natalini, B. & Popov, V., An alternative approach for calculation of the first and higher order derivatives in the DRM-MD. Engineering Analysis with Boundary Elements, 28(1), pp. 61–78, 2004. Partridge, P.W., Brebbia, C.A. & Wrobel, L.C., The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications: Southampton and Boston, 1992. Do Rêgo Silva, J.J., Acoustic and Elastic Wave Scattering Using Boundary Elements, Computational Mechanics Publications: Southampton and Boston, 1994. Telles, J.C.F., A self-adaptative co-ordinate transformation for efficient numerical evaluation of general boundary element integrals. International Journal for Numerical methods in Engineering, 24, pp. 959–973, 1987. Saad,Y., Iterative Methods for Sparse Linear Systems, PWS Publishing: New York, 1996. Paige, C.C. & Saunders, M.A., Algorithm 583 LSQR: sparse linear equations and least squares problems. ACM Transactions on Mathematical Software, 8(2), pp. 195–209, 1982. Schmidt, G. & Strese, H., BEM for Poisson equation. Engineering Analysis with Boundary Elements, 10, pp. 119–123, 1992. Levesley, J., Pointwise estimates for multivariate interpolation using conditionally positive functions, approximation theory. Wavelets and Applications, ed. S.P. Singh, Kluwer Academic Publishers: Netherlands, pp. 381– 401, 1997. Golberg, M.A., The numerical evaluation of particular solutions in the BEM – a review. Boundary Elements Communications, 6, pp. 99–106, 1995. Wendland, H., Piecewise polynomial, positive definite and compactly supported radial function of minimal degree. Advances in Computational Mathematics, 4, pp. 389–396, 1995. Buhmann, M.D., A new class of radial basis functions with compact support. Mathematics of Computations, 70(233), pp. 307–318, 2000. Bear, J. & Verruijt, A., Modeling Groundwater Flow and Pollution, Reidel Publishing Company: Dordrecht, 1987. Van Genuchten, M.T. & Alves, W.J., Analytical solutions of the onedimensional convective-dispersive solute transport equation, Technical Bulletin No. 1661, US Department of Agriculture, 1982.

CHAPTER 7 Iterative schemes for the solution of systems of equations arising from the DRM in multidomains M.I. Portapila1 & H. Power2 1Wessex Institute of Technology, UK. 2 Department of Mechanical Engineering,

University of Nottingham, UK.

Abstract The aim of this work is to carry out a systematic experimental study of the use of iterative techniques in the context of solving the linear system of equations arising from the solution of the diffusion–convection equation with variable velocity field through the use of the dual reciprocity method in multidomains (DRMMD). We analyse the efficiency and accuracy of the computed solutions obtained from the DRM-MD integral equation numerical approach applying various iterative algorithms. For every iterative method tested, we consider a set of different preconditioners, depending on the features of the input matrix to be solved with the chosen method. To check the accuracy of the solutions obtained through the selected iterative methods, they are contrasted against the solutions obtained applying some direct methods such as singular value decomposition, Golub’s method and Cholesky decomposition. The numerical results are also compared with a benchmark analytical solution. Furthermore, we present a comparative analysis of the linear systems of algebraic equations obtained from DRM-MD considering two approximating functions: the conical function r plus a constant, i.e. (1 + r), and the augmented thin plate spline, both of them radial basis functions.

1 Introduction Intensive research has taken place on the behaviour of the system of equations arising from boundary element techniques and its solution applying iterative methods.

238 Domain Decomposition Techniques for Boundary Elements The matrices of coefficients originated from standard single-zone boundary element method (BEM) are unsymmetric, fully populated and very large. This fact represents a considerable disadvantage when comparing with the classical domain methods, such as the finite element method (FEM) or finite difference method, since more computer operations are involved. To overcome this drawback when dealing with the BEM for large problems, it is usual to use the method of domain decomposition, in which the original domain is divided into subregions, and on each of them the full integral representation formulae are applied. At the interfaces of the adjacent subregions, the corresponding full matching conditions are imposed. The domain decomposition approach is itself a very powerful and popular scheme in numerical analysis that has recently increased its popularity due to its use in parallel computing algorithms (for more details see [1]). While the BEM matrices, which arise in the single domain formulation, are fully populated, the subregion formulation leads to block-banded matrix systems with one block for each subregion and overlaps between blocks when subregions have a common interface. When using continuous elements of high orders (more than constant), the application of the matching conditions at common interfaces, i.e. the matrix assembly, leads to an overdetermined system of algebraic equations (as more subregions are defined the bigger the overdetermination). Several schemes are known that reduce the overdetermined system to a closed system (for more details see [2]). The simplest possible scheme is obtained by expressing the derivatives of the field variables at the common nodes between more than two subregions in terms of the variables themselves by using the interpolation functions or by a finite difference approximation. A way of avoiding this problem is by using discontinuous elements in such common nodes, obtaining in this way a closed matrix system at the expense of having a larger number of unknown variables. Alternatively, one can solve the complete overdetermined system. In this regard, several algorithms are known in the mathematical literature which can numerically solve this kind of system, the most popular being the singular value decomposition (SVD) and the least-square approaches. In the limit of a very large number of subregions, the resulting internal mesh pattern looks like a finite element grid. Kane et al. [3], and Guru Prasad et al. [4] report that multi-zone problems use considerably less storage that single-zone applications. They noted that the performance of the conjugate gradient (CG) iterative method for the normal equation approach was improved by multi-zone modelling, but it did not improve the performance of the GMRES approach. In all the studied cases the converged solutions were as accurate as those derived using the direct equation. They remarked that for both single-zone and multi-zone problems, the preconditioned iterative approaches were generally faster than the direct methods. Further increase in the number of applications of the BEM has been hampered by the need to operate with relatively complex fundamental solutions or by the difficulties encountered when these solutions cannot be expressed in a closed form, and also when the technique is applied to nonlinear and time-dependent problems. In the BEM for this kind of problems, it is common to use an integral representation formula based on a partial differential equation for which a closed form

Iterative Schemes for Solution of Linear Systems Using DRM-MD

239

expression of the fundamental solution is known, and express the remaining terms of the original equation as domain integrals. In the early BEM analysis the evaluation of domain integrals was done using cell integration, a technique, which while effective and general, made the method lose its boundary-only nature, which is one of its attractive features, introducing additional internal discretization. Although good results can be obtained using the cell integration technique, this approach, for certain applications, is several orders of magnitude more time consuming than domain methods. This computational cost mainly depends on the fact that the solution at each surface or internal point must involve the evaluation of the complete surface integrals. An implementation of the subregion BEM formulation in this limiting case, i.e. with a very large number of subregions, including cell integration at each subregion has been called by Taigbenu and collaborators the Green element method (GEM) (see [5, 6]), in which a finite difference approximation is used to reduce the overdetermined system of equations. In these conditions, the resultant coefficient matrix is as sparse as that encountered in FEM and, therefore, its solution is as efficient as in the domain approach, and the results are as accurate as those of the BEM. Using a idea similar to the GEM, Ramsak and Skerget [7] solved viscous fluid flow problems governed by the complete Navier–Stokes equation in terms of a mixed boundary-domain integral approach and a subdomain technique. In this way they reach a final discretized matrix system that is overdetermined, sparse and block banded, which they solve using the iterative linear least squares solver (LSQR) proposed by Paige and Saunders [8], accelerating the convergence through a diagonal preconditioning. Several methods have been developed to take domain integrals to the boundary in order to eliminate the need for internal cells (boundary-only BEM formulations). One of the most popular to date is the dual reciprocity method (DRM) introduced by Nardini and Brebbia [9]. This method is closely related to the particular integrals technique (PIT) introduced by Ahmad and Banerjee [10], which is also used to transform domain integrals into boundary integrals. In the latter method a particular solution satisfying the non-homogeneous partial differential equation (PDE) is first found and then the remaining part of the solution satisfying the corresponding homogeneous PDE is obtained by solving the corresponding integral equations. The boundary conditions for the homogeneous PDE must be adjusted to ensure that the total solution satisfies the boundary conditions of the original problem. The DRM also uses the concept of particular solutions, but instead of solving for the particular solution and the homogeneous solution separately, it applies the divergence theorem to the domain integral terms and converts the domain integrals into equivalent boundary integrals (for more information on some other techniques for the evaluation of domain integrals see [11]). A major problem encountered with the PIT and the DRM is that the resulting algebraic system consists of a series of matrix multiplications of fully populated matrices. When only few internal points are required in the PIT or the DRM, the resulting computing time is in general smaller than the one required by the cell integration scheme, but is still costly in comparison with domain approaches.

240 Domain Decomposition Techniques for Boundary Elements Besides, in complex problems these two approaches have been limited only to small values of the nonlinear terms, or small Peclet numbers in the case of the convection– diffusion equation. From these limitations, it appears that these boundary-only formulations cannot be extended to solve highly domain-dominant problems. As in the case of the standard BEM technique, the algebraic system of equations resulting from the DRM in a single domain are large, fully populated and unsymmetric. Bulgakov et al. [12] showed that the matrices of these systems have properties that allow the efficient use of Krylov iterative solvers such as CG squared and generalized minimal residual (GMRES) methods with Jacobi preconditioning. The GMRES method appeared to be more efficient than the CG squared method. Popov and Power [13] found that the DRM approach can be substantially improved by using domain decomposition. The idea of using domain decomposition to improve the accuracy of the DRM approach was inspired by the work of Kansa and Carlson [14] on the radial basis function (RBF) data approximations, where they observed that the best approximation is obtained when the original domain is split into matching subdomains. The aim of this work is to carry out a systematic experimental study of the use of iterative techniques in the context of solving the linear system of equations arising from the solution of the diffusion–convection equation with variable velocity field through the use of the dual reciprocity method in multidomains (DRM-MD). Besides, considering different iterative solutions, we present a comparative analysis of the performance of two different RBFs used in the interpolation of the DRM approach, i.e. the conical function plus a constant (1 + r) and the augmented thinplate spline (ATPS).

2 Preliminary remarks on the BEM The BEM is a numerical approach that involves the transformation of original boundary integral equations into an equivalent algebraic form suitable to be solved numerically. This approach consists of the discretization of the boundary into a series of elements, this procedure being what gives the name to the method. The boundary integrals then are evaluated numerically over these segments or elements in which the boundary is divided. The BEM, therefore, consists of the following steps 1. The boundary is discretized into a series of elements over which the potential and its normal derivative are assumed to vary according to interpolation functions. The geometry of the elements can be represented by constant, linear, quadratic or even higher-order elements, depending on how accurately we may want to represent the geometry of the domain. 2. The discretized integral equation is applied to a number of nodes within each element (collocation technique), where the values of the potential and its normal derivative are associated. 3. The integrals over each boundary element are carried out by using an adequate numerical quadrature.


241

4. By imposing the prescribed boundary conditions of the problem, a system of linear algebraic equations is obtained, which can be solved either using direct or iteratives methods. The solution of this system of equations will produce the remaining boundary data. It is interesting to note that the unknowns are a mixture of the potential and its normal derivative, rather than only the potential as it would be the case in finite elements. 5. At last, if values of the function u and its derivative are required at internal points, it is possible to calculate them taking into account that the fundamental solution will be acting inside the domain and all values u and the derivatives on the boundary, present on the new system to be solved are already known. 2.1 The BEM and the domain integrals Domain integrals in the BEM may arise as a consequence of different situations, such as sources inside the domain, initial states, nonlinear terms, etc. When we have a Poisson-type equation (∇ 2 u = b), the integral equation (1) has a domain integral term which needs to be performed. Different methods are available to compute the domain integrals. A simple way of computing the domain term + + ∂u∗ (ξ , y) ∂u(y) c(ξ )u(ξ ) + u(y) b(y)u∗ (ξ , y) d = u∗ (ξ , y) d + d ∂n ∂n (1) is by subdividing the domain into a series of internal cells over which a numerical integration formula can be applied. By doing so, for each boundary point ξ we can write the domain integral as ) K *

∗ ∗ bu d = wk bu k e (2) dξ =

Ne

k=1

where wk are the numerical integration weights, the function (bu∗ ) must be evaluated at the k integration points on each cell, Ne is the number of cells into which has been divided, e is the area of cell e, and K is the number of integration points on each cell. The term dξ is the result of the numerical integration and we calculate it for each position of the N boundary nodes. The whole set of equations for the N nodes in matrix form can be expressed as d + Hu = Gq

(3)

Another way of integrating the domain terms consists of using the Monte Carlo technique [15, 16] based on the integration over a system of random points rather than applying a regular integration grid as proposed in the cell integration method. We shall also mention an alternative way of including internal sources into the formulation for cases when the function b is harmonic in [17], by transforming the corresponding domain integral into equivalent boundary integrals. This is done

242 Domain Decomposition Techniques for Boundary Elements by means of a new function w∗ such that ∇ 2 w∗ = u∗ , and through the Green’s second identity the domain integral is reduced to

∗

bu d =

2 ∗

∗

+

(b∇ w − w ∇ b)d = 2

∂w∗ ∗ ∂b b −w d ∂n ∂n

(4)

the function w∗ is given in [18] as w∗ =

r2 1 ln +1 8π r

(5)

The multiple reciprocity method [19] is another technique for transforming domain integrals to the boundary which can be seen as a generalization of the just mentioned approach. It makes use of a set of higher-order fundamental solutions in order to apply the Green’s second identity to each term of the sequence. As a result the method can lead in the limit to the exact boundary-only formulation of the problem. Finally we may consider the use of particular solutions as another way of solving eqn (1) without having to compute any domain integrals. This can be achieved by splitting the function u into a particular solution and the solution of the associated homogeneous equation. The main disadvantage of this methodology is that particular solutions are in many cases difficult to obtain when not impossible. The dual reciprocity approximation introduced by Nardini and Brebbia [9] is a generalization of the use of particular solutions where the basic idea is to expand the non-homogeneous terms as a series of known interpolating functions. This method is briefly described in the following section.

3 The dual reciprocity approximation The dual reciprocity approximation, first proposed in 1982 [9] can be summarized as a technique that enables a ‘boundary only’ solution to problems arising from the BEM with domain integrals. Basing its most remarkable feature on the fact that it does not depend on obtaining a new particular solution for each case under consideration. We consider here, for simplicity, the mathematical formulation of the DRM for the Poisson equation, extending this methodology later on to the nontransient diffusion–convection equation, for which we will be presenting numerical analyses. Hence, we consider ∇ 2u = b (6) where b is a known function of position. The solution to eqn (6) can be expressed as the sum of the solution of the homogeneous part plus a particular solution uˆ , such that ∇ 2 uˆ = b (7)


243

Since a particular solution uˆ is not easy to be found, the DRM proposes an approximation for b in terms of a series of particular solutions uˆ j . The number of uˆ j will be as many as the nodes we will prescribe to study the problem. Therefore, the approximation will take place at N boundary nodes (the same as in the BEM) along with L internal nodes, where the term b is interpolated as follows b

N+L

αj fj

(8)

j=1

Here the αj are unknown coefficients and the fj are approximating functions. The particular solutions uˆ j and the approximating functions fj are related through the Laplacian operator ∇ 2 uˆ j = fj

(9)

and there will be N + L values of uˆ j (Fig. 1). The functions fj are geometry dependent, and when we substitute eqn (9) into (8) it yields b=

N+L

αj (∇ 2 uˆ j )

(10)

j=1

Combining eqns (6) and (10) we find ∇ 2u =

N+L

αj (∇ 2 uˆ j )

(11)

j=1

N + L collocation nodes

field point k

N boundary nodes rik

rkj

source point i L internal nodes

Figure 1: Type of nodes in the DRM approach.

244 Domain Decomposition Techniques for Boundary Elements Applying the Green’s second identity to eqn (11) and using the Laplacian fundamental solution u∗ as the auxiliary potential, the following surface-only integral representation formula, in terms of the interpolation coefficients αj , is obtained cu −

u

∗

∂u d + ∂n

N+L

"

∗

q ud =

∗

uˆ j q d −

αj

j=1

∗

#

qˆ j u d + cûij

(12) where the coefficient c, the fundamental solution u∗ and the normal derivative q∗ are those defined in the standard BEM technique, and qˆ j = ∂ uˆ j /∂n. In eqn (1) Green’s second identity was applied only to the left-hand side of the expression. Now this operation (weighted residual technique, Green’s identity or a reciprocity principle) is applied to both sides of the equation in order to take all terms to the boundary. This procedure gives its name to the DRM. The discretized form of eqn (12) is $ N %& N N N+L N

αj Hik uk − Gik qk = Hik uˆ kj − Gik qˆ kj + ci uˆ ij ci ui + k=1

k=1

j=1

k=1

k=1

(13) where Hik and Gik are the standard influence coefficients coming from the integrations of u∗ and q∗ at each boundary element and the functions uˆ and qˆ for a given interpolant function f can be found analytically. In this work, the linear boundary element is used in the discretization of the above integral equation. After collocating in all boundary nodes i, we can write eqn (13) in matrix notation, i.e. N+L

Hu − Gq = αj (Huˆ j − Gqˆ j ) (14) j=1

where the terms ci have been included in the diagonal of H. Furthermore, considˆ and Q ˆ we can rewrite eqn (14) as ering uˆ j and qˆ j as columns of the matrices U ˆ − GQ)α ˆ Hu − Gq = (HU

(15)

In the above equation the coefficients α are determined by the inversion of the interpolation matrix F, i.e. (16) α = F−1 b where F is the matrix resulting from the above interpolation with f as the approximating function. After applying the boundary conditions to expression (15) we will be able to get a boundary solution to problems governed by a Poisson equation. To obtain the solution at internal nodes we calculate ui from eqn (13), recalling that in this case ci = 1, and this procedure will involve the multiplication of known vectors and matrices.


245

3.1 Analysis of the convection–diffusion equation using DRM In this section we will focus on a boundary-only integral method for the solution of non-transient convection–diffusion problems. The general form of the convection– diffusion equation can be written as Dx

∂ 2u ∂ 2u ∂u ∂u − vy − ku = 0 + Dy 2 − vx 2 ∂x ∂y ∂x ∂y

(17)

where u is a concentration of a substance, a temperature, etc., Dx and Dy are dispersion coefficients, vx and vy are velocities and k is a decay parameter. When vx = vy = k = 0 then eqn (17) reduces to the diffusion equation commonly used in heat transfer problems. If Dx = Dy the problem is isotropic, if not it is orthotropic. Thus far, we have considered the DRM for handling Poisson equations where the right-hand side of eqn (6) is a known function of position, namely ∇ 2 u = b (x, y). In such applications the solution is divided into a boundary solution and an internal solution. The use of the DRM for solving convection–diffusion problems by means of the fundamental solution available for the Laplace equation requires that all terms in eqn (17) not including the Laplacian, are transferred to the right-hand side as forcing terms, to obtain an equation of the type ∇ 2 u = b(x, y, u)

(18)

In the isotropic case, where Dx = Dy = D, the non-homogeneous term b is then given by 1 ∂u ∂u b= vx + vy + ku (19) D ∂x ∂y For this class of problems it is no longer possible to separate boundary and interior solutions, resulting in a coupled system of equations in which both sets of values are calculated at once. 3.1.1 The decay term This term is b1 =

k u D

(20)

thus from eqn (16) we have in matrix notation b1 = αF α = F−1 b1 = F−1

k u D

(21)

and we are able to recast eqn (15) as follows ˆ − GQ)F ˆ −1 Hu − Gq = (HU

k u D

(22)

246 Domain Decomposition Techniques for Boundary Elements Since α cannot be obtained explicitly, as it was the case for Poisson-like equations, expression (21) will always be necessary. Matrix F depends only on the geometry of the problem and has no relation to either governing equations or boundary conditions. In practice, it can be computed once and stored to save time and computing resources, in order to be ready to use in subsequent analyses involving the same discretization. If we define ˆ − GQ)F ˆ −1 S = (HU (23) which calculation involves only known matrices, we will be able to rewrite eqn (22) as follows k Hu − Gq = Su (24) D It is important to note that this equation involves N values of u and q on the boundary that are unknown, along with L values of u at interior nodes also unknown. While q is not defined for internal nodes. 3.1.2 The convective term First-order space derivatives are rather frequent in engineering problems, and adapting them into the DRM treatment allows us to recover the boundary-only nature of the BEM. This term is of the form ∂u ∂u 1 b2 = vx + vy (25) D ∂x ∂y To complete the DRM formulation of this term an algorithm must be established to relate the values of b to the nodal values of the concentration u. This is usually done by expressing the concentration at a domain point in terms of the approximation functions fi , i.e. u = Fβ; (i = 1, . . . , N + L) (26) where α = β. Differentiation of eqn (26) produces: ∂u ∂F = β ∂xl ∂xl

l = 1, 2

(27)

Rewriting eqn (26) as β = (F)−1 u, eqn (27) becomes: ∂u ∂F −1 = F u ∂xl ∂xl

l = 1, 2

(28)

Equation (28) states the fact that the nodal values of the derivatives are in terms of the product of two known matrices which depend only on the geometry and the nodal values of u.


247

Returning to eqn (15) and replacing ∂u/∂xl from expression (28) into (16) we arrive to ∂F −1 ∂F −1 F−1 ˆ ˆ u (29) Hu − Gq = (HU − GQ) vx F + vy F D ∂x ∂y or using expression (23) Hu − Gq = S

∂F ∂F 1 vx F−1 + vy F−1 u D ∂x ∂y

(30)

Finally, the complete DRM equation for the non-transient convection–diffusion problem, with expression (19) as forcing term, results in a matrix system for the normal derivative of the concentration at the surface points, and for the concentration at the surface and internal points, in terms of only the surface integral of the influence coefficients H and G, i.e. ∂F −1 ∂F −1 F−1 ˆ ˆ (31) vx F + vy F + k u Hu − Gq = (HU − GQ) D ∂x ∂y or considering eqn (23)

∂F −1 ∂F −1 1 vx F + vy F + k u Hu − Gq = S D ∂x ∂y

(32)

3.2 Overview of DRM in multidomains This treatment of the domain integral (eqn (31)), allows us to handle a boundaryonly integral formulation of this type of problems. However, to improve the accuracy of the solution we need to increase the number of internal nodes in the domain. As the number of internal nodes increases, the multiplication of the above set of fully populated matrices requires higher computational effort. A technique to prevent this drawback is to subdivide the domain into subregions where the DRM can be applied one by one. This multiple region technique is named the dual reciprocity method in multidomains (DRM-MD), and has proved to possess better convergence and better approximation properties than the application of the DRM directly to the single domain [20, 21] . The subdomains will have common interfaces with adjacent subdomains where compatibility conditions, also called matching conditions, have to comply. At each of these interfaces, the flux leaving one subregion has to be equal to the flux entering the other. Therefore, it is necessary that the following flux-matching conditions hold at the mth interface of the subregions i and i + 1: ∂ui ∂ui+1 i i i+1 i+1 D (33) − u v l nl = D − u v l nl ∂n ∂n m m Besides the above conditions, the concentration at each interface needs to be continuous, i.e. (34) ui |m = ui+1 |m

248 Domain Decomposition Techniques for Boundary Elements After applying the boundary conditions of the problem, imposing the compatibility conditions between subdomains, rearranging all the unknowns on the left-hand side and defining a vector y on the right-hand side obtained by multiplying matrix elements by the known surface values of the concentration and its normal derivative, the following overdetermined banded linear system of equations is obtained: Ax = y

(35)

where x is the vector of unknowns us and qs. We shall mention here, that for standard DRM the approximation defined in eqn (26) estimates only the values of u at the interpolation nodes, no information about the derivatives is used unless a Hermitian interpolation were implemented. Besides, the derivatives approximated by eqn (28) can differ from their exact values because the interpolation defined in eqn (26) does not use any information about the derivatives of u. In the DRM-MD approach, even when non-Hermitian interpolation is used, the compatibility conditions reinforce the information on the derivatives, yielding more accurate numerical results. The compatibility conditions are expressed numerically in the way the matrices in eqn (31) are assembled. For instance, for a domain subdivided into 4 subdomains defined by four linear continuous boundary elements (Fig. 2), the continuity of u is reflected in the assembly of the H matrix as follows H1I

H2I

0

H4I

H5I

0

0

0

0

0

H2II

H3II

0

H5II

H6II

0

0

0

H5III H5IV

0

H7III

0

H6IV

0

H8III H8IV

0

0

0

H4III

0

0

0

0

(36)

H9IV

The G matrix will be arranged in a similar fashion to comply with the antisymmetry of q. 4

1

I

2

7

III

5

II

IV

8

9

3 6

Figure 2: Decomposition of the entire domain in four subdomains.


249

The sparsity pattern of the resulting matrix A is sparse and, therefore, similar to those obtained with classical domain techniques. The more subdomains we consider, the more banded blocks are included (one block per subdomain) in the coefficient matrix. Along with this, more common interfaces are involved and in such cases the blocks overlap due to the matching conditions. Consequently the system of equations tends to be block-banded with a large number of zero submatrices. This fact leads to systems of equations with an increased sparsity pattern, as can be seen in Fig. 3. As mentioned above, the use of the DRM-MD will increase the accuracy of the solution, compared to the single DRM, since for those internal nodes placed on

(a)

(b)

0

0

20

20

40

40

60

60

80

80 0

20

40 nz = 840

60

0

(c) 0

20

40 nz = 880

60

(d) 0 50

50

100 150

100

200 250

150

300 0

50 100 nz = 1840

0

100 200 nz = 3760

Figure 3: Sparsity pattern of coefficient matrices related to one domain subdivided into (a) 8 × 2, (b) 4 × 4, (c) 8 × 4 and (d) 16 × 4 subdomains (subdomains made up of four linear continuous boundary elements).

250 Domain Decomposition Techniques for Boundary Elements the edges of the subdomains we will be enforcing continuity and equilibrium laws. Moreover, on each subdomain the approximation will be more local, reducing the risk of propagating numerical errors. In addition, the overall system of equations is suitable to be solved by iterative methods more efficiently due to the large number of zero entries currently present in the coefficient matrices, using similar techniques to those used in the classical domain approaches. We will further analyse this topic when we thoroughly investigate the performance of different iterative techniques together with several preconditioners. When using continuous linear or higher order elements, the assembling procedure of the final coefficient matrix will yield to an overdetermined system of equations. In the numerical cases we will analyse, where the subdomains will all be rectangular, one degree of overdetermination is found per node shared by four subdomains. A way to avoid this overdetermination is to make use of discontinuous boundary elements at the expense of having more equations plus more unknowns. When the complexity of the problem requires a high number of subregions, the resulting system of equations will be extremely large.

4 Approximating functions fj As pointed out above, the key feature of the DRM is the transference of domain integrals to the boundary. For this purpose it is necessary to interpolate the nonhomogeneous term of the equation (forcing term) in terms of prescribed basis functions fj , which are usually recommended to be RBFs. Given a set of nodes X = {xj , j = 1, 2, . . . , N} ∈ ⊂ n a radial basis function fj (x) ≡ f (||x − xj ||) where ||x − xj || is the Euclidean distance between two points or interpolation nodes. Because of RBFs’ spherical symmetry about the centres xj , they are called radial. The primary advantage of RBFs is that they produce approximate functions that vary smoothly and are differentiable. Another advantage is that they involve a single independent variable regardless of the dimensionality of the problem. The RBFs more commonly used are φ(r) = r 2m−2 log r

(generalized thin-plate splines),

φ(r) = (r + c )

(generalized multiquadric),

φ(r) = e−βr

(Gaussian)

2

2 m/2

(37)

where m is an integer number and r = ||x − xj ||. The Gaussian and the inverse multiquadric (MQ) are positive definite functions, while the thin-plate splines (TPSs) and MQ are conditionally positive definite functions of order m, which require the addition of a polynomial term Pm−1 of order m − 1 in conjunction with some homogeneous constrained conditions in order to obtain an invertible interpolation matrix.


251

In a typical interpolation problem we have N pairs of data points {(xj , u(xj ))N j=1 }, which are assumed to be samples of some unknown function u that is to be interpolated by the function f, i.e. u(x) =

N

αj f (x − xj ) + Pm (x)

x ∈ 2

(38)

j=1

in the sense that u(xi ) =

N

αj f (x − xj ) + Pm (x)

(39)

j=1

along with the constraints N

αj Pk (xj ) = 0

1≤k≤m

(40)

j=1

Here the numbers αj , j = 1, 2, . . . , N, are real coefficients and f is a radial basis function. The matrix formulation of the above interpolation problem can be written as Ax = y with F Pm A= (41) T Pm 0 x T (α, β) and yT = (u, 0), where β are the coefficients of the polynomial. Micchelli [22] proved that for the case when the nodal points are all distinct, the matrix resulting from the above radial basis function interpolation is always nonsingular. Numerical experiments carried out by Schaback [23] proved that the condition number of matrices following on the previous interpolation, for smooth RBFs such as Gaussian or MQs, are extremely large when compared to those resulting from non-smooth RBFs such as the TPSs. In 1982, Franke [24] published a review article assessing nearly all the interpolation methods for scattered data sets available by that time. Among the methods tested, RBFs outperformed all the other methods regarding accuracy, stability, efficiency, memory requirement and simplicity of implementation. In 1984, Stead [25] examined the accuracy of partial derivative approximations over scattered data sets, also concluding that RBFs performed more accurately compared to other considered methods. Of the RBFs tested by Franke, Hardy’s MQs [26] were positioned as the best regarding accuracy, followed by Duchon’s TPS, i.e. the generalized MQ and the TPS with m = 1 and m = 2, respectively. Duchon [27] derived the TPS as an optimum solution to the interpolation problem in a certain Hilbert space through the construction of a reproducing kernel. Therefore, they are the natural generalization of cubic splines in n > 1 dimension. Even though the TPS have been considered as optimal interpolating multivariate functions [27] they converge linearly. The MQ functions, on the other hand, converge exponentially and always produce a minimal seminorm error [28]. Nevertheless, in

252 Domain Decomposition Techniques for Boundary Elements spite of this excellent performance, the MQ function contains a free parameter c2 whose choice is not an easy one, and certain values can greatly affect the accuracy of the approximation. As a result of what we discussed in this section we will present in our numerical results a comparison of the performance of the DRM-MD for both (1 + r) and 2 r log r + P1 with P1 = ax + by + c, as interpolating functions for the transference of domain integrals to the boundary. To increase the accuracy of domain schemes which involve a domain discretization, as is the case for finite difference, finite elements, finite volumes and even the DRM-MD, a refinement of the computational mesh is required. However, in the DRM-MD accuracy can also be increased adding internal nodes to the subdomains, without the additional and non-trivial problem of further refinement of the mesh. Numerical examples comparing efficiency and accuracy of the approximated solutions for mesh refinement along with an increment of the internal DRM nodes will also be presented.

5 Iterative solution methods for sparse linear systems 5.1 Krylov subspace methods The Krylov subspace methods are considered, currently, to be among the most important iterative techniques available for solving large linear systems. These techniques are based on projection processes, both orthogonal and oblique, onto Krylov subspaces, which are subspaces spanned by vectors of the form p(A)v where p is a polynomial. These techniques approximate A−1 b by p(A)b, where p is a good polynomial. The Krylov subspace methods are derived from, or related to, the Arnoldi orthogonalization – and this is the case for CG and GMRES – or based on Lanczos biorthogonalization. A general projection method for solving the linear system Ax = b is a method that seeks an approximate solution xm from an affine subspace x0 + Km of dimension m by imposing the condition b − Axm ⊥ Lm , where Lm is another subspace of dimension m, x0 being an arbitrary initial guess to the solution. A Krylov subspace method is a method for which the subspace Km is the Krylov subspace Km (A, r0 ) = span{r0 , Ar0 , A2 r0 , . . . , Am−1 r0 }, where r0 = b − Ax0 . The different versions of Krylov subspace methods arise from different choices of the subspace Lm and from the ways in which the system is preconditioned. Viewed from the angle of approximation theory, the approximations obtained from a Krylov subspace method are of the form A−1 b ≈ xm = x0 + qm−1 (A)r0 , in which qm−1 is a certain polynomial of degree m − 1. In the simplest case where x0 = 0, then A−1 b ≈ qm−1 (A)r0 . In other words, A−1 b is approximated by qm−1 (A)b. Although all techniques provide the same type of polynomial approximations, the choice of Lm , i.e. the constraints used to build these approximations, will have an important effect on the iterative technique. Two broad choices for Lm give rise to the best-known techniques. The first is Lm = Km and the minimum residual variation Lm = AKm . CG and GMRES are among the methods in this category.


253

The second class of methods is based on defining Lm to be a Krylov subspace method associated with AT , namely, Lm = Km (AT , r0 ). 5.2 CG-type methods CG methods can be seen as iterative solution methods to solve linear systems of equations Ax = b by minimizing quadratic functionals, such as f (x) = ( 21 x T Ax) − bT x, if A is symmetric and positive definite, or the residual functional f (x) = (Ax − b)T (Ax − b) in the general case. The minimization takes place on a sequence of subspaces Vk of increasing dimension that are constructed recursively by adding a new basis vector Ak r 0 to those of the previous subspace Vk−1 , where V0 = {r 0 } with r 0 = Ax 0 − b as the residual for the initial vector x 0 which can be an arbitrary approximation of x. Hence, Vk is spanned by the vectors r 0 , Ar 0 , . . . , Ak r 0 . As the Krylov vectors eventually span the whole space (or the space spanned by the eigenvectors of A, represented in r0 ), the methods will give the exact solution after at most n steps, where n is the order of A. Therefore, the method can be seen as a direct solution method. However, in the presence of roundoff errors, the generated vectors will not be exactly (conjugately) orthogonal and the method may need more iterations to reach machine number precision. Even the conjugate gradient methods can generate approximations to the solution vector x that are accurate after many fewer steps than n. The method is then used as an iterative solution method. Conjugate gradient methods can be seen as generalized least square methods where the minimization takes place on a particular vector subspace, the Krylov subspace. Compared with other types of iterative methods, it turns out that the conjugate gradient method can converge with a faster rate, at least, when a proper preconditioning is used. To find a minimizer of f , we will use an iterative method where at each stage we construct a new search direction d k (which will be conjugately orthogonal to the previous search directions). We compute the local minimizer along this search direction, i.e. given x k , the approximation at stage k, we compute τ = τk such that f (x k + τ d k ),

−∞ < τ < ∞

(42)

is minimized by τk and then let x k+1 = x k + τk d k

(43)

be the new approximation. When the search directions are conjugately orthogonal, the residuals (or gradients) become orthogonal to the previous search directions. This property implies that the method computes the best approximation x k+1 = x k + d of all vectors d in Vk . The method whereby the search directions are computed is called the CG method. There is a somewhat simpler, but less efficient, method of computing the minimizer of f , called the steepest descent method. In this one the search of a local minimum takes place along the current gradient vector. On the other hand, in the CG method

254 Domain Decomposition Techniques for Boundary Elements we move along a plane spanned by the gradient at the most recent point and the most recent search direction. Because of the conjugate orthogonality property (d i , Ad j ) = 0, i = j, the method was named the CG method [29]. Since it is actually the search directions that are conjugate orthogonal, a more proper name might be the conjugate direction method. However, for the inner product (x, y) = x T Ay, the residuals (or gradients) indeed become conjugate orthogonal. The CG method has the property of finite termination in the absence of roundoff errors. Nevertheless, it turns out that in practice, there is a tendency towards increasing roundoff error and loss of orthogonality. The rate of convergence depends on the distribution of eigenvalues and with a proper preconditioning of the matrix, this distribution can be such that the method converges much faster with preconditioning than without. 5.3 Generalized minimum residual method The GMRES is a projection method based on taking K = Km and L =AKm , in which Km is the mth Krylov subspace. In the CG method, the residuals form an orthogonal basis for the space span {r 0 , Ar 0 , A2 r 0 , . . .}. In GMRES, this basis is formed explicitly wi = Av i do k = 1, . . . , i wi = wi − (wi , v k )v k end v i+1 = wi /wi and all previously computed vectors in the orthogonal sequence have to be retained. The GMRES iterates are constructed as x i = x 0 + y1 v 1 + . . . + y i v i where the coefficients yk have been chosen to minimize the residual norm b − Ax i . A difficulty with the basic GMRES algorithm is that it does not provide the approximate solution x m explicitly at each step.As a result, it is not easy to determine when to stop. One remedy is to compute the approximate solution xm at regular intervals and check for convergence by a test on the residual. Another solution is related to the way in which the least-squares problem is solved. This approach will allow us to obtain the residual norm at every step, and a stopping criterion can be applied. If we carefully examine the GMRES algorithm, we may observe that the only possibilities of breakdown in GMRES are in the Arnoldi loop. In this situation, the algorithm stops because the next Arnoldi vector cannot be generated. However in this situation, the residual vector is zero, i.e. the algorithm will deliver the exact


255

solution at this step. Namely, if we have a matrix A which is non-singular, then the GMRES algorithm breaks down at step j if and only if the approximate solution x j is exact. The GMRES algorithm becomes impractical when m is large because of the growth of memory and computational requirements as m increases. There are two remedies for this situation. One is based on restarting and the other on truncating the Arnoldi orthogonalization. The restarting GMRES consists in restarting the algorithm periodically. There are many variations to this basic scheme. One that is generally more economical in practice is based on the observation that sometimes a small m is sufficient for convergence and sometimes the largest possible m is necessary. A well-known difficulty with the restarted GMRES algorithm is that it can stagnate when the matrix is not positive definite. The full GMRES algorithm is guaranteed to converge in at most n steps, but this would be impractical if there were many steps required for convergence. A preconditioner for the linear system can be used to reduce the number of steps, or a better preconditioner if one is already in use. Both algorithms, GMRES and CG, are strongly related to, as well as defined by, the choice of a basis of the Krylov subspace. The GMRES algorithm uses an orthogonal basis. In the CG algorithm, the auxiliary vectors are A-orthogonal, i.e. conjugate. The CG-type algorithms, i.e. algorithms defined through short-term recurrences, are more desirable than those algorithms which require storing entire sequences of vectors as in the GMRES process. On the other hand, these latter algorithms require less memory and operations per step.

5.3.1 Convergence of the CG and GMRES algorithms and eigenvalue distribution The CG and GMRES algorithms, each generate the optimal approximate solution from a Krylov subspace, where optimal means to have an error with minimal Anorm in the case of CG or having a residual with minimal 2-norm in the case of GMRES. To describe the behaviour of the algorithms it is appropriate to derive a sharp upper bound on the reduction in the A-norm of the error for CG or in the 2-norm of the residual for GMRES, i.e. an upper bound that is independent of the initial vector but that is actually attained for certain initial vectors. CG-type methods consist mainly of matrix vector multiplication and vector operations. If the matrix is normal, then the norm of the residual can be estimated by the error in a best polynomial approximation problem. The rate of convergence depends on the distribution of eigenvalues and, to some extent, also on the initial residual. If the matrix is not normal, as it is the case in DRM-MD, the estimate also involves the condition number of the matrix. In general, we are unable to give a sufficiently accurate estimate, but we can derive various upper bounds. For Hermitian problems, the A-norm of the error in the CG algorithm is minimized over the space e0 + span{Ae0 , A2 e0 , . . . , Ak e0 }. The CG error vector at step k can be written as a function of e0 and kth-degree polynomials with value 1 at the origin.

256 Domain Decomposition Techniques for Boundary Elements In other words, the error ek in the CG approximation satisfies ek A = min pk (A)e0 A

(44)

pk

where the minimum is taken over all polynomials pk of degree k or less with pk (0) = 1. A sharp upper bound is derived involving all of the eigenvalues of A. This sharp upper bound can be written in the form ek A /e0 A ≤ min max |pk (λi )| pk

i=1,...,n

for CG

(45)

The problem of describing the convergence of this algorithm, therefore, reduces to one in approximation theory – how well can we approximate zero on the set of eigenvalues of A using a kth-degree polynomial with value 1 at the origin. This minimax polynomial can be calculated if the eigenvalues of A are known; more importantly, this sharp upper bound provides an intuition as to what constitutes good and bad eigenvalue distributions. Eigenvalues tightly clustered around a single point (away from the origin) are good. Widely spread eigenvalues, especially if they lie on both sides of the origin, are bad, because a low-degree polynomial with value 1 at the origin cannot be small at a large number of such points. When we have only limited information about the eigenvalues of A, it is useful to have error bounds that involve only a few properties of the eigenvalues. For example, in the CG algorithm for Hermitian positive definite problems, knowing only the largest and smallest eigenvalues of A, we can obtain an error bound by considering the minimax polynomial on the interval from λmin to λmax . Knowing only the largest and smallest eigenvalues of a Hermitian positive definite matrix A, the best possible bound is $ √ $√ %k $ √ %k −1 %k k − 1 k + 1 k−1 ek A   ≤2 + √ ≤2 √ √ e0 A k+1 k+1 k−1

(46)

where k = λmax /λmin is the ratio of the largest to smallest eigenvalue of A. If additional information is available about the interior eigenvalues of A, we can often improve on the estimate while maintaining a simpler expression than the sharp bound (45). For example, when A has one eigenvalue much larger than the others, say, λ1 ≤ · · · ≤ λn−1 λn , that is, λn /λn−1 1, it follows that √ k−1 ek A kn−1 − 1 ≤2 √ , e0 A kn−1 + 1

kn−1 =

λn−1 λn

(47)

Similarly if the matrix A has just a few large outlying eigenvalues, i.e. λ1 ≤ · · · ≤ λn−l λn−l+1 ≤ · · · ≤ λn (λn−l+1 /λn−l 1) k−l √ ek A kn−l − 1 ≤2 √ , e0 A kn−l + 1

kn−l =

λn−l λn

(48)


257

The rate of convergence can be measured in various norms. When a relative measure is used – say, the ratio of the norm of the current error and the initial error – the condition number of the matrix can frequently be used to give a sufficiently accurate estimate. More generally, if the eigenvalues are contained in an ellipse not covering the origin and with sufficiently small eccentricity, then similar estimates can be derived. However, when the eigenvalues are distributed non-uniformly in this interval or ellipse, the condition number alone gives too rough an overestimate of the necessary number of iterations. Therefore, it is possible to find various improved estimates of the number of iterations required to get a sufficiently small relative residual, based on some additional assumptions made of the eigenvalue distribution. In particular, that could be the case of isolated eigenvalues at one or both ends of the spectrum. Regarding GMRES, for xk being the approximate solution from the kth step of the algorithm, and the kth step residual rk = b − Axk . Then, the approximation obtained is xk = x0 + qk (A)r0 where qk is a polynomial of degree k − 1 such that rk 2 = (I − Aqk (A))r0 2 = min (I − Aq(A))r0 2 q∈Pk−1

(49)

Unfortunately, it is not possible to get a simple result for this expression unless the matrix A is normal. If we assume that A is a diagonalizable matrix A = X!X −1 , where ! = diag {λ1 , λ2 , . . . , λn } is the diagonal matrix of eigenvalues, and we define (k) =

min

max |p (λi )|

p∈Pk , p(0)=1, i=1,...,n

Then, again, the rate of convergence of the GMRES algorithm can be measured through the residual norm achieved by the kth step of the algorithm. Satisfying the inequality rk 2 ≤ κ2 (X) (k) r0 2 , where κ2 (X) ≡ X2 X −1 2 , κ2 is the condition number of the matrix of eigenvectors. In this case, as in the Hermitian case, the problem of describing the convergence of GMRES reduces to a problem in approximation theory – how well we can approximate zero on the set of complex eigenvalues using a kth-degree polynomial with value 1 at the origin. We do not have simple estimates based on the ratio of largest to smallest eigenvalues, but we can still apply some notions about good and bad eigenvalue distributions in the complex plane. Eigenvalues tightly clustered about a single point, away from the origin are good, since, for instance, the polynomial (1 − z/c)k is small at all points close to c in the complex plane. Eigenvalues all around the origin are bad because, by the maximum principle, it is impossible to have a polynomial that is 1 at the origin and small in absolute value at many points distributed all around the origin. In this case, it is A’s eigenvalue distribution that essentially determines the behaviour of GMRES. From what we have said we find a close relationship between condition numbers, distribution of eigenvalues and rate of convergence of the CG-like and GMRES algorithms. For this reason we will present in the numerical analysis a report of the eigenvalue (singular values when A is overdetermined) distribution of the different

258 Domain Decomposition Techniques for Boundary Elements A matrices arising from different discretizations of the domain after the treatment of the studied problem with the DRM-MD. In addition we will describe the performance of CG-like and GMRES methods with and without preconditioners. 5.3.2 Methods related to the normal equation As we stated in Section 3.2, the system of equations arising from the DRMMD approach is overdetermined. There are a number of techniques for converting an overdetermined and non-symmetric linear system of equations into a closed and symmetric one. One such technique solves the equivalent linear system AT Ax = AT b, called the normal equations. Often, this approach is avoided in practice because the coefficient matrix AT A is much worse conditioned than A. However, the normal equations approach may be adequate in some situations. There are certain applications in which it is preferred to the usual Krylov subspace techniques. A general consensus is that solving the normal equations can be an inefficient approach in the case when A is poorly conditioned. Indeed, the 2-norm condition number of AT A is given by Cond2 (AT A) = AT A2 (AT A)−1 2 . 2 (A) where σ AT A2 = σmax max (A) is the largest singular value of A, which, incidentally, is also equal to the 2-norm of A. Thus, using a similar argument for the inverse (AT A)−1 yields Cond2 (AT A) = A22 A−1 22 = Cond22 (A). The 2-norm condition number for AT A is exactly the square of the condition number of A, which could cause difficulties. For example, if originally Cond2 (A) = 108 , then an iterative method may be able to perform reasonably well. However, a condition number of 1016 can be much more difficult to handle by a standard iterative method. That is because any progress made in one step of the iterative procedure may be annihilated by the noise due to numerical errors. On the other hand, if the original matrix has a good 2-norm condition number, then the normal equation approach should not cause any serious difficulties. The normal equations approach might be better, because of its robust quality (symmetric and positive definite matrices) which outweighs the additional cost due to the slowness of the method in the generic case. For ill-conditioned problems, these methods will simply fail to converge, unless a good preconditioner is available. In the numerical results we will present singular-eigenvalue analysis to determine whether it is suitable to use the normal equation as input data for the iterative techniques, as well as those iterative algorithms where the Krylov subspace is associated with AT . 5.4 LSQR, algorithm for sparse least-squares problems The LSQR is an iterative method for computing a solution x to either unsymmetric equations Ax = b or linear least squares min Ax − b2 . Developed by Paige and Saunders in 1982 [8], it is analytically equivalent to the standard method of conjugate gradients, but with more favourable numerical properties. CG-like methods are characterized by their need for only a few vectors of working storage and by their theoretical convergence within at most n iterations (if exact


259

arithmetic could be performed). And they are, in general, more useful when A is well-conditioned and has many nearly equal singular values. The LSQR is based on the bidiagonalization procedure of Golub and Kahan [30]. It generates a sequence of approximations {xk } such that the residual norm rk 2 decreases monotonically, where rk = b − Axk . Analytically, the sequence {xk } is identical to the sequence generated by the standard CG algorithm. The algorithm is derived by applying the Lanczos process to a particular symmetric system. Given a symmetric matrix B and a starting vector b, the Lanczos process is a method for generating a sequence of vectors {vi } and scalars {αi }, {βi } such that B is reduced to tridiagonal form. After applying k steps of the Lanczos process we reach BVk = Vk Tk + βk+1 vk+1 eTk (50) where Tk ≡ tridiag(βi , αi , βi+1 ) and Vk ≡ [v1 , v2 , . . . , vk ]. In the event we wish to solve a symmetric system Bx = b, multiplying (50) by an arbitrary k-vector yk gives BVk yk = Vk Tk yk + βk+1 vk+1 ηk , where ηk is the last element of yk . If yk and xk are defined by the equations Tk yk = β1 e1 xk = Vk yk then we shall have Bxk = b + ηk βk+1 vk+1 . Hence, xk may be taken as the exact solution to a perturbed system and will solve the original system whenever ηk βk+1 is negligibly small. 5.4.1 The least-squares system When the Lanczos process is applied to the symmetric system that satisfies the least-squares problem, it results in two forms of a bidiagonalization procedures, depending on the starting vector. • •

Bidiag 1: starting vector b; reduction to lower bidiagonal form AT Uk+1 = Vk BkT + αk+1 vk+1 eTk+1 Bidiag 2: starting vector AT b; reduction to upper bidiagonal form AT Pk = Vk RkT + θk+1 vk+1 eTk+1

The principal connection between the two bidiagonalization procedures is that the matrices Vk are the same for each, and that the identity BkT Bk = RkT Rk holds. This follows from the fact that v1 is the same in both cases, and Vk is the result of applying the Lanczos process 50 with B = AT A. The rather surprising conclusion is that Rk must be identical to the matrix that would be obtained from the conventional QR factorization of Bk . Thus Qk Bk = Rk 0

(51)

260 Domain Decomposition Techniques for Boundary Elements The quantities generated from A and b by Bidiag 1 is used to solve the leastsquares problem, min b − Ax. Computationally, it is advantageous to solve this problem using the standard QR factorization of Bk , that is, the same factorization 51 that links the two bidiagonalizations. LSQR needs to save only the most recent iterates. The stopping criteria is set in terms of three dimensionless quantities, which the user is required to specify. These rules are based on allowable perturbations in the data, and in an attempt to regularize ill-conditioned systems. Stability is more apparent for LSQR than for the standard method of conjugate gradients. After having performed various numerical comparisons, the authors of LSQR recommend the application of the symmetric CG to the normal equations AT Ax = AT b only if it would produce a satisfactory estimate of x in very few iterations. For ill-conditioned matrices, LSQR should be more reliable than the least-squares adaption of symmetric CG, at the expense of more storage and work per iteration. 5.4.2 Preconditioned iterations Although iterative methods are well founded theoretically, they are all likely to suffer from slow convergence for problems which arise from typical applications. Preconditioning is a key factor for the success of the Krylov subspace methods in these applications. A weakness of iterative solvers, compared to direct solvers, is the lack of robustness they may have. Both the efficiency and robustness of iterative techniques can be improved by using preconditioning. Preconditioning is a means of transforming the original linear system into one which has the same solution, but which is likely to be easier to solve with an iterative solver. In general, the reliability of the iterative techniques, when dealing with various applications, depends much more on the quality of the preconditioner than on the particular Krylov subspace used. Finding a good preconditioner to solve a given sparse linear system is rarely based on theoretical results, and some methods work surprisingly well, often despite expectations. A preconditioner can be defined as any subsidiary approximate solver which is combined with an outer iteration technique, typically one of the Krylov subspace iterations. We have to note that there are virtually no limits to available options for obtaining good preconditioners. For example, preconditioners can be derived from the knowledge of the original physical problems from which the linear system arises. However, a common feature of the preconditioners is that they are built from the original coefficient matrix. In general, we can say that a preconditioner is any form of implicit or explicit modification of an original linear system which makes it easier to solve by a given iterative method. For example, scaling all rows of a linear system to make the diagonal elements equal to one is an explicit form of preconditioning. The resulting system can be solved by a Krylov subspace method and may require fewer steps to converge than with the original system (although this is not guaranteed). Then a preconditioner M is a matrix which approximates the system matrix A in some sense. One of the simplest ways of defining a preconditioner is to assume that M is a diagonal matrix with A’s main diagonal as entries. This preconditioner is


261

called Jacobi preconditioner. From a practical point of view, the only requirement for M is that it is inexpensive to solve linear systems Mx = b. This is because the preconditioned algorithms will all require a linear system solution with the matrix M at each step. Then, the following preconditioned system could be solved: M −1 Ax = M −1 b or AM −1 u = b, x = M −1 u (left or right preconditioning, respectively). In theory, any general splitting in which M is nonsingular can be used. In the case of preconditioned CG, left and right preconditioning lead to two systems of equations which are no longer symmetric in general. When M is available in the form of an incomplete factorization, i.e. when M = LL T , then a simple way to preserve symmetry is to split the preconditioner between left and right, i.e. to solve L −1 AL −T u = L −1 b, x = L −T u, which involves a symmetric positive definite matrix. However, it is not necessary to split the preconditioner in this manner in order to preserve symmetry. We can observe that M −1 A is self-adjoint for the M-inner product, (x, y)M ≡ (Mx, y) = (x, My), since (M −1 Ax, y)M = (Ax, y) = (x, Ay) = (x, M(M −1 A)y) = (x, M −1 Ay)M . Therefore, an alternative is to replace the usual Euclidean inner product in the CG algorithm by the M-inner product. It is interesting to observe that M −1 A is also self-adjoint with respect to the Ainner product. Indeed, (M −1 Ax, y)A = (AM −1 Ax, y) = (x, AM −1 Ay) = (x, M −1 Ay)A and a similar algorithm can be written for this dot product. When M is a Cholesky product M = LL T , two options are available, namely, the split preconditioning option, which is the one we present in the numerical results; or the algorithm using the M-inner product (left preconditioned system). If we consider a right preconditioned system, the matrix AM −1 is not Hermitian with either the standard inner product or the M-inner product. However, it is Hermitian with respect to the M −1 -inner product. If we introduce this M −1 -inner product in the CG algorithm we get the right-preconditioned CG. The implication is that the left-preconditioned CG algorithm with the M-inner product is mathematically equivalent to the right-preconditioned CG algorithm with the M −1 -inner product. In the case of preconditioned GMRES the same three options for applying the preconditioning operation as for the CG (namely, left, split and right preconditioning) are available. However, there will be one fundamental difference – the right preconditioning versions will give rise to what is called a flexible variant, i.e. a variant in which the preconditioner can change at each step. In many cases, M is the result of a factorization of the form M = LU. Then there is the option of using GMRES on the split-preconditioned system L −1 AU −1 u = L −1 b, x = U −1 u. In this situation the residual norm available is that of L −1 (b − Axm ). The question arising on the differences between the right, left and split preconditioning is the fact that different versions of the residuals are available in each case. And this may affect the stopping criterion and may cause the algorithm to stop either prematurely or with delay. When comparing the left, right and split preconditioning options, a first observation to make is that the spectra of the three associated operators M −1 A, AM −1

262 Domain Decomposition Techniques for Boundary Elements and L −1 AU −1 are identical. Therefore, in principle one should expect convergence to be similar, although eigenvalues do not always govern convergence. In most practical situations, the comparison between the difference in the convergence behaviour of right and left preconditioning is not significant. The only exception is when M is ill-conditioned which could lead to substantial differences if we consider that in the right preconditioning case the residual norm is ||b − Axm ||2 , and the preconditioned residual norm in the left preconditioning case is ||M −1 (b − Axm )||2 . It is for this reason that we will be solving the system with right preconditioning in the numerical results. In Table 1 we present a comparison of the condition numbers of the normal equation AT A, and the condition number (cn) of the preconditioner matrix ILU of the numerical examples presented in Section 6, using different subregions and interpolation functions. For some of the numerical examples considered, as can be observed, the cns of the normal equation are not significantly large, but when the cns of M are too large it is not possible to solve the preconditioned systems of equations. When M is available in the form of an incomplete factorization of the original matrix A, of the form A = LU − R where L and U have the same non-zero structure as the lower and upper parts of A, respectively, and R is the residual or error of the factorization. This incomplete factorization is rather easy and inexpensive to compute. On the other hand, it often leads to a crude approximation which may result in the Krylov subspace accelerator requiring many iterations to converge. To remedy this, several alternative incomplete factorizations have been developed by allowing more fill-in in L and U. In general, the more accurate ILU factorizations require fewer iterations to converge, but the preprocessing cost to compute the factors is higher. Table 1: Condition numbers of ILU preconditioners and their applicability. Mesh – A(m, n) – RBF

Condition number AT A

Condition number M

A/NA

Mesh I (400, 343)(1 + r) Mesh I (400, 343)(r 2 log r + p) Mesh II (472, 415)(1 + r) Mesh II (472, 415)(r 2 log r + p) Mesh III (512, 454)(1 + r) Mesh III (512, 454)(r 2 log r + p) Mesh IV (720, 663)(1 + r) Mesh IV (720, 663)(r 2 log r + p) Mesh V (1000, 853)(1 + r) Mesh V (1000, 853)(r 2 log r + p) Mesh VI (1000, 853)(1 + r) Mesh VI (1000, 853)(r 2 log r + p)

3.2921E+05 6.9687E+05 7.2188E+05 1.6265E+06 8.3502E+05 2.2670E+06 7.0963E+05 4.5261E+06 1.4099E+06 2.4351E+06 4.4126E+05 1.0155E+06

4.1928E+07 5.6078E+08 1.6735E+13 6.4587E+19 2.2552E+18 4.0696E+20 5.0890E+19 1.3739E+20 4.1119E+28 6.3813E+21 1.1470E+09 9.7157E+10

A A A NA A NA NA NA NA NA A A

A, applicable; NA, not applicable.


263

When the preconditioning matrix is of the form M = LU where L and U have the same pattern as the L-part and the U-part of A, respectively, the question is whether or not it is possible to find L and U that yield an error that is smaller in some sense than the one we get from the same method without any preconditioning. We can, for example, try to find such an incomplete factorization in which the residual matrix A − LU has zero elements in locations where A has non-zero entries. This turns out to be possible in general and yields the ILU(0) factorization. Generally, a pattern for L and U can be specified and L and U may be sought so that they satisfy certain conditions. This leads to the general class of incomplete factorization techniques.

5.4.3 Level of fill and ILU threshold strategies A general Incomplete LU factorization can be derived by performing Gaussian elimination and dropping some elements in predetermined non-diagonal positions. We can consider any lower triangular matrix L which has the same structure as the lower part of A, and any matrix U which has the same structure as that of the upper part of A. If the product LU were performed, the resulting matrix would not have the same pattern of A, due to the extra diagonals in the product. It is impossible, in general, to match A with this product for any L and U. The entries in these extra diagonals are called fill-in elements. In domain-dominant problems, as are the problems to be solved by the DRMMD approach, the order of the system matrix is very large (being dependant on the number of subdomains and the number of DRM nodes in every subdomain). Still, the matrix has a sparse structure, meaning that most entries aij are zero. However, during the factorization of A, entries that are zero can be replaced by non-zero at later stages (fill-in elements). If these fill-in elements are ignored, then it is possible to find L and U so that their product is equal to A in the other diagonals. This defines the ILU(0) factorization, in general terms: any pair of matrices L (unit lower triangular) and U (upper triangular) so that the elements of A − LU are zero in the locations of NZ(A). The accuracy of the ILU(0) incomplete factorization may be insufficient to yield an adequate rate of convergence. More accurate Incomplete LU factorizations are often more efficient as well as more reliable. These more accurate factorizations will differ from ILU(0) by allowing some fill-in. Incomplete factorizations which rely on the levels of fill are blind to numerical values because elements that are dropped depend only on the structure of A. A generic ILU algorithm with threshold (ILUT) can be derived by including a set of rules for dropping small elements [31]. Applying a dropping rule to an element may only mean replacing the element by zero if it satisfies a set of criteria. The dropping rule for ILU(0) is to drop elements that are in positions not belonging to the original structure of the matrix. In the factorization ILUT(p, τ ), the following procedure is accomplished: an element wk is dropped (i.e. replaced by zero) if it is less than the relative tolerance τi obtained by multiplying τ by the original norm of the ith row (e.g. the 2-norm). Then, a dropping rule of a different type is applied. First, drop again any element in the row with a magnitude that is below the relative tolerance τi . Then, keep only

264 Domain Decomposition Techniques for Boundary Elements the p largest elements in the L part of the row and the p largest elements in the U part of the row in addition to the diagonal element, which is always kept. The goal of the second dropping step is to control the number of elements per row. Here, p can be viewed as a parameter that helps control memory usage, while τ helps to reduce computational cost. There are several possible variations on the implementation of the second dropping step. For example we can keep a number of elements equal to nu(i) + p in the upper part and nl(i) + p in the lower part of the row, where nl(i) and nu(i) are the number of non-zero elements in the L part and the U part of the ith row of A, respectively. This variant is adopted in the ILUT code used for the numerical examples. The changes in the sparsity pattern from a matrix A(400, 343) arising from DRMMD, towards the preconditioned ILUT matrix used as input for the GMRES algorithm are shown in Fig. 4. (a)

(b)

0

0

50

50

100

100

150 150 200 200

250 300

250

350

300

400

0

100

200

300

0

100

nz = 4720

(c)

(d) 0

50

50

100

100

150

150

200

200

250

250

300

300 100

200

nz = 11713

300

nz = 7321

0

0

200

300

0

100

200

300

nz = 23806

Figure 4: (a) Matrix A(m, n); (b) matrix AT A(n, n); (c) preconditioning matrix M = ILUT(n, n); (d) left preconditioned matrix [M −1 (AT A)](n, n).


265

5.4.4 Preconditioned LSQR The number of iterations required for the LSQR algorithm to reach a certain accuracy depends strongly on the scaling of the problem. Poor scaling of the rows or columns of A would lead to inefficient convergence. For example if a row of A is very small or large compared to the other rows of A, the corresponding row of A and b should be scaled up or down. Unless better information is known, the nonzero columns of A should be scaled so that they all have the same Euclidean norm (e.g. 1.0). Another way to reduce the number of iterations is to solve a related system Mx = b efficiently, where M approximates A in some helpful way (e.g. M − A has low rank or its elements are small relative to those of A). LSQR may converge more rapidly on the system AM −1 u = b, after which x can be recovered by solving x = M −1 u. In the numerical analysis we will report the behaviour of LSQR with Jacobian preconditioner and with column scaling.

6 Numerical analysis To test the performance of the different iterative solvers as well as to observe the behaviour of the numerical solution when the two different interpolation functions are used, we will consider a one-dimensional convection–diffusion problem with variable velocity. This equation has been used before as a test example of different implementations in the DRM literature (see [20, 32–34]). In the problem under consideration the convective velocity is assumed to be a linear function of the longitudinal direction and in the governing equation the constant decay parameter is related to the gradient of the velocity in the following way: D

d2 u du − (A + kx) − ku = 0 dx 2 dx

(52)

Dirichlet boundary conditions at the inlet and outlet of the domain are imposed, i.e. u(0) = U0 and u(L) = U1 . In the particular case when D = 1, L = 1 and A = ( ln (U1 /U0 ) − k/2), a simple analytical solution can be obtained: " u = U0 exp

# k 2 U1 k x + ln x − 2 U0 2

(53)

This simplified case is the one that has been used before as a benchmark test in the DRM literature. In our numerical analysis we will deal with this one-dimensional problem as a two-dimensional one in a rectangular domain with dimension 1 × 0.2, where 0 ≤ x ≤ 1 and −0.1 ≤ y ≤ 0.1, the value considered for the decay parameter is k = 40.

266 Domain Decomposition Techniques for Boundary Elements The boundary conditions of our numerical problem are u(0, y) = U0 = 300; ∂u |y=−0.1 = 0; ∂n

u(1, y) = U1 = 10

(54)

∂u |y=0.1 = 0 ∂n

(55)

In Fig. 5 we show the analytical solution for the concentration for the studied problem with the above boundary conditions. In the analysis of the results we will present the profiles of the concentration in semi-logarithmic scale to be able to appreciate in more detail the differences between the analytical and the approximated solutions. For the numerical comparison of the results, accuracy and efficiency are of primary interest. Accuracy is related to several concepts, consistency will be tested through the results for six different subdivisions of the domain (approximation power) and stability will be considered as we report the relative errors for the different meshes and different iterative methods. Regarding efficiency we will study the performance of the iterative methods, along with several chosen preconditioners to accelerate their rate of convergence, in terms of the CPU time and number of iterations. To show the influence of the distribution of the internal nodes in the DRM-MD we present results for four meshes with the same grid but varying the distribution of internal nodes in each subdomain. Besides, we also study two further meshes with a bigger number of subdomains, one with a uniform distribution of subdomains and the other with non-uniform ones. Mesh I (Fig. 6a): with 1 central internal node per subdomain, resulting in an overall mesh of 80 cells and 185 nodes. 350 300

Analytical Solution

Concentration

250 200 150 100 50 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x coord.

Figure 5: Analytical solution for the concentration.

0.9

1


267

0.1

0.05

0 0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

-0.05

-0.1

(a) 0.1

0.05

0 0

0.2

0.4

-0.05

-0.1

(b)

Figure 6: (a) Mesh I and (b) mesh II.

Mesh II (Fig. 6b): 1, 2 or 4 internal nodes per subdomain distributed in a way that more internal nodes are used in those subregions where the bigger errors were obtained with the previous mesh. Resulting in an overall mesh of 80 cells and 257 nodes. Mesh III (Fig. 7a): 1, 3 or 5 internal nodes per subdomain. Here every subdomain has the same distribution of internal nodes as mesh II, plus its central point as an internal node, resulting in an overall mesh with 80 cells and 297 nodes. Mesh IV (Fig. 7b): 5 internal nodes per subdomain. An overall mesh with 80 cells and 505 nodes.


0.05

0 0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

-0.05

-0.1

(a) 0.1

0.05

0 0

0.2

0.4

-0.05

-0.1

(b)

Figure 7: (a) Mesh III and (b) mesh IV.

Mesh V (Fig. 8a): a mesh with 200 cells, increasing the density of the grid in the centre of the domain (opposite to the previous meshes, where the grid was more dense towards the ends of the domain), and with 1 central internal node per subdomain, resulting in an overall mesh of 200 cells and 455 nodes. Mesh VI (Fig. 8b): a mesh with 200 cells, uniformly distributed in a regular grid; with 1 central internal node per subdomain. Resulting in an overall mesh of 200 cells and 455 nodes. As we have stated in Section 4, the six mentioned meshes are tested for (1 + r) and (r 2 log r + P1 ) as interpolating radial functions.


269

0.1

0.05

0 0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

-0.05

-0.1

(a) 0.1

0.05

0 0

0.2

0.4

-0.05

-0.1

(b)

Figure 8: (a) Mesh V and (b) mesh VI.

6.1 Computer implementation If two iterative methods are both capable of generating a sufficiently accurate approximation to a system of linear equations, then we usually compare the two methods by counting operations (how many additions, subtractions, multiplications and divisions each method requires). If the number of operations per iteration is about the same for the two methods, then we might just compare the number of iterations. We should be very careful about iteration count comparisons, to be sure that the algorithms being compared really do require the same amount of work per

270 Domain Decomposition Techniques for Boundary Elements iteration. The bulk of the time for the iterative solver lies in the matrix–vector multiplication and in the preconditioning step. All the algorithms that we implemented perform matrix–vector multiplications, as well as vector inner products during the course of the iteration. This is why all the matrices were stored using the same storage scheme, which allows the operations to be comparable. In the tables we present regarding efficiency we report CPU time, in addition to the number of iterations. All the algorithms were implemented in Fortran using double precision arithmetic. Test runs were performed on a Pentium IV with one 2.53 GHz processor. 6.2 Performance of Krylov iterative solvers 6.2.1 CG scheme A CG algorithm for non-symmetric systems (CGN) was tested for the overdetermined matrix obtained from the DRM-MD approach. As the original residual ri = b − Ax must be available at every step, we compute the residual in two parts: rj+1 = rj − αApj and then AT ri+1 which is the residual for the normal equation AT Ax = AT b. In this case, xm minimizes the function f (x) = (AT A(x∗ − x), (x∗ , x)) (the difference with the GMRES algorithm is the subspace in which the residual norm is minimized), while AT A is never formed in practice. The convergence behaviour of CGN was often quite irregular, leading to a loss of accuracy. The performance improved when a Jacobi preconditioner was applied, after a partial pivoting being implemented to get rid of the zero entries in the main diagonal. As for efficiency, the values of CPU time were one (and sometimes even two) orders of magnitude higher than the values obtained for the CG algorithm used with the explicit normal system of equations. As the results do not compare well against the analytical solution we only present those obtained with the symmetric CG method. To determine whether the use of the explicit normal equation on Krylov solvers is suitable, rather than the direct equation, we perform a singular/eigenvalue analysis. From the comparison of the eigenvalue distribution of AT A against the distribution of singular values of A for the different meshes, we observe that the eigenvalues follow the singular values quite closely. This tendency is increased when the mesh has more internal nodes per subdomain, as we can see in Fig. 9 (mesh I with one internal node per subdomain, mesh IV with five internal nodes per subdomain). In both Fig. 9a and b we can see some scattered eigenvalues in the upper end of the spectrum, but still most of the eigenvalues have a good matching with the singular values. The spectra for meshes II and III are in between those shown in Fig. 9, while the distributions for meshes V and VI follow the same pattern as mesh I (due to also having one internal node per subdomain). In Fig. 9 we show results for the TPS interpolation function, similar behaviour is found for the interpolation function (1 + r). Thus, when we apply the CG algorithm to the normal equation (the algebraic system is symmetric and positive definite), it performs well when we compare the results with the analytical solution of the studied problem. However, when we tested preconditioned versions of CG, the algorithm fails to converge in less than

Iterative Schemes for Solution of Linear Systems Using DRM-MD 9

271

60

sing.v. eigenv.

sing.v. eigenv.

8 50 7

6

40

5 30 4

3

20

2 10 1

0

0

100

200

(a)

300

400

0

0

200

400

600

800

(b)

Figure 9: Singular-eigenvalue distribution for (a) mesh I (r 2 log r + P1 ) and (b) mesh IV (r 2 log r + P1 ). the expected steps for the applied preconditioners. Failures of this type, characterized by very slow convergence, are rather common for CG. One reason of the poor performance of preconditioned CG is that even if M and A are symmetric and positive definite, generally M −1 A is not. Another drawback with CG is the effect of finite precision. In fact, the basis vectors generated by this algorithm in finite precision arithmetic often loses orthogonality completely and may even become linearly dependent, especially when the system is ill-conditioned as it usually happens with normal equations. This situation frequently takes place with the CG algorithm, which uses a short recurrence to generate orthogonal basis vectors for the Krylov subspace [35, 36]. In Fig. 10 we show the concentration profile obtained for the set of six meshes already described; Fig. 10a corresponds to the interpolation function (1 + r) and Fig. 10b to (r 2 log r + P1 ). The corresponding relative errors are presented in Fig. 11. In Table 2 we report the maximum relative errors (%) for the concentration for the two interpolation functions. As can be observed, in this case the best solution is obtained with the mesh IV using the ATPS function and without preconditioner. Notice that the maximum relative error of 11% is found in the region where the analytical solution of the problem is almost zero. The accuracy reached by meshes V and VI (200 subdomains) deteriorate with respect to mesh IV (80 subdomains).

272 Domain Decomposition Techniques for Boundary Elements 1000 Analytical CG Mesh I CG Mesh II

Concentration

100

CG Mesh III CG Mesh IV CG Mesh V

10

CG Mesh VI

1 0

0.2

0.4

0.6

0.8

1

x coord. 0.1 (a) 1000 Analytical CG Mesh I CG Mesh II

100 Concentration

CG Mesh III CG Mesh IV CG Mesh V 10

CG Mesh VI

1 0

0.2

0.4

0.6

0.8

1

x coord. 0.1 (b)

Figure 10: CG semi-log concentration profile per mesh for the interpolation functions (a) (1 + r) and (b) (r 2 log r + P1 ).

To analyse the efficiency of the CG algorithm we will compare the number of iterations and the cpu time required to reach the approximate solution for the six different meshes. In terms of storage requirements, the model size grows with the number of total nodes of the mesh, and consequently the number of iterations and cpu time does (see Table 3). Comparing running time for the same mesh and different interpolation functions, we observe that the cpu time and number


273

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Relative Error (%)

-100

-200

-300

Mesh I Mesh II Mesh III Mesh IV Mesh V Mesh VI

-400 x coord. -500

(a) 40


30

Relative Error (%)

20 10 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-10 -20 x coord. -30

(b)

Figure 11: Relative error for the concentration for the interpolation functions (a) (1 + r) and (b) (r 2 log r + P1 ). of iterations are higher for the function (r 2 log r + P1 ) (Table 3). As we stated in Section 5.3.1, the convergence of CG is related to the condition number and the distribution of the eigenvalues of AT A. Analyzing then the eigenvalue distribution for the six meshes and for the two interpolation functions, we find that for the function (1 + r) the eigenvalues are evenly distributed between λmin and λmax , whereas for (r 2 log r + P1 ) the interval spanned by λmin and λmax is wider (meaning a higher condition number) and the eigenvalues are more irregularly distributed in the upper end of the spectrum, with more scattered values at the top of the interval (see Fig. 12). Accordingly, those systems of equations corresponding to the interpolation function (1 + r) converge more quickly than those obtained from the TPS.

274 Domain Decomposition Techniques for Boundary Elements Table 2: Maximum relative errors for the concentration for the interpolation functions (1 + r) and (r 2 log r + P1 ).

Mesh

No. of Total no. subdomains of nodes

No. of internal Maximum relative error (%) nodes per subdomain (1 + r) (r 2 log r + P1 )

Mesh I Mesh II Mesh III Mesh IV

80

185 257 297 505

1 1–2–4 1–3–5 5

−200.9 −145.9 −149.8 −123.0

16.3 13.6 12.9 11.0

Mesh V Mesh VI

200

455 455

1 1

−333.6 −403.9

35.6 29.6

Table 3: Efficiency in terms of number of iterations and cpu time per mesh, for the interpolation functions (1 + r) and (r 2 log r + P1 ). (1 + r)

(r 2 log r + P1 )

Matrix

No. of iterations

cpu time (s)

No. of iterations

cpu time (s)


AT A AT A AT A AT A

824 1361 1928 1510

1.1678E−01 2.4046E−01 3.5987E−01 4.0822E−01

1031 1802 2787 2481

1.5296E−01 3.1579E−01 5.1908E−01 6.5132E−01

Mesh V Mesh VI

AT A AT A

1240 1275

3.7500E−01 3.7829E−01

1533 1739

4.4441E−01 5.0296E−01

Mesh

Nevertheless, the cpu time remains in the same order of magnitude for the same mesh, and, moreover, the relative error obtained for the concentration through the interpolation function (r 2 log r + P1 ) is one order of magnitude lower than that obtained with (1 + r) (see Fig. 11). 6.2.2 GMRES scheme As we mentioned in the previous section, solving the normal equation is not a bad option when the eigenvalue distribution agrees fairly well with the singular values of the original system. Thus, we will study the performance of the GMRES algorithm applied to AT A for all the six meshes. We will also present the results for two preconditioning techniques, Jacobi and Incomplete LU (ILUT), for accelerating the convergence of the iterative solver. An explicit preconditioner was tested scaling the columns of the matrix in order to make all of them have the same norm. We are not reporting those results, due to a lack of accuracy in the attained solutions.


60

1+r 2 r logr+pol.

275

1+r r2logr+pol.

8 50 7

6

40

5 30 4

3

20

2 10 1

0

0

50

100

150

200

(a)

250

300

350

0

0

100

200

300

400

500

600

700

(b)

Figure 12: Eigenvalue distribution for the two interpolation functions (a) mesh I and (b) mesh IV.

In Fig. 13 we present the concentration profile per mesh for the interpolation functions (1 + r) in part (a) and (r 2 log r + P1 ) in part (b). In Fig. 14 the corresponding relative errors (%) on the concentration are shown. Since we run the system of equations related to every mesh for different preconditioners, we only plot the results for the most efficient preconditioned matrix. We should point out here, that the accuracy attained for the different preconditioners for a certain mesh is nearly the same, with the exception of mesh V for the augmented TPS interpolation function. We will comment on this matter below. Again, for this method, the most accurate solution is obtained for mesh IV (with five internal nodes per subdomain) using as before the augmented TPS function. As regards the relative error for the concentration, we observe that for the function (1 + r) the performance of the GMRES algorithm reaches the same level of accuracy as the CG method (see Figs 13a and 14a). Whereas for the function (r 2 log r + P1 ) we find some small differences in comparison with CG. While the general shape of the error function remains the same, it is possible to observe some irregularities, especially for mesh V (see Fig. 14b). The range of x coordinates in which we spot those inconsistencies matches the interval of the mesh where the boundary nodes of the subdomains have their maximum proximity, namely for 0.3m ≤ x ≤ 0.5m (see Fig. 8a). Some irregularities found for meshes II, III and IV can also be related to the closeness of the mesh nodes, this time between internal

276 Domain Decomposition Techniques for Boundary Elements 1000 Analytical Prec. ILU M I GMRES M II

100 Concentration

Prec. ILU M III Prec. ILU M IV GMRES M V

10

GMRES M VI

1 0

0.2

0.4

0.6

0.8

1

x coord. 0.1 (a) 1000 Analytical Prec. ILU M I Prec. Diag. M II

100 Concentration

Prec. Diag. M III Prec. Diag. M IV Prec. Diag. M V

10

Prec. ILU M VI

1 0

0.2

0.4

0.6

0.8

1

x coord. 0.1 (b)

Figure 13: GMRES semi-log concentration profile per mesh for the interpolation functions (a) (1 + r) and (b) (r 2 log r + P1 ).

nodes and boundary nodes, as well as between internal nodes with each other. We have to mention at this point that in the limit of a mesh refinement process, the inconsistencies found (as far as stability is concerned) due to the proximity between boundary nodes outweigh those related to the vicinity of internal nodes, proving GMRES to be very sensitive to the mesh width parameter. In Fig. 15 we show the differences in the attained accuracy for the concentration, when different preconditioners are applied to the system of equations corresponding to mesh V. This fact highlights the influence of the mesh width parameter in the


277

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Relative Error (%)

-100

-200

-300


-400

x coord. -500

(a) 40


Relative Error (%)

30

20

10

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-10 x coord.

-20

(b)

Figure 14: Relative error for the concentration for the interpolation functions (a) (1 + r) and (b) (r 2 log r + P1 ). convergence of the GMRES algorithm, as it is for mesh V where this parameter takes the smallest values for coordinates 0.3m ≤ x ≤ 0.5m. In Table 4 we report the maximum relative errors (%) of the concentration for the different preconditioning schemes and the two interpolation functions. The irregularities mentioned above for mesh V can also be observed in this table. As mentioned for CG, a lack of convergence is found for the DRM-MD when the approximation power is increased (meshes V and VI). In Table 5 the number of iterations and cpu time are presented. The numbers in parenthesis in these two tables mean either the relative error was attained in too many iterations or for those number of iterations and cpu time the obtained relative error was not acceptable.


Analytical GMRES Prec.Diag.

Concentration

100

Prec.ILU

10

1 0

0.2

0.4

0.1

0.6

0.8

1

x coord.

Figure 15: Semi-log concentration profile, mesh V, interpolation function (r 2 log r + P1 ).

Table 4: Maximum relative errors for the concentration for the interpolation functions (1 + r) and (r 2 log r + P1 ). Maximum relative error (%) (1 + r)

Mesh

Without preconditioner

(r 2 log r + P1 )

With preconditioner Jacobi

ILU

Without preconditioner

With preconditioner Jacobi

ILU


−201.2 −146.1 −150.2 −123.0

−202.1 −146.1 (−149.5) −123.5

−200.8 – −151.2 −124.8

16.8 13.8 13.0 11.0

− 13.4 12.9 13.9

16.3 – – –

Mesh V Mesh VI

−332.4 −404.7

(−335.4) (−409.0)

– −403.3

49.6 27.4

59.5 31.9

34.3 30.2

From the results in Tables 4 and 5 we observe that the GMRES method presents an erratic behaviour. There is not a single GMRES scheme which will perform better for all the meshes, or for the same interpolation function. The running time for unpreconditioned GMRES is similar for the two RBFs. The cpu time increases with the total number of nodes per mesh, this increment is more


279

Table 5: Efficiency in terms of number of iterations and cpu time per mesh for the interpolation functions (1 + r) and (r 2 log r + P1 ). (1 + r) No. of iterations

cpu time (s)

No. of iterations

cpu time (s)

AT A AT A Jacobi preconditioner AT A ILU preconditioner

232 76

2.6053E−01 4.9342E−02

224 92

2.4934E−01 5.0658E−02

12

7.5658E−03

14

1.7763E−02


310 3500

5.5329E−01 2.5612E+00

305 135

5.2928E−01 1.1974E−01

–

–

(356)

(1.3230E+00)


358 –

7.9605E−01 –

357 165

7.8026E−01 1.8651E−01

209

4.8092E−01

(265)

(6.8980E−01)


401 147

1.5730E+00 2.4836E−01

453 174

2.0115E+00 3.3849E−01

276

9.8026E−01

–

–


502 –

2.9967E+00 –

483 186

2.7980E+00 5.0461E−01

–

–

187

7.6414E−01


506 –

3.0444E+00 –

487 205

2.8270E+00 5.7039E−01

21

6.4803E−02

37

7.9276E−02

Mesh

Matrix

I

II

III

IV

V

VI

(r 2 log r + P1 )

280 Domain Decomposition Techniques for Boundary Elements distinctive when there are more subdomains per mesh (meshes V and VI), rather than more internal nodes per subdomain (meshes I to IV). Preconditioning softens this mesh size dependence, at the expense of losing accuracy. In an attempt to understand the performance of the incomplete LU preconditioner, we will briefly study the eigenvalue distribution of this preconditioner. The spectrum of the ILU matrices lies on the complex plane. For those cases in which the distribution of the eigenvalues remains on the positive part of the real axis, and furthermore their distribution takes the shape of an ellipse symmetrically oriented along the real right axis, the incomplete factorization method can be used as a good smoother for multidomain methods, improving the rate of convergence. The incomplete factorization scheme applied to mesh I (Fig. 16) turns out to be the most efficient accelerator, reducing the convergence time by one order of magnitude. When the same factorization is applied to mesh VI (Fig. 17) again the performance of GMRES is improved by dropping the cpu time by one order of magnitude compared to the unpreconditioned results (Table 5). For meshes II to V the incomplete factorization is either a poor option or it is merely not applicable

400

400

(r2 log r + pol.)

(1+r) 300

300

200

200

100

100

0

0

−100

−100

−200

−200

−300

−300

−400

0

0.5

1

1.5

2

2.5

3

−400

0

0.5

1

1.5

4

(a)

2

2.5

3 4

x 10

x 10

(b)

Figure 16: Eigenvalue distribution of the incomplete LU preconditioner for the interpolation functions (a) (1 + r) and (b) (r 2 log r + P1 ) (mesh I).


4000 (r2 log r + pol.)

(1+r)

3000

3000

2000

2000

1000

1000

0

0

−1000

−1000

−2000

−2000

−3000

−3000

−4000

281

0

1

2

3

4

5

6

−4000

0

1

2

3

4

(a)

x 10

4

5 4

(b)

x 10

Figure 17: Eigenvalue distribution of the incomplete LU preconditioner for the interpolation functions (a) (1 + r) and (b) (r 2 log r + P1 ) (mesh VI).

(see Table 5). This is a consequence of the high condition number of the resulting preconditioner (as mentioned in Section 5.4.2 and shown in Table 1), and also to the eigenvalues distributed on both the positive and negative parts of the real axis. The fact that the numerical results improve when the subdomains have got more number of internal nodes can be related to the intrinsic properties of the system of equations arising from different discretizations of the domain. For this reason we study the distribution of eigenvalues for the six meshes and the two interpolation functions. When using a very simple interpolation function as is (1 + r), the accuracy of the approximated solution improves when we increase the number of internal nodes per subdomain. The gap in the distribution of eigenvalues for mesh I (Fig. 18a) is closed when more internal nodes are considered and this is reflected in a more uniform distribution of the eigenvalues along with many nearly equal singular values clustered around the number 1. On the other hand, when we use (r 2 log r + P1 ) as the interpolation function, the results are more accurate and the increment of the number of internal nodes per subdomain has a less dramatic influence on the improvement of the approximated solution (Fig. 18b): there is not


mesh I mesh II mesh III mesh IV

10

9

mesh I mesh II mesh III mesh IV 25

8

7

20

6 15

5

4 10 3

2 5 1 1 100

200

300

(a)

400

500

600

100

200

300

400

500

600

(b)

Figure 18: Eigenvalue distribution for meshes I to IV for the interpolation functions (a) (1 + r) and (b) (r 2 log r + P1 ). such a gap in the spectrum of mesh I, still we can notice a more uniform distribution of eigenvalues clustered around the number 1 as the number of internal nodes increases. In Fig. 18 we present the changes in the eigenvalue distribution for the coefficient matrices that arise from DRM-MD when the domain is divided into 80 subdomains, and the number of internal nodes per subdomain is increased from one to five, namely from mesh I to mesh IV, for the two RBFs used. 6.2.3 LSQR scheme In this section we present the numerical results obtained by the LSQR algorithm applied to the overdetermined system of equations obtained from DRM-MD. To study the performance of this method we solve the tested problem for the six meshes previously considered. Besides, we check the behaviour of three different preconditioners. First we preprocessed the matrix in order to place large entries on the main diagonal, which improves the stability of the method (due to the way in which the assembly of the whole system of equations is done in DRM-MD, we get zero values on the main diagonal). Although this practice may destroy some useful sparsity structure of A (see Fig. 19), it allows us to use the new diagonal as a Jacobi preconditioner. For this purpose we take the diagonal of the square part of

Iterative Schemes for Solution of Linear Systems Using DRM-MD nz = 4720

283

nz = 4720

0

0

50

50

100

100

150

150

200

200

250

250

300

300

350

350

400

400 0

50

100

150

200

250

300

0

50

100

(a)

150

200

250

300

(b)

Figure 19: Matrix A (a) before and (b) after preprocessing. Table 6: Condition numbers of the matrix A before and after scaling. Condition number A

Condition number A-scaled

Mesh I (400, 343)(1 + r) Mesh I (400, 343)(r 2 log r + p) Mesh II (472, 415)(1 + r) Mesh II (472, 415)(r 2 log r + p) Mesh III (512, 454)(1 + r) Mesh III (512, 454)(r 2 log r + p) Mesh IV (720, 663)(1 + r) Mesh IV (720, 663)(r 2 log r + p)

5.7377E+02 8.3479E+02 8.4964E+02 1.2753E+03 9.1379E+02 1.5056E+03 8.4240E+02 2.1275E+03

3.2280E+01 3.8935E+01 4.4604E+01 5.8059E+01 5.3534E+01 8.4657E+01 6.2815E+01 7.8179E+01

Mesh V (1000, 853)(1 + r) Mesh V (1000, 853)(r 2 log r + p) Mesh VI (1000, 853)(1 + r) Mesh VI (1000, 853)(r 2 log r + p)

1.1874E+03 1.5605E+03 6.6428E+02 1.0077E+03

9.6298E+01 1.3422E+02 7.6106E+01 1.0086E+02

Mesh – A(m, n) – RBF

the preprocessed matrix in order to use a diagonal matrix D(n, n). For the diagonal preconditioning we perform [A(m, n) · D−1 (n, n)], obtaining the preconditioned matrix with dimensions (m, n). Column scaling is another preconditioning strategy that we experimented with, so as to obtain an equivalent system with a matrix that will be better conditioned (see Table 6), thereby speeding up the solution process and having a beneficial effect on the accuracy of the computed solution.

284 Domain Decomposition Techniques for Boundary Elements 1000 Analytical Scale Mesh I Scale Mesh II Scale Mesh III Scale Mesh IV

Concentration

100

Scale Mesh V Scale Mesh VI

10

1 0

0.2

0.4

0.6

0.8

1

x coord. 0.1 (a)

1000 Analytical Scale Mesh I Scale Mesh II

Concentration

100

Scale Mesh III Scale Mesh IV Scale Mesh V Scale Mesh VI

10

1 0

0.2

0.4

0.6

0.8

1

x coord. 0.1 (b)

Figure 20: LSQR semi-log concentration profile per mesh for the interpolation functions (a) (1 + r) and (b) (r 2 log r + P1 ).

The results for the concentration of the test problem are reported in Fig. 20, for the six meshes and for the two interpolation functions. Again here we are plotting the most efficient solution which turned out to be, for all the cases, the scaling scheme. When we observe the distribution of the relative errors, in Fig. 21, we can see that no irregularities are present for either the six meshes or the two interpolation functions, as was the case for the numerical results we got for the


285

0 0

0.2

0.4

0.6

0.8

1

Relative Error

-100

-200

-300 Mesh I Mesh II Mesh III Mesh IV Mesh V Mesh VI

-400

x coord.

-500

(a) 50 Mesh I Mesh II Mesh III Mesh IV Mesh V Mesh VI

40

Relative Error

30 20 10 0 0

0.2

0.4

0.6

0.8

1

-10 -20

x coord. (b)

Figure 21: Relative error for the concentration for the interpolation functions (a) (1 + r) and (b) (r 2 log r + P1 ).

CG and GMRES methods (Figs 11 and 14). Moreover, the LSQR algorithm performs better for the same preconditioned scheme, regardless of the mesh tested or the interpolation function used. In Table 7 we report the maximum relative errors (%) for the concentration for the two interpolation functions. We only show the numbers for the scaling scheme because all the other schemes reached the same level of accuracy, the difference between them being in the solving time. As in the

286 Domain Decomposition Techniques for Boundary Elements Table 7: Maximum relative errors for the concentration for the interpolation functions (1 + r) and (r 2 log r + P1 ).

Mesh

No. of Total no. subdomains of nodes

No. of internal Maximum relative error (%) nodes per subdomain (1 + r) (r 2 log r + P1 )


80

185 257 297 505

1 1–2–4 1–3–5 5

−200.9 −145.9 −149.8 −123.0

16.3 13.6 12.9 11.0

Mesh V Mesh VI

200

455 455

1 1

−333.6 −403.9

35.6 29.6

previous cases the best result is found with the mesh IV and the use of the ATPS function. CPU time results are presented in Table 8, as well as the number of iterations to converge. It is clear from these results that scaling the matrix is the most reliable preconditioned scheme, which greatly enhances the efficiency of the LSQR solver. Even when the cpu time, reported for a specific mesh and interpolation function, remains at the same order of magnitude as the most efficient implementations of the CG and GMRES methods (Tables 3 and 5), the LSQR convergence method is always more stable than the others, yielding to a more uniform distribution of the relative error. In Fig. 22 we present the changes in the distribution of the singular values of the coefficient matrices when the number of internal nodes per subdomain is increased from one to five, i.e. for meshes I to IV. Figure 22a corresponds to the RBF (1 + r) and Fig. 22b to the ATPS. As in the case of the distribution of eigenvalues (Section 6.2.2, Fig. 18), the systems of equations with more nearly equal singular values, clustered around 1, attain better solutions. It is worth, at this point, to take an insight into the distribution of the singular values of the original matrices and their scaled versions. In Figs 23–25 we observe that the singular values of the scaled matrices are in all cases distributed in the interval (0, 3); besides, the curves for both (1 + r) and r 2 log r + P1 have nearly the same distribution. As a consequence of this we find a very regular behaviour of the scaled LSQR algorithm, not showing the high sensitivity to the mesh width parameter we reported for GMRES. Scaling also makes the matrix norm smaller, resulting in a good attempt to reduce the inaccuracy contributed by roundoff errors. Up to now we have reported the computed solutions for the concentration. In Figs 26 and 27 we show the numerical results obtained for the derivatives in the x direction for the ATPS interpolation function. We should recall that Meshes I to IV have the same grid with different distributions of internal points per subdomain.


287

Table 8: Efficiency in terms of number of iterations and cpu time per mesh, for the interpolation functions (1 + r) and (r 2 log r + P1 ). (1 + r) Mesh Matrix

(r 2 log r + P1 )

No. of No. of iterations cpu time (s) iterations cpu time (s)

I

A A preprocessor A Jacobi preconditioner A scale

1162 1159 624 100

3.4507E−01 3.4572E−01 1.7961E−01 3.1250E−02

1555 1559 919 202

4.4474E−01 4.5789E−01 2.6612E−01 5.9539E−02

II


1776 1766 860 120

6.3684E−01 6.3059E−01 3.0855E−01 4.4408E−02

2484 2491 1265 300

8.7467E−01 8.9408E−01 4.5263E−01 1.0954E−01

III


3982 3255 1761 413

1.5681E+00 1.2924E+00 7.2237E−01 1.6579E−01

4064 4064 1797 548

1.5658E+00 1.6247E+00 7.1250E−01 2.1579E−01

IV


2131 2143 – 178

1.2533E+00 1.2543E+00 – 1.0757E−01

3577 3569 – 351

2.1247E+00 2.0720E+00 – 2.1480E−01

V


2595 2591 1871 245

1.9059E+00 1.9224E+00 1.3997E+00 1.8421E−01

3479 3487 1503 434

2.5480E+00 2.5911E+00 1.0842E+00 3.1612E−01

VI


2067 2070 1122 249

1.5171E+00 1.5243E+00 8.3026E−01 1.8388E−01

3015 3022 1193 369

2.2359E+00 2.2566E+00 8.7829E−01 2.7664E−01

While the three iterative algorithms performed similarly, as far as accuracy in the computed concentration is concerned, we find LSQR to be more reliable than the others for the results on the derivatives, reaching solutions with smaller relative errors. When we focus on the computed results for the x-derivatives for meshes I, V and VI – whose common pattern is that they all have one internal point per subdomain, while there is a refinement in the mesh width parameter – we still find that the DRM-MD approach loses convergence with the proximity of the boundary nodes.

288 Domain Decomposition Techniques for Boundary Elements mesh I mesh II mesh III mesh IV

3

mesh I mesh II mesh III mesh IV

7

6

2.5

5 2

4 1.5 3

1 2

0.5

1

100

200

300

400

500

100

600

(a)

200

300

400

500

600

(b)

Figure 22: Singular value distribution for meshes I to IV: (a) (1 + r) and (b) (r 2 log r + P1 ). A (1+r) A (r2log r+p ) 1 ASc. (1+r) ASc. (r2log r+p ) 1

2.5

A (1+r) A (r2logr+p ) 1 ASc. (1+r) 2 ASc. (r logr+p1)

4

3.5

3

2

2.5 1.5 2

1.5

1

1 0.5 0.5

50

100

150

200

(a)

250

300

350

400

50

100 150 200 250 300 350 400 450

(b)

Figure 23: Singular value distribution of the matrices A and A-scaled, for the interpolation functions (1 + r) and (r 2 log r + P1 ): (a) mesh I and (b) mesh II.

289


8 A (1+r) 2 A (r logr+pol.) ASc.(1+r) ASc.(r2logr+pol.)

A (1+r) A (r2logr+pol.) ASc.(1+r) 2 ASc.(r logr+pol.)

7

5 6 4 5

3

4

3 2 2 1 1

0

0

100

200

300

400

500

0

0

100

200

(a)

300

400

500

600

700

(b)

Figure 24: Singular value distribution of the matrices A and A-scaled, for the interpolation functions (1 + r) and (r 2 log r + P1 ): (a) mesh III and (b) mesh IV.

3.5

3 A (1+r) A (r2logr+pol.) ASc.(1+r) ASc.(r2logr+pol.)

A (1+r) A (r2logr+pol.) ASc.(1+r) ASc.(r2logr+pol.)

3

2.5

2.5 2 2 1.5 1.5

1 1

0.5

0

0

0.5

200

400

(a)

600

800

0

0

200

400

600

800

(b)

Figure 25: Singular value distribution of the matrices A and A-scaled, for the interpolation functions (1 + r) and (r 2 log r + P1 ): (a) mesh V and (b) mesh VI.

290 Domain Decomposition Techniques for Boundary Elements 100 80

Relative Error (%)

60 40 20 0 -20

0

0.2

0.4

0.6

0.8

1

-40

Mesh I

-60

Mesh II Mesh III

-80

Mesh IV

x coord.

-100

(a) 100 80 60 Relative Error (%)

40 20 0 -20 0

0.2

0.4

0.6

0.8

1

-40 -60

Mesh I

-80

Mesh II Mesh III

-100 -120

x coord.

Mesh IV

(b)

Figure 26: Relative errors for the longitudinal derivative for meshes with 80 subdomains. Interpolation function (r 2 log r + P1 ): (a) CG and (b) GMRES. Despite this fact, we observe that the LSQR algorithm attains more stable results (see Figs 28 and 29). 6.3 Direct methods The efficient solution of a linear system is largely a function of the proper choice of the iterative method. A direct method may sometimes be preferable to an iterative


291

100 80

Relative Error (%)

60 40 20 0 -20

0

0.2

0.4

0.6

0.8

1

-40

Mesh I

-60

Mesh II Mesh III

-80 -100

x coord.

Mesh IV

LSQR

Figure 27: Relative errors for the longitudinal derivative for meshes with 80 subdomains obtained with LSQR. Interpolation function (r 2 log r + P1 ).

method. The performance of direct methods, both for dense and sparse systems, is to a great extent that of the factorization of the matrix. This operation is absent in iterative methods (although preconditioners may require a setup phase), and with it, iterative methods do not require dense matrix suboperations, which involve high computational cost. Furthermore, the basic operations in iterative methods often use indirect addressing, depending on the data structure, having a low cost of execution. However, this does not imply anything about the total solution time for a given system. On the other hand, iterative methods will always retain advantages for certain applications in which we can predict a quick convergence, and they will also have lower storage requirements than direct methods. In spite of this, it is still worth analyzing the performance of some direct methods, in particular to compare their results with indirect solutions. When we solve an overdetermined system of equations in the least squares sense it always leads us to matrices of the form AT A, which is then needed to be inverted. If we are ready to use AT A, this matrix being symmetric and positive definite, then it has a special and efficient triangular decomposition. Thus, Cholesky decomposition is a useful tool. When we can use it, this decomposition is about a factor of two faster than alternative direct methods for solving linear equations. It is extremely stable numerically and once the matrix is decomposed, the triangular factor can be used to solve the linear system by back substitution. SVD is amongst a set of very powerful techniques for dealing with sets of equations or matrices that are either singular or else numerically very close to singular. SVD is also the method of choice for solving most linear least-squares problems.

292 Domain Decomposition Techniques for Boundary Elements 100 80

Relative Error (%)

60 40 20 0 -20

0

0.2

0.4

0.6

0.8

1

-40 Mesh I

-60

Mesh V

-80

x coord.

Mesh VI

-100 (a) 100 80

Relative Error (%)

60 40 20 0 -20

0

0.2

0.4

0.6

0.8

1

-40 Mesh I

-60

Mesh V

-80 x coord.

Mesh VI

-100 (b)

Figure 28: Relative errors for the longitudinal derivative for meshes with 1 internal node per subdomain. Interpolation function (r 2 log r + P1 ): (a) CG and (b) GMRES.

We are not presenting results for this method as its time consumption is highly uncompetitive. Golub’s method is a direct method that works directly with A and has the advantage of being considerably more accurate than methods that invert AT A. It is reckoned that about twice as much precision is required to invert AT A than the


293

100 80

Relative Error (%)

60 40 20 0 -20

0

0.2

0.4

0.6

0.8

1

-40

Mesh I

-60 -80

Mesh V x coord.

Mesh VI

-100

LSQR

Figure 29: Relative errors for the longitudinal derivative for meshes with 1 internal node per subdomain obtained with LSQR. Interpolation function (r 2 log r + P1 ).

one needed to deal directly with A. Golub’s method uses Householder transformations to convert any general overdetermined matrix into an upper triangular form [30]. In this section we describe the results for Golub’s method. The solutions reached by this direct method are in good agreement with those obtained with the indirect methods. The estimated solutions for the concentration and for the derivatives attain the same results as those obtained by the LSQR algorithm. In particular, for mesh IV and the ATPS function (where the lowest relative errors where obtained for the iterative techniques) both direct and indirect techniques yielded the same relative error, the direct solutions being more time-consuming. In Table 9 the solving time for different preconditioning schemes is shown, where it can be observed that preconditioning has no effect upon the computed solutions for direct solvers. The preconditioning schemes are the same as those applied for the LSQR method. A comparison of the CPU times for all the implemented iterative techniques and for Golub’s method is presented in Table 10.

7 Conclusions This work presents a thorough analysis of the performance and reliability of several preconditioned iterative techniques for the solution of linear systems arising from the DRM-MD integral equation approach. In the experiments shown in this work the shape functions used for every boundary element for the approximation of the

294 Domain Decomposition Techniques for Boundary Elements Table 9: Efficiency for different preconditioners per mesh for Golub’s method. Interpolation functions (1 + r) and (r 2 log r + P1 ). cpu time (s) (1 + r)

(r 2 log r + P1 )

A A preprocessor A Jacobi A scale

4.4967E−01 4.5132E−01 4.5921E−01 4.6776E−01

4.4901E−01 4.4934E−01 4.5757E−01 4.6678E−01

II


7.6579E−01 7.6809E−01 7.8059E−01 7.9145E−01

7.6645E−01 7.6612E−01 7.7928E−01 7.9112E−01

III


9.8243E−01 9.7826E−01 1.0172E+00 1.0112E+00

9.9178E−01 9.9638E−01 1.0388E+00 1.0211E+00

IV


2.9707E+00 3.0257E+00 3.0158E+00 3.0822E+00

2.9655E+00 2.9674E+00 – 3.0319E+00

V


7.2030E+00 7.3148E+00 7.4895E+00 7.3217E+00

7.2049E+00 7.3717E+00 7.9793E+00 7.3168E+00

VI


7.3095E+00 7.3155E+00 7.8112E+00 7.4164E+00

7.1974E+00 7.1970E+00 7.9428E+00 7.3151E+00

Mesh

Matrix

I

geometry and for the considered governing equations are linear functions. For the interpolation functions used by the DRM we test the results for two different radial basis interpolation functions, namely (1 + r) and (r 2 log r + P1 ). Besides, we run all these series of tests for six different meshes, in which the whole domain is subdivided, refining both the grid and the number of internal nodes per subdomain. This allows us to understand the convergence behaviour of the DRM-MD. From the analysis per method with different preconditioners we observed that the most accurate situation was consistently obtained for (r 2 log r + P1 ). The most accurate results for a single mesh and different preconditioning schemes turned out to be also the most efficient.


295

Table 10: Efficiency per method with the best preconditioner for every mesh. Interpolation functions (1 + r) and (r 2 log r + P1 ). Mesh

Method

(1 + r) cpu time (s)

(r 2 log r + P1 ) cpu time (s)

I

Golub Scaled CGN CG ILU GMRES Scaled LSQR

4.4967E−01 2.6217E−01 1.1678E−01 7.5658E−03 3.1250E−02

4.4901E−01 2.0526E−01 1.5296E−01 1.7763E−02 5.9539E−02

II


7.6579E−01 6.5000E−01 2.4046E−01 1.6875E−01 4.4408E−02

7.6612E−01 8.3355E−01 3.1579E−01 5.2928E−01 1.0954E−01

III


9.8243E−01 1.0882E+00 3.5987E−01 4.8092E−01 1.6579E−01

9.9178E−01 1.5582E+00 5.1908E−01 1.8651E−01 2.1579E−01

IV

Golub Scaled CGN CG Jacobi GMRES Scaled LSQR

3.2375E+00 2.0766E+00 4.0822E−01 2.4836E−01 1.0625E−01

3.1941E+00 1.8967E+00 6.5132E−01 3.3849E−01 2.1131E−01

V

Golub Scaled CGN CG GMRES Scaled LSQR

7.2030E+00 1.8171E+00 3.7500E−01 2.9967E+00 1.8421E−01

7.2049E+00 1.4372E+00 4.4441E−01 5.0461E−01 3.1612E−01

VI


7.3095E+00 2.8345E+00 3.7829E−01 6.4803E−02 1.8388E−01

7.1970E+00 2.0191E+00 5.0296E−01 7.9276E−02 2.7664E−01

In relation to the CG and GMRES Krylov subspace methods, we solved the explicit normal system of equations after having found a good agreement between the singular eigenvalue distributions of A and AT A. GMRES outperforms CG, but proves to be too sensitive to the mesh width parameter, resulting in an erratic behaviour of the method. With reference to preconditioning schemes, GMRES improves its convergence rate through different preconditioners, although the

296 Domain Decomposition Techniques for Boundary Elements accelerator happened to be different for every test carried out, showing irregularities in its performance once more. LSQR yields the most accurate and efficient solutions when used in combination with a scaling of the input matrix. This not only has a stabilizing effect on the computed solutions but also alleviates the calculations of roundoff errors. As we can notice from the numerical results, not every method will work on every problem type, and knowledge of matrix properties is the main criterion for selecting an iterative technique. For this reason a study on the intrinsic properties of the systems is presented. A concluding remark on this subject is the regularization on the distribution of the singular (eigen) values of the matrices when the number of the internal nodes per subdomain is increased. With reference to the convergence of the DRM-MD, it is clear that the computed solutions improve when we increase the number of internal nodes per subdomain for a single grid in which the domain is divided. On the other hand, when the refinement of the mesh consists in increasing the number of subdomains, the solution deteriorates losing accuracy, and also efficiency due to a higher storage requirement. We should point out here that the proximity of interior nodes also generates a reduction in accuracy, but this is a local problem and far less important than what is observed due to the proximity of boundary nodes. Therefore, it might be worth seeking an adaptive meshing technique capable of considering the increment of internal nodes per subdomain prior to the refining of the mesh itself.

References [1]

[2] [3]

[4]

[5] [6]

[7] [8]

Berzins, M. & Walshaw, C., Tutorial on domain decomposition for unstructured meshes. Proc. of Domain Decomposition for CFD on Message Passing Machines, University of Leeds, UK, 1992. Banerjee, P.K., The Boundary Element Methods in Engineering, McGraw– Hill: London, 1981. Kane, J.H., Keyes, D.E. & Guru Prasad, K., Iterative solution techniques in boundary element analysis. International Journal for Numerical Methods in Engineering, 31, pp. 1511–1536, 1991. Guru Prasad, K., Kane, J.H., Keyes, D.E. & Balakrishna, C., Preconditioned Krylov solvers for BEA. International Journal for Numerical Methods in Engineering, 37, pp. 1651–1672, 1994. Taigbenu, A.E., The Green element method. International Journal for Numerical Methods in Engineering, 38, pp. 2241–2263, 1995. Taigbenu, A.E. & Onyejekwe, O.O., Green element simulations of the transient nonlinear unsaturated flow equation, Applied Mathematical Modelling, 19, pp. 675–684, 1995. Ramsak, M. & Skerget, L., Mixed boundary elements for high re laminar flows. BEM 21, 1999. Paige, C., Saunders, M., LSQR sparse linear equations and least squares problems. ACM Transactions on Mathematical Software, 8(2), pp. 195–209, 1982.


[9]

[10] [11] [12]

[13] [14]

[15]

[16]

[17] [18]

[19]

[20]

[21]

[22] [23]

297

Nardini, D. & Brebbia, C.A., A new approach to free vibration analysis using boundary elements. Boundary Element Methods in Engineering, Computational Mechanics Publications: Southampton and Springer Verlag: Berlin, 1982. Ahmad, S. & Banerjee, P.K., A new method in vibration analysis by BEM using particular integrals, J. Eng. Mech. Div., ASCE, 113, 1986. Golberg, M.A. & Chen, C.S., Discrete Projection Methods for Integral Equations, Computational Mechanics Publications: Southampton, 1997. Bulgakov, V., Sarler, B. & Kuhn, G., Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation. International Journal for Numerical Methods in Engineering, 43, pp. 713– 732, 1998. Popov, V. & Power, H., A domain decomposition in the dual reciprocity approach. Boundary Element Communications, 7(1), pp. 1–5, 1996. Kansa, E.J. & Carlson, R.E., Radial basis functions: a class of grip-free, scattered data approximations. Computational Fluid Dynamics Journal, 3(4), pp. 479–496, 1995. Gipson, G.S., Boundary Element Fundamentals: Basic Concepts and Recent Developments in the Poisson Equation. Computational Mechanics Publications: Southampton, 1987. Partridge, P.W., Brebbia, C.A. & Wrobel, L.C., The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications: Southampton, 1992. Brebbia, C.A., Telles, J.C.F. & Wrobel, L.C., Boundary Element Techniques. Springer-Verlag: Berlin, 1984. Fairweather, G., Rizzo, F.J., Shippy, D.J. & Wu, Y.S., On the numerical solution of two-dimensional potential problems by an improved boundary integral equation method. Journal of Computational Physics, 31, pp. 96–112, 1979. Nowak, A.J. & Brebbia, C.A., The multiple reciprocity method: a new approach for transforming BEM domain integrals to the boundary. Engineering Analysis with Boundary Elements, 6(3), pp. 164–167, 1989. Popov, V. & Power, H., The DRM-MD integral equation method: an efficient approach for the numerical solution of domain dominant problems. International Journal for Numerical Methods in Engineering, 44(3), pp. 327–353, 1999. Popov, V. & Power, H., The DRM-MD integral equation method for the numerical solution of convection-diffusion equation. Boundary Element Research in Europe, Computational Mechanics Publications: Southampton, pp. 67–81, 1999. Micchelli, C.A., Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx., 2, pp. 11–22, 1986. Schaback, R., Multivariate interpolation and approximation by translates of a basis function. Approximation Theory VIII, eds. C.K. Chui & L.L. Schumaker, pp. 1–8, 1995.

298 Domain Decomposition Techniques for Boundary Elements [24] [25] [26]

[27]

[28] [29]

[30] [31] [32] [33] [34]

[35] [36]

Franke, R., Scattered data interpolation: tests of some methods. Math. Comp., 38, pp. 181–200, 1982. Stead, S., Estimation of gradients from scattered data. Rocky Mount. J. Math., 14, pp. 265–279, 1984. Hardy, R.L., Theory and applications of the multiquadric-biharmonic method: 20 years of discovery. Comput. Math. Applic., 19, pp. 163–208, 1990. Duchon, J., Spline minimizing rotation-invariant seminorms in Sobolev spaces. Constructive Theory of Functions of Several Variables, Lectures Notes in Mathematics, Vol. 571, ed. W. Schempp & K. Zeller, pp. 85–100, Springer Verlag: Berlin, 1977. Madych, W.R. & Nelson, S.A., Multivariable interpolation and conditionally positive definite functions-II. Math. Comput., 54, pp. 211–230, 1990. Hestenes, M.R. & Steifel, E., Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards, 49, pp. 409–436, 1952. Golub, G.H. & Kahan, W., Calculating the singular values and pseudoinverse of a matrix. SIAM J. Numer. Anal. Ser. B 2, 2, pp. 205–224, 1965. Saad, Y., Iterative Methods for Sparse Linear Systems. PWS: Boston, 1996. De Figueiredo, D.B., Boundary element analysis of convection-diffusion problems, PhD Thesis, Wessex Institute of Technology: Southampton, 1990. Partridge, P.W., Transport analysis using boundary elements. Software Package, Computational Mechanics Publications: Southampton, 1993. Qiu, Z.H., Wrobel, L.C. & Power, H., An evaluation of boundary element schemes for convection-diffusion problems. Boundary Elements XV, eds. C.A. Brebbia & J.J. Remcis, Computational Mechanics Publications: Southampton, 1993. Greenbaum Anne, Iterative Methods for Solving Linear Systems. SIAM: Philadelphia, PA, 1997. Axelsson, O., Iterative Solution Methods, Cambridge University Press: Cambridge, 1994.

Trefftz and Collocation Methods A. H-D. CHENG, University of Mississippi, USA, Z-C. LI, National Sun Yat-sen University, Taiwan, T-T. LU, National Center for Theoretical Science, Taiwan, HY. HU, National Tsing Hua University, Taiwan This book covers a class of numerical methods that are generally referred to as “Collocation Methods”. Different from the Finite Element and the Finite Difference Method, the discretization and approximation of the collocation method is based on a set of unstructured points in space. This “meshless” feature is attractive because it eliminates the bookkeeping requirements of the “element” based methods. This text discusses several types of collocation methods including the radial basis function method, the Trefftz method, the Schwartz alternating method, and the coupled collocation and finite element method. Governing equations investigated include Laplace, Poisson, Helmholtz and biharmonic equations. Regular boundary value problems, boundary value problems with singularity, and eigenvalue problems are also examined. Rigorous mathematical proofs are contained in these chapters, and many numerical experiments are also provided to support the algorithms and to verify the theory. A tutorial on the applications of these methods is also provided. ISBN: 978-1-84564-153-5 2007 apx 400pp apx £135.00/US$240.00/€202.50

All prices correct at time of going to press but subject to change. WIT Press books are available through your bookseller or direct from the publisher.

Boundary Elements and Other Mesh Reduction Methods XXVIII Edited by: C. A. BREBBIA, Wessex Institute of Technology, UK, J. T. KATSIKADELIS, National Technical University of Athens, Greece This volume contains papers from the World Conference on Boundary Elements and other Mesh Reduction Methods, an internationally recognized forum for the dissemination of the latest advances on Mesh Reduction Techniques and their applications in sciences and engineering. The book publishes articles dealing with computational issues and software developments in addition to those of a more theoretical nature. Engineers and scientists within the areas of numerical analysis, boundary elements and meshless methods will find the text invaluable. Topics include: Advances in Mesh Reduction Methods; Meshless Techniques; Advanced Formulations; Dual Reciprocity Method; Modified Trefftz Method; Fundamental Solution Method; Damage Mechanics and Fracture; Advanced Structural Applications; Dynamics and Vibrations; Material Characterization; Acoustics; Electrical Engineering and Electromagnetics; Heat and Mass Transfer; Fluid Mechanics Problems; Wave Propagation; Inverse Problems and Computational Techniques. ISBN: 978-1-84564-164-1 2006 360pp £115.00/US$195.00/€165.00

WITPress Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK. Tel: 44 (0) 238 029 3223 Fax: 44 (0) 238 029 2853 E-Mail: [email protected]

Computer Aided Design of The Trefftz Finite and Boundary Element Wire Structures Method Frequency and Time Domain Analysis

Q.-H. QIN, University of Sydney, Australia

D. POLJAK, V. DORIC & S. ANTONIJEVIC, University of Split, Croatia

“…a much needed unique systematic treatment of the subject.”

As an introduction to the integral equation analysis of wire structures, this book and enclosed software packages contain the user friendly version of the boundary element software for modelling the straight thin wire arrays in both frequency and time domain. This package is designed as a step by step guide for postgraduate students, researchers and also practising engineers to learn CAD of wire antennas immersed in inhomogeneous media. Some electromagnetic compatibility (EMC) applications can be also handled using this package. The package contains detailed description of antenna theory, integral equation modelling and full manuals for software packages. Series: Advances in Electrical and Electronic Engineering Vol 7

““…a good up-to-date account of some modern methods in numerical analysis.…should be of interest to researchers in finite and boundary element methods and…accessible to graduate students interested in these topics as well. Presentation, style and layout…are all very good. This review can warmly recommend Trefftz Finite and Boundary Element Method to anyone looking for a clear introduction to the subject…”

ISBN: 978-1-85312-884-4 2006 160pp+CD-ROM £79.00/US$139.00/€118.50

We are now able to supply you with details of new WIT Press titles via E-Mail. To subscribe to this free service, or for information on any of our titles, please contact the Marketing Department, WIT Press, Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: +44 (0) 238 029 3223 Fax: +44 (0) 238 029 2853 E-mail: [email protected]

ZENTRALBLATT FÜR MATHEMATIK

APPLIED MECHANICS REVIEWS

This text is designed for researchers, postgraduate students and professional engineers requiring an accessible introduction to this field, and little mathematical knowledge beyond the usual calculus is needed. For convenience matrix presentation is used throughout. ISBN: 978-1-85312-855-4 2000 296pp £118.00/US$183.00/€177.00

WIT Press is a major publisher of engineering research. The company prides itself on producing books by leading researchers and scientists at the cutting edge of their specialities, thus enabling readers to remain at the forefront of scientific developments. Our list presently includes monographs, edited volumes, books on disk, and software in areas such as: Acoustics, Advanced Computing, Architecture and Structures, Biomedicine, Boundary Elements, Earthquake Engineering, Environmental Engineering, Fluid Mechanics, Fracture Mechanics, Heat Transfer, Marine and Offshore Engineering and Transport Engineering.

Viscous Incompressible Flow For Low Reynolds Numbers M. KOHR and I. POP, Babes-Bolyai University, Cluj-Napoca, Romania This book presents the fundamental mathematical theory of, and reviews stateof-the-art advances in, low Reynolds number viscous incompressible flow. The authors devote much of the text to the development of boundary integral methods for slow viscous flow pointing out new and important results. Problems are proposed throughout, while every chapter contains a large list of references. A valuable contribution to the field, the book is designed for research mathematicians in pure and applied mathematics and graduate students in viscous fluid mechanics. Contents: Introduction; Fundamentals of Low Reynolds Number Viscous Incompressible Flow; The Singularity Method for Low Reynolds Number Viscous Incompressible Flows; The Theory of Hydrodynamic Potentials with Application to Low Reynolds Number Viscous Incompressible Flows; Boundary Integral Methods for Steady and Unsteady Stokes Flows; Boundary Integral Formulations for Linearized Viscous Flows in the Presence of Interfaces; List of Symbols; Index. Series: Advances in Boundary Elements, Vol 16 ISBN: 978-1-85312-991-9 2004 448pp £148.00/US$237.00/€222.00

Find us at http://www.witpress.com Save 10% when you order from our encrypted ordering service on the web using your credit card.

Boundary Elements XXVI Editor: C.A. BREBBIA, Wessex Institute of Technology, UK Featuring the results of state-of-the-art research from many countries, this book contains papers from the Twenty-Sixth World Conference on Boundary Elements and Other Mesh Reduction Methods. Over 40 contributions are included and these cover specific topics within areas such as: Advanced Formulations; Advances in DRM and Radial Basis Functions; Inverse Problems; Advances in Structural Analysis; Fracture and Damage Mechanics; Electrical and Electromagnetic Problems; Fluid and Heat Transfer Problems; and Wave Propagation. Series: Advances in Boundary Elements, Vol 19 ISBN: 978-1-85312-708-3 2004 488pp £172.00/US$275.00/€258.00

WIT eLibrary Home of the Transactions of the Wessex Institute, the WIT electronic-library provides the international scientific community with immediate and permanent access to individual papers presented at WIT conferences. Visitors to the WIT eLibrary can freely browse and search abstracts of all papers in the collection before progressing to download their full text. Visit the WIT eLibrary at http://library.witpress.com

Adaptive Meshing with Boundary Elements J.C. MIRANDA-VALENZUELA, ITESM Campus Toluca, Mexico and K.H. MUCI-KUCHLER, University of Detroit Mercy, USA This book provides an introduction and review of adaptive meshing techniques in the BEM. A brief summary of the mathematical and engineering background necessary to understand most topics is followed by chapters covering a range of problems and techniques in detail. Series: Topics in Engineering, Vol 41 ISBN: 978-1-85312-888-2 2002 304pp £110.00/US$169.00/€165.00

Boundary Elements XXVII Incorporating Electrical Engineering and Electromagnetics Edited by: A. J. KASSAB, University of Central Florida, USA, C. A. BREBBIA, Wessex Institute of Technology, UK, E. A. DIVO, Institute for Computational Engineering (ICE), USA, D. POLJAK, University of Split, Croatia This book contains the edited proceedings of the 27th World Conference on Boundary Elements together with papers presented at the associated International Seminar on Computational Methods in Electrical Engineering and Electromagnetics. The Boundary Element Conference series continues to attract original contributions on theoretical and fundamental developments, as well as innovative applications. Its scope has also recently been expanded to include other mesh reduction

methods. The presentations from the Computational Methods in Electrical Engineering and Electromagnetics Seminar cover a wide variety of theoretical and applied topics. Over 65 papers are included and these are divided under the following headings: BOUNDARY ELEMENTS AND OTHER MESH REDUCTION METHODS Meshless Methods; Dual Reciprocity Method; Advanced Formulations; Inverse Problems; Stress Analysis; Plates and Shells; Damage Mechanics; Wave Propagation; Fluid Problems; Electrostatics and Electromagnetics; Computational Problems. ELECTRICAL ENGINEERING AND ELECTROMAGNETICS - Interaction of Humans with Electromagnetic Fields; High Frequency Electromagnetic Field Coupling to Transmission Lines; Numerical and Computational Methods; Electrical Engineering and Electronics. Series: Advances in Boundary Elements Vol 20 ISBN: 978-1-84564-005-7 2005 768pp £266.00/US$425.00/€399.00

Singular Integrals in Boundary Element Methods Editors: V. SLADEK and J. SLADEK, Slovak Academy of Sciences, Slovak Republic “...a good source...for...researchers investigating advanced BIE formulations.” AIAA JOURNAL

Series: Advances in Boundary Elements, Vol 3 ISBN: 978-1-85312-533-1 1998 448pp £125.00/US$195.00/€187.50



Domain Decomposition Techniques for Boundary Elements : Application to Fluid Flow (Advances in Boundary Elements)

Introduction to boundary elements

Boundary elements: an introductory course

Boundary elements: theory and applications

Boundary elements: Theory and applications

Boundary Methods Elements Contours and Nodes

Boundary Methods: Elements, Contours, and Nodes

Boundary methods: elements, contours, and nodes

Boundary

Boundary

Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements

Boundary

Boundary

Boundary

Boundary

Boundary

Boundary

Boundary Collocation Techniques and Their Application in Engineering