Editor
G. R. Liu
ADVANCES IN
ESHFREE AND X - F E M
ETHODS Proceedings of the 1st Asian Workshop on Meshfree Methods
World Scientific
ADVANCES IN
MESHFREE AND X-FEM METHODS Proceedings of the 1st Asian Workshop on Meshfree Methods
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ADVANCES IN
MESHFREE ANDX-FEM
METHODS Proceedings of the 1 st Asian Workshop on Meshfree Methods
Singapore
1 6 - 1 8 December 2002
Editor
G. R. Liu National University of Singapore
V f e World Scientific wb
Jersey'London'Singapore* New Jersey'London • Singapore • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
ADVANCES IN MESHFREE AND X-FEM METHODS Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-238-247-X(pbk)
This book is printed on acid-free paper. Printed in Singapore by Mainland Press
V
PREFACE This volume collects the Proceedings of the 1st Asian Workshop on Meshfree Methods held as a part of the 2 nd International Conference on Structural Stability & Dynamics (ICSSD02) onl6 - 18 December 2002 in Singapore. The workshop proceedings contain 36 papers covering a large number of the aspects on meshfree and the extended finite element methods (X-FEM). The aim of the workshop is to provide researchers working with meshfree methods an opportunity to exchange freely their new ideas, concepts, and techniques in this rapidly developing area of research. The idea of organizing this workshop was conceived among Professors Yagawa, W. Kanok-Nukulchai, H. Noguchi and G. R. Liu during the first AsianPacific Congress on Computational Mechanics held in Sydney in 2001. Profs. W. KanokNukulchai, H. Noguchi and G. R. Liu have then worked together to materialize this idea. Since G. R. Liu was at that time helping to organize the ICSSD02 in Singapore, the workshop is then naturally held as a part of the ICSSD02, so that other ICSSD02 dedicates can participate in the meshfree workshop, and the workshop dedicates can also benefit from all the practical engineering examples presented in the ICSSD02 as well as other workshops held under the umbrella of ICSSD02. I wish to express my sincere appreciation to Professors Yagawa, W. Kanok-Nukulchai, H. Noguchi for their strong support. Without their guidance, help and support, we will not be able to receive this excellent response to this workshop. I would like to also offer my sincere thanks to all the invited speakers, technical session speakers and participants. They are the ones who made this workshop possible and built up this volume through their hard works in preparing their manuscripts. My special thanks go to Profs. Ohtsubo and Suzuki who provided very strong support to the workshop. In addition, they have put up a very good session of an interesting topic of the XFEM methods. I would also like to mention a very encouraging finding in the process of editing this volume: many young students and researchers have contributed a lot of papers on meshfree methods to this workshop. I am very happy to find out about this, as they are the ones to take the meshfree and other advanced methods to a new height in the future. I would also like to express my sincere appreciation to all the conference sponsors for all the support provided by them. Last but not least, I would like to thank Suwamo, Ms. Lim Hui Leng and other PAC members at the National University of Singapore for their assistance in the preparation of this proceedings and the management of the workshop.
G. R. Liu
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vii
The Editor
Dr. G. R. Liu
Dr. Liu received his PhD from Tohoku University, Japan in 1991. He was a Postdoctoral Fellow at Northwestern University, USA. He is currently the Director of the Centre for Advanced Computations in Engineering Science (ACES), National University of Singapore. He is also an Associate Professor at the Department of Mechanical Engineering, National University of Singapore. He authored more than 250 technical publications including more than 150 international journal papers and 5 books. He is the author of the book entitled "Mesh Free Method: Moving Beyond the Finite Element Method".
He is the recipient of the Outstanding University Researchers Awards
(1998), the Defence Technology Prize (National award, 1999), and the Silver Award at CrayQuest 2000 Nationwide competition. His research interests include Computational Mechanics, Element Free Methods, Nano-scale Computation, Micro Bio-system computation, Vibration and Wave Propagation in Composites, Mechanics of Composites and Smart Materials, Inverse Problems and Numerical Analysis.
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IX
Organization
Conference Organization Department of Civil Engineering National University of Singapore
Meshfree Workshop Organization Centre for Advanced Computations in Engineering Science (ACES) Department of Mechanical Engineering National University of Singapore
Sponsors Institution of Structural Engineers, Singapore Branch Singapore Structural Steel Society Centre for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering, National University of Singapore Nanyang Centre for Super Computing and Visualization Nanyang Technological University Army Research Office-Far East (ARO-FE) Army Research Office (ARO) Asian Office of Aerospace Research and Development (AOARD) Office of Naval Research International Field Office Asia (ONR IFOA)
X
Conference Organizing Committee Chairman:
C. M. Wang, NUS
Co-Chairman:
N. E. Shanmugam, NUS
Hon. Secretary:
K. K. Ang, NUS
Technical Comm. Chairman: G. R. Liu, ACES, NUS Members:
Y. S. Choo, NUS C. G. Koh, NUS J. Y. R. Liew, NUS K. M. Liew, NTU Q. Wang, NUS
Advisors:
Y. B. Yang, National Taiwan University J. N. Reddy, Texas A &M University
Meshfree Workshop Organizing Committee Chairman: Co-Chair men:
G. R. Liu, National University of Singapore, Singapore W. Kanok-Nukulchai Asian Institute of Technology, Pathumthani,Thailand H. Noguchi, Keio University, Japan
XI
Contents Preface The Editor Organization
v vii ix
Section 1. Keynote Paper
1
1.1 Seamless and Parallel Computing by Using Free Mesh Method: A Kind of Meshless Technique G. Yagawa
3
Section 2. Meshfree Formulations
5
2.1 Analysis of 3D Solid with Complicated Geometry using Finite Cover Method (Invited Paper) K. Suzuki and H. Ohtsubo
7
2.2 A Meshless Method Using Radial Basis Functions for Solving Wave Equations (Invited Paper) C.S. Chen, Jichun Li and D.W. Pepper
15
2.3 Meshless Computational Method By using Radial Basis Functions (Invited Paper) Benny Y.C. Hon
16
2.4 Recent Advances in the Method of Fundamental Solutions A. Karageorghis, G. Fairweather and P.A. Martin
17
2.5 A Study on the Patch Test of Point Interpolation Methods Y.TGu
23
2.6 A Comparison Between Radial Point Interpolation Method (RPIM) and Kriging Based Meshfree Method G.R. Liu, K.Y. Dai, Y.T. Gu and KM Lint
29
2.7 Radial Basis Point Interpolation Collocation Method For 2-D Solid Problem Xin Liu, G.R. Liu, Kang Tai and K. Y. Lam
35
XII
Section 3. Meshfree Methods for Smart Materials/ Structures
41
3.1 Point Interpolation Mesh Free Method for Static and Frequency Analysis of Two-dimensional Piezoelectric Structures K.Y Dai, G.R. Liu and KM. Lim
43
3.2 A Hybrid Meshless-Differential Order Reduction ( H M - D O R ) Method for Deformation Control of Smart Circular Plate by Sensors/Actuators /. Q. Cheng, Hua Li, K.Y. Lam, T. Y. Ng and Y.K. Yew
49
Section 4. Meshfree Methods for Fracture Analysis
55
4.1 Application of 3D Free Mesh Method to Fracture Analysis of Concrete Hitoshi Matsubara, Shigeo Iraha, Jun Tomiyama and Genki Yagawa
57
4.2 Meshless Analysis Integrate System for Structural and Fracture Mechanics Analysis Seiya Hagihara, Mitsuyoshi Tsunori, Torn Ikeda, Noriyuki Miyazaki, Takayuki Watanabe and Chaunrong Jin
63
4.3 Application of 2- Dimensional Crack Propagation Problem using Free Mesh Method /. Imasato and Y. Sakai
69
Section 5. Meshfree Methods for Membranes, Plates & Shells
75
5.1 Analysis of Membrane Structures with Large Sliding Cable using Mixed Displacement Formulation and EFGM (Invited Paper) Hirohisa Noguchi, Yoshitomo Sato and Tetsuya Kawashima
77
5.2 The Effects of the Enforcement of Compatibility in the Radial Point Interpolation Method for Analyzing Mindlin Plates X. L. Chen, G.R. Liu and S.P Lim
84
5.3 A Mesh Free Method for Dynamic Analysis of Thin Shells L. Liu and V.B.C. Tan 5.4 A Conforming Point Interpolation Method for Analyzing Spatial Thick Shell Structures L. Liu, G.R. Liu, V.B.C. Tan and Gu, Y.T.
90
96
Section 6. Meshfree Methods for Soil
107
6.1 Characteristics of Localized Behavior of Saturated Soil with Pore Water via Mesh-Free Method S. Arimoto, A. Murakami
109
6.2 Radial Point Interpolation Method for Interface Problems /. G. Wang, T. Nogami and Md. Rezaul Karint
115
Section 7. Meshfree Methods for CFD
121
7.1 Application of Free Mesh Method to Viscoplastic Flow Analysis of Fresh Concrete Jun Tomiyama, Yoshitomo Yamada, Shigeo Iraha and Genki Yagawa
123
7.2 A Meshless Local Radial Point Interpolation Method (LRPIM) for Fluid Flow Problems y. L. Wu
129
73 Application of Meshless Point Interpolation Method with Matrix Triangularization Algorithm to Natural Convection G. R. Liu and Y.L Wu
135
7 A The Solution for Convection-Diffusion Equations using the Quasi-Interpolation Scheme with Local Polynomial Reproduction Based on Moving Least Squares Xin Liu, G.R. Liu, Kang Tai and K.Y Lam
140
Section 8. Boundary Meshfree Methods
149
8.1 Regular Hybrid Boundary Node Method (Invited Paper) J.M Zhang and Z.H. Yao
151
8.2 Radial Boundary Node Method for Elastic Problem H. Xie, T. Nogami and J.G. Wang
161
8.3 A Hybrid Boundary Point Interpolation Method (HBPIM) and its Coupling with EFG Method Y.T. Gu and G.R. Lin
167
Section 9. Coding, Error Estimation, Parallisation
177
9.1 Error Regulation in EFGM Adaptive Scheme (Invited Paver) W. Kanok-Nukulchai and X.P. Yin
179
XIV
9.2 Object Oriented Development of FMM3D: Foundation Software for Parallel 3D Free Mesh Method Yutaka Nakama, Akio Shimada, Yasuhiro Kanto, Tomoaki Ando and Genki Yagawa
194
9.3 An Approach for Nodal Selection in MFree2D® G.R. Liu, Edgar Frijters and Y.T. Gu
200
Section 10. Meshfree Particle Methods
209
10.1 Coupling Meshfree Particle Method with Molecular Dynamics Novel Approach for Multdscale Simulations M.B. Liu, G.R. Liu and K.Y. Lam
211
10.2 Adaptive Smoothed Particle Hydrodynamics with Strength of Materials, Part I G.L. Chin, K.Y. Lam and G.R. Liu
217
10.3 Adaptive Smoothed Particle Hydrodynamics with Strength of Materials, Part II G.L. Chin, K.Y. Lam and G.R. Liu
223
10.4 Numerical Simulation of Perforation of Concrete Slabs by Steel Rods using SPH Method H.F. Qiang and S.C. Fan
229
Section 11. X-FEM
237
11.1 Three Dimensional Crack Growth Analysis using Overlaying Mesh Method and X-FEM (Invited Paper) S. Nakasumi, K. Suzuki and H. Ohtsubo
239
11.2 Buckling Analysis of Composite Laminates with Delaminations using X-FEM T. Nagashima and H. Suemasu
245
11.3 Boundary Condition Enforcement in Voxel-Type FEM T. Nagashima
251
Author Index
257
SECTION 1 Keynote Paper
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3
Advances in Meshfree andX-FEMMethods, G.R. Liu, editor, World Scientific, Singapore 2002
SEAMLESS AND PARALLEL COMPUTING BY USING FREE M E S H METHOD: A KIND OF MESHLESS TECHNIQUE
G. Yagawa School of Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, JAPAN yagawa@q. t. u-tokyo. ac.jp
With the amazing progress of computers, the finite element method (FEM) has become widely used in many practical situations. There exists, however, a large gap between its industrial applications and academic fundamental studies. One of its reasons is due to the difficulty in the pre-processing for FEM, i.e. mesh generation. The generation of mesh is a very difficult task if the degree-of-freedom of the analysis model is extremely large, or if the geometry of the domain is much complex. In addition, mesh generation is usually carried out in a sequential algorithm, although parallel computing is becoming available for large-scale numerical. Therefore, we need to develop a new CAE algorithm, where parallel computing is employed throughout the process. Especially, the sequential procedure of mesh generation becomes a serious bottleneck in the whole computational processing, if frequent mesh refinement is required in the problems, such as moving boundary problem, compressible flow involving shocks, crack propagation, large deformation problem. On the other hand, in order to avoid the troublesome processing of mesh generation for the FEM analysis, various 'meshless methods' have been proposed, in which analysis domain is discretized without employing any 'mesh' or 'elements'. The smooth particle hydrodynamics (SPH), the diffuse element method (DEM), the element-free Galerkin method (EFGM), the reproducing kernel particle method (RKPM), the moving-particle semi-implicit method (MPS) are among others. However, it seems that these meshless methods have not succeeded in replacing the FEM analysis completely, while they show excellent performance in several special fields. On the other hand, new finite element approaches are proposed in order to overcome the difficulty of mesh generation, such as the manifold method the voxel finite element method, the generalized finite element method (GFEM), the extended finite element method (X-FEM) , the finite cover method (FCM), the free mesh method (FMM) and the node-by-node finite element method (NBN-FEM). Although these methods employ elements or mesh, they are not given a priori, so that they are categorized as 'meshless method' in a wide sense or 'mesh free methods'. The FMM or NBN-FEM aims at seamless finite element computing from GAD models to the final numerical solutions in the parallel environments. In the method, both pre-processing and main-processing of finite element analysis can be parallelized in terms of nodes, where the pre-preprocessing involves the local mesh generation and the construction of system of equations, and the main-processing indicates the solution of system of equations. The method is quite suitable for massively parallel environments, while the commercial parallel mesh generator can use only ten processors at most. Here,
4 we discuss the algorithm of the method with several numerical examples. Reference G. Yagawa, T. Yamada, Free mesh method: A new meshless finite element method, Comp. Mech. 1996;18:383-386. G. Yagawa, T. Furukawa, Recent developments of free mesh method, Int. J. Numer. Meth. Engrg. 2000;47:1419-1443. M. Shirazaki, G Yagawa, Large-Scale parallel flow analysis based on free mesh method: a virtually meshless method, Comput. Methods Appl. Mech. Engrg. 1999;174:419-431. G. Yagawa, Parallel computing of local mesh finite element method, Proc. The First Asian-Pacific Congress on Computational Mechanics, Sydney, 2001;17-26. T. Fujisawa, G. Yagawa, Node-based parallel mesh generation and finite element solver for high Speed compressible flows, Proc. the Fifth world Congress on Computational Mechanics (WCCM V), July 7-12, 2002, Vienna, http://wccm.tuwien.ac.jp G. Yagawa, Node-by-node parallel finite elements: A virtually meshless method, ibid.
SECTION 2 Meshfree Formulations
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7 Advances in Meshfree and X-FEM Methods, G.R. Liu,editor World Scientific, Singapore,2002
ANALYSIS OF 3D SOLID WITH COMPLICATED GEOMETRY USING FINITE COVER METHOD
K. Suzuki Department of Environmental Studies, Graduate School of Frontier University of Tokyo, Japan
Sciences,
katsu@k. u-tokyo. ac.jp
H. Ohtsubo Department of Environmental and Ocean Engineering, University of Tokyo, Japan ohtsubo@nasl. t. u-tokyo.ac.jp
Abstract The new meshless method based on the cover least square approximation, which utilize cover instead of points that are used in the moving least square approximation, is proposed. For the cover distribution, multi scale voxel data is used for the ease of analysis in 3D solid. Several 3D examples are shown for demonstration. The method is applied to adaptive analysis of seepage flow problem of rock including complicated cracks. The rock model with 3 cracks and 1000 cracks are analyzed.
Introduction The Finite element method has been widely used in the design process of industrial products as CAE tools. As the design process moves to 3D CAD and as the computational power increase, the FE analysis also have been moving to 3D. In the 3D analysis the most of the time is consumed in the generation of FEM model. Especially for the analysis of 3D solid with complicated geometry the generation of model often takes several weeks to months, or sometimes impossible to make model. Under these circumstances, the meshless analysis method has been emerging which does not require the mesh in the analysis, but still meshless approach has several disadvantage compared to FEM, and has not been widely used in the engineering practice. Especially in the analysis of 3D solid, the integration of 3D domain and increase of computational costs prevents the meshless method from becoming popular. The authors have been proposing the alternative meshless approach, and named Finite Cover Method (FCM). The FCM aims at 3D solid analysis and utilizes voxel concept.
8
Single scale voxel is used in the PU version of FCM (FCM-PU), which employs voxel cover for the cover of Manifold Method. By changing the polynomial order of cover functions it is possible to control the accuracy locally (p-type adaptive). Also we extended the method to utilize multi-level sized voxel to control the accuracy by mesh size (h-type adaptive) using Cover Least Square Approximation (CLSA, FCM-CLSA). The model generation of the voxel analysis is so simple and effective that it is possible to mesh any complex structure.
Formulation Voxel Analysis The voxel analysis was proposed by Kikuchi et. al. (Hollister S and Kikuchi N (1994), Terada K and Kikuchi N (1996)). The model generation of the voxel analysis is so simple and effective that it is possible to mesh any complex structure either from CAD data or real object by CT scanner (Figure 1). They used voxel data as HEXA element in FEM. However, because all the elements are uniform, any local refining requirement causes a global refining. When small voxels are necessary to improve the accuracy in a local area, a numerous number of elements and degrees of freedom are required. How to get reasonable accuracy and avoid the sharp increase of degrees of freedom is a problem of the voxel analysis. Suzuki et.al. (1998) used boundary shape voxel (Figure 2) that subdivide the boundary shape voxel into smaller voxels for better geometric representation of original shape without increasing the analysis voxel, and used the boundary shape voxel for the domain integration and applying boundary conditions (displacement and foruce).
Figure 1. Voxel Analysis (Kikuchi and Diaz (1996))
9
Figure 2. Voxel analysis with Boundary Shape Voxel Finite Cover Method Since voxel analysis divide the domain into same size cube, it is impossible to control the accuracy locally, which is commonly used in the FEM by changing the size of the element. In the Finite Cover Method we have developed 2 methods to control the accuracy. In the FCM-PU, (Figure 3 left) by keeping the size of the analysis voxel same, the degree of polynomial for approximation function is changed. In the FCM-CLSA, the size of the voxel is changed to accept multi scale voxel subdivision (Figure 3 right) and Cover Least Square Approximation is proposed for constructing shape function.
^^ V 1-'
•
•
"
^' / \
.'
•
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-
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\ ,> 1 i,^ pi \ ii m \
) \.^^/ \s
•
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^
y'
.
^ \ .
'
/
/
,-' " 1
^
\ ,> N
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. * ' •^, ___ -"
- ' -"
Sample Mathematical Cover field
Sample Mathematical Cover field
Boundary of analysis domain
Boundary of analysis domain
'\
\ \ ^
Figure 3 FCM-PU and FCM-CLSA Finite Cover Method-PU In FCM-PU, the concept of cover in the Manifold Method (Shi, (1991)) is used, which separate the definition domain for approximation function (cover) and physical domain where basic equation should be satisfied. Since the cover can be defined arbitrary independent from physical domain, the flexibility of the model generation is increased considerably. FCM utilize voxel shape cover for the mathematical cover in Manifold Method. As shown in Figure 3 left, the each physical domain is covered by 4 voxel
10
covers (in 3D, 8 covers) in FCM-PU. The displacement functions are approximated as follows. k
u(x)
{" f#(x)w,(x)
(1)
i 1
where f (x) is cover function and w,.(x) is weight function. The weight function needs to satisfy following conditions, where U, is common domain of mathematical cover and physical domain. Equation (3) is called Partition of Unity condition that guarantees the reproducibility of the function in cover functions. Suzuki et. al. (1998) also discuss the weight function that guarantee the linear independency of approximation functions for arbitrary degree of polynomial order of cover functions, which allow accuracy control by changing the order of polynomial. The displacement function (1) is substituted into Galerkin formulation to derive the linear set of equations. Figure 5 is the example of the analysis of gear. The gear is divided into 40x40x4 analysis voxel. Tw,( x ) x Ox W
X
4 ,( )
Ox
f »,C0
1
(2)
(3)
K* W l#
Lo; I
^ .1
U, U,
•»*••
* A *' m
"*- **
' ixed Figure 4 Analysis of Gear
Finite Cover Method-CLSA When the size of voxel is changed as in Figure 3 right, it is not easy to find the weight function that satisfies PU condition. Jin et. al. (2000) proposed Cover Least Square Approximation (CLSA) that is similar to the Moving Least Square Approximation (MLSA) but instead of evaluating the function by nodes, CLSA evaluate the function by cover. In CLSA, the approximation function "(*) is defined as equation (4), which is same function as the one used in MLSA. In the CLSA, to derive approximation function the functional J in equation (5) is minimized. The evaluation of the functional to be minimized is carried out on each cover, while in MLSA the evaluation is evaluated at
11
each node. It has been proved that by CLSA, the linear independency between approximation function can be guaranteed. Most meshless approaches do not have the guaranteed linear independency of functions. H
u(x) = u'(x,x~) = 2_iaj(x)0J(x-x~) = &
(4)
J(a(x)) = £ w , ( x ) j"ft>,0)e,2(x,x>ft
where e.(x,x) =
(5)
O,(x,O,)-u'(x,x) = cp,d, -cpa
(6)
and w,(5c) is weight function, a>Xx)is localization factor function
Numerical Example Constant Stress Cube The following conditions are imposed on a cube of 1 x 1 x 1. Displacement conditions ux=0
on X = 0 uy=0
on y = 0 «2 = 0 on Z = 0
Traction condition P z = l on z = l The covers are distributed based on a two-level voxel data as shown in Figure 5. The computed results of displacement is linear and the stress is constant everywhere, and it is proven that the CLSA can give an exact solution for constant stress problem.
V \ A
Figure 5 A Cube with Constant Stress
Figure 6 1/8 plate with a hole
12
Plate with a Hole Consider a 20x20x4 plate with a hole at its center, whose radius is 2. Uniform traction acts on a pair of opposite sides. Figure 6 shows 1/8 of it. The boundary conditions are given as a, = 0 on X = 0 uz=0
^=0
on z = 0
on y = 0
ffw=lony
= 10
Three Models as shown in Figure 7 are calculated. In the Model a, four-level voxel data is employed to create the cover distribution. The sizes of the four-level voxels are respectively 10/2x10/2x2/2, 10/4x10/4x2/4, 10/16x10/16x2/8 and 10/64x10/64x2/8. In the Model b, three-level voxel data is employed to create the cover distribution. The sizes of the three level voxels are respectively 10/8x10/8x2/4, 10/32x10/32x2/8, and 10/128x10/128x2/16. In the Model c, four-level voxel data is employed to create the cover distribution. The sizes of the four level voxels are respectively 10/4x10/4x2/4 10/16x10/16x2/8, 10/64x10/64x2/16, and 10/256x10/256x2/32. The computed results of ayy are shown in Figure 8. The stress values at point A and B (see Figure 6) are listed in Table 1 together with the ANSYS results and the numbers of DOF are listed in Table 2.
Model a
Model b
Figure 7
Model c
3D Plate with Circular Hole Model
Figure 8 Computed results of a
13
Table 1 Stress values °"*r
Point A
Point B
Cyy
^zz
Von Mises stress
Model a
—
3.269
0.382
3.092
Model b
—
3.339
0.405
3.116
Model c
—
3.464
0.420
3.189
ANSYS
—
3.578
0.398
3.389
Model a
-1.287
—
-0.387
—
Model b
-1.458
—
-0.416
—
Model c
-1.502
—
-0.428
—
ANSYS
-1.488
—
-0.405
—
Seepage Flow Problem of Rock with Cracks Figure 9 is the model of rock with 1000 circular cracks, which is made for the seepage flow problem for the safety evaluation of nuclear waste disposal. For this kind of model of nature, it is impossible to make FEM mesh. Also to evaluate the size of crack, the mesh size of the small area need to be reasonably small, and it is impossible to generate model by voxel with uniform size. By using multiscale voxel subdivision, the model is generated with about 200,000 elements with minimum voxel size is 1/1000 of one side as shown in Figure 10, while 1000 million elements are required if the domain is divided into uniform voxel of the size.
Figure 9 Rock with 1000 cracks (birds view and sectional view)
14
Figure 10 Multi-scale voxel Model Conclusions In this paper, the Cover Least Square Approximation method is implemented for linear structure analysis. The covers are distributed using multi-resolution voxel data. Numeric examples show that multi-resolution voxel data based cover distribution can conveniently guarantee the linear independence of CLSA shape functions and the approximation accuracy can be flexibly controlled by locally justifying the density of cover distribution. References Fish J and Markolefas (1993), "Adaptive s-method for linear elastostatics", Computer Methods in Applied Mechanics and Engineering, 104, 363-396. Hollister S and Kikuchi N (1994), Homogenization theory and digital imaging: a basis for studying the mechanics and design principles of bone tissue, Biotechnology and Bioengineering, Vol.43, No. 7, pp.586-596 Jin C, Suzuki K, Fujii D, and Ohtsubo H (2000), "Methodology and Property of Cover Least Square Approximation", Transaction of the Japan Society for Computational Engineering and Science Vol. 2 pp 213-218 Jin C, Suzuki K and Ohtsubo H (2000), " Linear Structural Analysis Using Cover Least Square Approximation ", Journal of Applied Mechanics, JSCE Vol.3 pp 167-176 Kikuchi N and Diaz A (1998), "CAD/CAE using Image Base Method", 14th Quint Seminar Texbook Shi, G H (1991), "Manifold Method of Material Analysis", Transactions of the 9th Army Conference On Applied Mathematics and Computing, Report No. 92-1. U.S. Army Research Office. Suzuki K etal. (1998) "The Analysis of 3D Solid Using Multi-scale Voxel Data", Computational Mechanics -New Trends and Applications VII, 2-15, CIMNE Terada K and Kikuchi N (1996), "Microstructural design of composites by using the homogenization method and digital images, Mat. Sci. Res. Int. , Vol.2, No.2, pp.73-81
15 Advances in Meshfree and X-FEM Methods, C.R. Liu, editor World Scientific, Singapore, 2002
A MESHLESS METHOD USING RADIAL BASIS FUNCTIONS FOR SOLVING WAVE EQUATIONS C. S. Chen, Jichun Li, and D.W. Pepper University of Nevada Las Vegas
[email protected] Abstract Using various time difference schemes or integral transforms, a given wave equation can be reduced to solving a series of inhomogogenous Helmholtz-type equations which can then be further split into evaluating particular solutions and solving the related homogeneous equations. Recent development of deriving closed-form particular solution for Helmholtztype equations using radial basis functions has made it possible to solve time-dependent problems efficiently. As a result, the domain integration can be avoided in the solution process. The method of fundamental solutions (MFS), a meshless and often spectrally accurate boundary method, will be further developed and adopted as the major numerical method to solve the corresponding homogeneous equations in this paper.
16 Advances in Meshfree andX-FEMMethods, G.R.Liu, editor, World Scientific, Singapore, 2002
MESHLESS COMPUTATIONAL METHOD BY USING RADIAL BASIS FUNCTIONS
Benny Y. C. Hon City University of Hong Kong Benny.Hon@cityu. edu.hk Abstract The recent development of a meshless method by using radial basis functions will be reported in this talk. Application to both multivariate interpolation and solving partial differential equations have demonstrated the spectral convergence of the method for some particular radial basis functions like multi quadric. This talk will also discuss some of the recent proposed techniques for solving the ill-conditioning problem resulted from solving the full resultant matrix.
17 Advances in Meshfree andX-FEMMethods, G.R. Liu.editor World ScientiflcSingapore, 2002
R E C E N T ADVANCES IN THE M E T H O D OF FUNDAMENTAL SOLUTIONS
Department
Department
of Mathematics
A. Karageorghis and Statistics, University 1618 Nicosia, Cyprus andreaskflucy.ac.cy
of Cyprus, P. 0. Box
G. Fairweather, P. A. Martin of Mathematical and Computer Sciences, Colorado School of Golden, Colorado 80401, USA gfairweaQmirtes.edu, pamartinflmines.edu
20537,
Mines,
Abstract The aim of this paper is to describe recent developments in the method of fundamental solutions (MFS) and related methods for the numerical solution of certain elliptic boundary value problems. K e y w o r d s : Method of Fundamental Solutions, Nonlinear Least Squares, Boundary Collocation. Introduction The method of fundamental solutions (MFS) is a meshless technique for the numerical solution of certain elliptic boundary value problems which falls in the class of methods generally called boundary methods. Like the boundary element method (BEM), it is applicable when afundamental solution of the differential equation in question is known, and shares the same advantages of the BEM over domain discretization methods. Moreover, it has certain advantages over the BEM. In the MFS, the approximate solution is expressed as a linear combination of fundamental solutions with singularities placed outside the domain of the problem. The locations of the singularities are either preassigned and kept fixed or are determined along with the coefficients of the fundamental solutions so that the approximate solution satisfies the boundary conditions as well as possible. This is usually achieved by a least squares fit of the boundary data. Early uses of the MFS were for the solution of various linear potential problems in two and three space variables. It has since been applied to a variety of situations such as plane potential problems involving nonlinear radiationtype boundary conditions, free boundary problems, biharmonic problems, problems in elastostatics and in the analysis of wave scattering in fluids and solids.
T h e M F S for H e l m h o l t z p r o b l e m s To illustrate the essential features of the MFS, we consider a two-dimensional exterior Helmholtz problem which is closely related to the external scattering problem for acoustic waves by a rigid obstacle. We let Q be an unbounded domain in IR2 and Q c its bounded complement in Ht2 with
18 boundary dU. We consider the problem
Au(P) + k2u(P)
= 0,
Bu(P) = o,
Pen,, Peda,
where A denotes the Laplacian, u is the dependent variable, k a real constant, and Q is a bounded domain in the plane with boundary dil. The operator B specifies the boundary conditions (BCs). The behaviour of u at infinity must also be specified. In the MFS, the solution u is approximated by a function of the form N
UA ,(c,P;