Institute of Mechanical Engineering Aalborg University Special Report no. 23
Finite Element Based Design Sensitivity Analysis and Optimization Ph.D. Dissertation by Erik Lund
c 1994 Erik Lund Copyright
Reproduction of material contained in this report is permitted provided the source is given. Additional copies are available at the cost of printing and mailing from Helle W. Mane, Institute of Mechanical Engineering, Aalborg University, Pontoppidanstraede 101, DK9220 Aalborg East, Denmark. Telephone +45 98 15 42 11 ext. 3506, FAX +45 98 15 14 11. Questions and comments are most welcome and can be directed to the author at the same adress or by electronic mail:
[email protected]. Printed at Aalborg University, April 1994.
ISBN 87-89206-01-0
Preface
This dissertation has been submitted to the Faculty of Technology and Science of Aalborg University, Aalborg, Denmark, in partial ful lment of the requirements for the technical Ph.D. degree. The project underlying this thesis has been carried out from August 1991 to April 1994 at the Institute of Mechanical Engineering at Aalborg University. The project has been supervised by Professor, Dr.techn. Niels Olho to whom I am most indebted for his vivid engagement, inspiring and supporting guidance, immense competence, and his time demanding e orts in providing good research conditions and facilities for our research group. I am also most indebted to my friend and collegue, Associate Professor, Ph.D. John Rasmussen for many invaluable suggestions, support, collaborations, and inspiring discussions. Furthermore, I would like to thank my colleagues and friends, M.Sc. Lars Krog & M.Sc. Oluf Krogh for many inspiring discussions and suggestions, Professor Alexander P. Seyranian, Moscow State Lomonosov University, Russia, for our joint work on multiple eigenvalues in structural optimization problems, and the other colleagues at the Institute of Mechanical Engineering. Finally, I want to thank my ance Dorte for her support and understanding during periods with much work and less spare time; her love has been an invaluable support. The present work has been supported by the Danish Technical Research Council's \Programme of Research on Computer Aided Engineering Design".
Erik Lund Aalborg, April 1994
1
2
Abstract
The objective of the present Ph.D. project is to develop, implement and integrate methods for structural analysis, design sensitivity analysis and optimization into a general purpose computer aided environment for interactive structural design, analysis, design sensitivity analysis, synthesis, and design optimization. This system is named \ODESSY" (Optimum DESign SYstem) and is integrated with commercially available CAD systems. The nite element library used for the analysis has facilities for solution of linear types of analysis problems, i.e., static stress analysis, natural frequency analysis, steady state thermal analysis, thermo-elastic analysis, eigenfrequency analysis with initial stress sti ening e ects due to mechanical or thermal loads, and linear buckling analysis with the possibility of including thermo-elastic e ects. Many di erent isoparametric nite elements are described and implemented in the system. A reliable tool for design sensitivity analysis is a prerequisite for performing interactive structural design, synthesis and optimization. General expressions for design sensitivity analysis of all implemented types of analysis problems are derived with respect to shape as well as sizing and material design variables. The method of design sensitivity analysis used is the direct approach, and the semi-analytical method where derivatives of various nite element matrices and vectors are approximated by rst order nite di erences is adopted. However, the traditional semi-analytical method may yield severely erroneous results for certain types of problems involving shape design variables. Therefore, a new semi-analytical method based on \exact" numerical di erentiation of element matrices is developed and implemented for all types of nite elements and design variables in ODESSY. It is demonstrated by several examples that this new approach of semi-analytical design sensitivity analysis is computationally ecient and completely eliminates the inaccuracy problem. A general and exible method of formulating problems of mathematical programming is developed. The method enables formulation and solution of problems involving local, integral, min/max, max/min and possibly non-di erentiable user de ned functions in any conceivable mix. The mathematical formulation is based on the bound formulation and the implementation involves a parser capable of interpreting and performing symbolic di erentiation of the user de ned functions. This database module can also be used for graphical visualization of user de ned mathematical expressions for design criteria. The special case of solving structural optimization problems involving multiple eigenvalues is discussed. The main diculty associated with multiple eigenvalues is the lack of usual Frechet di erentiability with respect to changes in design, i.e., multiple eigenvalues can only be expected to be directionally di erentiable. Necessary optimality conditions are discussed and iterative numerical algorithms for solution of such design problems are developed and used for solution of a design problem. Several examples illustrate how ODESSY can be used very e ectively for interactive engineering design with focus on design sensitivity analysis, synthesis, and optimization.
3
Abstrakt
Formlet med dette Ph.D. projekt er at udvikle, implementere og integrere metoder for strukturel analyse, design somhedsanalyse og optimering i et generelt, datamatassisteret system for interaktiv strukturel design, analyse, design somhedsanalyse, syntese og designoptimering. Dette system er blevet dbt \ODESSY" (Optimum DESign SYstem) og er integreret med kommercielt tilgngelige CAD-systemer. Finite element biblioteket, der bruges til analysen, har faciliteter for lsning af linere typer analyseproblemer, dvs., statisk spndingsanalyse, analyse af frie svingninger, steady state termisk analyse, termo-elastisk analyse, egenfrekvensanalyse med initiale spndingsafstivende e ekter pga. mekaniske/termiske belastninger og liner bulingsanalyse med mulighed for at inkludere termo-elastiske e ekter. Mange forskellige isoparametriske elementtyper er beskrevet og implementeret i systemet. Et plideligt vrktj til design somhedsanalyse er en forudstning for at kunne udfre interaktiv strukturel design, syntese og optimering. Generelle udtryk for somhedsanalyse af de implementerede typer af analyseproblemer er udledt for form-, tykkelse- og materialedesignvariable. Den direkte metode er valgt til somhedsanalysen, og den semianalytiske metode, hvor a edede af diverse elementmatricer og -vektorer approksimeres med frste ordens di erenser, anvendes. Imidlertid kan den traditionelle semi-analytiske metode give fuldstndig forkerte resultater for visse typer af problemer, der involverer formdesignvariable. Derfor er en ny semi-analytisk metode, der er baseret p \eksakt" numerisk di erentiation af elementmatricer, blevet udviklet og implementeret for alle typer af elementer og designvariable i ODESSY. Denne nye semi-analytiske metode for design somhedsanalysen er beregningsmssig e ektiv og eliminerer unjagtighedsproblemet, hvilket illustreres ved adskillige eksempler. En generel og eksible metode til at formulere matematiske programmeringsproblemer er udviklet. Metoden gr det muligt at formulere og lse problemer, der involverer lokale, integrale, min/max, max/min og mulige ikke-di erentiable brugerde nerede funktioner i enhver tnkelig kombination. Den matematiske formulering er baseret p bound-formuleringen, og implementeringen indeholder en fortolker, der kan forst og udfre symbolsk di erentiation af brugerde nerede funktioner. Dette databasemodul kan ogs bruges til interaktiv, gra sk visualisering af brugerde nerede matematiske udtryk for designkriterier. Det specielle til de vedrrende lsning af strukturelle optimeringsproblemer med multiple egenvrdier behandles. Den strste vanskelighed forbundet med multiple egenvrdier er manglen p normal Frechet-di erentiabilitet med hensyn til designndringer, dvs. multiple egenvrdier kan kun forventes at vre retningsdi erentiable. Ndvendige optimalitetskriterier diskuteres og iterative numeriske algoritmer til lsning af sdanne designproblemer prsenteres og anvendes til lsning af et designproblem. Adskillige eksempler illustrerer, at ODESSY er et srdeles e ektivt vrktj for interaktiv strukturel design med fokus p designflsomhedsanalyse, syntese og optimering.
4
Chapter Introduction
1
involved endeavours towards optimization, and E this particularly holds true foralways the eld of engineering design. Earlier, engineering ngineering activity has
design was conceived as a kind of \art" that demanded great ingenuity and experience of the designer, and the development of the eld was characterized by gradual evolution in terms of continual improvement of existing types of engineering designs. The design process generally was a sequential \trial and error" process where the designer's skills and experience were most important prerequisites for successful decisions for the \trial" phase. However, nowaday's strong technological competition which requires reduction of design time and costs of products with high quality and functionality, and current emphasis on saving of energy, saving and re-use of material resources, consideration of environmental problems, etc., often involve creation of new products for which prior engineering experience is totally lacking. Development of such products must naturally lend itself on application of scienti c methods such as structural analysis, design sensitivity analysis, and optimization. The scienti c research in the eld of structural optimization has increased rapidly during recent decades. The increasing interest in this eld has been strongly boosted by the advent of reliable general analysis methods like the nite element method, methods of design sensitivity analysis, and methods of mathematical programming, along with the exponentially increasing speed and capacity of digital computers. The rapid development of the eld of optimum design also re ects a natural shift of emphasis from analysis to synthesis. The research activities in the elds of design sensitivity analysis and optimum design can be found in, e.g., the review papers by Haug (1981), Schmit (1982), Olho & Taylor (1983), Haftka & Grandhi (1986), Ding (1986), and Haftka & Adelman (1989) and the proceedings from various conferences and symposia published by Haug & Cea (1981), Morris (1982), Eschenauer & Olho (1983), Atrek, Gallagher, Ragsdell & Zienkiewicz (1984), Bennett & Botkin (1986), Mota Soares (1987), Rozvany & Karihaloo (1988), Eschenauer & Thierauf (1989), Eschenauer, Mattheck & Olho (1991), Bends e & Mota Soares (1993), Rozvany (1993), Haug (1993), Pedersen (1993), Herskovits (1993) and Gilmore, Hoeltzel, Azarm & Eschenauer (1993). 5
6 Rational engineering design and optimization based on the concept of integration of nite element analysis, design sensitivity analysis, and optimization by mathematical programming, was early undertaken by Zienkiewicz & Campbell (1973), Kristensen & Madsen (1976), Pedersen (1981, 1983), Pedersen & Laursen (1983), Esping (1983), Braibant & Fleury (1984), Eschenauer (1986) and Santos & Choi (1989), and this work has paved the road for development of large, practice-oriented optimization systems, see, e.g., Braibant & Fleury (1984), Sobieszanski-Sobieski & Rogers (1984), Bennett & Botkin (1985), Esping (1986), Stanton (1986), Eschenauer (1986), Botkin, Yang & Bennett (1986), Hornlein (1987), Arora (1989), Rasmussen (1989, 1990), Choi & Chang (1991), and Rasmussen, Lund, Birker & Olho (1993). However, until recently, these methods for rational engineering design and optimization have not, in general, been adopted by designers in industry as stated by Santos & Choi (1989). Only major companies in elds like aeronautical, aerospace, mechanical, nuclear, civil, and o -shore engineering have adopted these tools for rational design and optimization, although their use have proved successful for many di erent kinds of applications concerning structural synthesis. The reasons for this lack of use are many. It may be dicult, and may even prove impossible, to formulate one or more simple performance criteria for optimization. The designer must be able to de ne the objective function, identify all important constraints, identify the levels of constraint limits, and enter this information into the structural optimization system. Very often, in applications, during the design optimization process, additional constraints may become important that the designer did not foresee in the initial stages of formulation of the design problem. The structural optimization system must also contain very general and exible facilities for de nition of the design problem in order to cover possible de nitions. If such facilities are not present, designers in industry will not use the optimization systems as their design criteria in many cases have been established during many years of practice. If designers in industry cannot de ne their usual design criteria, they will typically rather abstain from the use of structural optimization than being forced into changing their well-tested design criteria just because of a limitation in the software system. Furthermore, another possible reason for the lack of adoptivity of structural optimization in the industry may be that developers of structural optimization systems have been too focused on selling the systems as automated \black boxes" for obtaining optimum designs of structures. In this way designers in industry have not become familiar with design sensitivity analysis which is used to calculate all necessary derivatives of criterion functions, e.g., derivatives of stresses, displacements, frequencies, user de ned functions, etc., with respect to some design parameters. This can also be seen from the fact that only a few nite element systems, for example MSC-NASTRAN, have facilities for design sensitivity analysis although this tool can be just as valuable as structural analysis in the design process. Of course, it is possible to employ structural synthesis procedures which do not need derivative information, but apart from the synthesis method, design sensitivity analysis provides the designer with valuable information about the sensitivity of the design with respect to various parameters which can be used for a revision of these parameters. Finally, a structural optimization system must be integrated in a general purpose, interac-
Chapter 1. Introduction
7
tive CAD environment as stated by Fleury & Braibant (1987), see also Braibant & Fleury (1984), Stanton (1986), Botkin, Yang & Bennett (1986), Santos & Choi (1989), Gu & Cheng (1990), Rasmussen (1990, 1991), and Botkin, Bajorek & Prasad (1992), if designers shall use the system. If the structural optimization system is integrated in a CAD environment that the designer is familiar with, the system will much easier be accepted for daily use in the design process. The above-mentioned necessary features for a general purpose computer aided environment for interactive rational design, structural analysis, design sensitivity analysis, and optimization involve many di erent topics and require expertice in quite di erent elds. The development of such a computer aided environment must therefore naturally be carried out as a collaboration between several people.
1.1 Programme of Research on Computer Aided Engineering Design The research on computer aided design and mechanical engineering was initiated at the Institute of Mechanical Engineering, Aalborg University, in the mid-eighties and further intensi ed by inclusion in a research framework programme under the Danish Technical Research Council, the \Programme of Research on Computer Aided Design". This research framework programme was carried out in the period 1989-1993 and extended for the period 1993-1997 under the name \Programme of Research on Computer Aided Engineering Design". The participants in this programme are the Institute of Mechanical Engineering, Aalborg University, and the Institute of Engineering Design, the Department of Solid Mechanics, and the Mathematical Institute, all of the Technical University of Denmark. This thesis has been carried out within this programme under the project \Engineering Design Optimization". The primary objective of this programme is to further the research and development of new methods, techniques and tools for concurrent (or integrated) computer aided design of mechanical products, systems and components which are critical in terms of cost, development time, functionality and quality. This research on computer aided design and mechanical engineering at the Institute of Mechanical Engineering has resulted in several Ph.D. theses in the eld of optimum design, see Rasmussen (1989), Kibsgaard (1991), and Thomsen (1992). In the beginning, work within the eld focused on formulating general strategies, gaining experience and implementing prototypes of the proposed concepts. My collegue John Rasmussen developed the prototype shape optimization system \CAOS" (Computer Aided Optimization System) which is a tool for interactive shape optimization of planar geometries subjected to static loads. The problems are de ned by means of the AutoCAD system, using a design element concept as described by Fleury (1987). CAOS was further developed by Birker & Lund (1991) to include three-dimensional structures and thermo-elastic problems, see also Rasmussen, Lund, Birker & Olho (1993). Kibsgaard developed a general shape optimization system based on the publicly available nite element program Modulef, see Kibsgaard, Ol-
8
1.1. Programme of Research on Computer Aided Engineering Design
ho & Rasmussen (1989) and Kibsgaard (1991). However, the computational organization in Modulef proved less suitable for optimization purposes and this led to the decision of writing our own analysis code. It might be advantageous to use a commercial nite element system for the analysis due to the generality of the analysis facilities but we wanted to develop a kind of computer engineering design laboratory where experiments concerning, for example, structural analysis, design sensitivity analysis, mathematical programming, and links between CAD environments and structural optimization systems can be carried out. Furthermore, the tools available for software developers have reached a high standard, making it easier to develop large and complex software systems. In 1991 the development of a general purpose fully three-dimensional computer aided environment for interactive structural design, analysis, design sensitivity analysis, synthesis, and engineering design optimization was initiated. This system is named \ODESSY" (Optimum DESign SYstem) and is based on the experiences obtained by developing the above-mentioned systems. ODESSY is being developed by several people at the Institute of Mechanical Engineering and is programmed in ANSI C, except for the postprocessing facilities which are written in C++. The ANSI C programming language has many advantages such as dynamic allocation of memory, facilities for de nition of complex data structures, and portability between di erent computer platforms. CAD integration
Topology optimization
Preprocessor ODESSY
Interactive shape, sizing, or material what-if study and design optimization
Finite element modules for structural analysis Modules for structural design sensitivity analysis
Database module for extraction of results Postprocessor
Figure 1.1: Main features of ODESSY. The main features of ODESSY can be seen in Fig. 1.1. The features shown in shaded boxes are all covered in this thesis whereas the other features are only described super cially. ODESSY is being integrated with the commercial CAD systems AutoCAD and Pro/Engineer, thereby setting rational design facilities directly at the disposal of the designer. In Fig. 1.2 it is illustrated how ODESSY can be used as a tool for interactive engineering design. The di erent tasks shown in Fig. 1.2 are not necessarily performed sequentially and several of them may be omitted, but the gure illustrates the facilities available for the design engineer. It is the hope of the people involved in the development of ODESSY that designers
Chapter 1. Introduction
9 Define parameterized design model in CAD environment.
Convert design model to a finite element analysis model using the preprocessor Perform structural analyses
Export improved design model to CAD environment Interactive engineering design using ODESSY
Monitor analysis results and user defined design criteria using database module Perform design sensitivity analyses w.r.t. specific parameters
Define optimization problem and perform automatic design optimization Perform what-if study and obtain improved design
Monitor design sensitivities of various design criteria
Figure 1.2: Interactive engineering design using ODESSY. in industry will adopt these methods for rational design, synthesis, and optimization.
1.2 Objective of Ph.D. Project and Contents of Thesis The objective of the present Ph.D. project is to develop, implement and integrate methods for structural analysis, design sensitivity analysis and optimization into the general purpose computer aided design system ODESSY. More precisely, a general nite element module for structural analysis must be developed (in cooperation with colleagues). The facilities for structural analysis have been limited to linear types of analysis problems. A number of reliable nite elements must be implemented in the system. A reliable tool for design sensitivity analysis is the basis for doing interactive structural design, synthesis, and optimization by means of ODESSY. Ecient and reliable methods for design sensitivity analysis of all implemented analysis types and nite element types must therefore be developed and implemented. Furthermore, it must be possible to evaluate user de ned design criteria and their derivatives w.r.t. speci c design parameters, and a general and exible module for de nition of structural optimization problems must be available. Chapter 2 of this thesis gives a general introduction to the facilities in ODESSY for parametric modelling. Basic concepts like design variables, design models, and modi ers are introduced and a brief overview of facilities for de nition of design models is given.
10
1.2. Objective of Ph.D. Project and Contents of Thesis
Chapter 3 provides a description of the analysis capabilities in ODESSY for which structural synthesis and optimization can be performed. Furthermore, the implemented nite elements are described. Chapter 4 is devoted to derivations of general expressions for design sensitivity analysis. Di erent approaches to design sensitivity analysis are discussed and the direct approach is chosen. This approach is computationally ecient and based on implicit di erentiation of the equations obtained when the continuum equations are discretized. The semi-analytical method where derivatives of various element matrices are approximated by rst order nite di erences is used due to its ease of implementation. Chapter 5 treats the problem of obtaining \exact" numerical derivatives of various nite element matrices as inaccuracy of the nite di erences involved in the semi-analytical method may lead to erroneous design sensitivities. A new and ecient method of \exact" semi-analytical design sensitivity analysis has been developed and implemented in ODESSY. \Exact" numerical derivatives are derived for element matrices of all implemented nite elements w.r.t. shape as well as sizing and material design variables. The eciency of the new semi-analytical method is also discussed. Chapter 6 contains several examples of design sensitivity analysis of structural problems in order to demonstrate the validity of the methods of design sensitivity analysis derived in the two preceding chapters. Some of the examples are used to illustrate that the di erent methods for design sensitivity analysis have been implemented correctly in ODESSY while other examples are used to emphasize the importance of using the new semi-analytical method of design sensitivity analysis that has been developed. Chapter 7 gives a description of the implementation of a general and exible method of formulating problems of mathematical programming in structural optimization systems. The mathematical formulation is based on the so-called bound formulation, and the implementation involves a parser capable of interpreting and performing symbolic di erentiation of the user de ned functions. This database module can also be used for interactive monitoring of user de ned design criteria. Chapter 8 is devoted to the problem of solving structural optimum design problems with multiple eigenvalues. When several eigenvalues of a structural problem coalesce and attain the same numerical value, special attention must be made due to lack of usual di erentiability properties of multiple eigenvalues with respect to changes in design. Necessary optimality conditions for optimum solutions are derived and iterative numerical algorithms for eigenvalue optimization problems involving multiple eigenvalues are described. Examples are given to illustrate the eciency of a proposed mathematical programming approach. Chapter 9 presents four examples of interactive engineering design with ODESSY. It is shown how design sensitivity display and what-if studies can be used to improve the design of a turbine disk. The use of the general and exible method for de nition of structural optimization problems as described in Chapter 7 is illustrated by optimizing the shape of the turbine disk, taking complex design constraints into account. Furthermore, the turbine disk is redesigned using a ceramic material where the probability of failure must be reduced. Finally, the hood of a Mazda 323 automobile is shape optimized with the
Chapter 1. Introduction
11
objective of maximizing the fundamental frequency of vibrations. The present work is summarized in the conclusions in Chapter 10 where possible future extensions also are discussed.
12
1.2. Objective of Ph.D. Project and Contents of Thesis
Chapter
2
Parametric Modelling for Optimum Design 2.1 Introduction
devoted to a general description of the parametric modelling facilities T available in ODESSY. This preprocessing part of ODESSY is mainly being developed his chapter is
by my collegue John Rasmussen, and the aim is therefore to give an overview of the geometric facilities available for de ning structural optimization problems and to introduce some basic concepts. The motivation behind developing a system like ODESSY is the current trend toward uni cation of engineering design tools which were previously developed and used separately, namely geometric modelling in the form of CAD systems, and mechanical analysis and design sensitivity analysis using the nite element method. The united capabilities of these techniques allow for a major step forward in mechanical engineering design, i.e., from tools aimed at analysis to tools directly aimed at synthesis. Before the parametric modelling facilities are described, some basic concepts for structural design optimization are introduced.
2.2 Basic Concepts The label structural design optimization identi es the type of design problem where the set of structural parameters is subdivided into so-called preassigned parameters and design variables, and the problem consists in determining optimum values of the design variables such that they maximize or minimize a speci c function termed the objective (or criterion, or cost) function, while satisfying a set of geometrical and/or behavioural requirements which are speci ed prior to design and are called constraints. 13
14
2.2. Basic Concepts
2.2.1 Design Variables In order to introduce the possible design variables of the problem it is convenient to de ne the speci cations necessary to perform an ordinary analysis using the nite element method. In general, such an analysis requires information about (see, e.g., Cook, Malkus & Plesha (1989) and Mouritsen (1992)): 1. Geometry (domain shape of the structure, division into nite elements, and kinematic boundary conditions). 2. Actions (loads acting on the structure). 3. Constitution (physical properties of the materials, and properties of the used nite elements). If a design sensitivity analysis and/or optimization is to be performed, a number of design variables must be de ned. The design variables will be denoted by
ai; i = 1; : : : ; I (2.1) and are assembled in the vector a. The design variables can be categorized as follows, see, e.g., Olho & Taylor (1983): 1. Geometrical design variables: (a) Sizing design variables: describe cross-sectional properties of structural components like dimensions, cross-sectional areas or moments of inertia of bars, beams, columns, and arches; or thicknesses of membranes, plates, and shells. (b) Con gurational design variables: describe the coordinates of the joints of discrete structures like trusses and frames; or the form of the center-line or mid-surface of continuous structures like curved beams, arches, and shells. (c) Shape design variables: govern the shape of external boundaries and surfaces, or of interior interfaces of a structure. Examples are the cross-sectional shape of a torsion rod, column, or beam; the boundary shape of a disk, plate, or shell; the surface shape of a threedimensional component; or the shape of interfaces within a structural component made of di erent materials. (d) Topological design variables: describe the type of structure, number of interior holes, etc., for a continuous structure. For a discrete structure like a truss or frame, these variables describe the number, spatial sequence, and mutual connectivity of members and joints. 2. Material design variables: represent constitutive parameters of isotropic materials, or, e.g., stacking sequence of lamina, and concentration and orientation of bers in composite materials.
Chapter 2. Parametric Modelling for Optimum Design
15
3. Support design variables: describe the support (or boundary) conditions, i.e., the number, positions, and types of support for the structure. 4. Loading design variables: describe the positioning and distribution of external loading which in some cases may be at the choice of the designer. 5. Manufacturing design variables: parameters pertaining to the manufacturing process(es), surface treatment, etc., which in uence the properties and cost of the structure. The topological type of design variables is not covered by the presentation in this report, but topology optimization using the homogenization method, see Bends e & Kikuchi (1988), Bends e (1989, 1994), Bends e & Rodrigues (1990, 1991), and Diaz & Kikuchi (1992), and theory on optimal orientations of orthotropic materials, see Pedersen (1989, 1990, 1991), is being implemented in ODESSY by my collegue Lars Krog, see Olho , Krog & Thomsen (1993). The topology optimization can be extremely ecient in nding a good topology for the structure which then can be further improved by subsequent sizing or shape optimization, see Olho , Bends e & Rasmussen (1991), Bends e, Rodrigues & Rasmussen (1991), Bremicker, Chirehdast, Kikuchi & Papalambros (1991), Rasmussen, Thomsen & Olho (1993), Olho , Lund & Rasmussen (1992), and Rasmussen, Lund & Birker (1992). Furthermore, the support and loading design variables are currently only available in the form of position type of variables. In the following the design variables of the structural design problem is, for simplicity, generally divided into the three groups
Generalized shape design variables. Sizing design variables. Material design variables. Here, the group of shape design variables include con gurational variables as well as position type of variables for support and loading. Furthermore, the types of manufacturing design variables covered can be included in the above-mentioned three groups.
2.3 Introduction to the Design Model Concept Having de ned the possible design variables of the structural design problem considered, a convenient way of linking them to the nite element analysis model must be available. In case of sizing and material design variables, the design variables may be linked directly to the analysis model, but this approach is not suitable for shape design variables. Geometrically, shape optimization is much more dicult to handle than sizing (and material) optimization, because it involves a successive shape change of the model, and the description in the following is therefore mainly with attention to shape optimization. In the early
16
2.3. Introduction to the Design Model Concept
days of structural shape design sensitivity analysis and optimization, attempts were made to use the nite element model directly as design model, i.e., to use node coordinates as design variables, see e.g., Zienkiewicz & Campbell (1973). It turns out that this method has at least four serious drawbacks (see, e.g., Ding (1986) and Rodrigues (1988)):
The number of design variables can become very large. It is dicult to ensure compatibility and slope continuity between boundary nodes.
It is dicult to maintain an adequate nite element mesh during the domain shape updating process when a shape optimization is performed.
The structural shape design sensitivities might not be accurate unless high order nite element types are used.
Based on these experiences, most computer aided environments for interactive structural shape design and optimization are founded on an important distinction between the design model and the analysis model as demonstrated by, e.g., Braibant & Fleury (1984), Esping (1984), Rasmussen (1990), and Olho , Bends e & Rasmussen (1991). The design model is a variable description of the domain shape of the structure but sizing as well as material design variables can also be linked to the design model. It can be closely connected with a CAD model as described by Rasmussen (1990), and Rasmussen, Lund, Birker & Olho (1993), and it is totally distinct from the nite element model that is used for the analysis. The design model may consist of so-called design elements as presented by Braibant & Fleury (1984) and Bennett & Botkin (1985). The boundaries (or surfaces in case of a three-dimensional model) of the design elements can be curves of almost any character, i.e., piecewise straight lines, arcs, b-splines with speci ed degree of continuity, bezier curves, Coons patches, etc. It is therefore very simple to generate relatively complicated geometries with a small number of design elements. The shapes of the boundaries are controlled by a number of control points, also often termed \master nodes". Then, shape design variables can be de ned to control the positions of these master nodes, and possible sizing as well as material design variables can be linked to each design element.
2.3.1 The Prototype Shape Optimization System CAOS The design model concept in ODESSY is based on the experience with the prototype shape optimization system CAOS which were developed by Rasmussen (1989) and further developed by Birker & Lund (1991), see also Rasmussen (1990), Olho , Bends e & Rasmussen (1991), Rasmussen (1991), Rasmussen, Lund & Birker (1992), Olho , Lund & Rasmussen (1992), and Rasmussen, Lund, Birker & Olho (1933). The design elements in CAOS are either topologically quadrilateral design elements in case of a two dimensional structure, or topologically hexahedral design elements in case of a three dimensional structure. This
Chapter 2. Parametric Modelling for Optimum Design
17
approach enables simple mapping mesh generation but may require quite many design elements even for simple geometries. The mapping mesh generation technique is illustrated on Fig. 2.1.
transformation
s
y
x
r t transformation
r
s
z x
y
Figure 2.1: Mapping mesh generation technique in 2D and 3D. In CAOS, the shape of boundaries or surfaces of design elements are controlled by a number of master nodes. In the case of de ning a shape optimization problem, this creates an evident connection between the generalized shape design variables (the positions of the master nodes) and the shape of the geometry, and thus provides a simple parametric model of the structure. At rst glance, the design element approach used in CAOS seems to be an adequate solution to the problem of parameterizing a geometry and coupling it with structural optimization. However, there are also a few inherent problems which manifests themselves when the geometries get just slightly complicated. To illustrate this, consider in Fig. 2.2 the CAOS design model of a, from a geometrical point of view, very simple 3D structure, the so-called wishbone which was rst presented by Brama & Rosengren (1990). In spite of the geometrical simplicity of the model, as many as 30 design elements with 202 boundaries were required to obtain an acceptable nite element modeling of the structure. Furthermore, the parametrization required as many as 308 generalized shape design parameters to control translation directions for master nodes. A subsequent linking of parameters reduced the number of independent variables to 112, but this is still far more than what would naturally be assigned to a structure of this complexity. Generating and checking such a design model manually is, even with the use of an interactive graphics environment, a time consuming and error prone task. For a detailed description of this example, refer to Birker & Lund (1991) and Rasmussen, Lund & Birker (1992).
18
2.4. The Design Model Concept in ODESSY
Figure 2.2: CAOS design model of wishbone structure comprising 30 design elements, 202 boundaries, and 308 translation directions.
2.4 The Design Model Concept in ODESSY Based on the experiences obtained by developing CAOS, a much more general and geometrically versatile design model concept is implemented in ODESSY, re ecting the desire of integrating the system with three dimensional CAD systems for advanced geometric modelling, i.e., including Constructive Solid Geometry (CSG) techniques. The problem of generating parameterized geometric models of even complicated geometries can be solved by state of the art CAD systems, and the challenge therefore lies in the creation of interfaces between the CAD system and the structural optimization system. My collegues John Rasmussen and Anders Kristensen are currently working on solutions to this problem, and a geometric formulation with solid modelling capability is being developed based on the commercial CAD system Pro/Engineer. A more simple interface to the AutoCAD system has also been implemented by my collegues Oluf Krogh & John Rasmussen. The design model concept in ODESSY is based on a hierarchical parametric design model because the desire to be able to parameterize virtually any property of the model has been the focal point in the development of ODESSY from the very beginning. ODESSY is based on a totally parametric modelling system in which data are initially divided into two categories or levels of increasing complexity, Level0 and Level1. Entities in both levels are always de ned by user de ned names, rather than by numbers. Level0 is the \ground" level. This level contains geometrical entities which are independent of other entities in the model. This would typically be points, vectors and scalar numbers. In the terminology of CAOS and other traditional shape optimization systems, master nodes would be Level0 entities. Other possible Level0 entities could be modules of elasticity or thicknesses. Level0 is characterized by the property that it is the only level containing real numbers.
Chapter 2. Parametric Modelling for Optimum Design
19
Level1 contains entities which depend only on Level0 information. Boundary curves and surfaces are typical examples. Their shapes are de ned solely by the position of the master nodes. Level1 could contain, for instance, circles, curves, boxes, tetrahedrons, cylinders, cones, etc. The actual shapes of all entities in Level1 are de ned by pointers to Level0 information. Level1 thus consists only of integer type information. Level0 and Level1 of the geometric model of ODESSY cannot handle the problem of nding implicitly de ned curves and surfaces which may result from the union of several solid components, i.e., when CSG techniques are used. Yet another level of information, Level2, is devised for this purpose. Level2 is not meant for processing by ODESSY itself but by an external solid modelling system. In this context, the solid modeler may be thought of as a geometric compiler which transforms high level geometric assemblies into lower level curves and surfaces that can be handled directly by the more primitive geometric system of ODESSY. For further descriptions of this approach of integration with solid modelling CAD systems, see Rasmussen, Lund & Olho (1993a, 1993b).
2.5 Design Variables and Modi ers The parametric nature of the design model comes automatically from the division of information into interdependent levels. We simply control the model by modifying the entities of Level0. Because of the hierarchical construction of the model, modi cations of Level0 entities automatically lead to corresponding changes in Level1. In order to improve the possibility of de ning the design space, data in Level0 are controlled by the use of so-called modi ers. As previously mentioned, traditional systems, like CAOS, usually contain only one type of modi er, i.e., translational transformations (also called move directions). Other possible modi ers may be point scaling, line scaling, and rotation. This way, Level0 entities may be subjected to one or several transformations thus controlling the overall design. The design variables of the problem will then simply be the amount of transformation, i.e., the rotation angle or the scale factor. In this framework, for instance the move directions of the ODESSY system are realized as translation modi ers coupled with design variables which specify the magnitude of the translation. Each master node is Level0 information, and the entire design model therefore changes with relocation of the master nodes. Similarly, Level0 information may be subjected to:
translation modi ers scaling modi ers rotation modi ers combinations of the three above-mentioned modi ers
In designing the structure of links between design variables, modi ers and Level0 information, the following points shall be observed:
20
2.5. Design Variables and Modi ers P1 S4
S3
P1 S3
design variable = 45° S4
S2
modifier: rotate around P1 S1
S1
S2
Figure 2.3: Example of rotation modi er and design variable a ecting the four points S1, S2, S3 , and S4.
It shall be possible to have the same design variable a ect several Level0 entities simultaneously through the same modi er as illustrated in Fig. 2.3.
It shall be possible to have the same design variable a ect several Level0 entities simultaneously through di erent modi ers.
It shall be possible to use the same modi er for several di erent links between
design variables and Level0 entities. In a problem comprising, say, 50 master nodes which can move freely in the plane, it would suce to de ne two modi ers, namely an x-translation and a y-translation, rather than having to de ne two modi ers for each individual master node.
It shall be possible to combine any number of modi ers into a new compound modi er, for instance simultaneous rotation and scale.
These requirements can be ful lled by creating link data structures of the following form: LINK: [one design variable] ! [any number of modi ers] ! [one Level0 entity] This construction ful ls any of the requirements above. For instance, given a design variable \angle" and a modi er \rotation", the example of Fig. 2.3 is realized by the links: LINK: angle ! rotation ! S1 LINK: angle ! rotation ! S2 LINK: angle ! rotation ! S3 LINK: angle ! rotation ! S4 In case of material or sizing design variables, translation modi ers may be used to a ect the current value of the chosen variable, e.g., Young's modulus or plate thickness. Currently, material and sizing parameters are assumed to be constant within each design element.
Chapter 2. Parametric Modelling for Optimum Design
21
2.6 Mesh Generation The implementation of good algorithms for mesh generation is very important in a structural shape optimization system. It must be possible to specify desired element sizes within each design element, and the mesh must remain adequate during the domain shape updating process when a shape optimization is performed. It may be advantageous to include adaptive mesh generation in a shape optimization system, see, e.g., Diaz, Kikuchi, Papalambros & Taylor (1983), Bennett & Botkin (1985), and Kikuchi, Chung, Torigaki & Taylor (1986), but such facilities are currently not implemented. Mapping mesh generation techniques are available in both two and three dimensions, but in order to overcome the problem of having to divide the design model into quadrangular or hexagonal design elements as necessary for mapping mesh generation, ODESSY also allows for the use of unstructured free meshing. Currently, a modi ed version of an algorithm by George (1988) is implemented. This algorithm can subdivide an arbitrary planar domain into triangles, and in ODESSY, the algorithm is extended to cover curved surfaces. Furthermore, another unstructured mesh generation algorithm can generate a mixed mesh predominantly of quadrangular elements, supplemented with a few triangular elements. Robust free meshing of arbitrary volumes, which is a very challenging area, is currently being implemented by my collegue Anders Kristensen. After this basic overview of the facilities for de nition of design models in ODESSY, the analysis facilities will be presented.
22
2.6. Mesh Generation
Chapter
3
Analysis Capabilities for Structural Optimization 3.1 Introduction
Isensitivity given with focus on the types of analysis problems for which both analysis and design analysis have been implemented. Furthermore, descriptions of the implemented n this chapter a general description of the nite element module in ODESSY is
nite elements are given. The nite element module in ODESSY has been developed by several people at the Institute of Mechanical Engineering, but the main code has been developed by my collegue Oluf Krogh and myself. The implementation of subroutines for solution of the nite element discretized equations has been performed by Oluf Krogh. When I started this project, Oluf Krogh had implemented a pro le skyline solver for solution of linear static equations based on algorithms in Dhatt & Touzot (1984). Furthermore, he had implemented the Subspace iteration method for determination of the lowest eigenvalues of a structural eigenvalue problem, see Bathe & Ramaswamy (1980), Bathe (1982), and Dhatt & Touzot (1984), and some simple elements were implemented. My contribution to the nite element module has mainly been
Implementation of a family of isoparametric nite elements with capabilities for both analysis and design sensitivity analysis.
Development of general and ecient storing schemes for strains and stresses for any combination of nite element types.
Extension of the analysis facilities to include thermal analysis, linear buckling analysis, and any mixture of the available types of analyses.
General system development. 23
24
3.2. Implemented Analysis Types for Structural Optimization
3.2 Implemented Analysis Types for Structural Optimization In the following the di erent types of analysis problems that are covered for both analysis and design sensitivity analysis in ODESSY are brie y decribed. The intention with this section is mainly to give an overview, as the di erent analysis problems are described more detailed in Chapter 4 where general expressions for design sensitivity analysis are developed. Furthermore, the notation of di erent matrices and vectors is introduced. Only linear analysis types have yet been implemented in ODESSY.
Static Stress Analysis The most common analysis type with the nite element method is static stress analysis. The global equilibrium equation of a nite element discretized structural problem with linearly elastic response is given by KD = F (3.1) where K is the global sti ness matrix, D the nodal displacement vector and F is the consistent nodal force vector. These global matrices and vectors are, as always in the nite element method, assembled from element matrices and vectors, i.e., X X X K = k; D = d; F = f (3.2) ne
ne
ne
Here, k is the element sti ness matrix, d the element nodal displacement vector, f the consistent element nodal force vector, and ne is the number of nite elements used to discretize the structure. In ODESSY, the solution of Eq. 3.1 is carried out by a Crout decomposition scheme as described in Dhatt & Touzot (1984).
Free Vibration Analysis The free vibration analysis problem covered is a real, symmetric, structural eigenvalue problem where the nite element formulation is K j = !j2M j ; j = 1; : : : ; n (3.3) K is the global sti ness matrix, M the global mass matrix, !j the eigenfrequency, j the corresponding eigenvector, and n denotes the dimension of the problem.
Thermal Analysis The nite element equilibrium equation for a steady state heat conduction problem is given by Kth T = Q (3.4) where Kth is the global thermal \sti ness matrix" involving contributions from element heat conduction matrices and coecients of the temperature vector T arising from convection boundary conditions, and Q is the thermal load vector involving forcing terms due to
Chapter 3. Analysis Capabilities for Structural Optimization
25
heat addition processes, e.g., heat ux. The solution of Eq. 3.4 is similar to the solution of a static stress analysis problem.
Thermo-Elastic Analysis This analysis type consists of a steady state thermal analysis followed by a static analysis. First the temperature eld T is found from the solution of Eq. 3.4, resulting in thermal strains "th . Then Eq. 3.1 is solved by taking the thermal e ects into account.
Eigenfrequency Analysis with Initial Stress Sti ening When calculating eigenfrequencies of a structure, it may be necessary to take into account initial stress sti ening e ects due to mechanical loading. First a static analysis is performed, resulting in element stresses . These stresses are used to generate element initial stress sti ness matrices k (also termed geometric sti ness matrices) which represent the initial stress sti ening e ects due to the loads applied to the structure. The conventionel global sti ness matrix K is then augmented with the global initial stress sti ening matrix K resulting in a modi ed form of Eq. 3.3, i.e., (K + K ) j = !j2M j ; j = 1; : : : ; n
(3.5)
Eigenfrequency Analysis with Thermal Loading and Initial Stress Sti ening The eigenfrequency analysis taking initial stress sti ening e ects into account can be extended taking thermal e ects into account, i.e., the stress sti ening e ects may originate from a thermo-elastic analysis taking both thermal and mechanical loading into consideration.
Linear Buckling Analysis It is possible to perform linear buckling analysis which is restricted to the assumptions that the structure fails suddenly and the structure has constant sti ness e ects for all loads up to the bifurcation point. These assumptions make this analysis type limited in its applications as results will be most often on the unconservative, unsafe side. In practice, linear buckling analysis can be used to inexpensively nd an upper bound of the load carrying capability of a structure. If buckling analyses are to be used safely for structural optimization of real-life structures, it is most often necessary to perform non-linear analyses resulting in a very complicated design sensitivity analysis, see, e.g., Ringertz (1992), and large displacement buckling analysis has therefore not been implemented in ODESSY. The nite element formulation of a linear buckling analysis can be written as (K + j K ) j = 0; j = 1; : : : ; n
(3.6)
26
3.3. Finite Element Types Implemented for Structural Optimization
where K is the global sti ness matrix, K the initial stress sti ness matrix established from an initial static stress analysis, n the dimension of the problem, j the buckling load factor, and j is the corresponding eigenvector of displacements.
Linear Buckling Analysis with Thermal Loading Finally, the linear buckling analysis can be extended by taking thermal loading into account, i.e., the buckling load factors found correspond to the load vector resulting from both thermal and mechanical loading.
3.3 Finite Element Types Implemented for Structural Optimization Although many di erent nite elements have been implemented in ODESSY, the following description will be restricted to the isoparametric family of nite elements that I have implemented and used for solving structural design and optimization problems. Besides the elements described here, di erent types of spring, beam, truss, frame, and sandwich plate and shell nite elements are currently available in the system. The isoparametric nite elements can be divided into the following three groups 1. 3D solid nite elements. 2. 2D solid nite elements. 3. Mindlin plate and shell nite elements. The element matrices and vectors necessary for performing the di erent types of analyses described in Section 3.2 have been implemented for all these isoparametric nite elements. The types of elements implemented will be only brie y described in the following while more detailed descriptions are given in Appendices A, B, and C. Furthermore, all elements are described in Chapter 5, where \exact" numerical derivatives of element matrices and vectors are found as a basis for a new method of design sensitivity analysis.
3.3.1 3D Solid Isoparametric Finite Elements An 8-node and a 20-node isoparametric nite element as illustrated in Fig. 3.1 have been implemented. A detailed description of element matrices and vectors for these two elements is given in Appendix A. It should be noted that the code for computing the sti ness matrix and the consistent nodal load vector for the 3D isoparametric nite elements has been implemented by my collegues Niels Kristian Bau-Madsen and Oluf Krogh, respectively, whereas I have implemented all other matrices and vectors for these elements.
Chapter 3. Analysis Capabilities for Structural Optimization
27
z,w
y,v
x,u
Figure 3.1: 8- and 20-node isoparametric nite elements.
3.3.2 2D Solid Isoparametric Finite Elements Five di erent 2D solid isoparametric nite elements have been implemented in ODESSY as illustrated in Fig. 3.2. They are all formulated in a uni ed way for both plane stress, plane strain, and axisymmetric situations.
of rotational (axissymmetry ) y,v (z,w)
x,u (r,u)
Figure 3.2: 3-, 4-, 6-, 8-, and 9-node 2D isoparametric nite elements. Text in parantheses refer to standard notations for problems with rotational symmetry. It should be noted that element matrices for the two 2D solid triangular elements can be formulated analytically. For the 6-node element it is then necessary that the element sides are straight and nodes at element sides are midside nodes. Such derivations for plane stress elements can be found in Pedersen (1973) and for axisymmetric elements in Ladefoged (1988). Although it is advantageous to have analytic expressions for element
28
3.3. Finite Element Types Implemented for Structural Optimization
matrices, I have nevertheless chosen the standard approach of using numerical integration because the 2D solid isoparametric nite elements can be formulated in a uni ed way for both plane stress, plane strain, and axisymmetric situations as shown in Appendix B. Furthermore, using the isoparametric formulation we have the advantage that the 6-node element can have straight as well as curved edges. The 2D solid nite elements are described in detail in Appendix B.
3.3.3 Isoparametric Mindlin Plate and Shell Finite Elements
y
y vi θyi
i z
x
wi z
θxi
ui
x
Figure 3.3: 3-, 4-, 6-, 8-, and 9-node isoparametric Mindlin plate nite elements. Six di erent Mindlin plate nite elements have been implemented in ODESSY. The elements have 3-, 4-, 6-, 8-, and 9-nodes, respectively, and are shown in Fig. 3.3. The 6-, 8-, and 9-node elements can have straight as well as curved boundaries in the element plane. The 9-node element is implemented both as the Lagrange plate element and as the \heterosis" plate nite element as described in Section C.9 in Appendix C, where a detailed description of the elements is given. These six plate elements can also be used as at shell elements as described in Section C.10 in Appendix C. This approach of generating at shell elements from plate elements has worked well for the problems studied, but may cause problems for strongly curved surfaces where it is necessary to use many elements in order to obtain a proper nite element model. This is also discussed in Section C.10 in Appendix C. After the presentation in this chapter of the analysis problems that can be solved and the type of nite elements that can be used, the subsequent chapter will focus on the design sensitivity analysis.
Chapter General Expressions for Design Sensitivity Analysis
4
4.1 Introduction
Ipressions and optimization, design sensitivity analysis is the basic enabling tool, and general exfor design sensitivity analysis will be given in this chapter. The types of design n a general purpose computer aided environment for interactive structural design
variable of the structural design problem can be shape as well as sizing or material variables and they are denoted by ai; i = 1; : : : ; I . It is now the aim to establish expressions for design sensitivities of various criteria with respect to these design variables in an accurate and ecient way. The overall nite di erence (OFD) approach to sensitivity analysis and the consequences of using nite di erence schemes are described rst. This approach to sensitivity analysis is not computationally ecient, but can be used as a reliable reference method. Next, three di erent methods for design sensitivity analysis are brie y described and it is chosen to use the so-called discrete version of the direct approach. Then the direct approach to design sensitivity analysis of static problems is discussed. The direct approach is based on di erentiation of the equations obtained when the continuum equations are discretized by the nite element method, and this method is very ecient. Using this approach it is necessary to determine derivatives of element sti ness matrices, and the semi-analytical (S-A) method where these derivatives are approximated by rst order nite di erences is chosen. Expressions for design sensitivities of displacements, stresses and compliance are found with respect to single and simultaneous change of design variables. Next, the sensitivity analysis of thermo-elastic problems is discussed and the direct approach is used again. The sensitivities of the thermal problem are taken into account for thermo-elastic problems. Finally, eigenvalue design problems are considered. The direct approach can be used 29
30
4.2. Overall Finite Di erence Approach to Sensitivity Analysis
to simple (distinct) eigenvalues but in the case of multiple (repeated) eigenvalues the sensitivity analysis is more complicated. Then it is necessary to use perturbation techniques in order to develop sensitivity expressions for multiple eigenvalues, as the eigenvalues are no longer di erentiable functions of the design in the normal (Frechet) sense. The sensitivity analysis again will be presented both for perturbations of a single design variable ai and for simultaneous change of several design variables.
4.2 Overall Finite Di erence Approach to Sensitivity Analysis The simplest nite di erence approximation is the rst order forward (or backward) difference approximation. If displacements, stresses, compliance, mass, or any other property calculated by the analysis module is denoted by a function fj (a) of the design variables ai , the overall forward nite di erence approximation fj =ai to the sensitivity @fj =@ai is given by @fj (a) ' fj (a1; : : : ; aI ) @ai ai f ( a ; = j 1 : : : ; ai + ai ; : : : ; aI ) fj (a1 ; : : : ; ai; : : : ; aI ) (4.1) ai If the derivatives of the function fj are sought for n design variables, the overall nite di erence (OFD) method requires n additional analyses. Therefore this method is computationally costly and mainly used as a reference method whose limit with regard to accuracy is known to be set only by the solution procedure, the discretization, and the usual accuracy capabilities of the applied nite element. Whenever a nite di erence scheme is used to approximate derivatives, there are two sources of error: truncation and condition errors. The truncation error is a result of the neglected terms in the Taylor series expansion of the perturbed function fj (a1; : : : ; ai + ai ; : : : ; aI ). This source of error can be reduced by using a small perturbation ai . The condition error is the di erence between the numerical evaluation of the function and its exact value. Contributions to the condition error are, e.g., computational round-o errors or errors due to an iterative solution process which is terminated early. The round-o errors are normally small for most computers unless the perturbation ai is very small. These opposite demands to the magnitude of the perturbation may give rise to the so-called \step-size dilemma", see, e.g., Haftka & Adelman (1989). If we select the perturbation to be small, so as to reduce the truncation error, the result may be an excessive condition error. In some cases there may not even be any perturbation which results in suciently small errors. As noted by Haftka & Adelman (1989), the \step-size dilemma" may be reduced if a higher order nite di erence approximation is used, e.g., a second order central di erence approximation, but this implies additional computational cost.
Chapter 4. General Expressions for Design Sensitivity Analysis
31
4.3 Selection of Method for Design Sensitivity Analysis As the overall nite di erence (OFD) method is computationally inecient for the design sensitivity analysis, another approach has to be used. The derivatives of structural response in principle can be calculated at three stages. We can (I) di erentiate the continuum equations de ning the response of the system, using the material derivative concept of continuum mechanics, then discretize the problem and solve it by using an adjoint variable technique as described by, e.g., Haug & Rousselet (1980a, 1980b), Cea (1981), Zolesio (1981), Choi (1985), Haug, Choi & Komkov (1986), Haug & Choi (1986), Haber (1987), Dems & Mroz (1983, 1984, 1993), and Dems & Haftka (1989). This continuum approach is known as the material derivative (or speed) method. Using this variational approach, the derivatives can be expressed in terms of boundary integrals which are computationally inexpensive to evaluate. Unfortunately, there are considerable numerical diculties associated with the evaluation of the boundary integrals, see Yang & Botkin (1986), especially for low-order elements which do not model a curved boundary well, see Yang & Choi (1985). This can be avoided by using domain instead of boundary integrals, see Choi & Haug (1983) and Choi & Seong (1986), but then the numerical eciency decreases. The main advantage of the continuum approach, as stated by Haftka & Grandhi (1986), seems to be the generality of its results as the method is equally applicable to nite element, boundary element (see Mota Soares, Rodrigues & Choi (1984)), or any other numerical or analytical solution technique. Furthermore, sensitivity calculations can be carried out outside existing nite element codes, using postprocessing data only. Thus, the design sensitivity analysis software does not have to be embedded in an existing nite element code, see Choi, Santos & Frederick (1985) and Santos & Choi (1989). This is a great advantage if the analysis module is a commercial nite element package like ANSYS, NASTRAN, COSMOS, etc. We can (II) use the direct approach which is based on di erentiation of the equations obtained when the continuum equations have been discretized (in this case by the nite element method). This approach, which is also known as the implicit di erentiation approach, thus has a reversed order of discretization and di erentiation compared to the continuum approach described above. For static design sensitivity analysis, we can distinguish between two types of the direct approach. In the (A) discrete version, the displacement sensitivity eld @ D=@ai is calculated for the whole structure for each design variable ai; i = 1; : : : ; I , while the (B) adjoint method calculates the derivative of a function g(D) of displacements D. The adjoint method requires the solution for each desired function g(D), i.e., if the number of stress and displacement functions needed for formulating and solving the optimization problem is less than the number of design variables, then the adjoint method is more ecient than the discrete version of the direct approach to design sensitivity analysis. The direct approach is, in both the discrete and adjoint version, very popular due to its ease of implementation compared to the continuum approach. Finally, we can (III) di erentiate directly the computer program used to solve the structural
32
4.4. Design Sensitivity Analysis of Displacements
response, see, e.g., Wexler (1987) and Masmoudi, Broudiscou & Guillaume (1993). This approach of automatic di erentiation of a function de ned by its program (in Fortran, C, etc.) by using Taylor expansion is very interesting, but has not been considered here as it seems to require substantial available memory in the computer, see Masmoudi, Broudiscou & Guillaume (1993). The issue of which approach of sensitivity analysis is better is a much debated subject and several authers, e.g., Yang & Botkin (1986), Choi & Twu (1988), and Haftka & Adelman (1989), have presented comparisons between the variational methods and the direct approach and their relative merits. The conclusive argument for selection of method for design sensitivity analysis has been that the continuum approach takes a lot of analytical work in order to develop the expressions for design sensitivities as can be seen in the references mentioned. I have wanted to implement many di erent kinds of nite element types and at the same time be able to handle many di erent kinds of design variables. For this purpose the direct approach seems much easier to implement and to be just as applicable to solve the problem as the continuum approach. If we were going to use a commercial nite element package for the analysis, the continuum approach might be advantageous as postprocessing techniques can be employed, but we have decided to write our own analysis code due to less promising experiences of implementing design sensitivity analysis in an existing nite element code, such as, e.g., Modulef, see Kibsgaard (1991). So, the direct approach to design sensitivity analysis has been chosen due to its ease of implementation, and it will be shown in the following that this method is very ecient. The direct approach will be only used in the discrete version as the adjoint method is not suited for the way optimization problems are formulated in ODESSY, cf. the description in Chapter 7.
4.4 Design Sensitivity Analysis of Displacements In the displacement based nite element method, the design sensitivity analysis of various criteria is based on sensitivities of the displacement eld. Thus, when the displacement sensitivities are known, e.g., stress and compliance sensitivities are easily computed. The global equilibrium equation of a nite element discretized structural design problem with linearly elastic response is given by
KD = F (4.2) where K is the global sti ness matrix, D is the nodal displacement vector and F is the
consistent nodal force vector. The solution of Eq. 4.2 is carried out by Gaussian elimination reformulated in a two phase process that does not require to modify K and F simultaneously. It is thereby possible to solve Eq. 4.2 for additional load cases, i.e. several right hand sides, without much additional computational e ort. The time consuming part of solving Eq. 4.2 is the factorization of the global sti ness matrix K, in which this matrix is basically decomposed into the product LU, where L is a lower triangular matrix (with
Chapter 4. General Expressions for Design Sensitivity Analysis
33
elements only on the diagonal and below) and U is a upper triangular matrix. Varieties of this basic decomposition are, for example, the LDU, Crout, and Cholesky decomposition schemes, see, e.g., Dhatt & Touzot (1984). In ODESSY, the Crout decomposition scheme has been implemented. When the sti ness matrix K has been decomposed into the product LU, Eq. 4.2 can be rewritten as LUD = F (4.3) Using this decomposed form of the sti ness matrix, it is only necessary to solve a triangular set of equations which is quite trivial. First a vector V is found by forward substitution
LV = F
(4.4)
and then the displacement vector D can be found by back substitution
UD = V (4.5) Having determined the displacement vector D, the design sensitivity analysis of displace-
ments now can be considered. The direct approach to obtain design sensitivities of the displacement eld is based on implicit di erentiation of the global equilibrium equation. If Eq. 4.2 is di erentiated with respect to a design variable ai and the terms are rearranged, the following discrete version of the direct approach for the displacement sensitivities @ D=@ai is obtained
@ F ; i = 1; : : : ; I K(a) @@aD = @ K@a(a) D + @a i i i
(4.6)
Eq. 4.6 is of the same form as Eq. 4.2, so the factorized sti ness matrix K in the form LU can be reused, and only the new right hand side which is termed the pseudo load vector need to be calculated before the sensitivities @ D=@ai for each design variable ai can be found by forward and back substitution. This approach is therefore much more ecient for the design sensitivity analysis than the OFD approach described in Section 4.2. It is seen that the pseudo load vector is the load that must be applied to the structure to produce the displacement sensitivity eld due to changes of a design variable ai. The derivatives @ F=@ai of the force vector are easily calculated (are zero for design independent loads), and then the determination of @ D=@ai in Eq. 4.6 only requires calculation of the design sensitivities @ K=@ai of the sti ness matrix. These derivatives are normally calculated at the element level, i.e. @ K = X @ k ; i = 1; : : : ; I (4.7) @ai ne @ai where k is the element sti ness matrix and ne is the number of nite elements. If the design sensitivities @ k(a)=@ai are determined analytically before their numerical evaluation, the approach is called analytical design sensitivity analysis, and if they are determined by numerical di erentiation, the method is called semi-analytical (S-A) design sensitivity analysis, cf. Zienkiewicz & Campbell (1973), Esping (1983), Cheng & Liu
34
4.4. Design Sensitivity Analysis of Displacements
(1987), and Haftka & Adelman (1989). That is, in semi-analytical design sensitivity analysis the derivatives of the element matrices are approximated by rst order forward nite di erences (or another nite di erence scheme) @ k(a) ' k(a1 ; : : : ; aI ) @ai ai = k(a1 ; : : : ; ai + ai ; : : : ; aI ) k(a1 ; : : : ; ai; : : : ; aI ) (4.8) ai The method of analytical design sensitivity analysis is very dicult to implement in a general purpose shape design system which contains many di erent kinds of shape design variables and nite element types. Thus, a large amount of analytical work and programming will be required in order to develop analytic expressions for derivatives of various sti ness matrices with respect to possible shape design variables. Some examples of determining analytical derivatives of an element sti ness matrix with respect to a speci c kind of shape design variable can be found in, e.g., Braibant & Fleury (1984), Wang, Sun & Gallagher (1985), and El-Sayed & Zumwalt (1991). It is much more attractive to use the method of semi-analytical (S-A) design sensitivity analysis in this context, as it is easy to implement for many di erent kinds of shape design variables and nite element types, because simple and computationally inexpensive rst order nite di erences are used. Therefore, the method of S-A sensitivity analysis is very popular and, in most cases, this method is very ecient and reliable. As a nite di erence approximation is involved in this method, both truncation and conditions errors may occur. Furthermore, for shape design variables the perturbation ai must be selected suciently small, so that the elements do not become distorted. A strongly distorted mesh may result in changing accuracy of the solution and thereby give the derivatives a spurious contribution, see Botkin (1988). This problem is avoided in ODESSY by selecting the perturbation ai of a shape design variable ai so that boundary nodes are perturbed less than 1=1000 of the smallest side length of the elements in the structure. It should be noted, however, that the approximate rst order nite di erence calculation of derivatives of element sti ness matrices used in the S-A sensitivity method may result in severe inaccuracy problems for shape design variables due to truncation and condition errors. This problem and solutions to it will be addressed in Chapter 5.
: perturbed elements
Figure 4.1: Perturbation of boundary elements for shape design variables. The eciency of the design sensitivity analysis can be increased by using a so-called \active element" strategy. When the domain shape is perturbed in case of a shape design
Chapter 4. General Expressions for Design Sensitivity Analysis
35
variable ai , only nite elements situated at the surface are perturbed (and named active) as illustrated in Fig. 4.1 for a two-dimensional domain where the perturbation has been strongly exaggerated. The design boundary in Fig. 4.1 is modelled by a quadratic b-spline and the master node shown is translated in the direction speci ed by the vector. The active (perturbed) nite elements are hatched in Fig. 4.1 where the unperturbed mesh also is shown. The active element strategy implies that in the assembly of element matrix derivatives, only active elements contribute, and the pseudo load vector in Eq. 4.6 can thereby be calculated at the element level in the following computationally ecient way
@K D + @F = X @ai @ai nae
! @ ka da + @ f a ; i = 1; : : : ; I @ai @ai
(4.9)
where nae is the number of active nite elements, ka is the element sti ness matrix, da is the element displacement vector, and f a is the element load vector for an active element. The choice of perturbing the entire nite element mesh or perturbing only boundary nodes has originated many a dispute. Braibant & Fleury (1984) and Botkin (1988), among others, have advocated that it is necessary to move the interior nodes because the direct object of the optimization is the nite element model, rather than the real structure. Pedersen (1988) has argued that only boundary nodes should be perturbed as relocation of interior nodes may lead to a substantial risk to \maximize the errors of the nite element model, rather than minimize the physical stress concentration". Kibsgaard (1991) has made several comparisons between these two methods of mesh perturbations, and he obtained best results perturbing only boundary nodes. The physical argument to support this observation, as noted by Kibsgaard (1991), is that only the structure in the vicinity of design boundaries are subject to changes as the design changes. The rest of the structure is unchanged and therefore can be kept unchanged as well. If the interior nodes are perturbed, thereby resulting in di erences in the sti ness matrix other than for the boundary elements, these di erences would not re ect changes in the mechanical properties but rather inaccuracies of the sensitivity analysis. Furthermore, it is dicult to perturb interior nodes using unstructured mesh generators as a new mesh topology for the perturbed design model may appear. This is not allowed as it would re ect the accuracy of a di erent analysis model. Based on numerical experience and the arguments above, this \design boundary layer" approach of using one-element-deep sensitivity calculations for shape design variables has been adopted in ODESSY due to the advantageous increase of numerical eciency.
4.5 Design Sensitivity Analysis of Stresses When the displacement design sensitivities have been calculated it is straight-forward to calculate stress design sensitivities as will be shown in the following. The nite element expression for the element stresses (x; y; z) = fx y z xy yz xz gT may be written (a) = E "(a) = E B(a) d(a) (4.10)
36
4.6. Design Sensitivity Analysis of Compliance
where the constitutive matrix E is independent of the design, B is the strain-displacement matrix, and d is the element nodal displacement vector. One way to calculate the stress design sensitivities is to use a rst order forward nite di erence approximation, i.e. @ ' (a + ai ) (a) (4.11) @ai ai where (a + ai) = E "(a + ai) = E B(a + ai ) d(a + ai ) (4.12) The perturbed strain-displacement matrix B(a + ai) is easily calculated and the element displacement vector for the perturbed design can be approximated by a rst order Taylor series expansion d(a + ai ) ' d(a) + @ d@a(a) ai (4.13) i where the element displacement sensitivities @ d=@ai are determined by the solution of Eq. 4.6.
4.6 Design Sensitivity Analysis of Compliance The compliance C can be calculated as
C = DT F
(4.14)
By di erentiating Eq. 4.14 the following expression for the compliance design sensitivity is obtained @C = @ DT F + DT @ F (4.15) @ai @ai @ai All terms on the right hand side are known from the calculation of displacement sensitivities, see Eq. 4.6, so the compliance design sensitivity is easily evaluated.
4.7 Simultaneous Change of Design Variables In the following, displacements, stresses, compliance, mass, or any other property (except for multiple eigenvalues) calculated by the analysis module is denoted by a function fj . If some or all of the design variables ai are changed simultaneously then the linear increment of the function fj can be found by using rst order Taylor series expansions fj =
r
rTfj a
(4.16)
where fj denotes the gradient vector of fj and a is the vector of changes of the design variables ai ! @f @f j j (4.17) fj = @a ; : : : ; @a ; a = (a1 ; : : : ; aI )
r
1
I
Chapter 4. General Expressions for Design Sensitivity Analysis
37
Eq. 4.16 is valid due to di erentiability of the above mentioned vector elds and functions fj with respect to the design variables. These notations are useful for parametric studies of functions fj as well as for formulation of optimization problems.
4.8 Inclusion of Thermo-Elastic E ects In many engineering examples, thermo-elastic e ects due to a temperature distribution in the structure have to be considered. The nite element equilibrium equation for a steady state heat conduction problem is given by
Kth T = Q
(4.18)
where Kth is the global thermal \sti ness matrix" involving contributions from element heat conduction matrices and coecients of the temperature vector T arising from convection boundary conditions, and Q is the thermal load vector involving forcing terms due to heat addition processes, e.g., heat ux. Eq. 4.18 can be solved for temperatures T by standard solution procedures as described in Section 4.4, and having calculated the temperature distribution, its in uence on the static nite element analysis can be included. The temperature T at a given point gives rise to thermally induced strains "th given by
n o n o "th = "thx "thy "thz xyth yzth xzth T = T^ T^ T^ 0 0 0 T ; T^ = T T0
(4.19)
where is a matrix containing thermal expansion coecients, T is the temperature at the given point, and T0 is the temperature at which the structure is free of thermally induced strains (typically 20C). It is not possible to add these thermally induced strains directly to the mechanical strains in order to calculate the induced stress eld. This is due to the necessity of taking structural boundary conditions and internal resistance into account, i.e., a structure is free of thermally induced stresses if its supports do not inhibit thermal expansion or contraction. Instead the thermally induced strains "th are used to calculate a consistent global nodal force vector Fth due to the thermally induced strains
Fth
=
XZ ne
T E "th d
B
(4.20)
where is the domain of the nite element and ne is the number of nite elements. This nodal force vector due to thermally induced strains is added to the mechanical nodal force vector in Eq. 4.2 when solving the static equilibrium equations. This results in a displacement vector D used to calculate the element strains ", see Eq. 4.10, and then the total element stresses , due to mechanical and thermal strains, are given by
= E(B d "th )
(4.21)
38
4.9. Design Sensitivity Analysis of Eigenvalues
where E is the constitutive matrix, B is the strain-displacement matrix and d is the element displacement vector. In this way boundary conditions are considered when calculating thermal stresses. In the design sensitivity analysis the sensitivities of the temperatures T can be found in the same way as described for displacement sensitivities in Section 4.4. Eq. 4.18 is di erentiated with respect to a design variable ai; i = 1; : : : ; I , and rearranging the terms, the following expression for the temperature sensitivities @ T=@ai is obtained @ T = @ Kth (a) T + @ Q Kth(a) @a (4.22) @ai @ai i The factorized global thermal \sti ness matrix" can be reused as in the case of sensitivity analysis of displacements, and the new right hand side which can be termed the thermal pseudo load vector can be obtained using the S-A approach as described previously for static design sensitivity analysis, see Eqs. 4.7, 4.8, and 4.9. When the new right hand side has been determined, the temperature sensitivities @ T=@ai can be calculated by forward and back substitution reusing the factorized global thermal \sti ness matrix". Having obtained the temperature sensitivities @ T=@ai , the perturbed thermally induced strains "th(a +ai ) and the perturbed nodal force vector Fth (a +ai ) are easily calculated, and nite di erence approximations of sensitivities of thermally induced strains and the corresponding nodal force vector can be evaluated. These sensitivities are included in the static design sensitivity analysis, whereby Eq. 4.6 becomes @ F + @ Fth K(a) @@aD = K@a(a) D + @a (4.23) @ai i i i In a similar way, the thermally induced strain sensitivities are included in the stress sensitivities in Eq. 4.11 by using Eqs. 4.12 and 4.21, and the thermo-elastic e ects are thereby included in the design sensitivity analysis.
4.9 Design Sensitivity Analysis of Eigenvalues It is a well known fact that the design sensitivity analysis of eigenvalues is problematic in the case of multiple eigenvalues, i.e., the case where two or more eigenvalues attain exactly the same value. In this case, the eigenvalues are no longer di erentiable functions of the design in the normal Frechet sense. In the following it is described how design sensitivities of both simple (distinct) and multiple (repeated) eigenvalues can be obtained. The eigenvalue analysis problem considered can be either free vibration frequency analysis, eigenfrequency analysis including initial stress sti ening e ects, or linear buckling analysis as described in Section 3.2. For such real, symmetric, structural eigenvalue problems, the nite element formulation, in general, can be written as
K j = j M j ; j = 1; : : : ; n
(4.24)
where K and M are symmetric, positive de nite matrices, j is the eigenvalue and j is the corresponding eigenvector. Depending on the type of analysis problem the global
Chapter 4. General Expressions for Design Sensitivity Analysis
39
K and M matrices consist of contributions from either element sti ness, mass or initial stress sti ness matrices. The dimension of the problem is denoted by n, so Eq. 4.24 has n solutions consisting of eigenvalues j and corresponding eigenvectors j . The eigenvalues are all real and represent eigenfrequencies or linear buckling load factors depending on the type of analysis problem. In ODESSY, Eq. 4.24 is solved by the Subspace iteration method, see Bathe (1982), for the lowest eigenvalues j . The eigenvalues can be ordered by magnitude as 0 < 1 2 : : : j : : : n (4.25) In the following it is assumed that the eigenvectors have been M-orthonormalized, i.e.,
Tj M k = jk ; j; k = 1; : : : ; n
(4.26)
where jk denotes Kronecker's delta. If Eq. 4.24 is premultiplied by Tj the following expression is obtained
Tj K k = j jk ; j; k = 1; : : : ; n
(4.27)
meaning that the eigenvectors are also K-orthogonal. So far multiple eigenvalues and corresponding eigenvectors have not been mentioned. In this case the eigenvectors are not unique. In fact, an in nite number of linear combinations of the eigenvectors corresponding to a multiple eigenvalue will satisfy Eqs. 4.24 and 4.26. However, a set of M-orthonormal eigenvectors which span the subspace that corresponds to a multiple eigenvalue can always be chosen. In other words, if it is assumed that j has multiplicity N (i.e., j = j+1 = : : : = j+N 1), then N eigenvectors j ; : : : ; j+N 1, which span the N -dimensional subspace corresponding to the eigenvalues of magnitude j and satisfy the orthogonality conditions in Eqs. 4.26 and 4.27, can be chosen.
4.9.1 Design Sensitivity Analysis of Simple Eigenvalues As before it is assumed that the design variables of the structural design problem is denoted by ai; i = 1; : : : ; I , and the goal is to obtain expressions for eigenvalue sensitivities with respect to these design variables. It is also assumed that the components of the K and M matrices are smooth functions of design variables ai. The direct approach to obtain the eigenvalue sensitivities is to di erentiate Eq. 4.24 with respect to a design variable ai assuming that j is simple @ K + (K M) @ j = @j M + @ M ; i = 1; : : : ; I (4.28) j @ai j @ai @ai j j @ai j By premultiplying Eq. 4.28 by Tj and making use of Eq. 4.24, the following expression is obtained for the eigenvalue sensitivity in case of simple eigenvalues j , see, e.g., Courant & Hilbert (1953) and Wittrick (1962)
! @j = T @ K @ M ; i = 1; : : : ; I j j @a @ai @ai j i
(4.29)
40
4.9. Design Sensitivity Analysis of Eigenvalues
where the term Tj M j = 1 due to the M-orthonormalization, Eq. 4.26, has been omitted. It appears that the only unknown quantities in Eqs. 4.29 are the derivatives of the K and M matrices. As in the case of static S-A design sensitivity analysis, these derivatives are calculated at the element level as in Eqs. 4.7 and 4.8, and then the eigenvalue design sensitivities can be determined. If all the design variables ai are changed simultaneously, then, as described in Section 4.7, due to the di erentiability of simple eigenvalues with respect to the design variables, the linear increment of the simple eigenvalue j can be found in the form j = Tj a (4.30) where j denotes the gradient vector of j and a is the vector of changes of the design variables ai ! @ @ j j j = @a ; : : : ; @a ; a = (a1 ; : : : ; aI ) (4.31)
r
r
r
I
1
These notations are useful for parametric studies of eigenvalues as well as for formulation of optimization problems.
4.9.2 Design Sensitivity Analysis of Multiple Eigenvalues When the solution of the generalized eigenvalue problem in Eq. 4.24 yields a N -fold multiple eigenvalue ~ = j ; j = 1; : : : ; N (4.32) where, for convenience, the repeated eigenvalues have been numbered from 1 to N , then the computation of the sensitivities of this eigenvalue is not straight-forward. This is due to the fact that the eigenvectors j ; j = 1; : : : ; N , of the repeated eigenvalues are not unique. Thus, any linear combination of the eigenvectors will satisfy the original eigenvalue problem, Eq. 4.24. In the following sensitivity analysis we shall use such eigenvectors ~ j which remain continuous with design changes, see Courant & Hilbert (1953). For this purpose linear combinations of eigenvectors k are introduced
~ j =
N X
k=1
jk k ; j = 1; : : : ; N
(4.33)
where jk are unknown coecients to be determined. Works by Courant & Hilbert (1953), Wittrick (1962), and Lancaster (1964) have provided a basis for calculating the sensitivities of multiple eigenvalues. It is shown that the design sensitivities of multiple eigenvalues can be found by formulation and solution of a subeigenvalue problem. Let us rst consider a small change "ai of a single, arbitrarily chosen design parameter ai where " is a small positive parameter. Due to the perturbation of this design variable the K and M matrix will be incremented, i.e., the new matrices become (4.34) K + " @@aK ai and M + " @@aM ai; i = 1; : : : ; I i i
Chapter 4. General Expressions for Design Sensitivity Analysis
41
Then multiple eigenvalues and corresponding eigenvectors for the perturbed design can be written as j (ai + "ai) = ~ + "j (ai; ai ) + o("); j = 1; : : : ; N (4.35) j (ai + "ai) = ~ j + " j (ai ; ai) + o("); j = 1; : : : ; N (4.36) where j and j are unknown eigenvalue and eigenvector sensitivities, respectively, and o(") represents higher order terms. Substituting Eqs. 4.34, 4.35, and 4.36 into the main eigenvalue problem in Eq. 4.24, we obtain in the rst approximation
! @ K ~ @ M ~ + (K ~M) = M ~ j j j @ai @ai j
(4.37)
Premultiplying this equation by Ts gives
! @ K ~ @ M ~ = T M ~ ; s = 1; : : : ; N (4.38) @ai @ai j j s j Here the term Ts (K ~M) j = Tj (K ~M) s drops out because s is the eigenvector corresponding to ~. Recalling that ~ j is the linear combination in Eq. 4.33 of the original eigenvectors k , from Eq. 4.38 the following system of linear algebraic equations of unknown coecients jk is obtained ! ! N X @ K @ M T ~ (4.39) jk s @a @a k j sk = 0; s = 1; : : : ; N i i k=1
Ts
where the M-orthonormalization, Eq. 4.26, has been used. A nontrivial solution to these equations only exists if the determinant of the system is equal to zero ! @ K @ M T det s @a ~ @a k sk = 0; s; k = 1; : : : ; N; i = 1; : : : ; I (4.40) i i This is the main equation for determining the coecients j ; j = 1; : : : ; N , of the power series in Eq. 4.35 which represent the sensitivities of the multiple eigenvalue ~ with respect to changes ai of a single design parameter ai . As in the case of simple eigenvalues, the derivatives of the K and M matrix, respectively, must be calculated rst, and then the subeigenvalue problem of Eq. 4.40 is easily formulated and solved. If the o -diagonal terms in the quadratic matrix of dimension N in Eq. 4.40 are equal to zero, the eigenvalues of this matrix, i.e., the directional derivatives of the multiple eigenvalue ~, are equal to the traditional Frechet derivatives obtained by using Eq. 4.29. Let us consider the general case when all the design variables ai; i = 1; : : : ; I , are changed simultaneously. It should be noted that multiple eigenvalues are not di erentiable in the common sense, i.e., not Frechet-di erentiable, see, e.g., Haug, Choi & Komkov (1986). This means that the expression for the eigenvalue increments in Eqs. 4.30 is no longer valid.
42
4.9. Design Sensitivity Analysis of Eigenvalues
Thus, to nd the sensitivities of multiple eigenvalues it is necessary to use directional derivatives in the design space. For this purpose, for the vector of design variables a = (a1; : : : ; aI ), a varied form a + "e is considered, q 2 where e is2 an arbitrary vector of variation e = (e1 ; : : : ; eI ) with the unit norm kek = e1 + : : : + eI = 1 and " is a small positive parameter. The vector e represents a direction in the design space along which the design variables ai are changed, and " represents the magnitude of the perturbation in this direction. As a result of perturbation of the vector a the matrices K and M are incremented and become I I X X K + " @@aK ei; M + " @@aM ei (4.41) i=1
i
Using expansions for j and j in the form
i=1
i
j = ~ + "j + o("); j = 1; : : : ; N (4.42) j = ~ j + " j + o("); j = 1; : : : ; N (4.43) and performing the same manipulations as earlier, instead of Eq. 4.40 the following N -th order equation for determining the sensitivities j of the eigenvalues j is obtained: X ! I @ K @ M T det s @a ~ @a k ei sk = 0; s; k = 1; : : : ; N (4.44) i i i=1 If the generalized gradient vectors fsk of dimension I are introduced
! ! ! @ K @ M @ K @ M T ~ ~ fsk = @a1 @a1 k ; : : : ; s @aI @aI k then Eq. 4.44 takes the form det fskT e sk = 0; s; k = 1; : : : ; N
Ts
(4.45) (4.46)
Note that fsk = fks due to the symmetry of the matrices K and M. Also note the notation used here for the generalized gradient vectors fsk . The subscripts refer to the modes from which the generalized gradient vector is calculated, i.e., fskT e is a scalar product. Thus, knowing the eigenvectors k ; k = 1; : : : ; N , corresponding to the multiple eigenvalue ~, the generalized gradient vectors fsk can be constructed and the sensitivities j ; j = 1; : : : ; N , for any vector of variation e can be determined, i.e. for any direction in the space of the design variables. The quantities j constitute the directional derivatives of the multiple eigenvalue ~, cf. Eq. 4.42. In this form Eq. 4.46 was obtained by Bratus & Seyranian (1983), and Seyranian (1987), see also Haug & Rousselet (1980b), Masur (1984, 1985), and Haug, Choi & Komkov (1986). In many cases it is expedient to eliminate the unit vector e from Eq. 4.46 and establish a formula for determining the increments j ; j = 1; : : : ; N , of the N -fold eigenvalue ~ subject to a given vector a = (a1; : : : ; aI ) of actual increments of the design variables
Chapter 4. General Expressions for Design Sensitivity Analysis
43
ai ; i = 1; : : : ; I . To this end, we multiply each of the components in Eq. 4.46 by ", note from the foregoing that "e = a and "j = j ; j = 1; : : : ; N , and obtain det f T a sk = 0; s; k = 1; : : : ; N (4.47) sk
If we solve this N -th order algebraic equation for , we obtain the increments = j ; j = 1; : : : ; N , of the N -fold eigenvalue corresponding to the vector a of actual increments of the design variables. As in the case of semi-analytical (S-A) design sensitivity analysis of static problems, possible inaccuracies in the approximate numerical di erentiation of the element sti ness matrices can lead to severe inaccuracy problems in the sensitivities of eigenvalues. This problem will be investigated in Chapter 5. The technique described above in principle solves the problem of design sensitivity analysis of eigenvalues. However, in the case of multiple eigenvalues, the problem of nondi erentiability continues to be a potential source of diculty in relation to the numerical optimization procedure based on common derivative information. Fortunately, in a numerically based optimization system, the case of exactly coalescing eigenvalues is very unlikely for most common structures, and in most cases this problem does not have a signi cant impact on the optimization procedure. However, for plate and shell structures, possibly reinforced by sti eners, multiple eigenvalues frequently occur and therefore Chapter 8 is devoted to the special case of optimization of multiple eigenvalues.
44
4.9. Design Sensitivity Analysis of Eigenvalues
Chapter
5
\Exact" Numerical Di erentiation of Element Matrices 5.1 Introduction
devoted to the problem of obtaining \exact" numerical derivatives T of various nite element matrices as the accuracy of the rst order nite di erence his chapter is
approximations in the semi-analytical (S-A) method, see Eq. 4.8, is strongly dependent on the chosen size of perturbation ai of a shape design variable ai . This dependency arises as the element matrices generally depend non-linearly on shape design variables, and in some cases, so small perturbations are needed that computational round-o errors become the problem. The inaccuracy problem associated with the S-A method in connection with shape design variables is described in Section 5.2. In order to avoid dependence on the chosen perturbation ai, the goal is to construct a method for \exact" numerical di erentiation of element matrices based on computationally inexpensive rst order nite di erences. Here and in the following, derivatives obtained by numerical di erentiation will be termed \exact derivatives" if they have no truncation error due to neglection of higher order terms in their Taylor series expansion and are exact except for computational round-o errors. This goal may seem unattainable, but a closer study of the functions that form the element matrices reveals that the same mathematical forms are common for large groups of nite elements. For instance, the element matrices for all isoparametric elements with translational degrees of freedom and isoparametric Mindlin plate and shell elements depend on the same class of functions. Similarly, the element matrices of a large class of nite elements comprising Bernoulli-Euler beam and Kirchho plate and shell elements have a similar mathematical structure. The members of these classes of matrices in general depend non-linearly on the design variables, but are de ned within a special mathematical form. These element functions will be described in Section 5.3 where it is shown that the mathematical form implies that their approximate numerical derivatives, computed by a
45
46
5.2. Problem of Inaccuracy in the Traditional Semi-Analytical Method
usual rst order nite di erence scheme, can be upgraded to \exact" derivatives by simple multiplication by appropriate correction factors. The values of these correction factors can be very easily pre-computed and be used throughout the procedure of design sensitivity analysis. It follows as a remarkable side-e ect of the \exactness" that the results become totally independent of the magnitude of the perturbation. In Section 5.4 \exact" numerical derivatives of various element matrices of 3D solid isoparametric nite elements will be found by using the results of Section 5.3. Similar derivations are given in Sections 5.5 and 5.6 for 2D isoparametric solid elements and isoparametric Mindlin plate nite elements, respectively. Due to simplicity, only shape design variables are considered in these sections. Thus, it is assumed that the element matrices of a particular nite element only depend on shape design variables, i.e., design variables related to nodal coordinates of elements. It is assumed that an element matrix depends on a given sub-set from among the total set of shape design variables ai; i = 1; : : : ; Is, which in this context is considered to be the global coordinates of the nite element nodal points. This sub-set of the design variables is assumed to be renumbered and denoted by aj ; j = 1; : : : ; J , where J < Is. Section 5.7 describes how derivatives of various element matrices can be found with respect to generalized shape design variables, e.g., positions of master nodes as described in Chapter 2, by using the results obtained in Sections 5.4, 5.5, and 5.6. In Section 5.8 \exact" numerical derivatives of various element matrices are given for all kinds of design variables, i.e., including sizing as well as material design variables. Finally, the numerical eciency of the new method of S-A design sensitivity analysis and the actual implementation is described in Section 5.9.
5.2 Problem of Inaccuracy in the Traditional SemiAnalytical Method Recent references have demonstrated that the method of S-A sensitivity analysis may su er serious accuracy drawbacks in particular types of problems involving shape design variables. A similar inaccuracy problem is not found if the analytical method or the overall nite di erence technique of sensitivity analysis are employed. The problem must therefore be attributed to the numerical di erentiation of the nite element sti ness matrix that is inherent in the S-A method, cf. Eqs. 4.7 and 4.8. The inaccuracy problem associated with the S-A method was rst discovered by Barthelemy, Chon & Haftka (1988) and the problem was rst encountered with a car model made of 3D beam elements. The authors had previously used the overall nite di erence (OFD) approach to design sensitivity analysis, see Section 4.2, but shifted to the S-A approach due to its superiority with respect to computational eciency. The S-A method worked excellently for sizing variables but for certain global dimensions of the car, i.e., for a few of the shape design variables, inaccuracy problems were detected. For these shape variables the S-A method proved to be very sensitive to the step size used in the nite di erence
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
47
approximation of the sti ness matrix derivative, cf. Eqs. 4.7 and 4.8. Because the car model was complex, Barthelemy, Chon & Haftka studied a cantilever beam subjected to a moment at the free end. For this model problem where the length of the beam was taken as a shape design variable they observed the same inaccuracy problem associated with the S-A method for the sensitivity of the lateral tip displacement. The OFD method was used as a reference method and the only problem encountered with this method was the usual \step-size dilemma" as discussed in Section 4.2. Two types of abnormal errors have been noticed by Barthelemy & Haftka (1988). For beam- and plate-like structures, (I) the numerical errors of displacement sensitivities with respect to shape design variables such as length increase quadratically with nite element mesh re nement. This observation would not be expected in advance as a traditionel nite element analysis is known to converge to a stationary point with mesh re nement if a proper nite element is used, so this abnormal error type was surprising. Another type of error problem appears in problems involving linearly elastic bending of long-span beam-like structures. Here, (II) the errors of displacement sensitivities with respect to beam length increase rapidly with the length of the beam, i.e., with the beam aspect ratio. Barthelemy & Haftka (1988) observed that the error problem is not symptomatic of only beam elements but can be expected for beam- and plate-like structures, no matter what kind of nite element is used to model them. Barthelemy & Haftka (1988) and Pedersen, Cheng & Rasmussen (1989) traced the inaccuracy problem for the beam model to the basic concept of the pseudo load vector, see Eq. 4.6, which is the load that must be applied to the structure to produce the displacement sensitivity eld. In many cases of shape variations, the sensitivity displacement eld is not a reasonable displacement eld for the structure and its boundary conditions. For example, for beam- or plate-like structures, the displacement sensitivity to a length dimension is dominated by shear rather than bending and in order to produce the unlikely sheardominated elds, the pseudo load must include large self-cancelling components. These components will contain small truncation errors due to the nite di erence scheme used and these errors are ampli ed into large errors in the displacement sensitivity. A local error index for the S-A truncation error was developed by Barthelemy, Chon & Haftka (1988) for beam and plane truss elements based on an examination of error contribution of each nite element. Using these error indices they found that elements which undergo large rigid body motions contribute the most to the total error. This observation was later analyzed in detail by Cheng & Olho (1991, 1993) who introduced a rigid body motion test for detection of errors in the S-A sensitivities. Pedersen, Cheng & Rasmussen (1989) studied the same model problem of a cantilever beam with a tip moment and observed that the components of the beam element sti ness matrix depend on the design variable considered, i.e., the element length, in three di erent powers. Thus, using a rst order forward nite di erence approximation of the sti ness matrix derivative, the derivatives of these components are of di erent accuracy. These uneven truncation errors result in relative errors of the pseudo forces that are di erent from those of the pseudo moments in the pseudo load vector. These di erent inaccuracies of the pseudo load components due to the forward (or backward) nite di erence scheme are shown by
48
5.2. Problem of Inaccuracy in the Traditional Semi-Analytical Method
Pedersen, Cheng & Rasmussen to be the reason for the two types of error problems for this model problem, i.e., they showed for this model problem that the sensitivity error is proportional to the relative length di erence, but unfortunately it is also proportional to the square of the number of nite elements. Furthermore, they showed that a second order central nite di erence scheme cannot remove the inaccuracy problem. However, using a central nite di erence approximation of the sti ness matrix derivative, the sensitivity error is proportional to the square of the relative length di erence and thereby much reduced. The dependence on the square of the number of elements is not removed. This important work of error analysis by Pedersen, Cheng & Rasmussen (1989) was extended by Olho & Rasmussen (1991a) by deriving the analytical solution to the global set of nite element equations for the S-A design sensitivity analysis problem for any degree of discretization. This enabled the authors to precisely identify and explain the source of the numerical inaccuracy problem to originate from the forward nite di erence approximation of the sti ness matrix. The non-uniform distribution of pseudo load errors is such that it has a very critical in uence on the displacement design sensitivities as the nite element mesh is re ned because the values of the latter, which should be mesh independent, result from subtractions and additions of an increasing number of increasingly large terms as the number of nite elements increases with the mesh re nement. They also showed mathematically that the error of the sensitivity of the lateral tip displacement increases quadratically with the number of nite elements used to model the beam, see also Fenyes & Lust (1991). Di erent approaches have been suggested for improvement of inaccurate S-A design sensitivities. Haftka & Adelman (1989) have advocated the use of central di erences instead of forward di erences which implies additional computational cost. The use of second order central nite di erences reduces the error problem as described above but it cannot remove it. Furthermore, central di erences involve a doubling of the computation time needed to calculate the sti ness matrix derivative. Therefore, the approach of using higher order nite di erences has not been chosen. Cheng, Gu & Zhou (1989) proposed an alternate forward/backward nite di erence scheme which preserves the computationally eciency of the S-A method. Cheng, Gu & Wang (1991) introduced a second order correction method but none of these methods can completely eliminate the inaccuracy problem. Olho & Rasmussen (1991b) developed an ecient method of error elimination by \exact" numerical di erentiation which eliminates errors associated with mesh re nement and design variable perturbation for a class of problems that comprises the beam model problem. Their initial approach to remove the inaccuracy problem forms the basis for the method of \exact" numerical di erentiation which will be described later in this chapter. A recent paper by Cheng & Olho (1991, 1993) is also devoted to the problem of error elimination, and a geometrical-physical interpretation of the in uence of the error associated with the rst order forward di erence approximation k=ai to the element sti ness derivative @ k=@ai , cf. Eq. 4.8, is presented. The contribution from the matrix derivative to the element pseudo loads are given by (@ k=@ai )d, see Eq. 4.6 and 4.9, where d is the vector of nodal displacements of the element. Now, any displacement vector d can be subdivided into three vectors: a vector dt for the rigid body translation, a vector dr for the
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
49
rigid body rotation, and a vector for the remaining part of d which is associated with the actual deformation of the nite element. Along the same lines as a proper nite element should possess the rigid body motion capabilities kdt = 0 and kdr = 0 (vanishing of the element nodal forces associated with dt and dr ), it is clear that all the components of the element pseudo loads associated with a rigid body translation and rotation, respectively, should vanish such that (@ k=@ai )dt = 0 and (@ k=@ai )dr = 0. Cheng & Olho show that the latter conditions are satis ed for both sizing and shape design variables when analytical design sensitivities @ k=@ai are used. Now, Cheng & Olho show that if the design sensitivities are replaced by their forward di erence approximations, then the conditions (k=ai )dt = 0 and (k=ai )dr = 0 are also both satis ed if ai is a sizing design variable. However, if ai is a shape design variable, only the former condition is satis ed, i.e., only the approximate element pseudo loads associated with a rigid body translation vanish. Hence the latter condition is generally not satis ed if ai is a shape variable which means that the components of the approximate element pseudo loads that correspond to a rigid body rotation do not vanish in general, i.e., (k=ai )dr 6= 0. This fact was also shown later by Mlejnek (1992). Thus, the conclusion from Cheng & Olho (1991, 1993) is that in S-A design sensitivity analysis with a shape design variable ai, the use of the rst order nite di erence approximation k=ai in Eq. 4.8 generally introduces an error in the form of an extra moment to the exact element pseudo loads, and that this extra moment is the resultant of the approximate pseudo loads associated with the rigid body rotation of the element. The extra (error) pseudo moments from each nite element become aggregated as a nonuniform distribution of errors of the system level pseudo loads in the process of assembling the system level, see Eq. 4.7 (with @ replaced by throughout), and are responsible for the errors of the approximate displacement design sensitivities D=ai . It is the conclusion of the above-mentioned papers concerning the accuracy of the S-A method of design sensitivity method, that the numerical di erentiation of the nite element sti ness matrix that is inherent in the S-A method in certain cases may result in serious inaccuracy problems. The problems may occur for design sensitivities with respect to structural shape design variables in problems where the displacement eld is characterized by rigid body rotations which are large relative to actual deformations of the nite elements, i.e., for example in problems involving linearly elastic bending of long-span, beam-like structures, and of plate and shell structures. However, it should be noted that the S-A method works excellently for most problems and the inaccuracy problem is only encountered for the above-mentioned type of design sensitivity analysis problems. In order to obtain a robust method for design sensitivity analysis for all possible design problems, the goal has been to construct a method for \exact" numerical di erentiation of the nite element matrices and in order to maintain the computationally eciency the method must be based on rst order nite di erences, i.e., higher order nite di erences as central di erences are not used. Such a method has been developed and published by Olho , Rasmussen & Lund (1992) and Lund & Olho (1993a, 1993b), and the remainder of this chapter is devoted to a presentation of this new method of S-A design sensitivity analysis.
50
5.3. \Exact" Numerical Di erentiation of Special Element Functions
5.3 \Exact" Numerical Di erentiation of Special Element Functions The foundation for the new S-A method using \exact" numerical di erentiation of nite element matrices is the observation, that element functions g, used in the de nition of element matrices for large groups of nite elements, have the same mathematical form. This observation and the results described in this section were made by Niels Olho in initial studies of the inaccuracy problem. For instance, the element matrices for all isoparametric elements with translational degrees of freedom and isoparametric Mindlin plate and shell elements depend on the same class of element functions g. Similarly, the element matrices of a large class of nite elements comprising Bernoulli-Euler beam and Kirchho plate and shell elements have a common mathematical structure. The latter kind of nite elements will not be discussed in this chapter but a description of \exact" numerical di erentiation of such elements can be found in Olho , Rasmussen & Lund (1992). It should be noted that in the present context, the design variables are assumed to be shape design variables in the form of the global coordinates of the nite element nodal points. The element matrices of a particular nite element only depend on a given sub-set from among the total set of shape design variables ai; i = 1; : : : ; Is. This sub-set of the design variables is renumbered and denoted by aj ; j = 1; : : : ; J , where J < Is. J is, at maximum, equal to the total number of nodal coordinates of the element. In the case of isoparametric nite elements, the element matrices turn out to depend on element functions g, which, in general, depend non-linearly on the design variables, i.e., on the nodal coordinates. The element functions g are incomplete polynomia and de ned by the following form: g(a1; : : : ; aJ ) = pj (a1 ; : : : ; aj 1; aj+1; : : : ; aJ ) + qj (a1 ; : : : ; aj 1; aj+1; : : : ; aJ ) (aj )rj ; rj 2 @ [ f0g; j = 1; : : : ; J (5.1) Thus, a function g is such that for any j , the term pj and the coecient qj 6= 0 are independent of aj . The design variable aj appears in one and only one power rj which belongs to the set @ [ 0 of non-negative integers. Although not stated explicitly, pj and qj , and therefore the element function g, will generally depend on the local coordinates within the nite element. Typically, g may represent the determinant or components of the Jacobian matrix or components of other matrices in the de nition of an element matrix. It should be noted that for isoparametric elements, the power rj is always 0 or 1, but for reasons of generality, the presentation in this section covers any power rj , thereby making the theory generally applicable to other types of elements, cf. Olho , Rasmussen & Lund (1992). Before numerical derivatives of an element function are considered, let us de ne the following standard nite di erence operators d(akj) for numerical di erentiation of a function f with respect to a design variable aj : d(0) aj f = 0
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
51
f 1 d(1) aj f = a = a (f (aj + aj ) f (aj )) j j 1 d(2) (5.2) aj f = 2a (f (aj + aj ) f (aj aj )) j 1 d(3) aj f = 6a ( f (aj + 2aj ) + 6f (aj + aj ) 3f (aj ) 2f (aj aj )) j 1 d(4) aj f = 12a ( f (aj + 2aj ) + 8f (aj + aj ) 8f (aj aj ) + f (aj 2aj )) j ::: In Eq. 5.2, the symbol k in d(akj) designates the order of the complete polynomial for which the standard nite di erence operator d(akj) yields the exact numerical derivative of the function f . An element function g as de ned by Eq. 5.1 is now substituted into the usual rst order forward di erence expression for numerical di erentiation with respect to aj , i.e., the formula for k = 1 in Eq. 5.2, g = 1 (g(a + a ) g(a )) (5.3) j j j aj aj = 1a (pj + qj (aj + aj )rj pj qj (aj )rj ) ; j = 1; : : : ; J j
Then, in Eq. 5.2, set k equal to the value of rj associated with aj , i.e., @g = d(k=rj )g; j = 1; : : : ; J @aj aj and introduce the symbol j for the relative perturbation of aj , i.e. j = aaj > 0; j = 1; : : : ; J j
(5.4) (5.5)
Substitute now g from Eq. 5.1 into the right hand side of Eq. 5.4, and consider the formulas in Eq. 5.2 for f = g and k = rj = 0; 1; 2; : : :. For each value of rj , the following proportional relationship is easily established between the analytical (and exact) derivative @g=@aj and its rst order nite di erence approximation g=aj : @g = c g ; r 2 @ [ f0g; j = 1; : : : ; J (5.6) @aj rj aj j The proportionality factors crj are called correction factors and are found to depend only on j (for crj > 1)
c0 = 0 c1 = 1 1 c2 = 1 + 21 j c3 = 1 + j + 13 j2
1
(5.7)
52
5.3. \Exact" Numerical Di erentiation of Special Element Functions
3 1 2 3 1 + j + j + j 2 4
1
c4 = ::: 0 k1 ! 11 X 1 k ck = @ k j(k 1) pA p p=0 where the latter expression for ck is given in terms of binomial coecients. g (a 1 ,...,a j ,...,aJ )
∂g ∂ aj
∆g = c rj ∆ a j
∆g ∆ aj aj
Figure 5.1: Illustration of relationsship between the analytical derivative @g=@aj and its rst order nite di erence approximation g=aj . In Fig. 5.1 this relationship of proportionality between the analytical derivative @g=@aj of an element function g and its rst order nite di erence approximation g=aj is illustrated. The following important points concerning the above development should be noted:
Eqs. 5.5 - 5.7 represent an ecient and simple computational scheme for deter-
mining \exact derivatives" of element functions g by application of correction factors to computationally inexpensive rst order di erences. Truncation errors are completely avoided.
The correction factors in Eq. 5.7 are independent of the actual values of the
design variables and can therefore be precomputed for a selected nite value of j > 0. These values of the correction factors are then applicable in all future sensitivity analyses, provided that the original value of j is observed when computing the value of aj = j aj to be used in Eq. 5.6.
Hence, using Eq. 5.5 the absolute perturbation j can be eliminated in Eq. 5.6 which may be rewritten as @g = c g = crj ( g((1 + )a ) g(a ) ) ; r 2 @ [ 0; j = 1; : : : ; J (5.8) j j j j @aj rj aj j aj
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
53
where the value of j > 0 has been preselected. Eqs. 5.7 and 5.8 are the main expressions to be used when calculating derivatives of various element matrices for isoparametric nite elements.
5.4 3D Solid Isoparametric Finite Elements In this section the method of \exact" numerical di erentiation described in Section 5.3 is used to determine \exact" numerical derivatives of element matrices and vectors for 3D isoparametric hexahedral (brick) nite elements with respect to shape design variables in the form of nodal coordinates. These elements are described in Appendix A where all element matrices and vectors are given. Local coordinates, domain, and nodal degrees of freedom for these isoparametric nite elements can be seen in Fig. 5.2. z,w
ζ
η ξ
y,v
x,u
Figure 5.2: Domain, node numbering, and nodal degrees of freedom of 3D isoparameric nite elements.
5.4.1 Derivative of Element Sti ness Matrix The element sti ness matrix k is given by
k=
Z
T E B jJj d
B
(5.9)
Here, is the domain of the nite element described in curvilinear, non-dimensional -- coordinates for the element, see Fig. 5.2, and jJj is the determinant of the Jacobian matrix J which at each point de nes the transformation of di erentials d , d, and d into dx, dy, and dz. Like J, the strain-displacement matrix B depends on coordinates of the nodal points, whereas the constitutive matrix E depends only on the constitutive parameters of the assumed linearly elastic material. The expressions for the Jacobian J and for the strain-displacement matrix B are given in the following.
54
5.4. 3D Solid Isoparametric Finite Elements
Let us recall that within the isoparametric formulation of a nite element with an arbitrary number n of nodal points, the same set of shape functions
Ni = Ni(; ; ); i = 1; : : : ; n
(5.10)
is used for interpolation of global x, y, and z coordinates from nodal values xi , yi, and zi and of displacements functions u, v, and w from nodal values ui, vi, and wi, i.e.,
x = u(x; y; z) =
n X i=1
n X i=1
Nixi ; y =
n X i=1
Niui; v(x; y; z) =
Niyi; z =
n X i=1
n X i=1
Nizi
Nivi; w(x; y; z) =
(5.11) n X i=1
Niwi
(5.12)
Shape functions Ni; i = 1; : : : ; n, for the two implemented 3D solid isoparametric nite elements are given in Appendix A in Tables A.1 and A.2. In terms of the vector di of nodal degrees of freedom
di = fui vi wigT ; i = 1; : : : ; n
(5.13)
the element nodal vector d containing nodal displacements is
d = fdT1 dT2 : : : dTi : : : dTn gT and the strain vector function " is "(x; y; z) = f"x "y "z xy yz xz gT
(5.14) (5.15)
with their mutual relationship de ned by
"=Bd
(5.16)
The strain-displacement matrix B is determined by operating on the shape functions Ni, and it is found that B = [ b1 b2 : : : bi : : : bn ] (5.17) where the submatrix bi , which is associated with the nodal point i of the nite element, has the form 2 3 N 0 0 i;x 66 7 66 0 Ni;y 0 777 bi = 6666 N0 N0 N0i;z 7777 ; i = 1; : : : ; n (5.18) i;y i;x 66 7 4 0 Ni;z Ni;y 75 Ni;z 0 Ni;x Here, the derivatives of the shape functions Ni with respect to x, y, and z are given by
8 9 2 9 38 > > > N ; ; ; N < i;x = 6 x x x 7 < i; > = 6 7 = N ; ; ; N i;y y y y i; = > : Ni;z > ; 4 ;z ;z ;z 5 > : Ni; > ;
8 9 > < Ni; > = N i; ; i = 1; : : : ; n > : Ni; > ;
(5.19)
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices where the matrix is the inverse of the Jacobian
2 66 x; y; z; J = 4 x; y; z; x; y; z;
=J
1
55 (5.20)
3 n 77 X 5=
2 3 N x N y N z i; i i; i i; i 66 7 (5.21) Ni; xi Ni; yi Ni; zi 75 4 i=1 Ni; xi Ni; yi Ni; zi Note that J is expressed in terms of the derivatives of Ni; i = 1; : : : ; n, with respect to the curvilinear element coordinates , , and and of the coordinates (xi ; yi; zi), i = 1; : : : ; n, of each of the n nodal points of the nite element.
Now all terms necessary for calculating the element sti ness matrix in Eq. 5.9 are de ned, and in order to determine its derivative, Eq. 5.9 is di erentiated with respect to any of the shape design variables aj ; j = 1; : : : ; J , leading to " # @ k = Z @ BT E B + BT E @ B jJj d + Z BT E B @ jJj d ; (5.22) @aj @aj @aj @aj
j = 1; : : : ; J Since the elasticity matrix E is symmetric, each of the two matrix terms in the rst integral in Eq. 5.22 is equal to the transpose of the other, and the matrix term in the second integral is symmetric in itself. Application of this observation can reduce Eq. 5.22 to a more simple expression. Introducing the notation [ ]S for the operation [C]S = 1 CT + C (5.23) 2 of symmetrization of a quadratic matrix C, Eq. 5.22 can be rewritten as " # " # @ k = Z 2 BT E @ B jJj d + Z BT E B @ jJj d ; j = 1; : : : ; J (5.24) @aj
@aj S @aj
S If a matrix B^ (j) is de ned as @ B + B @ jJj B^ (j) = @a (5.25) 2jJj @a j
j
then Eq. 5.24 can be written in compact form as @ k = 2 Z BT E B^ (j) jJj d ; j = 1; : : : ; J (5.26) @aj
S It is seen that Eq. 5.26 is of the same form as Eq. 5.9 so existing subroutines for computation of k can be used to compute the derivative @ k=@aj , provided that the straindisplacement matrix B can be substituted by B^ (j) determined from Eq. 5.25 and the full matrix k is calculated so that the symmetrization operation, Eq. 5.23, of the matrix in Eq. 5.26 can be carried out. However, in the following the form of Eq. 5.22 will be preferred instead of the compact form of Eq. 5.26 as it makes it easier to follow the derivations given. It is seen from Eq. 5.22 that the derivatives of the determinant jJj and of the components in the strain-displacement matrix B must be determined before the derivatives @ k=@aj , j = 1; : : : ; J , of the element sti ness matrix can be calculated.
56
5.4. 3D Solid Isoparametric Finite Elements
5.4.2 \Exact" Numerical Di erentiation of J and B j
j
In this section it will be shown that \exact" numerical derivatives of the Jacobian J and of the strain-displacement matrix B can be obtained on the basis of \exact" numerical di erentiation by applying Eqs. 5.1, 5.7, and 5.8 in Section 5.3. Note rst that all the shape functions Ni; i = 1; : : : ; I , of the nite element depend only on the non-dimensional curvilinear coordinates , , and within the element and thus are independent of the actual geometry of the element. The shape functions and their derivatives with respect to local coordinates are therefore independent of the shape design variables aj ; j = 1; : : : ; J . Consider rst the Jacobian matrix J in Eq. 5.21. The determinant jJj of this matrix can be found as jJj = J11cof (J11) + J22 cof (J22) + J33 cof (J33), where cof denotes cofactor. It is thereby seen that any coordinate x, y, or z will appear only linearly in the expression for the determinant jJj. Thus, the scalar jJj will be either independent or a linear function of any of the shape design variables aj , and the derivative of jJj can therefore be found by applying Eqs. 5.8 where the correction factor crj = 1, cf. Eq. 5.7. The following expression for the derivative @ jJj=@aj ; j = 1; : : : ; J , is obtained:
@ jJj = J = 1 ( jJ((1 + )a )j jJ(a )j ) ; j = 1; : : : ; J j j j @aj aj j aj
(5.27)
The computation of derivatives of components of the strain-displacement matrix B requires di erentiation of bi with respect to aj and hence of the derivatives of Ni;x, Ni;y , and Ni;z , see Eqs. 5.17, 5.18, and 5.19. This involves di erentiation of the matrix , and since the components qp of this matrix are given by qp = jJj 1cof (Jpq ), these components cannot be di erentiated exactly on the basis of a simple polynomial approximation. This diculty can be circumvented by di erentiating the identity J = I, where I is the identity matrix, which gives @ = @ J ; j = 1; : : : ; J (5.28) @a @a j
j
From Eq. 5.21 it is seen that each of the components of the Jacobian matrix J is either independent or a linear function of any of the shape design variables aj ; j = 1; : : : ; J , from among the set of coordinates xi , yi, and zi of the element nodal points. Hence, the derivative of each of the components in the Jacobian matrix J can be determined using Eq. 5.8. The derivative @ B=@aj can be written as
" # @ B = @ b1 @ b2 : : : @ bi : : : @ bn @aj @aj @aj @aj @aj
(5.29)
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices where
57
2 66 66 66 66 66 @ bi = 66 @aj 666 66 66 66 64
3 @Ni;x 0 0 77 @aj 77 i;y 77 0 @N 0 @aj 7 @N 77 i;z 7 0 0 @aj 77 ; i = 1; : : : ; n (5.30) @Ni;y @Ni;x 0 77 7 77 @aj @aj @N @N i;y 7 i;z 7 0 @aj @aj 777 @Ni;z 0 @Ni;x 5 @aj @aj The derivatives of Ni;x, Ni;y , and Ni;z , i.e., the components in the strain-displacement matrix B, can then be found using Eqs. 5.8, 5.19, and 5.28: 8 9 8 9 > > > Ni; > N = @ < Ni;x = = @ < N i;y i; @aj > @aj > : Ni;z > ; : Ni; > ; 8 9 > < Ni; > = @ J = (5.31) N i; @aj > : Ni; > ; 8 9 > Ni;x > < = @ J = J 1 @a > Ni;y > j :N ; i;z 8 9 > Ni;x > < = 1 = J 1 [ J((1 + j )aj ) J(aj )] > Ni;y > ; j aj : Ni;z ; i = 1; : : : ; n; j = 1; : : : ; J Now all terms needed for the \exact" numerical derivative of the sti ness matrix, cf. Eq. 5.22, are found using rst order nite di erences, i.e., using \exact" semi-analytical design sensitivity analysis. In fact, only a comparatively small amount of additional work is required to derive the corresponding expressions for analytical design sensitivity analysis. Such derivations can be found in Wang, Sun & Gallagher (1985), El-Sayed & Zumwalt (1991), and Olho , Rasmussen & Lund (1992). These analytical expressions have not been implemented in ODESSY because the method of \exact" numerical di erentiation is just as accurate within computational round-o errors. Furthermore, as many di erent kinds of generalized shape design variables are available in ODESSY, it would take a lot of work to obtain analytical sensitivities of the relations between derivatives of generalized shape design variables Am ; m = 1; : : : ; M (governing positions of master nodes controlling the shape of, e.g., b-splines), and derivatives of shape design variables aj ; j = 1; : : : ; J , which represent element nodal coordinates. If analytical sensitivity analysis is not carried out in all evaluation steps, then there is no reason to introduce it here, so we will stay within the semi-analytical
58
5.4. 3D Solid Isoparametric Finite Elements
approach to design sensitivity analysis. In Section 5.7 it is described how derivatives with respect to generalized shape design variables Am can be related to derivatives with respect to shape design variables aj .
5.4.3 Stress Sensitivities The stress components for a particular nite element are assembled in the stress vector function (x; y; z) = fx y z xy yz xz gT which is given by
= E " "th = E Bd "th
(5.32)
in terms of the elasticity matrix E, strain-displacement matrix B, element nodal displacement vector d, and possible initial thermal strains "th. These thermally induced strains are given by
n o n o "th = "thx "thy "thz xyth yzth xzth T = T^ T^ T^ 0 0 0 T ; T^ = T T0
(5.33)
where is a matrix containing thermal expansion coecients, T is the temperature at the given point, and T0 is the temperature at which the structure is free of thermally induced strains. The stress design sensitivities @ =@aj with respect to any shape design variable aj ; j = 1; : : : ; J , becomes
@ = E @B d + B @d @aj @aj @aj
! @ "th ; j = 1; : : : ; J @aj
(5.34)
where
( ) @ "th = @T @T @T 0 0 0 T ; j = 1; : : : ; J (5.35) @aj @aj @aj @aj The sensitivities @ B=@aj are given by Eqs. 5.29, 5.30, and 5.31 and the element nodal displacement sensitivities @ d=@aj and the temperature sensitivities @T=@aj for the element are known from the solutions of Eqs. 4.6 and 4.22, respectively. The stress sensitivities can be calculated in a similar way for 2D solid isoparametric elements and isoparametric Mindlin plate and shell elements, so Eqs. 5.34 and 5.35 can be adopted immediately for these elements.
5.4.4 Derivative of Element Mass Matrix The consistent element mass matrix m is given by
m=
Z
% NT N jJj d
(5.36)
Here, is the domain of the nite element in its local coordinate system, see Fig. 5.2, % the mass density, N contains shape functions Ni , and jJj is the determinant of the Jacobian matrix J.
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
59
The derivative of the mass matrix can be found by di erentiating Eq. 5.36 with respect to any of the shape design variables aj ; j = 1; : : : ; J , i.e., @ m = Z % NT N @ jJj d ; j = 1; : : : ; J (5.37) @aj
@aj The derivative of the determinant jJj of the Jacobian matrix is given by Eq. 5.27 so all terms needed for calculating the \exact" numerical derivative of the mass matrix are known.
5.4.5 Derivative of Element Initial Stress Sti ness Matrix Next derivatives of the element initial stress sti ness matrix, also called element geometric sti ness matrix, are considered. In the derivation of element initial stress sti ness matrices it is convenient to reorder nodal degrees of freedom by introducing the element displacement vector d, where translational d.o.f. are reordered so that rst all x-direction d.o.f. are given, then y, and then z as follows d = fu1 u2 : : : ui : : : un v1 v2 : : : vi : : : vn w1 w2 : : : wi : : : wngT (5.38) Relating d.o.f. to the reordered element vector d the element initial stress sti ness matrix k for the 3D isoparametric nite elements is given by
k =
Z
GT S G jJj d
(5.39)
Here, is the domain of the nite element described in curvilinear, non-dimensional -- coordinates for the element, see Fig. 5.2, G a matrix obtained by appropriate di erentiation of shape functions Ni, S a matrix of initial stresses, and jJj is the determinant of the Jacobian matrix J. The matrix G is given by 2 3
66 g 0 0 77 G=4 0 g 0 5
where each submatrix g is given by
0 0 g
2 Ni;x g = 664 Ni;y Ni;z
The stress matrix S is given by
3 77 5 ; i = 1; : : : ; n
2 3 s 0 0 S = 664 0 s 0 775
and each submatrix s is de ned as
0 0 s
2 3 x xy xz s = 664 xy y yz 775 xz yz z
(5.40)
(5.41)
(5.42)
(5.43)
60
5.4. 3D Solid Isoparametric Finite Elements
Here x, xy , etc., are stresses found by an initial static stress analysis. If Eq. 5.39 is di erentiated with respect to any of the design variables aj ; j = 1; : : : ; J , the following expression for the \exact" numerical derivative of the initial stress sti ness matrix is obtained " # @ k = Z @ GT S G + GT @ S G + GT S @ G jJj d
@aj @aj @aj
@aj Z + GT S G @@ajJj d ; j = 1; : : : ; J (5.44)
j Thus, it is necessary to nd the derivatives of the components in the stress matrix S, of the determinant jJj of the Jacobian, and of the components of the matrix G. The derivatives of the components in the stress matrix S, i.e., the stress sensitivities, are given by Eq. 5.34, and the derivative of the determinant jJj is given by Eq. 5.27. The matrix G contains the components Ni;x, Ni;y , and Ni;z , i.e., the same components as involved in the de nition of strain-displacement matrix B, see Eqs. 5.17, 5.18, 5.40, and 5.41. The derivatives of these components are given by Eq. 5.31. Thus, all terms necessary for evaluating the \exact" numerical derivative of the element initial stress sti ness matrix, cf. Eq. 5.44, are now found.
5.4.6 Derivative of Thermal Element \Sti ness Matrix" The thermal element \sti ness matrix" consists of contributions from the heat conduction matrix kth given by Z th k = BthT Bth jJj d
(5.45) Here, is the domain of the nite element in its local coordinate system, see Fig. 5.2, Bth a matrix obtained by appropriate di erentiation of shape functions Ni, the thermal conductivity matrix, and jJj is the determinant of the Jacobian matrix J. If the material is isotropic, can simply be replaced by the scalar , the conductivity coecient. The matrix Bth is given by
h
Bth = bth1 bth2 : : : bthi : : : bthn
i
(5.46)
where the submatrix bthi, which is associated with the nodal point i of the nite element, has the form 2 3 N i;x bthi = 664 Ni;y 775 ; i = 1; : : : ; n (5.47) Ni;z In case of boundary conditions in terms of convection heat transfer, the thermal \sti ness matrix" receives additional contributions given by the element matrix h
h=
Z
!2
NT h N jJj d!
(5.48)
Here, !2 is the surface of the nite element described in curvilinear, non-dimensional , , or coordinates for the element, for which the convection boundary condition
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
61
is applied. N contains shape functions Ni, h is the convection coecient speci ed, and jJj is the determinant of the Jacobian matrix J for the surface !2 . If Eqs. 5.45 and 5.48 are di erentiated with respect to any shape design variable aj ; j = 1; : : : ; J , the following expressions for the derivatives of the thermal \sti ness matrices" are obtained 2 3 @ kth = Z 4 @ Bth T Bth + BthT @ Bth 5 jJj d + Z BthT Bth @ jJj d
(5.49) @aj @aj @aj @aj
and @ h = Z NT h N @ jJj d!; j = 1; : : : ; J (5.50) @aj !2 @aj The derivative of the determinant jJj is given by Eq. 5.27, and derivatives of the components Ni;x, Ni;y , and Ni;z in the matrix Bth, see Eqs. 5.46 and 5.47, are given by Eq. 5.31. All terms required for evaluating the \exact" numerical derivatives in Eqs. 5.49 and 5.50 are therefore found.
5.4.7 Derivative of Consistent Load Vector In this section \exact" numerical derivatives will be found for the consistent element load vector f which is given by Z Z f = NT FB jJj d + ! NT FS jJj d! (5.51) where is the domain of the nite element in its local coordinate system, FB represents body forces, ! the surface described in curvilinear, non-dimensional , , or coordinates for the element at which surface forces FS are applied, and N contains shape functions Ni. In the surface integral, N and jJj are evaluated on !. If initial thermally induced strains have to be taken into account, the consistent nodal force vector f th due to thermally induced strains is calculated as Z f th = BT E "th jJj d
(5.52)
"th
where is an element vector containing thermally induced strains as given by 5.33. Taking the same approach as previously, the load vectors in Eqs. 5.51 and 5.52 is di erentiated with respect to any of the design variables aj ; j = 1; : : : ; J , leading to the following expressions for the \exact" derivatives @ f = Z NT F @ jJj d + Z NT F @ jJj d!; j = 1; : : : ; J (5.53) B S @aj
@aj @aj ! and " # @ f th = Z @ BT E "th + BT E @ "th jJj d + Z BT E "th @ jJj d ; (5.54) @aj @aj @aj @aj
j = 1; : : : ; J Here, the derivative @ "th=@aj of the element thermally induced strains is given by Eq. 5.35, @ jJj=@aj is given by Eq. 5.27, and \exact" numerical derivatives of the strain-displacement matrix B are given by Eqs. 5.29, 5.30, and 5.31.
62
5.5. 2D Isoparametric Finite Elements
5.4.8 Derivative of Consistent Thermal Flux Vector Finally, \exact" numerical derivatives of the thermal ux vector will be determined for the 3D isoparametric nite elements. The consistent thermal nodal ux vector q is given by
q=
Z
!1
NT qS jJj d! +
Z
!2
NT h Te jJj d!
(5.55)
where the rst term derives from speci ed ux at the surface !1 and the latter term from a speci ed convection boundary condition at surface !2. The surfaces !1, !2 are described in curvilinear, non-dimensional , , or coordinates for the element. The scalar qS is prescribed ux normal to the surface !1, N contains shape functions Ni that are evaluated on the surface !, jJj the determinant of the Jacobian matrix for the surface !, h the convection coecient speci ed, and Te is the environmental temperature speci ed for the convection boundary condition. The derivative @ q=@aj then can be determined by
@ q = Z NT q @ jJj d! + Z NT h T @ jJj d! (5.56) S e @aj !1 @aj @aj !2 where @ jJj=@aj is given by Eq. 5.27. Now \exact" numerical derivatives have been determined for all implemented element matrices and vectors of the 3D isoparametric nite elements.
5.5 2D Isoparametric Finite Elements Next, \exact" numerical derivatives of element matrices and vectors of the implemented 3-, 4-, 6-, 8-, and 9-node 2D isoparametric nite are considered. These elements are formulated for both plane stress, plane strain, and axisymmetric situations and they are described in Appendix A where all element matrices and vectors are given. Local coordinates, domain, and nodal degrees of freedom for these 2D isoparametric nite elements can be seen in Fig. 5.3. Shape functions for the implemented 3-, 4-, 6-, 8-, and 9-node 2D elements are given in Apppendix B in Tables B.4, B.1, B.5, B.2, and B.3, respectively. In case of triangular isoparametric elements, the interpolation functions are de ned conveniently in terms of non-dimensional area coordinates 1, 2, and 3 within the element as shown in Fig. 5.3. Only two of the three dimensionless area coordinates are mutually independent, due to the area constraint relation
1 + 2 + 3 = 1
(5.57)
If 1 and 2 are selected as independent coordinates the following relations are obtained
1 = ;
2 = ;
3 = 1
(5.58)
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices of rotational (axissymmetry ) y,v (z,w)
63
η ξ2
ξ1 ξ3
ξ
x,u (r,u)
Figure 5.3: Domain, node numbering, and nodal degrees of freedom of 2D solid isoparameric nite elements. Local, non-dimensional coordinates , and area coordinates 1, 2, 3 are shown for the quadrilateral and triangular elements, respectively. Text in parantheses refer to standard notations for problems with rotational symmetry. Derivatives with respect to and , with the constraint in Eq. 5.57 taken into account, can be found as @ = @ @ ; @ = @ @ (5.59) @ @1 @3 @ @2 @3 The element matrices for the triangular elements can then be formulated similarly to the quadrilateral elements by using Eqs. 5.57 - 5.59.
5.5.1 Derivative of Element Sti ness Matrix The element sti ness matrix k is given by Z k = ! BT E B jJj d! (5.60) Here, ! is the domain of the nite element in its local coordinate system, see Fig. 5.3, and jJj is the determinant of the Jacobian matrix J which at each point de nes the transformation of di erentials d and d into dx and dy. Like J, the strain-displacement matrix B depends on coordinates of the nodal points, whereas the constitutive matrix E depends only on the constitutive parameters of the assumed linearly elastic material. For the plane stress and strain situations, d! = t d d, where t is the thickness of the nite element, and for axisymmetric structures we have d! = 2 r d d, where r is the radius at the integration point. The axis of rotational symmetry is assumed to be parallel with the y-axis as shown in Fig. 5.3. The expressions for the Jacobian J and for the strain-displacement matrix B can be found in a uniform way for plane stress, plane strain and axisymmetric nite elements, if the strain vector " is de ned in the following way "(x; y) = f"x "y xy "z gT (5.61)
64
5.5. 2D Isoparametric Finite Elements
In order to have the relations between standard notations for axisymmetric problems and the notations used here, it should be noted that r = x, z = y, w = v, "r = "x, "z = "y ,
rz = xy , and " = "z as indicated on Fig. 5.3. The element nodal vector d containing nodal displacements is given by
d = fdT1 dT2 : : : dTi : : : dTn gT
(5.62)
where the vector di of nodal degrees of freedom is
di = fui vigT ; i = 1; : : : ; n (5.63) The relation between the strain vector " and the displacement vector d is "=Bd (5.64) The strain-displacement matrix B can be found to have the following form B = [ b1 b2 : : : bi : : : bn ] (5.65) where the submatrix bi , which is associated with the nodal point i of the nite element, has the form
2 66 Ni;x 0 bi = 6666 Ni;y 4 Ni r
3
0 7 Ni;y 77 (5.66) Ni;x 777 ; i = 1; : : : ; n 5 0 The fourth row in bi is used only in case of an axisymmetric problem. The derivatives of the shape functions with respect to x and y are given by ( ) " #( ) ( ) Ni;x = ;x ;x Ni; = Ni; ; i = 1; : : : ; n (5.67) Ni;y ;y ;y Ni; Ni; where the matrix is the inverse =J 1 (5.68) of the Jacobian " # X n " N x N y # x; y; i; i i; i J = x; y; = (5.69) N x N y
i=1
i; i
i; i
Now all terms necessary for calculating the element sti ness matrix in Eq. 5.60 are de ned. The \exact" numerical derivative of the element sti ness matrix is found, as before, by di erentiating Eq. 5.60 with respect to any of the shape design variables aj ; j = 1; : : : ; J , and because the derivative for the axisymmetric case is di erent from the plane stress or strain situations, expressions are given for both situations. For the plane stress or strain case we have: " # @ k = Z Z @ BT E B + BT E @ B jJj t d d (5.70) @aj @aj @aj ZZ + BT E B @@ajJj t d d; j = 1; : : : ; J (5.71) j
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
65
and for the axisymmetric case:
" # @ k = Z Z @ BT E B + BT E @ B jJj 2 r d d @aj @aj @aj " # ZZ @ j J j @r T + B E B @a r + jJj @a 2 d d; j = 1; : : : ; J j
j
(5.72)
Hence, it is necessary to determine the derivative @ B=@aj of the strain-displacement matrix, the derivative @ jJj=@aj of the determinant of the Jacobian matrix, and the derivative @r=@aj of the radius r. The derivative of the determinant of the Jacobian matrix J is given by Eq. 5.27 and the derivatives of Ni;x and Ni;y needed for the evaluation of the derivative @ B=@aj are given by 5.31. However, for completeness of the derivations given here, Eqs. 5.29, 5.30, and 5.31 are rewritten to the 2D case. The derivative @ B=@aj can be written as
" # @ B = @ b1 @ b2 : : : @ bi : : : @ bn @aj @aj @aj @aj @aj
where
2 66 66 6 @ bi = 666 @aj 66 66 64
3 @Ni;x 0 77 @aj @Ni;y 777 0 @aj 7 @Ni;y @Ni;x 777 ; i = 1; : : : ; n 7 @a Nji @aj 777 @ r 5 0 @aj The derivatives @Ni;x =@aj and @Ni;y =@aj are given by Eq. 5.31, i.e., ( ) ( ) @ Ni;x = J 1 1 [ J((1 + )a ) J(a )] Ni;x ; j j j @aj Ni;y j aj Ni;y i = 1; : : : ; n; j = 1; : : : ; J
(5.73)
(5.74)
(5.75)
and the derivative of the fourth row in the submatrix bi can be found by applying Eqs. 5.1 and 5.8, i.e.,
@
Ni
r @aj
@ 1r = Ni j 8 @a > < N2i @r = > r @aj :0
if aj is a x-coordinate if aj is a y-coordinate
(5.76)
Here it is assumed that the axis of rotational symmetry is parallel with the y-axis as shown in Fig. 5.3.
66
5.5. 2D Isoparametric Finite Elements
Finally, the derivative @r=@aj , see Eq. 5.76 can be calculated as @r = r = 1 (r((1 + )a ) r(a )) ; j = 1; : : : ; J j j j @aj aj j aj
(5.77)
Using Eqs. 5.27, 5.73, 5.74, 5.75, 5.76, and 5.77, the derivative @ k=@aj of the element sti ness matrix k, cf. Eqs. 5.71 and 5.72, can now be determined.
5.5.2 Derivative of Element Mass Matrix Next \exact" numerical derivatives of the consistent element mass matrix m are derived. The mass matrix m is given by
m=
Z
!
% NT N jJj d!
(5.78)
Here, ! is the domain of the nite element described in curvilinear, non-dimensional coordinates for the element, see Fig. 5.3, % the mass density, N contains shape functions Ni , and jJj is the determinant of the Jacobian matrix J. Taking the same approach as before, Eq. 5.78 is di erentiated with respect to a design variable aj leading to the following result for the plane stress or strain case:
@ m = Z Z % NT N @ jJj t d d @aj @aj and for the axisymmetric case: " # @ m = Z Z % NT N @ jJj r + jJj @r 2 d d; j = 1; : : : ; J @aj @aj @aj
(5.79)
(5.80)
The derivative @ jJj=@aj is given by Eq. 5.27 and the derivative @r=@aj is given by Eq. 5.77.
5.5.3 Derivative of Element Initial Stress Sti ness Matrix As in the case of 3D isoparametric nite elements, it is convenient to de ne the element initial stress sti ness matrix k in terms of the reordered displacement vector d
d = fu1 u2 : : : ui : : : un v1 v2 : : : vi : : : vngT (5.81) Relating d.o.f. to the reordered element vector d , the element initial stress sti ness matrix k for the 2D isoparametric nite elements is given by k =
Z
!
GT S G jJj d!
(5.82)
Here, ! is the domain of the nite element in its local coordinate system, see Fig. 5.3, G a matrix obtained by appropriate di erentiation of shape functions Ni , S a matrix of initial stresses, and jJj is the determinant of the Jacobian matrix J.
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices The matrix G is given by
"
where each submatrix g is given by
#
G = g0 g0
(5.83)
"
# N i;x g = N ; i = 1; : : : ; n i;y
The stress matrix S is given by where each submatrix s is de ned as
"
S = 0s 0s "
#
s = x xy xy y
67
(5.84) (5.85)
#
(5.86)
Here x, xy , etc., are stresses determined by an initial static stress analysis. Di erentiating Eq. 5.82 with respect to any of the shape design variables aj ; j = 1; : : : ; J , leads to the following expression for the \exact" numerical derivative in the plane stress or strain case: " # @ k = Z Z @ GT S G + GT @ S G + GT S @ G jJj t d d @aj @aj @aj @aj ZZ (5.87) + GT S G @@ajJj t d d; j = 1; : : : ; J j and for the axisymmetric case:
" # @ k = Z Z @ GT S G + GT @ S G + GT S @ G jJj 2 r d d @aj @aj @aj # @aj " ZZ @r 2 d d; j = 1; : : : ; J + GT S G @@ajJj r + jJj @a j
j
(5.88)
The derivative @ jJj=@aj is given by Eq. 5.27, the derivatives of the components in the matrix G, see Eqs. 5.83 and 5.84, are given by Eq. 5.75, the sensitivities @ S=@aj are given by Eq. 5.34, and the derivative @r=@aj is given by Eq. 5.77.
5.5.4 Derivative of Thermal Element \Sti ness Matrix" The thermal element \sti ness matrix" consists of contributions from the heat conduction matrix kth given by Z th k = ! BthT Bth jJj d! (5.89) Here, ! is the domain of the nite element described in curvilinear, non-dimensional coordinates for the element, see Fig. 5.3, Bth a matrix obtained by appropriate di erentiation of shape functions Ni, the thermal conductivity matrix, and jJj is the determinant
68
5.5. 2D Isoparametric Finite Elements
of the Jacobian matrix J. If the material is isotropic, can simply be replaced by the scalar , the conductivity coecient. The matrix Bth is given by
h
Bth = bth1 bth2 : : : bthi : : : bthn
i
(5.90)
where the submatrix bthi, which is associated with the nodal point i of the nite element, has the form " # N i;x th bi = N ; i = 1; : : : ; n (5.91) i;y
In case of boundary conditions in terms of convection heat transfer, the thermal \sti ness matrix" receives additional contributions given by the element matrix h
h=
Z
2
NT h N jJj d
(5.92)
Here, 2 is the boundary of the nite element described in curvilinear, non-dimensional or coordinates for the element, for which the convection boundary condition is applied. N contains shape functions Ni, h is the convection coecient speci ed, and jJj is the determinant of the Jacobian matrix J for the boundary 2 . For the plane stress and strain situations, d = t d , where t is the thickness, and for axisymmetric structures we have d = 2 r d , where r is the radius. Di erentiating Eqs. 5.89 and 5.92 with respect to a shape design variable aj ; j = 1; : : : ; J , leads to the following expressions for the \exact" numerical derivatives for the plane stress or strain case:
2 3 @ kth = Z Z 4 @ Bth T Bth + BthT @ Bth 5 jJj td d + Z Z BthT Bth @ jJj td d (5.93) @aj @aj @aj @aj
and
@ h = Z NT h N @ jJj t d ; j = 1; : : : ; J @aj 2 @aj For the axisymmetric case we obtain: 2 3 @ kth = Z Z 4 @ Bth T Bth + BthT @ Bth 5 jJj 2 r d d @aj @aj @aj " # ZZ @ j J j @r T th th + B B @a r + jJj @a 2 d d; j j and " # @ h = Z NT h N @ jJj r + jJj @r 2 d ; j = 1; : : : ; J @aj @aj @aj 2
(5.94)
(5.95)
(5.96)
Here, the derivative @ jJj=@aj is given by Eq. 5.27, the derivative of the components in the matrix Bth, see Eqs. 5.90 and 5.91, are given by Eq. 5.75, and the derivative @r=@aj is given by Eq. 5.77.
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
69
5.5.5 Derivative of Consistent Load Vector Next, \exact" numerical derivatives will be derived for the consistent element load vector f which is given by Z Z T f = N FB jJj d! + NT FS jJj d (5.97) !
where ! is the domain of the nite element in its local coordinate system, FB represents body forces, the boundary described in curvilinear, non-dimensional or coordinates for the element at which boundary forces FS are applied, and N contains shape functions Ni . In the boundary integral, N and jJj are evaluated on . If initial thermally induced strains have to be taken into account, the consistent nodal force vector f th due to thermally induced strains is calculated as
f th =
Z
T E "th jJj d! B !
(5.98)
where "th is an element vector containing thermally induced strains, i.e.,
n o n o "th = "thx "thy xyth "thz T = T^ T^ 0 T^ T ; T^ = T T0
(5.99)
where is a matrix containing thermal expansion coecients, T is the temperature at the given point, and T0 is the temperature at which the structure is free of thermally induced strains. Taking the same approach as previously, the load vectors in Eqs. 5.97 and 5.98 are di erentiated with respect to any of the design variables aj ; j = 1; : : : ; J , leading to the following expressions for the \exact" derivatives for the plane stress or strain case: @ f = Z Z NT F @ jJj t d d + Z NT F @ jJj t d ; j = 1; : : : ; J (5.100) B S @aj @aj @aj and " # @ f th = Z Z @ BT E "th + BT E @ "th jJj t d d @aj @aj @aj ZZ + BT E "th @@ajJj t d d; j = 1; : : : ; J (5.101) j For the axisymmetric case we have: " # @ f = Z Z NT F @ jJj r + jJj @r 2 d d B @aj @aj # " @aj Z + NT FS @ jJj r + jJj @r 2 d ; j = 1; : : : ; J (5.102) @aj @aj and " # @ f th = Z Z @ BT E "th + BT E @ "th jJj 2 r d d @aj @aj @aj # " ZZ @r 2 d d; j = 1; : : : ; J (5.103) + BT E "th @@ajJj r + jJj @a j j
70
5.6. Isoparametric Mindlin Plate and Shell Finite Elements
Here, the derivative @ "th =@aj of the element thermally induced strains is found by combining Eqs. 5.35 and 5.99, @ jJj=@aj is given by Eq. 5.27, \exact" numerical derivatives of the strain-displacement matrix B are given by Eqs. 5.73, 5.74, and 5.75, and the derivative @r=@aj is given by Eq. 5.77.
5.5.6 Derivative of Consistent Thermal Flux Vector Now \exact" numerical derivatives of the thermal ux vector q will be determined for the 2D isoparametric nite elements. The consistent thermal nodal ux vector q is given by
q=
Z
1
NT
qS jJj d +
Z
2
NT h Te jJj d
(5.104)
where the rst term derives from speci ed ux at the boundary 1 and the latter term from a speci ed convection boundary condition at boundary 2. The boundaries 1, 2 are described in curvilinear, non-dimensional or coordinates for the element. The scalar qS is prescribed ux normal to the boundary 1 , N contains shape functions Ni that are evaluated on the boundary , jJj the determinant of the Jacobian matrix J for the boundary , h the convection coecient speci ed, and Te is the environmental temperature speci ed for the convection boundary condition. The \exact" numerical derivative for the plane stress or strain case is given by: @ q = Z NT q @ jJj t d + Z NT h T @ jJj t d ; j = 1; : : : ; J (5.105) S e @aj 1 @aj @aj 2 and for the axisymmetric case: " # @ q = Z NT q @ jJj r + jJj @r 2 d S @aj @a"j @aj # 1 Z @r 2 d ; j = 1; : : : ; J (5.106) + NT h Te @@ajJj r + jJj @a 2 j j Here, the derivative @ jJj=@aj is given by Eq. 5.27 and the derivative @r=@aj is given by Eq. 5.77.
5.6 Isoparametric Mindlin Plate and Shell Finite Elements In this section the method of \exact" numerical di erentiation is used to determine \exact" numerical derivatives of element matrices and vectors for isoparametric Mindlin plate and shell nite elements. These elements are described in Appendix C where all element matrices and vectors are given. The four main assumptions of the Mindlin plate theory, which is a \thick" plate theory where transverse shear strains are accounted for, are also given in Appendix C.
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
71
These Mindlin plate elements have in-plane membrane capability, where the elements are formed by combining a plane membrane element, i.e., a 2D solid isoparametric element, with a standard Mindlin plate bending element. These plate elements can be transformed into at shell elements as described in Section C.10 in Appendix C. Using this approach of generating at shell elements, all element matrices for the shell elements are established in the local coordinate system for the Mindlin plate element, see Fig. 5.4, and all derivations in this section are therefore only given for the Mindlin plate elements. Local coordinates, domain, and nodal degrees of freedom for these isoparametric nite plate elements can be seen in Fig. 5.4. y
η ξ2
y
ξ1 ξ3
ξ
vi θ yi i
z
wi
x
z
θ xi
ui
x
Figure 5.4: Domain, node numbering, and nodal degrees of freedom of isoparametric Mindlin plate nite elements. Local, non-dimensional coordinates , and area coordinates 1, 2, 3 are shown for the quadrilateral and triangular elements, respectively. It is important to notice that all element matrices for the Mindlin elements are given in terms of standard right-hand-rule rotations x, y as illustrated in Figs. 5.4. When the Mindlin plate theory is derived it is normally done using the rotations 1 , 2 which are de ned as 1 = y (5.107) = 2
x
The use of rotations 1 , 2 greatly simplify the algebra when developing the Mindlin plate theory but as the use of standard right-hand-rule rotations is most common in nite element programs, standard right-hand-rule rotations x, y are used here. Shape functions for the implemented 3-, 4-, 6-, 8-, and 9-node Mindlin plate nite elements are similar to those used for the 2D solid elements, i.e., the shape functions given in Apppendix B in Tables B.4, B.1, B.5, B.2, and B.3, respectively. In case of triangular isoparametric elements, Eqs. 5.57, 5.58, and 5.59 in Section 5.5 are used as relations between the non-dimensional coordinates , and area coordinates 1, 2, 3.
72
5.6. Isoparametric Mindlin Plate and Shell Finite Elements
5.6.1 Derivative of Element Sti ness Matrix The element sti ness matrix k for a Mindlin plate nite element is given by
k=
Z
!
BT DM B jJj d!
(5.108)
where DM is the elasticity matrix, B the generalized strain-displacement matrix, jJj the determinant of the Jacobian matrix J, and ! is the domain of the nite element described in curvilinear, non-dimensional coordinates for the element as shown in Fig. 5.4. The matrices J and B depend on the coordinates of the nodal points, whereas DM depends only on the thickness t of the plate and on the constitutive parameters of the assumed linearly elastic material. The elasticity matrix DM is given in Appendix C in Eqs. C.7, C.8, and C.9. In terms of the vector di of nodal degrees of freedom, cf. Fig. 5.4
di = fui vi wi xi yi gT ; i = 1; : : : ; n (5.109) the relationship between the vector d of generalized displacements of element nodal points d = fdT1 dT2 : : : dTi : : : dTn gT (5.110) and the generalized strain vector " takes the standard form " = B d where B is given by B = [ b1 b2 : : : bi : : : bn ] (5.111) If the generalized strain vector " is de ned as
8 9 8 9 > > > > u; u; x x > > > > > > > > v; v; > > > > y y > > > > > > > u;y +v;x > > u;y +v;x > > > > > < 1;x > = > < = y;x "(x; y) = > = (5.112) > x;y > 2;y > > > > > > > > > > > 1;y + 2;x > y;y + x;x > > > > > > > > > w; w; 2 y x y > > > > > > > : 1 w;x ; : y w;x > ; then the submatrix bi , which is associated with the nodal point i of the nite element, has the form 2 3 N 0 0 0 0 i;x 66 0 N 7 0 0 0 77 i;y 66 66 Ni;y Ni;x 0 0 0 777 6 0 0 Ni;x 777 ; i = 1; : : : ; n bi = 666 00 00 (5.113) 0 Ni;y 0 77 66 66 0 0 0 Ni;x Ni;y 777 64 0 0 Ni;y Ni 0 75 0 0 Ni;x 0 Ni
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
73
It is seen that the rst three rows in bi originate from the membrane part, the next three from the bending terms and the last two rows from the shear part. The derivatives of the shape functions Ni with respect to x and y are given by Eq. 5.67 in Subsection 5.5.1. Having de ned the matrices involved in the element sti ness matrix k, the \exact" numerical derivative is, as before, found by di erentiating Eq. 5.108 with respect to any shape design variable aj ; j = 1; : : : ; J , which leads to " # @ k = Z @ BT D B + BT D @ B jJj d! + Z BT D B @ jJj d!; (5.114) M M @aj ! @aj M @aj @aj ! j = 1; : : : ; J Here, the derivative @ jJj=@aj is given by Eq. 5.27 and the derivative @ B=@aj of the generalized strain-displacement matrix B can be found using the results in Subsection 5.5.1. The derivatives @Ni;x=@aj and @Ni;y =@aj are given by Eq. 5.75, and as the shape functions Ni only depend on the local, non-dimensional coordinates and , we have @Ni = 0; j = 1; : : : ; J (5.115) @aj Using these equations the \exact" numerical derivative @ k=@aj in Eq. 5.114 can now be determined.
5.6.2 Derivative of Element Mass Matrix The consistent element mass matrix m is given by
m=
Z
!
% NT N jJj d!
(5.116)
Here, ! is the domain of the nite element in its local coordinate system, % the mass density, N contains shape functions Ni , and jJj is the determinant of the Jacobian matrix J. Di erentiating Eq. 5.116 with respect to any of the shape design variables aj leads to @ m = Z % NT N @ jJj d!; j = 1; : : : ; J (5.117) @aj ! @aj where the derivative @ jJj=@aj is given by Eq. 5.27.
5.6.3 Derivative of Element Initial Stress Sti ness Matrix In the derivation of element initial stress sti ness matrices it is convenient to reorder and omit some nodal degrees of freedom by introducing the reordered, condensed element displacement vector d that only contains translational degrees of freedom. These translational d.o.f. are reordered so that rst all x-direction d.o.f. are given, then y, and then z as follows
d = fu1 u2 : : : ui : : : un v1 v2 : : : vi : : : vn w1 w2 : : : wi : : : wngT
(5.118)
74
5.6. Isoparametric Mindlin Plate and Shell Finite Elements
Relating d.o.f. to the reordered, condensed element vector d , the element initial stress sti ness matrix k for an isoparametric Mindlin plate nite element is given by
k =
Z
!
GT S G jJj d!
(5.119)
where ! is the domain of the nite element described in curvilinear, non-dimensional coordinates for the element, G a matrix obtained by appropriate di erentiation of shape functions Ni , S a matrix of initial membrane stresses, and jJj is the determinant of the Jacobian matrix J. The matrix G for the isoparametric Mindlin plate nite element is given by
2 3 g 0 0 G = 664 0 g 0 775
0 0 g
(5.120)
where each submatrix g is de ned as
"
# N i;x g = N ; i = 1; : : : ; n i;y
(5.121)
The stress matrix S has the following form
2 3 s 0 0 S = 664 0 s 0 775
0 0 s
where each submatrix s is de ned as
"
s = x xy xy y
#
(5.122)
(5.123)
Here x, xy , etc., are membrane stresses in the plate found by an initial static stress analysis. Now, Eq. 5.119 is di erentiated with respect to any of the design variables aj ; j = 1; : : : ; J :
" # @ k = Z @ GT S G + GT @ S G + GT S @ G jJj d! @aj @aj @aj ! @aj Z + GT S G @ jJj d!; j = 1; : : : ; J @a
j
(5.124)
Here, the derivatives of the components in the stress matrix S, i.e., the stress sensitivities, are given by Eq. 5.34, and the derivative of the determinant jJj is given by Eq. 5.27. The matrix G contains the components Ni;x and Ni;y , and the derivatives of these components are given by Eq. 5.75. Using these equations, all terms necessary for evaluating the \exact" numerical derivative of the element stress sti ness matrix for an isoparametric Mindlin plate nite element, cf. Eq. 5.124, are now found.
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
75
5.6.4 Derivative of Thermal Element \Sti ness Matrix" Next, the \exact" numerical derivative of the thermal element \sti ness matrix" is derived for isoparametric Mindlin plate nite elements. The thermal element sti ness matrix consists of contributions from the heat conduction matrix kth given by
kth
=
Z
th T Bth jJj d! B !
(5.125)
Here, ! is the domain of the nite element in its local coordinate system, Bth a matrix obtained by appropriate di erentiation of shape functions Ni, the thermal conductivity matrix, and jJj is the determinant of the Jacobian matrix J. If the material is isotropic, can be simply replaced by the scalar , the conductivity coecient. The matrix Bth is given by Eqs. 5.90 and 5.91. In case of boundary conditions in terms of convection heat transfer, the thermal \sti ness matrix" receives additional contributions given by the element matrix h
h=
Z
!2
NT h N jJj d! +
Z
2
NT h N jJj t d
(5.126)
The convection boundary condition is applied to either the surface !2 or the boundary 2 of the nite element, both described in local coordinates. N contains shape functions Ni , t is the thickness of the element, h the convection coecient speci ed, and jJj is the determinant of the Jacobian matrix J for the surface !2 or the boundary 2 . Di erentiating Eqs. 5.125 and 5.126 with respect to any shape design variable aj ; j = 1; : : : ; J , leads to
2 3 @ kth = Z 4 @ Bth T Bth + BthT @ Bth 5 jJj d! @aj @aj @aj ! Z + BthT Bth @@ajJj d! !
j
(5.127)
and
@ h = Z NT h N @ jJj d! + Z NT h N @ jJj t d ; j = 1; : : : ; J (5.128) @aj !2 @aj @aj 2 Here, the derivative of the determinant jJj is given by Eq. 5.27, and derivatives of the components Ni;x and Ni;y in the matrix Bth, see Eqs. 5.90 and 5.91, are given by Eq. 5.75. All terms required for evaluating the \exact" numerical derivatives in Eqs. 5.127 and 5.128 are therefore found.
5.6.5 Derivative of Consistent Load Vector The consistent element load vector f is given by
f=
Z
!
NT
FB jJj d! +
Z
!
NT
Z
FS jJj d! + NT FS jJj t d
(5.129)
76
5.6. Isoparametric Mindlin Plate and Shell Finite Elements
where ! is the domain of the nite element in its local coordinate system, FB represents body forces, is the boundary described in curvilinear, non-dimensional or coordinates for the element, t the thickness, FS represents surface forces, and N contains shape functions Ni . In the surface and boundary integrals, N and jJj are evaluated on ! and , respectively. If initial thermally induced strains have to be taken into account, the consistent nodal force vector f th due to thermally induced strains is calculated as
f th
=
Z
T E "th jJj d! B !
(5.130)
where "th is an element vector containing thermally induced strains and it is given by Eq. 5.33, where "thz is omitted. Di erentiating Eqs. 5.129 and 5.130 with respect to any shape design variable aj ; j = 1; : : : ; J , leads to @ f = Z NT F @ jJj d! + Z NT F @ jJj d! + Z NT F @ jJj t d (5.131) B S S @aj ! @aj @aj @aj ! and " # @ f th = Z @ BT E "th + "th E @ BT jJj d! + Z BT E "th @ jJj d!; (5.132) @aj ! @aj @aj @aj ! j = 1; : : : ; J Here, the derivative @ "th=@aj of the thermally induced element strains can be found using Eqs. 5.35, @ jJj=@aj is given by Eq. 5.27, and \exact" numerical derivatives of the straindisplacement matrix B are found by combining by Eqs. 5.111, 5.113, 5.75, and 5.115.
5.6.6 Derivative of Consistent Thermal Flux Vector Finally, \exact" numerical derivatives of the thermal ux vector will be determined for the isoparametric Mindlin plate nite elements. The consistent thermal nodal ux vector q is given by
q =
Z
2
qS jJj d! +
Z
T q jJj t d N S !Z Z + NT h Te jJj d! + NT h Te jJj t d ! 1
NT
1
2
(5.133)
where the rst two terms derive from speci ed ux at either surface !1 or boundary 1 and the two latter terms from speci ed convection boundary conditions at surface !2 or boundary 2 . The surfaces !1 , !2 and boundaries 1 , 2 are described in curvilinear, non-dimensional coordinates for the element, and t is the thickness of the element. The scalar qS is prescribed ux normal to the surface !1 or the boundary 1 , N contains shape functions Ni that are evaluated on the surface ! or the boundary , jJj the determinant of the Jacobian matrix J for the surface ! or the boundary , h the convection coecient
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
77
speci ed, and Te is the environmental temperature speci ed for the convection boundary condition. Not surprisingly, we di erentiate Eq. 5.133 with respect to any of the shape design variables aj ; j = 1; : : : ; J , leading to
@ q = Z NT q @ jJj d! + Z NT q @ jJj t d S S @aj @aj @aj !1 1 Z Z + NT h Te @ jJj d! + NT h Te @ jJj t d ; j = 1; : : : ; J (5.134) @a @a !2
j
2
j
Here, the derivative @ jJj=@aj is given by Eq. 5.27. Now \exact" numerical derivatives with respect to any shape design variable aj ; j = 1; : : : ; J , have been found for element matrices and vectors of all the implemented isoparametric nite elements.
5.7 Element Derivatives w.r.t. Generalized Shape Design Variables In this section it will be shown how the element derivatives obtained in Sections 5.4, 5.5, and 5.6 with respect to shape design variables aj ; j = 1; : : : ; J , which represent nodal coordinates of the individual nite element, can be associated with generalized shape design variables Am ; m = 1; : : : ; M , which may represent, e.g., positions of master nodes controlling the shape of a Coons patch surface, a b-spline, etc. Di erent modi ers like translation, scaling, and rotation types as described in Chapter 2 will be covered, and it will be shown how to treat combinations of these modi ers when they are linked to a single shape design variable. As described in Chapter 2, the coordinates of nite element nodes are never used as individual shape design variables as this may lead to a very large number of design variables and other major drawbacks as poor convergence properties and diculties in ensuring compatibility and slope continuity between boundary nodes. Therefore, it is advantageous to introduce a comparatively small number M of shape design variables Am ; m = 1; : : : ; M , which, through a suitable set of shape functions for the design boundary, control the large number of design variables ai; i = 1; : : : ; Is, which are all element nodal coordinates that are selected as design variables. In the description given here it is assumed that the \design boundary layer technique", where only nite element nodes at the design boundary are subjected to perturbation, see Fig. 4.1 in Section 4.4, is adopted. This implies that the computation of element matrices at perturbed values of the design variables is as limited as possible. Introducing the set of generalized shape design variables Am; m = 1; : : : ; M , it is only necessary to perform M sensitivity analyses at a given step of redesign. For a static design sensitivity analysis, for example, Eq. 4.6 can be rewritten in terms of any of the generalized
78
5.7. Element Derivatives w.r.t. Generalized Shape Design Variables
shape design variables Am; m = 1; : : : ; M : @ D = @ K D + @ F ; m = 1; : : : ; M K @A (5.135) @Am @Am m @K D + @F . which is restricted to the pseudo load vector @A @Am m Thus, instead of solving Eq. 4.6 for each pseudo load associated with a design variable ai ; i = 1; : : : ; Is, in the form of a nodal coordinate, the single pseudo loads are superposed a priori, resulting in a pseudo load vector that corresponds to the generalized shape design variable Am . Mechanically, this may be conceived as the superposition principle, in terms of pseudo loads and corresponding displacement sensitivities. Although a particular Am may not be related to all ai; i = 1; : : : ; Is, for the sake of generality it is assumed that Am = Am (a1; : : : ; aIs ). Application of the chain rule then yields Is @ K @a @K = X i (5.136) @Am i=1 @ai @Am ; m = 1; : : : ; M with summation over all element nodal coordinates that are selected as design variables ai ; i = 1; : : : ; Is. The derivative @ F=@Am is established in a similar way. Consider now an element sti ness matrix k in global coordinates of any of the nite elements of the discretized structure. The possible nodal coordinates of a particular element that play the role of shape design variables are denoted by aj ; j = 1; : : : ; J , and generally constitute a small subset of the total set of shape design variables ai; i = 1; : : : ; Is, in terms of nodal coordinates. Thus, the derivative @ K=@ai in Eq. 5.136 can be calculated as
@ K = X @ k @ai nae @aj aj =ai
(5.137)
where nae is the number of active (perturbed) elements in the design boundary layer as described in Section 4.4. Each of these elements may contain one or more of the parameters ai ; i = 1; : : : ; Is, as a nodal point coordinate design variable aj . Eqs. 5.136 and 5.137 are applicable to all derivatives of element matrices and vectors de ned in the preceding sections, and the formulas for \exact" numerical di erentiation of element matrices and vectors are now applicable to generalized shape design variables when the mesh sensitivities @ai =@Am have been established.
5.7.1 Boundary Shape Representation In order to determine the derivatives @ai =@Am , a general description of the boundary shape representation is given. This is done for the general case of a design boundary surface of a solid structural domain. It is straight-forward to specialize from the general case to the more commonly treated cases of boundary curves for planar structural domains. Fig. 5.5 illustrates the design boundary surface, which may be conceived to be a part, or the full surface, of a solid continuum structure. The shape of the boundary surface is assumed to be controlled by M master nodes Sm; m = 1; : : : ; M , and the nite element
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
79
z z1 x1
Q1
zt y1
Sm
xt
Qt yt
zT
S1
QT
yT
xT SM
y
x
Figure 5.5: Design boundary surface of solid body. nodal points on the boundary surface are denoted by Qt ; t = 1; : : : ; T . In the general case, the surface may have been generated by some kind of mesh generator which do not contain analytical descriptions of the surface representation. In ODESSY, unstructured mesh generators may be used to generate the nite elements on the surface, and this mesh generation may include a smoothing process of the boundary surface nodes. It is thereby not possible to have an explicit relation between the geometric design element description and the nite element nodes at the surface. Nevertheless, it will be assumed that the design boundary can be described by some smooth shape interpolation functions Rm , m = 1; : : : ; M , that are functions of some non-dimensional curvilinear coordinates k that are embedded in the surface, but explicit expressions for these shape interpolation functions Rm are not available in the general case. If mapping techniques are used to generate the design surface, it is possible to calculate analytical derivatives @Rm (k )=@Am . This has been done by, e.g., Yao & Choi (1989) for Bezier surfaces, by Choi & Chang (1991) for geometric surfaces and other surface representations, and for B-spline curves by Braibant & Fleury (1984), but it is not possible in the general case to determine the derivatives of these shape interpolation functions Rm (k ). This has also been realized by Botkin (1992) and Botkin, Bajorek & Prasad (1992) in their approach of using feature-based structural design and fully automatic mesh generation techniques in shape optimization. Each of the master nodes Sm; m = 1; : : : ; M , which controls the shape of the design boundary surface, can be characterized by the vector
som = fxom ymo zmo gT
(5.138)
which contains the global coordinates of the initial position of the m-th master node Sm. The coordinate vector c of any nite element nodal point Qt; t = 1; : : : ; T , on the design
80
5.7. Element Derivatives w.r.t. Generalized Shape Design Variables
boundary surface can now be written symbolically as
c=
M X m=1
Rm(k ) [som + G(Am)]
Here, the vector
(5.139)
c = fx y zgT
(5.140) contains the set of global coordinates for any nite element point on the design boundary surface and the vector G(Am) represents some transformation of the m-th master node Sm as a function of the generalized shape design variable Am . The master node transformation G(Am ) may, in ODESSY, be controlled by
translation modi ers scaling modi ers rotation modi ers combinations of the three above-mentioned modi ers
For simplicity, it is assumed here that only one shape design variable Am controls the position of a master node Sm, but in reality one design variable may control many master nodes or the position of a speci c master node may depend on several shape design variables. Now, let the set of shape design variables ai ; i = 1; : : : ; Is, be associated with the design boundary surface under study and assume, for reasons of generality, that the set ai ; i = 1; : : : ; Is, comprises all xt , yt, and zt , t = 1; : : : ; T , coordinates of the nite element nodal points Qt; t = 1; : : : ; T , that belong to the design boundary as shown in Fig. 5.5. The problem is now to establish the derivative of the coordinate vector, i.e., the mesh sensitivities @ai =@Am for the implemented master node transformations G(Am ).
Translation Modi ers If the master node transformation G(Am ) represents a translation modi er, the coordinate vector c of any nite element nodal point Qt ; t = 1; : : : ; T , on the design boundary surface can be written as M X c = Rm (k ) [som + Am nm ] (5.141) m=1
Here, nm is a unit vector in global coordinates that represents the translation direction, i.e., the move direction for the m-th master node. The generalized shape design variable Am represents the movement of the m-th master node in the direction nm relative to its initial location given by som. This is illustrated in Fig. 5.6. Now, from Eq. 5.141 it is seen that the coordinate vector c, and thereby the shape design variables ai; i = 1; : : : ; I , I = 3T , depend linearly on the parameters Am ; m = 1; : : : ; M .
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
81
nm
z Am n m
z1
n1 x1
A1 n1
Q1
y1
Sm
zt xt
Qt yt
zT
S1
Q xT T
yT
y
SM AM n M
x nM
Figure 5.6: Translation modi ers for design boundary surface. Hence, the mesh sensitivities @ai =@Am in case of a translation modi er can be computed by means of \exact" numerical di erentiation using Eq. 5.8, with crj = 1; i.e.,
@ai = ai = 1 (a ((1 + )A ) a (A )) ; m m i m @Am Am mAm i i = 1; : : : ; I; I = 3T; m = 1; : : : ; M
(5.142)
Having determined the mesh sensitivities @ai =@Am , it is now possible to determine \exact" numerical derivatives of element matrices and vectors with respect to generalized shape design variables Am; m = 1; : : : ; M , cf. Eqs. 5.136 and 5.137, if Am is linked to a translation modi er.
Scaling Modi ers Next, master node transformations G(Am) in terms of scaling modi ers are considered. In this case, the coordinate vector c of any nite element nodal point Qt ; t = 1; : : : ; T , on the design boundary surface can be written as
c=
M X m=1
Rm(k ) [pm + (som pm )(1 + Am)]
(5.143)
Here, the generalized shape design variable Am represents the scaling of the distance between the m-th master node and the point Pm, which is characterized by the coordinate vector pm. The value of Am is assumed to be equal to zero in the initial geometry. Scaling modi ers are illustrated in Fig. 5.7.
82
5.7. Element Derivatives w.r.t. Generalized Shape Design Variables z
o p ) (1+ A ) (s m - m m
z1
(s1o - p1) (1+ A1) x1
zt
Q1
y1
Sm
xt
Qt yt
zT
S1
QT
yT
xT P1
Pm
SM
(soM - p M ) (1+ AM )
y
PM
x
Figure 5.7: Scaling modi ers for design boundary surface. As in case of translation modi ers, the coordinate vector c, and thereby the shape design variables ai; i = 1; : : : ; I , I = 3T , depend linearly on the parameters Am ; m = 1; : : : ; M , and Eq. 5.142 can be used to compute the mesh sensitivities @ai =@Am .
Rotation Modi ers In case of rotation modi ers, here illustrated for the two-dimensional case but implemented for the general three-dimensional case in ODESSY, the coordinate vector c of any nite element nodal point Qt ; t = 1; : : : ; T , on the design boundary surface can be written as
c=
M X m=1
h i Rm (k ) pm + (som pm )T T(Am)
(5.144)
Here, the generalized shape design variable Am represents the rotation of the m-th master node around the point Pm, which is characterized by the coordinate vector pm . Am denotes the angle of rotation in the x y plane, see Fig. 5.8, and T(Am ) is the transformation matrix given by 2 3 cos ( A ) sin ( A ) 0 m m T(Am ) = 664 sin(Am ) cos(Am) 0 775 (5.145) 0 0 1 It is seen from Eqs. 5.144 and 5.145 that the coordinate vector c does not depend linearly on the parameters Am; m = 1; : : : ; M , and furthermore, the method of \exact" numerical di erentiation cannot yield \exact" numerical derivatives of the transformation matrix T. Alternatively, one could di erentiate Eq. 5.144 and use analytical expressions for the derivative of the transformation matrix T. However, such an evalution of derivatives @ c=@Am is very dicult to implement in general in the mesh generation routines, and it
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
83
(smo - p m )T T ( A m ) y Sm Am smo - p m Pm z
boundary curve x
Figure 5.8: Rotation modi er for design boundary. is therefore decided to use rst order nite di erence approximations, i.e., @ai ai ; i = 1; : : : ; I; I = 3T; m = 1; : : : ; M @Am Am
(5.146)
Combinations of Modi ers If several modi ers are linked to the same generalized shape design variable Am , and thereby evaluated sequentially, it is chosen to use rst order nite di erences to approximate the mesh sensitivities @ai =@Am , i.e., @ai ' ai ; i = 1; : : : ; I; I = 3T; m = 1; : : : ; M (5.147) @Am Am Eq. 5.147 may yield the \exact" numerical derivative, depending on the types of modi ers combined, cf. the preceding subsections. In general, the derivatives @ai =@Am ; m = 1; : : : ; M , are evaluated by the preprocessor using the following steps:
84
5.8. Element Derivatives for Arbitrary Design Variables
Perturb the generalized design variable Am in question. Call design-model-building routines which update the design model. Call boundary surface remeshing routines to update nite element mesh. Subtract the new boundary surface mesh from the original mesh to obtain mesh sensitivities @ai =@Am , cf. Eqs. 5.147. Now the relations @ai =@Am , m = 1; : : : ; M , have been determined for the possible types of generalized shape design variables Am implemented in ODESSY, and the new semianalytical method of design sensitivity analysis using \exact" numerical di erentiation is thereby applicable for all types of shape variables implemented. 1. 2. 3. 4.
5.8 Element Derivatives for Arbitrary Design Variables The aim of this section is to describe how the new S-A method for design sensitivity analysis can be applied for arbitrary types of design variables, i.e., shape as well as sizing and material design variables. The description is given for 2D solid isoparametric nite elements but is easily applied to 3D solid and Mindlin plate and shell nite elements. Each of the design variables ai ; i = 1; : : : ; I , can be divided into the three following groups: 1. Generalized shape design variables Am ; m = 1; : : : ; M , which are transformed to shape design variables in the form of nodal coordinates ai ; i = 1; : : : ; Is. 2. Sizing design variables ai ; i = 1; : : : ; It. 3. Material design variables ai; i = 1; : : : ; Im . Element derivatives w.r.t. the rst group containing shape design variables have been covered in the preceding sections and the new S-A method for design sensitivity analysis using \exact" numerical di erentiation of element matrices for the two other types of design variables will be described in the following.
5.8.1 Element Derivatives w.r.t. Sizing Design Variables For a 2D solid isoparametric nite element as described in Section 5.5, the thickness t may be chosen as a sizing design variable ai; i = 1; : : : ; It, in case of plane stress or strain situations. As an example, take the sti ness matrix k given by Eq. 5.60, where d! = t d d. The derivative @ k=@ai is easily computed as @ k = @ k = Z BT E B jJj d d; i = 1; : : : ; I (5.148) t @ai @t ! The derivatives of other element matrices and vectors for 2D solid and Mindlin plate and shell nite elements w.r.t. sizing design variables ai; i = 1; : : : ; Is, in the form of element thicknesses are easily established in a similar way.
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
85
5.8.2 Element Derivatives w.r.t. Material Design Variables All types of material parameters like Young's modulus, mass density, conductivity coecients, shear modulus, and components of the constitutive matrix and orientation of anisotropy for a composite material can be chosen as material design variables ai; i = 1; : : : ; Im. Using the same example as for sizing variables, the sti ness matrix k given by Eq. 5.60 for a 2D solid element depends on material parameters in the constitutive matrix E, and choosing one of these material parameters as a material design variable, the derivative @ k=@ai is found as @ k = Z BT @ E B jJj d!; i = 1; : : : ; I (5.149) m @ai ! @ai where the derivative @ E=@ai is easily computed analytically. Derivatives of all other element matrices and vectors w.r.t. material design variables are established using the same approach as above. Now derivatives of all implemented element matrices and vectors have been calculated in an accurate way for a variety of design variables. The approach of \exact" numerical di erentiation of element matrices is used for shape design variables whereas analytical derivatives can be used for sizing and material design variables. The actual implementation of the new S-A method is described in the following section.
5.9 Implementation and Numerical Eciency of New S-A Method In this section the numerical eciency and the actual implementation of the new S-A method are discussed. The description is exempli ed by means of 3D solid nite elements and is similar for all other types of implemented elements. First derivatives w.r.t. shape design variables are discussed as these are most complicated and then sizing as well as material design variables are covered. The actual implementation of the new S-A method is a comprimise between accuracy, numerical eciency, and ease of implementation so the traditional S-A method using rst order nite di erences is applied for some sensitivities if it leads to improved eciency or easier implementation without signi cant loss of accuracy. In Chapter 6 several examples will demonstrate the accuracy of the new S-A method.
5.9.1 Derivatives w.r.t. Shape Design Variables Derivative of Sti ness Matrix When the derivative of the global sti ness matrix K is computed, it is done at the element level as described by Eqs. 4.7 and 4.9, restricting the summation of element matrix derivatives to the active (perturbed) nite elements. The \design boundary layer" approach of using one-element-deep sensitivity calculations for shape design variables is adopted.
86
5.9. Implementation and Numerical Eciency of New S-A Method
For 3D solid elements, the derivative of the element sti ness matrix k w.r.t. element shape design variables aj ; j = 1; : : : ; J , in the form of nodal coordinates is given by Eqs. 5.22 and 5.26, i.e., @ k = 2 Z BT E B^ (j) jJj d ; j = 1; : : : ; J (5.150) @aj
S where the operation [ ]S of symmetrization of a quadratic matrix, cf. Eq. 5.23, has been employed and the matrix B^ (j) is de ned by Eq. 5.25, i.e., @ B + B @ jJj B^ (j) = @a (5.151) j 2jJj @aj The element derivatives @ k=@aj yield the \exact" numerical derivatives and they only need to be calculated once for each possible perturbed nodal coordinate ai ; i = 1; : : : ; Is, of which aj ; j = 1; : : : ; J , is a small subset. This leads to the following procedural steps for establishing the derivatives @ K=@Am in the design sensitivity analysis w.r.t. generalized shape design variables Am; m = 1; : : : ; M : 1. If design sensitivity analysis w.r.t. rst generalized shape design variable A1 : (a) The preprocessor evaluates all generalized shape design variables Am ; m = 1; : : : ; M , in order to determine the set ai; i = 1; : : : ; Is, of possible perturbed nodal coordinates. Then it perturbs the variable A1 and updates the surface mesh. Mesh sensitivities @ai =@A1 are then available, see Eqs. 5.147. (b) The nite element module evaluates all element derivatives @ k=@ai , i = 1; : : : ; Is, cf. Eqs. 5.137 and 5.150, and store them on disk. (c) The global sti ness matrix derivative @ K=@A1 is evaluated using Eqs. 5.136, 5.137, and 5.147. 2. Else design sensitivity analysis w.r.t. Am ; m = 2; : : : ; M : (a) The preprocessor perturbs shape design variable Am leading to mesh sensitivities @ai =@Am . (b) The nite element module evaluates the derivative @ K=@Am based on the element derivatives @ k=@ai stored on disk and mesh sensitivities @ai =@Am . The perturbation Am of a generalized shape design variable Am is set to 1=1000 of the smallest side length of the nite elements in the structure as described in Section 4.4 in order to obtain accurate mesh sensitivities @ai =@Am . It should be noted that the mesh sensitivities are \exact" numerical derivatives in case of translation or scaling modi er links to the generalized shape design variable Am , cf. Eq. 5.142. The eciency of the new S-A approach to obtain the derivative @ K=@Am of the global sti ness matrix K compared with the traditional method is very much problem dependent.
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
87
If the number M of generalized shape design variables Am is small, then the traditional S-A method, where the element sti ness matrix derivatives are approximated by rst order nite di erences k=Am for all nae perturbed elements, is more ecient as the number Is of element derivatives @ k=@ai , see Eq. 5.150, involved in the new S-A method will be substantially larger than nae . However, increasing the number M of generalized design variables, the new S-A method becomes more ecient because the element derivatives @ k=@ai only needs to be calculated once and the derivative @ K=@Am is quickly evaluated from these element derivatives whereas the traditional S-A method needs to evaluate k=Am for nae perturbed elements for all design variables Am ; m = 1; : : : ; M . Furthermore, the ratio of eciency between the traditional and the new S-A method is very much dependent on the computer used. The new S-A method needs to store more element derivatives on disk than the traditional method, so the disk access speed is also an important factor. In general, for a few generalized shape design variables Am the new S-A method may be twice as slow to determine the global sti ness matrix derivative @ K=@Am (or even slower in special cases), but having a larger number of design variables, the two methods are comparable in eciency and for a large number of generalized shape design variables, say 50, the new S-A method may be considerably faster. For most problems, the traditional S-A method yields suciently accurate derivatives of sti ness matrices, but for many beam- and plate-like structures, it is absolute necessary to use the new S-A method in order to obtain satisfactory results as will be demonstrated by numerical examples in Chapter 6.
Derivatives of Mass Matrix and Thermal \Sti ness Matrix" The implementation of \exact" numerical derivatives for the global mass matrix M and the global thermal \sti ness matrix" Kth is very similar to the implementation for the sti ness matrix as described above. The description of the eciency of the new S-A method given above can also be applied directly to these element derivatives but it should be noted that the traditional S-A method based on rst order nite di erences has worked well for these matrix derivatives for all the design sensitivity analysis problems that I have studied.
Stress Sensitivities The element stress sensitivities are calculated using \exact" numerical di erentiation as described in Subsection 5.4.3, see Eqs. 5.34 and 5.35, which are easily rewritten to the situation of design sensitivity analysis w.r.t. a generalized shape design variable Am; m = 1; : : : ; M : ! @ = E @ B d + B @ d @ "th ; m = 1; : : : ; M (5.152) @Am @Am @Am @Am
88
5.9. Implementation and Numerical Eciency of New S-A Method
where
( ) @ "th = @T @T @T 0 0 0 T ; m = 1; : : : ; M (5.153) @Am @Am @Am @Am The element nodal displacement vector d, the displacement sensitivities @ d=@Am and the temperature sensitivities @T=@Am are known from the solutions of Eqs. 4.2, 5.135, and 4.22, respectively, and the strain-displacement matrix B is given by Eqs. 5.17, 5.18, and 5.19. Finally, the derivative @ B=@Am can be computed by application of the chain rule, i.e., Is @ B @a @B = X i (5.154) @A @a @A ; m = 1; : : : ; M m
i=1
i
m
The derivative @ B=@Am will be non-zero only for elements in the perturbed \boundary layer", i.e., elements that have nodal coordinates at the design boundary surface. Moreover, since B is independent of all design variables other than those associated with the particular nite element, i.e., aj ; j = 1; : : : ; J , then
8X > J @ B @a @ B = < @a @Aj for elements in the \boundary layer" j m @Am > : j=1 0 for all other elements, m = 1; : : : ; M
(5.155)
where the mesh sensitivities @aj =@Am are available from Eq. 5.147. It should be noted that the summation in Eq. 5.155 is only carried out over those shape design variables aj ; j = 1; : : : ; J , that are associated with nodal points of the element on the design boundary surface. In this way \exact" numerical derivatives of stresses are obtained. This approach of obtaining stress sensitivities is slightly slower than the traditional S-A method because the derivative @ B=@Am must be established for elements in the \design boundary surface layer", but the time it takes to perform these additional computations is not perceptible in a design sensitivity analysis.
Derivatives of Load Vectors Many numerical experiments have shown that usual nite di erence approximations of the derivatives of the nodal load vector f , see Eqs. 5.51 and 5.52, and the thermal ux vector q, see Eqs. 5.55, are suciently accurate, and therefore the expressions for the derivatives of these element vectors using \exact" numerical di erentiation are currently not implemented. Furthermore, it has been easier reusing the traditional S-A method for these computations when implementing the new S-A method in ODESSY.
Derivative of Initial Stress Sti ness Matrix There is a minor di erence between calculating the derivative of the initial stress sti ness matrix and of the sti ness matrix because part of the initial stress sti ness matrix derivative is dependent on the stress sensitivities calculated.
Chapter 5. \Exact" Numerical Di erentiation of Element Matrices
89
The \exact" numerical derivative of the element initial stress sti ness matrix is given by Eqs. 5.44, and using the operation [ ]S of symmetrization of a quadratic matrix, cf. Eq. 5.23, the \exact" numerical derivative @ k =@aj can be written as
@ k = 2 Z GT S G^ (j) jJj d + Z GT @ S G jJj d ; j = 1; : : : ; J @aj @aj
S where the matrix G^ (j) is de ned as @ G + G @ jJj G^ (j) = @a j 2jJj @aj
(5.156)
(5.157)
It is seen that the rst part of Eq. 5.156 only needs to be calculated once whereas the second part needs to be evaluated for the each sensitivity analysis w.r.t. a generalized shape design variable Am . The new S-A method is therefore substantially slower for calculating the initial stress sti ness matrix derivative than the traditional S-A method, but Eq. 5.156 always leads to accurate sensitivities. This will be exempli ed by numerical examples in Chapter 6.
5.9.2 Derivatives w.r.t. Sizing or Material Design Variables The implementation of the new S-A method for sizing and material design variables is very similar to the implementation for shape design variables as described above, and therefore only the di erences will be pointed out. Analytical derivatives similar to Eqs. 5.148 and 5.149 in Section 5.8 are implemented for derivatives of sti ness, mass, thermal \sti ness", and for initial stress sti ness matrices and stress sensitivities, whereas the traditional S-A method is applied for derivatives of load vectors. The analytical derivatives of element matrices are computed and stored on disk during the design sensitivity analysis w.r.t. the rst design variable ai as described in the preceding subsection for calculation of \exact" numerical derivatives of element sti ness matrices w.r.t. shape design variables. These derivatives are calculated only once and then reused for all other design variables, when necessary. This approach to design sensitivity analysis is in most cases more ecient than the traditional S-A method, because the element derivatives only need to be calculated once, and they are faster to evaluate than the rst order nite di erences used in the traditional S-A method. The actual implementation and the eciency of the new S-A method has now been discussed, and the accuracy of this approach to design sensitivity analysis will be demonstrated by several examples in the following chapter.
90
5.9. Implementation and Numerical Eciency of New S-A Method
Chapter Examples of Design Sensitivity Analysis
6
6.1 Introduction
Ianalysis examples are used to illustrate the implementation of the methods for design sensitivity that have been described in preceding Chapters 4 and 5. Some of the examples are n this chapter several examples of design sensitivity analysis are presented. The
used to illustrate that the di erent methods for design sensitivity analysis have been implemented correctly in ODESSY while other examples are used to emphasize the importance of using the new semi-analytical (S-A) method of design sensitivity analysis described in Chapter 5 for certain types of problems. In Section 6.2 the initial design for the classical problem of shape optimization of a planar llet is subjected to static design sensitivity analysis w.r.t. a generalized shape design variable. Both the OFD method, the traditional S-A method, and the new S-A method are shown to give accurate stress sensitivities for this problem. Next the cantilever beam example introduced by Barthelemy & Haftka (1988) for study of the inaccuracy problem associated with the traditional S-A method, as described in Section 5.2, is presented in Section 6.3. The results presented for this model problem illustrate the accuracy of the new S-A method and demonstrate the shortcoming of the traditional S-A method. In Section 6.3 a two-material cantilever beam subjected to thermal loading is presented. This example again witnesses inaccuracy problems for the traditional S-A method when the displacement eld is characterized by large rigid body rotations relative to actual deformations of the nite elements. This is also illustrated by a plate example in Section 6.5 where a thin plate clamped at all edges is subjected to a point load at the midpoint. The next example in Section 6.6 concerns design sensitivity analysis of a plate reinforced by sti neners. The objective here is to illustrate that the sensitivity analysis of both simple and multiple eigenfrequencies can be carried out in an accurate way based on the results 91
92
6.2. Static Design Sensitivity Analysis of Classical Fillet Problem
of Sections 4.9.1 and 4.9.2. In Section 6.7 the thickness of the reinforced plate is reduced, resulting in a dense spectrum of eigenfrequencies. By this example it is illustrated how dicult it can be to decide from numerical results the multiplicity of a speci c eigenvalue, with the risk of computing erroneous sensitivities in case of a wrong estimation of the multiplicity. Finally, two plate examples presented in Sections 6.8 and 6.9 illustrate that the inaccuracy problem associated with the traditional S-A method for static design sensitivity analysis also manifests itself in sensitivity analysis of eigenvalues. The sensitivity analysis of free transverse vibration frequencies of a thin clamped square plate is shown to give erroneous results for the traditional S-A method, and the inaccuracy problem is even worse for sensitivities of eigenfrequencies of a vibrating plate with in-plane loads, i.e., a problem where initial stress sti ening e ects are taken into account. Altogether, the examples in this chapter demonstrate the accuracy of the implemented methods for design sensitivity analysis of static, thermo-elastic, and dynamic problems. The examples also emphasize the superiority of the new S-A method to the traditional S-A method for sensitivity analysis of beam- and plate-like structures.
6.2 Static Design Sensitivity Analysis of Classical Fillet Problem The rst example is used to illustate that both the OFD, the traditional S-A, and the new S-A method have been implemented properly in ODESSY for static design sensitivity analysis w.r.t. generalized shape design variables. The initial design for the classical problem of shape optimization of a planar llet in plane stress is used as a test example. Fig. 6.1 shows schematically the loading and boundary conditions for the problem after use has been made of symmetry conditions. The oblique llet design boundary is modelled by a quadratic B-spline controlled by ve master nodes, each of which is assigned a translation modi er as shown in Fig. 6.1. A free mesh generation algorithm is used for discretization of the structural domain and has divided it into 173 nite elements. The mesh is a mixture of 6- and 8-node isoparametric nite elements for plane stress and the side length ` of the smallest element is speci ed to be 1 mm. The llet has unit thickness and is modelled by an isotropic material. The sensitivity of the von Mises reference stress at the nite element nodal point Q1 is studied here with respect to perturbation of the generalized shape design variable A5 which represents translation of the master node S5 in the direction n5 , see Fig. 6.1. With the discretization shown in Fig. 6.1, the von Mises reference stress vM at the point Q1 is 241.0 MPa, and the sensitivity vM =A5 of this stress component is calculated by the OFD method, by the traditional S-A method using rst order nite di erence approximations for derivatives of element matrices and vectors, and by the new S-A method using \exact" numerical di erentiation of element matrices and vectors. The computationally expensive OFD technique, cf. Eq. 4.1, is chosen as a reference method whose limit with regard to design sensitivity accuracy is known to be set only by the solution procedure, the
Chapter 6. Examples of Design Sensitivity Analysis 30
93
this design boundary is modelled by a quadratic B-spline controlled by five master nodes S1 translation modifier is linked to master node S5 n5 S5 Q1
45
100 MPa 20
100
20
Figure 6.1: Classical llet example. discretization, and the usual analysis accuracy capabilities of the applied nite element. The perturbation A5 is varied from 10 1 to 10 9 of the smallest side length ` of the nite elements in the structure and the results obtained are given in Table 6.1. The stress sensitivities obtained by the three di erent methods are seen to agree within at least four digits when the relative perturbation `=` of the boundary elements is less than 10 2, whereas the sensitivities obtained by the OFD and the traditional S-A method for larger perturbations di er slightly due to the large perturbations used. The relative perturbation is normally, as default in ODESSSY, set to 1=1000 of the smallest side length ` of the nite elements in the structure as described in Section 4.4, i.e., for this example A5 would be set to 10 3 as default. All three methods are seen to give accurate results for this perturbation, but it is worth noticing that the new S-A method is completely independent of the perturbation used. The present example indicates correct implementation of the three di erent methods for design sensitivity analysis of static problems with respect to generalized shape design variables, but does not endow the new S-A method with priority over the traditional S-A approach. This is because a problem has been considered in which the displacement eld is characterized by very small rigid body rotations relative to actual deformations of the nite elements, cf. the discussion in Section 5.2. It is very important to notice that under such conditions, the traditional S-A method is completely reliable.
6.3 Static Design Sensitivity Analysis of Long Cantilever Beam Consider next an example in which, still within the usual linear theory of elasticity, the displacement eld entails dominance of rigid-body rotation relative to actual deformation
94
6.3. Static Design Sensitivity Analysis of Long Cantilever Beam
Table 6.1: Computed von Mises reference stress sensitivity vM =A5 at the nite element nodal point Q1. Boundary element length perturbation ` ` 10 10 10 10 10 10 10 10 10
1 2 3 4 5 6 7 8 9
OFD method
Traditional S-A method
New S-A method
A5
vM A5
vM A5
vM A5
1 2 3 4 5 6 7 8 9
23:47 23:61 23:62 23:62 23:62 23:62 23:62 23:62 23:61
29:56 23:61 23:62 23:62 23:62 23:62 23:62 23:62 23:61
23:62 23:62 23:62 23:62 23:62 23:62 23:62 23:62 23:62
Perturbation
10 10 10 10 10 10 10 10 10
of the nite elements in a subdomain of the structure. The example is the cantilever beam problem used by Barthelemy & Haftka (1988) to study the inaccuracy problem associated with the traditional S-A method as described in Section 5.2. The problem pertains to a slender cantilever beam of given length L and aspect ratio 50 as shown in Fig. 6.2. The beam is subjected to a given tip load P at the free end and is modelled by a regular pattern of 8-node isoparametric serendipity nite elements. The length of the element sides are equal and denoted by `. The design sensitivity vL =L of the tip displacement vL with respect to the length L of the beam is studied, and the \design boundary layer" approach of using one-element-deep sensitivity information is adopted. It has been demonstrated by numerical experiments in Barthelemy & Haftka (1988) that the displacement design derivative vL =L determined by the traditional S-A method for this beam problem is subject to severe inaccuracy problems. This fact is illustrated by the results presented in Table 6.2 for di erent values of the relative perturbation `=` of the lengths ` of the nite elements in the \design boundary layer". The results are based on a discretization of the beam into 200 x 4 eight-node isoparametric nite elements as indicated in Fig. 6.2, the beam has length L = 100 and unit thickness, the load P has unit value, Young's modulus is set to 2.1105 MPa, and Poisson's ratio = 0.3. Since the beam is long, sensitivity results in Table 6.2 may be compared with analytical results for It Lis=@L seenstated from the results in Table 6.2 that foravalues of the relative perturbation `=` as @v in the caption of Table 6.2 for corresponding Bernoulli-Euler beam. 5 small as 10 , even the sign is wrong for the sensitivities obtained by the traditional S-A
Chapter 6. Examples of Design Sensitivity Analysis
95
y,v P
vL x,u L
Figure 6.2: Finite element model of long cantilever beam with aspect ratio 50. Table 6.2: Computed displacement design sensitivities vL =L for cantilever beam modelled by 200 x 4 eight-node isoparametric serendipity nite elements for plane stress. (Computed displacement vL = 2.3807. Bernoulli-Euler comparison beam has vL = 2.3809 and @vL =@L = 0.1429.) Boundary element length perturbation
Beam length perturbation
OFD method
Traditional S-A method
New S-A method
` `
L L
vL L
vL L
vL L
10 10 10 10 10 10
1 2 3 4 5 6
2:5 10 2:5 10 2:5 10 2:5 10 2:5 10 2:5 10
2 3 4 5 6 7
0:1429 0:1428 0:1428 0:1436 0:1523 0:2006
3235: 324:8 32:25 3:096 0:1771 0:1344
0:1428 0:1428 0:1428 0:1428 0:1428 0:1428
method. Furthermore, for perturbations less than 10 3 the OFD method starts to diverge due to truncation errors, showing that it is dicult to obtain accurate sensitivities for this beam problem. As expected, sensitivities obtained by use of the new S-A method are independent of the perturbation `=` and attain values that agree within 0:02% with the sensitivity value of the corresponding Bernoulli-Euler comparison beam. This illustrates the \exactness" of the new S-A method, and furthermore, it demonstrates that the \design boundary layer" approach of using one-element-deep sensitivity information gives accurate results.
96
6.4. Thermo-Elastic Sensitivity Analysis of Two-Material Beam
It should be noted that the traditional S-A method would yield more accurate sensitivities if a second order central di erence approximation of the sti ness matrix derivative is used, as the sensitivity error then is proportional to the square of the length perturbation. When rst order nite di erences are used the sensitivity error is linearly dependent on the length perturbation. This can also be seen from Table 6.2. However, in case of both rst and second order nite di erence approximations, the sensitivity error for the traditional S-A method for this problem will be proportional to the square of the number of elements used to model the beam in the length direction. This has been shown by Pedersen, Cheng & Rasmussen (1989) and Olho & Rasmussen (1991a).
6.4 Thermo-Elastic Sensitivity Analysis of Two-Material Beam Next, a two-material cantilever beam subjected to thermal loading is considered. The previous cantilever beam example illustrated the inaccuracy problem associated with the traditional S-A method when the displacement eld contains large rigid body rotations relative to usual strains, and because such displacement elds occur often in thermo-elastic problems, inaccuracy problems can be expected for the traditional S-A method. y,v
T = 373 K aluminium steel T = 573 K
interface
vL x,u
L
Figure 6.3: Two-material cantilever beam with aspect ratio 25. The cantilever beam consists of two material domains of equal heights and di erent temperatures are speci ed at the upper and lower boundary of the beam as shown in Fig. 6.3. The beam aspect ratio is 25 whereas the previous beam example in Section 6.3 had an aspect ratio of 50. The length of the beam is set to L = 50, i.e., the height of the beam is 2. As in the previous beam example, the derivative of the tip displacement, vL =L, is studied for di erent values of relative perturbations `=` of the lengths ` of the nite elements in the \design boundary layer". The nite element model consists of 400 eight-node elements, giving the tip displacement The materials the following properties: vL =two 0:6714 due to have the thermal loading. Displacement sensitivities vL =L are presented in Table 6.3. The results in Table 6.3 emphasize the possible need for using the new S-A method for thermo-elastic design sensitivity analysis. The S-A method needs so small relative pertur-
Chapter 6. Examples of Design Sensitivity Analysis Young's modulus Poisson's ratio Conductivity coecient Expansion coecient
: : : :
Steel 210000 MPa 0.3 27 10 6 W/(mm2K) 1:2 10 5 K 1
97
Aluminium 70000 MPa 0.3 202 10 6 W/(mm2K) 2:5 10 5 K 1
Table 6.3: Computed displacement design sensitivities vL =L for two-material cantilever beam modelled by 100 x 4 eight-node isoparametric serendipity nite elements for plane stress. The beam aspect ratio is 25. Boundary element length perturbation
Beam length perturbation
OFD method
Traditional S-A method
New S-A method
` `
L L
vL L
vL L
vL L
1 2 3 4 5 6
5:0 10 5:0 10 5:0 10 5:0 10 5:0 10 5:0 10
10 10 10 10 10 10
2 3 4 5 6 7
0:02686 0:02684 0:02684 0:02683 0:02658 0:02534
310:3 31:05 3:081 0:2840 0:004314 0:02292
0:02684 0:02684 0:02684 0:02684 0:02684 0:02684
bations as 10 6 just in order to obtain sensitivities of correct sign, and if a more slender beam is studied, this error problem is increased. The OFD method produces accurate results unless very small perturbations are used, and the new S-A method yields very accurate results for all perturbations. It should be noted that in the current implementation of the new S-A method in ODESSY, a nite di erence approximation is involved in the calculation of the derivatives of the load vectors as described in Subsection 5.9.1, but this approximation is seen not to a ect the accuracy of the sensitivities obtained. It should be noted that no inaccuracy problem has been detected in the thermal sensitivity analysis computing temperature sensitivities and thereby thermally induced strain sensitivities using the traditional S-A approach. The inaccuracy problem encountered here is entirely associated with the displacement sensitivities due to a thermal load.
98
6.5. Static Design Sensitivity Analysis of Thin Clamped Square Plate
6.5 Static Design Sensitivity Analysis of Thin Clamped Square Plate Next let us consider another example where the traditional method of S-A static design sensitivity analysis may result in severe errors, cf. Barthelemy & Haftka (1988). The example concerns a square plate clamped at all edges and subjected to a concentrated load at the midpoint. As indicated in Fig. 6.4, one quarter of the plate is modelled by 100 4-node isoparametric Mindlin plate nite elements. P
L
L
Figure 6.4: Finite element model of one quarter of square plate. The length/thickness ratio is 100, and the sensitivity of the lateral displacement at the point of load application with respect to the side length L is computed for di erent perturbations. The perturbations are con ned to the elements adjacent to the lines of symmetry. The results are presented in Table 6.4 and con rm that the traditional S-A method is strongly dependent on the size of the relative perturbation L=L, whereas the new S-A method gives accurate sensitivities. The inaccuracies of the OFD method for the perturbations 10 1 and 10 2 are due to these large values. Other numerical tests of this plate example with di erent length/thickness ratios have shown that, as expected, the errors of the traditional S-A method increase rapidly with increasing length/thickness ratios as the displacement eld becomes more and more characterized by large rigid body rotations relative to actual deformations of the nite elements.
6.6 Design Sensitivity Analysis of Reinforced Vibrating Plate Next a square plate reinforced by ribs as shown in Fig. 6.5 is considered. The aim of this example is to illustrate how the results of Sections 4.9.1 and 4.9.2 can be used in eigenfrequency design sensitivity analysis, that is, to show that sensitivities of both simple and multiple eigenvalues can be computed accurately. The new S-A method constitutes the basis for the design sensitivity analysis but the traditional S-A method may be applied as well.
Chapter 6. Examples of Design Sensitivity Analysis
99
Table 6.4: Computed derivative of lateral displacement wmid at the point of load application with respect to side length L. Length/thickness ratio is 100. Plate length perturbation
OFD method
Traditional S-A method
New S-A method
L L
wmid L
wmid L
wmid L
0:6815 0:7997 0:8162 0:8179 0:8181 0:8181 0:8181 0:8181
254:8 46:14 4:304 0:3011 0:7663 0:8129 0:8176 0:8180
0:8181 0:8181 0:8181 0:8181 0:8181 0:8181 0:8181 0:8181
1 2 3 4 5 6 7 8
a
10 10 10 10 10 10 10 10
a
b
a
b
plate
a
a b
vertical ribs
horizontal ribs
a b
Figure 6.5: Square plate reinforced by ribs. The plate is clamped at all edges and reinforced by two horizontal and two vertical ribs. The length a=0.5 m, b=0.05 m, the thickness of the ribs is 0.05 m, and the thickness of the plate is 0.005 m. Both the plate and the ribs are made of steel with the following material properties Young's modulus = 210000 MPa Poisson's ratio = 0.3 Mass density = 7800 Kg/m3 The structure is symmetric and therefore multiple eigenfrequencies are expected. The goal
100
6.6. Design Sensitivity Analysis of Reinforced Vibrating Plate
is to nd sensitivities of both the simple and the multiple eigenfrequencies with respect to 6 di erent design variables. The nite element model consists of 1156 4-node isoparametric Mindlin plate elements and the model has 5445 d.o.f. All eigenvectors are M-orthonormalized, see Eq. 4.26, and the 4 lowest eigenfrequencies will be considered. From the analysis it is found that the lowest eigenfrequency is simple and has the value 92.20 Hz, but the second and third eigenfrequency are identical and have the value 161.71 Hz. In this example, the eigenfrequencies are considered to be identical when the relative di erence between the values is 10 4. The fourth eigenfrequency is simple and equal to 175.03 Hz. The eigenmodes corresponding to the 4 eigenfrequencies can be seen in Figs. 6.6 - 6.9, and the in uence of the ribs is very clear. As the second eigenfrequency is multiple, an in nite number of linear combinations of the eigenvectors shown in Figs. 6.7 and 6.8 corresponding to the multiple eigenfrequency will satisfy the general eigenvalue problem in Eq. 4.24 and the M-orthonormalization condition in Eq. 4.26. The design sensitivities with respect to changes of single design variables are computed by two di erent methods, namely the overall nite di erence (OFD) method and the new semi-analytical (S-A) method. The OFD method which is used as a reference method implies that the design is perturbed, a new eigenfrequency analysis is performed, and then the eigenfrequency sensitivities j are found from j ' fj (a1 ;a: : : ; aI ) = fj (a1; : : : ; ai + ai; : : : ;aaI ) fj (a1 ; : : : ; ai; : : : ; aI ) (6.1) i i The new semi-analytical (S-A) method is based on Eq. 4.29 for the simple eigenfrequencies and Eq. 4.40 for the multiple eigenfrequencies. Eqs. 4.7, 5.114, 5.117, 5.136, 5.137, and 5.142 yield the \exact" numerical derivatives of sti ness and mass matrices. Furthermore, based on the S-A method we shall calculate the sensitivities of all four eigenfrequencies regarding them as simple, i.e., using only Eq. 4.29 in order to see the consequences of this erroneous assumption.
Chapter 6. Examples of Design Sensitivity Analysis
Figure 6.6: 1. eigenmode. f1 = 92.20 Hz.
Figure 6.7: 2. eigenmode. f2 = 167.7 Hz.
Figure 6.8: 3. eigenmode. f3 = 167.7 Hz.
101
102
6.6. Design Sensitivity Analysis of Reinforced Vibrating Plate
Figure 6.9: 4. eigenmode. f4 = 175.0 Hz. The rst design sensitivity analysis is with respect to the thickness of the plate as shown in Fig. 6.10. design variable no. 1: plate thickness
Figure 6.10: Design variable no. 1. The sensitivities are shown in Table 6.5 and it is seen that the same results are obtained by the OFD method and the S-A method using Eqs. 4.29 and 4.40. Increasing the plate thickness is a symmetric design change and therefore we may expect that the multiple eigenfrequency remains multiple. Note that while the sensitivities of the other eigenfrequencies are positive then the sensitivity of the lowest eigenfrequency is negative, i.e., the rst eigenfrequency will decrease if the thickness of the plate is increased. These results require some explanation. The eigenmode corresponding to the lowest eigenfrequency can be characterized as a global mode while all higher order eigenmodes can be regarded as nearly-local modes for each of the nine subdomains of the plate, see Figs. 6.6 - 6.9. Thus, increasing the plate thickness will have little e ect on the overall sti ness of the structure as it is mainly governed by the ribs whereas the thickness of the plate has a large in uence on the total mass, i.e., the inertia forces. Therefore, increasing the plate thickness will decrease the lowest eigenfrequency which mainly depends on the overall sti ness and total mass of the structure. The higher order eigenmodes, on the other hand, mainly depend on the local sti ness and mass of each of the nine subdomains, and increasing the plate thickness has a larger e ect on the local sti ness than on the local mass of each subdomain. This is the reason why the higher eigenfrequencies will increase with increasing plate thickness. For this design change, the erroneous assumption of using Eq. 4.29 to the double eigenfre-
Chapter 6. Examples of Design Sensitivity Analysis
103
Table 6.5: Eigenfrequency sensitivities with respect to design variable no. 1: plate thickness. Frequency no. j 1 2 3 4
OFD method fj a1
Eq. 4.29 & 4.40 j
1552: 14093: 14093: 31407:
1552: 14094: 14094: 31407:
Eq. 4.29 @fj @a1 1552: 14094: 14094: 31407:
quency, see last column in Table 6.5, in fact gives the same sensitivities as obtained by Eq. 4.40. This shows that the in uence of the o -diagonal terms f12T e = f21T e in the sensitivity matrix in Eq. 4.40 is very weak for this symmetric design change. Next, the eigenfrequency sensitivities are found with respect to the thickness of the ribs as shown in Fig. 6.11. The results are shown in Table 6.6. design variable no. 2: rib thickness
Figure 6.11: Design variable no. 2. Table 6.6: Eigenfrequency sensitivities with respect to design variable no. 2: rib thickness. Frequency no. j
OFD method fj a2
Eq. 4.29 & 4.40 j
1 2 3 4
1879: 1714: 1714: 304:7
1879: 1714: 1714: 305:1
Eq. 4.29 @fj @a2 1879: 1714: 1714: 305:1
104
6.6. Design Sensitivity Analysis of Reinforced Vibrating Plate
It is seen in Table 6.6 that again the same results are obtained by the OFD method and the S-A method using Eqs. 4.29 and 4.40. All the sensitivities are positive, and again the multiple eigenfrequency remains multiple with this design change. Next we want to determine sensitivities of the eigenfrequencies when the position of the horizontal ribs is changed. The design variable is the distance between the horizontal ribs, see Fig. 6.12.
design variable no. 3: position of horizontal ribs
Figure 6.12: Design variable no. 3. The results are shown in Table 6.7. Table 6.7: Eigenfrequency sensitivities with respect to design variable no. 3: position of horizontal ribs. Frequency no. j
OFD method fj a3
Eq. 4.29 & 4.40 j
Eq. 4.29 @fj @a3
1 2 3 4
40:86 380:6 186:9 169:5
40:86 380:6 186:9 168:6
40:86 287:5 93:75 168:6
It is seen from Table 6.7 that the multiple eigenfrequency f~ = f2 = f3 will split when the distance between the horizontal ribs is increased. Now we can also see di erences between the two last columns in Table 6.7 showing the in uence of o -diagonal terms in Eq. 4.40 which implies that calculation of sensitivities of the double eigenfrequency by means of the single-modal formula in Eq. 4.29 gives erroneous results. Next, the distance between the vertical ribs is used as a design variable as shown in Fig. 6.13, and the results are presented in Table 6.8. The sensitivities for this design variable should be the same as obtained for design variable no. 3, the distance between the horizontal ribs, except for that the sensitivities for the multiple eigenfrequency f2 = f3 should be interchanged when using Eq. 4.40. It is seen that the OFD method does not display this situation as the eigenfrequencies are ordered
Chapter 6. Examples of Design Sensitivity Analysis
105
design variable no. 4: position of vertical ribs
Figure 6.13: Design variable no. 4. Table 6.8: Eigenfrequency sensitivities with respect to design variable no. 4: position of vertical ribs. Frequency no. j
OFD method fj a4
Eq. 4.29 & 4.40 j
Eq. 4.29 @fj @a4
1 2 3 4
40:86 380:6 186:9 166:4
40:86 186:9 380:6 167:0
40:86 93:75 287:5 167:0
by magnitude in each analysis when using the OFD method. More importantly, we again note that application of the single-modal Eq. 4.29 yields erroneous sensitivities of the multiple eigenfrequency f2 = f3. The next design variable is the width of the horizontal ribs, see Fig. 6.14, and the results are shown in Table 6.9.
design variable no. 5: width of horizontal ribs
Figure 6.14: Design variable no. 5. Again, the results obtained by the OFD method and the S-A method using Eqs. 4.29 and 4.40 are very similar, and it is seen that the multiple eigenfrequency splits with this design change. Furthermore, the sensitivities of the double eigenfrequency obtained by using Eq. 4.29 even have a wrong sign, so using the erroneous assumption of regarding the
106
6.6. Design Sensitivity Analysis of Reinforced Vibrating Plate
Table 6.9: Eigenfrequency sensitivities with respect to design variable no. 5: width of horizontal ribs. Frequency no. j
OFD method fj a5
Eq. 4.29 & 4.40 j
Eq. 4.29 @fj @a5
1 2 3 4
273:4 866:2 26:38 401:4
273:4 866:2 26:38 400:9
273:4 719:7 120:1 400:9
eigenfrequencies as simple leads to completely wrong results for this design change. The last design variable is the width of the vertical ribs as shown in Fig. 6.15.
design variable no. 6: width of vertical ribs
Figure 6.15: Design variable no. 6. Table 6.10: Eigenfrequency sensitivities with respect to design variable no. 6: width of vertical ribs. Frequency no. j
OFD method fj a6
Eq. 4.29 & 4.40 j
Eq. 4.29 @fj @a6
1 2 3 4
273:4 866:2 26:38 399:5
273:4 26:38 866:2 399:8
273:4 120:1 719:7 399:8
As before, the sensitivities with respect to this design variable should be the same as obtained for design variable no. 5, the width of the horizontal ribs, except for that the
Chapter 6. Examples of Design Sensitivity Analysis
107
sensitivities for the multiple eigenfrequency f2 = f3 should be interchanged when using the S-A method and Eqs. 4.29 and 4.40. Again it is seen that very similar results are obtained by the OFD method and the SA method when Eqs. 4.29 and 4.40 are used properly, whereas the last column again witnesses shortcoming of Eq. 4.29 when applied to the double eigenfrequency f2 = f3 . Up to now the eigenfrequency sensitivities have been found with respect to single design changes of each of the 6 design variables thickness of plate, thickness of ribs, position of horizontal ribs, position of vertical ribs, width of horizontal ribs, and width of vertical ribs. Let us nally show that it is possible to determine sensitivities for any direction in the space of the 6 design variables when some of them are changed simultaneously. The position of both the horizontal and vertical ribs, see Figs. 6.12 and 6.13, will be changed simultaneously and again the OFD method is used as a reference. The two design variables will be given unit increments. All the generalized gradient vectors fsk in Eq. 4.46 have been calculated and Eq. 4.47 is used for determining the increments f =f2 and f =f3 of the multiple eigenfrequency f2 = f3 . Eq. 4.30 is used for determining increments of the simple eigenfrequencies f1 and f4 . The sensitivities for this simultaneous design change are shown in Table 6.11. Table 6.11: Eigenfrequency sensitivities for unit increments of design variable no. 3: position of horizontal ribs and design variable no. 4: position of vertical ribs. Frequency no. j 1 2 3 4
OFD method fj 81:72 193:8 193:8 335:5
Eq. 4.30 & 4.47 fj 81:72 193:8 193:8 335:5
It is seen that very accurate results are obtained by using Eqs. 4.30 and 4.47 for determining sensitivities of single and bimodal eigenfrequencies, respectively, in any direction in the space of design parameters. It has now been demonstrated that design sensitivities of simple as well as multiple eigenvalues with respect to single or simultaneous change of design variables can be calculated very accurately using the S-A approach.
108
6.7. Ribbed Plate with Cluster of Eigenfrequencies
6.7 Ribbed Plate with Cluster of Eigenfrequencies Next, we shall illustrate how important it is for the design sensitivity analysis to decide correctly from the numerical results whether some eigenvalues coalesce and become multiple, or remain distinct. We consider the same example as before, i.e., the square plate with ribs shown in Fig. 6.5, but now the thickness of the plate is 2 21 times less, i.e., 0.0020 m. This causes the 9 lowest eigenfrequencies to become very close as their corresponding eigenmodes can be regarded as local modes for each of the nine subdomains of the plate, see Figs. 6.16 - 6.25.
Chapter 6. Examples of Design Sensitivity Analysis
109
Figure 6.16: 1. mode. f1 = 70.40 Hz.
Figure 6.24: 9. mode. f9 = 72.32 Hz.
Figure 6.18: 3. mode. f3 = 72.17 Hz.
Figure 6.17: 2. mode. f2 = 72.17 Hz.
Figure 6.20: 5. mode. f5 = 72.28 Hz.
Figure 6.19: 4. mode. f4 = 72.25 Hz.
Figure 6.22: 7. mode. f7 = 72.32 Hz.
Figure 6.21: 6. mode. f6 = 72.31 Hz.
110
6.7. Ribbed Plate with Cluster of Eigenfrequencies
Figure 6.23: 8. mode. f8 = 72.32 Hz.
Figure 6.25: 10. mode. f10 = 110.0 Hz.
Chapter 6. Examples of Design Sensitivity Analysis
111
The rst eigenfrequency is 70.404 Hz and the second through the ninth eigenfrequency are close to 72.2 Hz. These 9 eigenfrequencies are so close that it is dicult to decide which of them are multiple. If we use as a criterion for identical eigenfrequencies that the relative di erence between the frequencies must be 10 3 then the 8 eigenfrequencies from the second to the ninth should be considered as multiple, but if we use a tighter criterion as 10 4 then the second and third should be considered as a double eigenfrequency and the sixth, seventh, eighth and ninth should be considered as a 4-fold multiple eigenfrequency. If the criterion 10 5 is used the second and third should be considered as a double eigenfrequency, and similarly with the seventh and eighth. We will determine sensitivities of the eigenfrequencies when the position of the horizontal ribs is changed as shown in Fig 6.12. The sensitivities are calculated (a) by the OFD method which is used as reference, (b) using Eq. 4.29 and 4.40 and assuming two double eigenfrequencies, (c) using Eq. 4.29 and 4.40 and assuming one double and one 4-fold multiple eigenfrequency, (d) using Eq. 4.29 and 4.40 and assuming one 8-fold multiple eigenfrequency, and (e) only using Eq. 4.29 which is just valid in cases of simple eigenvalues. The results are shown in Table 6.12. Table 6.12: Eigenfrequency sensitivities with respect to design variable no. 3: position of horizontal ribs. Freq. OFD Eq. 4.29 & 4.40, Eq. 4.29 & 4.40, Eq. 4.29 & 4.40 and Eq. 4.29 no. method f2 = f3 and f2 = f3 and f2 = f3 = f4 = f5 f7 = f8 f6 = f7 = f8 = f9 = f6 = f7 = f8 = f9 fj @fj j j j j a5 @ai 1 2 3 4 5 6 7 8 9 10
203:2 257:1 143:5 71:65 116:8 143:8 114:4 143:8 123:5 13:60
203:2 257:1 143:5 71:65 116:8 143:7 114:4 143:8 123:5 13:60
203:2 257:1 143:5 71:65 116:8 143:7 114:4 143:8 123:5 13:60
203:2 286:5 143:5 78:36 287:1 143:7 143:7 143:8 143:8 13:60
203:2 256:8 143:1 71:65 116:8 143:7 127:8 130:3 123:5 13:60
It is seen that we obtain the same results by the S-A method assuming either two double eigenfrequencies f2 = f3 and f7 = f8, or one double eigenfrequency f2 = f3 and one 4-fold multiple f6 = f7 = f8 = f9. These two columns are in excellent agreement with the OFD
112
6.8. Vibration Frequencies of Thin Clamped Square Plate
method but it is not a general situation that di erent choices of multiplicity lead to the same results. This is also illustrated by the next column in Table 6.12 because if we assume having an 8-fold multiple eigenfrequency f2 = f3 = f4 = f5 = f6 = f7 = f8 = f9, wrong sensitivities are obtained for several of the eigenfrequencies, i.e., for f2, f4, f5 , f7 , and f9. At last, if we consider all eigenfrequencies as simple, i.e., use Eq. 4.29, then the results for the multiple eigenfrequencies f2 = f3 and f7 = f8 only di er slightly from the results obtained by the OFD method while the other sensitivities are correct. This illustrates that the in uence of the o -diagonal terms in the sensitivity matrix in Eq. 4.40 is quite small for this design change. This is also the reason why the same sensitivities are obtained for f6 , f7, f8 , and f9 using Eq. 4.40 independently of whether the assumption f7 = f8 or the assumption f6 = f7 = f8 = f9 is used. In both cases, the o -diagonal terms in the sensitivity matrix in Eq. 4.40 are small compared to the diagonal terms for this design change, and therefore the same sensitivities are obtained for both assumptions. These results show the importance for the design sensitivity analysis of deciding correctly whether an eigenfrequency is multiple or simple. Thus, wrong assumptions concerning the multiplicity of an eigenfrequency may lead to erroneous results. The importance of deciding correctly the multiplicity of a multiple eigenvalue can be seen from the sensitivity matrix in Eqs. 4.40 and 4.46. If a wrong multiplicity N is assumed, a subeigenvalue problem of wrong dimension is posed, and if the o -diagonal terms fskT e; s 6= k, in the sensitivity matrix are non-zero, the eigenvalues of the sensitivity matrix, i.e., the directional derivatives, in general will be incorrect. A structural design with a dense spectrum of eigenfrequencies often occur for plate and shell structures, and special care must be taken in the sensitivity analysis as observed in this example.
6.8 Vibration Frequencies of Thin Clamped Square Plate This example deals with the subject of semi-analytical design sensitivity analysis of vibration frequencies of a thin clamped square plate, and it will be shown that application of the traditional S-A method can result in severe errors. The example concerns a square plate which is clamped at all edges and modelled by 100 9-node \heterosis" isoparametric Mindlin plate elements, see Fig. 6.26.
L
L
Figure 6.26: Finite element model of square plate.
Chapter 6. Examples of Design Sensitivity Analysis The plate is made of steel with the following material properties Young's modulus : 210000 MPa Poisson's ratio : 0.3 Mass density : 7800 Kg/m3
113
114
6.8. Vibration Frequencies of Thin Clamped Square Plate
The side length L of the square plate is 1:0 m and the thickness 1:0 10 3 m, i.e. the length/thickness ratio is 1000. The lowest eigenfrequency f1 = 8.994 Hz is simple, while f2 = f3 =18:36 Hz is a double. The shape design variable is the side length L of the plate, and the sensitivities of the three lowest eigenfrequencies will be computed for di erent perturbations. The overall nite di erence (OFD) method, see Eq. 6.1, is used as a reference method, and furthermore, the eigenfrequency sensitivities will be calculated by the traditional S-A method, cf. Eqs. 4.7 and 4.8, and by the new semi-analytical method using \exact" numerical di erentiation of element matrices. Table 6.13: Computed derivative of lowest eigenfrequency f1 with respect to side length L. Plate length perturbation L L 10 10 10 10 10 10 10 10 10
2 3 4 5 6 7 8 9 10
OFD method
Traditional S-A method
New S-A method
f1 L
f1 L
f1 L
17:71 17:96 17:99 17:99 17:99 17:99 18:00 17:97 18:13
12188: 1308: 115:7 4:605 16:65 17:86 17:98 17:99 17:97
17:99 17:99 17:99 17:99 17:99 17:99 17:99 17:99 17:99
The data presented in Table 6.13 are the sensitivities of the lowest eigenfrequency while Table 6.14 presents sensitivities of the multiple eigenfrequency. As f2 and f3 remain multiple with the design change considered, the same values are obtained for sensitivities 2 and 3, and therefore only results for 2 are listed in Table 6.14. The results in Table 6.13 and 6.14 con rm that the traditional S-A method is strongly dependent on the size of the relative perturbation, and that even the sign of the sensitivity is wrong unless a very small perturbation is used. The new S-A method, however, yields accurate sensitivities. The OFD method is used as a reference, and it should be noted that the inaccuracies of the OFD method for the perturbations 10 1 and 10 2 are due to these large values. Furthermore, the OFD method becomes inaccurate for very small perturbations due to numerical round-o errors. The traditional S-A method for static design sensitivity analysis is prone to large errors for problems involving large rigid body rotations relative to actual deformations of the nite
Chapter 6. Examples of Design Sensitivity Analysis
115
Table 6.14: Computed sensitivities 2(=3) of double frequency f2 = f3 with respect to side length L. Plate length perturbation L L 10 10 10 10 10 10 10 10 10
2 3 4 5 6 7 8 9 10
OFD method
Traditional S-A method
New S-A method
2
2
2
36:10 36:63 36:71 36:71 36:71 36:72 36:72 36:80 36:19
19415: 2074: 176:2 15:40 34:58 36:50 36:69 36:71 36:70
36:71 36:71 36:71 36:71 36:71 36:71 36:71 36:71 36:71
elements as the components of the approximate element pseudo loads that correspond to to a rigid body rotation dr do not vanish in general, i.e., (k=ai )dr 6= 0 as described in Section 5.2. As the design sensitivity expressions for eigenvalues involve multiplication of sti ness matrix derivatives by the eigenvector, see Eqs. 4.29 and 4.40, the same type of inaccuracy problem will arise when the S-A method is applied to design sensitivity analysis of eigenvalues for such structures. In the current example, the small length/thickness ratio will result in eigenvector displacement elds where the rigid body rotations are comparatively large relative to the actual deformations of the nite elements, and therefore the traditional S-A method yields completely wrong results unless very small perturbations are used.
6.9 Eigenfrequencies of Vibrating Square Plate with In-plane Loads The next example concerns design sensitivity analysis of a square plate which is clamped at one edge and simply supported at two opposite edges. The plate is subjected to a uniformly distributed in-plane load of magnitude 200 N/m at the fourth edge. The nite element model consists of 100 9-node \heterosis" isoparametric Mindlin plate elements, see Fig. 6.27. The plate has the same dimensions and material properties as the plate in the previous example in Section 6.8.
116
6.9. Eigenfrequencies of Vibrating Square Plate with In-plane Loads sim
ply
d
pe clam
sim
ply
L
supp
supp
orte
d
orte
200
d
N/m L
Figure 6.27: Finite element model of square plate with in-plane load. Table 6.15: Computed derivative of lowest eigenfrequency f1 with respect to side length L for plate without in-plane load. Plate length/thickness ratio is 1000. Plate length perturbation L L 10 10 10 10 10 10 10 10 10
2 3 4 5 6 7 8 9 10
OFD method
Traditional S-A method
New S-A method
f1 L
f1 L
f1 L
6:231 6:315 6:324 6:325 6:325 6:320 6:271 4:948 8:880
73253: 7951: 796:3 7:401 1:709 5:521 6:184 5:804 6:837
6:325 6:325 6:325 6:325 6:325 6:325 6:325 6:325 6:325
For comparison purposes it is rst assumed that the in-plane load is absent and then the lowest frequency is obtained as f1 = 3.167 Hz. The shape design variable is again taken to be the side length L of the plate, and the sensitivities of the lowest eigenfrequency are given in Table 6.15. It is seen from Table 6.15 that there is a good agreement between the results obtained by the OFD method and the new S-A method for the perturbations 10 4 - 10 6, but for smaller values of the perturbation the OFD method starts to diverge due to numerical round-o errors. The traditional S-A method starts to diverge for perturbations smaller than 10 8 and does not reach the same sensitivity value as the two other methods. The errors in the sensitivities obtained by the traditional S-A method are seen to be larger for this example than for the previous one. This is to be expected as the rigid body
Chapter 6. Examples of Design Sensitivity Analysis
117
rotations relative to actual deformations of the nite elements are larger for this example due to the boundary conditions considered. Now, consider the case where the plate is subjected to the in-plane load. This implies a lowering of the rst eigenfrequency from 3.167 Hz to 2.690 Hz. Again design sensitivities are calculated, and the results are given in Table 6.16. Table 6.16: Computed derivative of lowest eigenfrequency f1 with respect to side length L for plate with in-plane load = 200 N/m. Plate length/thickness ratio is 1000. Plate length perturbation L L 10 10 10 10 10 10 10 10 10
2 3 4 5 6 7 8 9 10
OFD method
Traditional S-A method
New S-A method
f1 L
f1 L
f1 L
6:511 6:583 6:597 6:598 6:598 6:570 6:322 5:477 0:384
103053: 11180: 1122: 106:4 4:714 5:451 6:383 5:847 7:325
6:587 6:587 6:587 6:587 6:587 6:587 6:587 6:587 6:587
This example emphasizes the importance of using our modi ed version of the S-A method. The new S-A method is independent of the perturbation used, whereas a comparison of the results obtained by the OFD method and the traditional S-A method for this example re ects diculties in obtaining reliable sensitivities. This is due to both comparatively larger rigid body rotations relative to actual deformations of the nite elements, and the necessity of performing two design sensitivity analyses. First the stress sensitivities are determined in a static sensitivity analysis and then the eigenfrequency sensitivities are calculated including the initial stress sti ening e ects. Inaccuracies in the static sensitivity analysis hereby become accumulated in the dynamic sensitivity analysis and this is the reason why there are small di erences in the results obtained by the OFD method and the new S-A method. The OFD method is based on rst order forward nite di erences, and as the problem is strongly non-linear this gives small inaccuracies although small perturbations are used. Finally, the same example is considered but now the thickness is increased to 20:0 10 3 m, i.e. the length/thickness ratio is 50, and the in-plane load is set to 1:5 106 N/m. Without the in-plane load the lowest eigenfrequency becomes f1 = 62.97 Hz, and with the in-plane load included, the lowest eigenfrequency f1 is reduced to 53.93 Hz. The shape
118
6.10. Concluding Remarks on Examples
design variable is still taken to be the side length L of the plate and the design sensitivities obtained are given in Table 6.17. Table 6.17: Computed derivative of lowest eigenfrequency f1 with respect to side length L for plate with in-plane load = 1:5 106 N/m. Plate length/thickness ratio is 50. Plate length perturbation L L 10 10 10 10 10 10 10 10 10
2 3 4 5 6 7 8 9 10
OFD method
Traditional S-A method
New S-A method
f1 L
f1 L
f1 L
127:9 129:5 129:6 129:7 129:7 129:7 129:7 129:7 131:1
5024: 430:0 72:93 123:7 128:8 129:3 129:4 129:4 129:5
129:5 129:5 129:5 129:5 129:5 129:5 129:5 129:5 129:5
For this example, there is seen to be good agreement between the OFD method and the new S-A method. The traditional S-A method is seen to converge to the same value as obtained by the other two methods but it is still dependent of the chosen value of the perturbation. The sensitivities obtained by the traditional S-A method are much better for this example than for the previous one, because the length/thickness ratio has been decreased. This results in smaller rigid body rotations relative to actual deformations of the nite elements, and therefore the errors associated with the fact that (k=aj )dr 6= 0 are reduced.
6.10 Concluding Remarks on Examples Several numerical examples of design sensitivity analysis have been presented in this chapter. The traditional semi-analytical method of design sensitivity analysis has been shown to give very accurate results in cases where the displacement eld is characterized by small rigid body rotations relative to actual deformations of the nite elements, cf. Section 6.2 where a classical llet example is studied. However, in cases where the displacement eld entails dominance of rigid body rotations relative to actual deformations of the nite elements, the traditional semi-analytical method
Chapter 6. Examples of Design Sensitivity Analysis
119
has been shown to be prone to yield erroneous sensitivities. This is demonstrated for static design sensitivity analysis of a long cantilever beam in Section 6.3 and of a thin clamped square plate in Section 6.5. Furthermore, the inaccuracy problem has been demonstrated to manifest itself in thermo-elastic design sensitivity analysis of a two-material cantilever beam in Section 6.4. Thus, the inaccuracy problem can also be expected in many thermoelastic design problems as displacement elds with dominance of rigid body rotations often occur in such problems. Design sensitivity analysis of simple as well as multiple eigenfrequencies of reinforced vibrating plates has been shown to give accurate results in Sections 6.6 and 6.7. One of the major diculties in computing sensitivities of multiple eigenvalues is to decide correctly from numerical results the multiplicity of a given eigenvalue in case of a dense spectrum of eigenvalues. A wrong estimation of the multiplicity of a repeated eigenvalue is shown to give erroneous sensitivities in Section 6.7 and this diculty associated with multiple eigenvalues, in addition to the lack of usual Frechet di erentiability, has to be taken into account when solving optimum design problems involving multiple eigenvalues. These questions will be further discussed in Chapter 8. Finally, the inaccuracy problem associated with the traditional semi-analytical method is shown to occur also in dynamic design sensitivity analyses. This is illustrated by computation of design sensitivies of simple as well as multiple free vibration frequencies of a thin clamped square plate in Section 6.8. Furthermore, if initial stress sti ening e ects are taken into account when computing eigenfrequencies of vibrating plates, in Section 6.9 the error problem is shown to be even worse because errors in stress sensitivities from the static design sensitivity analysis become accumulated in the dynamic design sensitivity analysis. The new approach to semi-analytical design sensitivity analysis based on \exact" numerical di erentiation of element matrices has been shown to yield accurate sensitivities for all studies made and must be considered a very reliable tool in a general purpose computer aided engineering design system.
120
6.10. Concluding Remarks on Examples
Chapter A General and Flexible Method of Problem De nition
7
7.1 Introduction
the implementation of a general and exible method of forT mulating problems of mathematical programming in structural optimization systems. his chapter describes
The method enables the formulation and solution of problems involving local, integral, min/max, max/min and possibly non-di erentiable user de ned functions in any conceivable mix. The mathematical formulation is based on the so-called bound formulation, and the implementation speci c details involve a parser capable of interpreting and performing symbolic di erentiation of the user de ned functions. The basic data involved in the user de ned functions are available through a class of database operations and many di erent mathematical operators and functions are implemented. The problem of mathematical programming consists in determining optimum values of the design variables such that they maximize or minimize a speci c function termed the objective function, while satisfying a set of geometrical and/or behavioural requirements which are called constraint functions. In Section 7.2 the traditional way of formulating problems of mathematical programming in general structural optimization systems is described. Usually, only standard formulations, such as to minimize the maximum stress with a volume constraint, can be chosen, but in practical design cases much more complicated problem de nitions may be necessary. The mathematical programming problems to be solved can involve criteria of various nature, e.g., integral type functions, max/min or min/max type functions, or local type functions like stress at a given point. A very general and elegant way of formulating and solving such mathematical programming problems is to make use of the bound formulation which is described in Section 7.3. Having formulated the problem of mathematical programming in a standard form using the bound formulation where the objective and constraint functions are given as vector 121
122
7.2. Background
functions, a description of how the user can de ne these vector functions is given. We have realized that the only way to obtain a exible system for de nition of structural optimization problems is to allow the user to de ne objective and constraint functions in a homemade language consisting of a set of operators and functions. Thus, a parser interprets the de nition of the user de ned objective and constraint functions as described in Section 7.4. This results in some database operations as described in Section 7.5 where basic data are read. The relations between these basic data must be de ned by some mathematical operators and functions described in Section 7.6. Having obtained the necessary data and their relations, the vector functions can be evaluated as described in Section 7.7 and symbolic di erentiation is easily implemented in each evaluation step. In this way analytical expressions for derivatives of vector functions are available. When all vector functions have been evaluated, the mathematical programming problem can be established and solved, resulting in improved values of the design variables as described in Section 7.8. The approach described in this chapter is valid for all types of optimization problems that can be solved by ODESSY, except the case of having multiple eigenvalues in the optimization problem. This special case is covered in Chapter 8. Documentation of this approach of de ning problems of mathematical programming in structural optimization systems can also be found in Lund & Rasmussen (1994).
7.2 Background In this section the traditional way of formulating problems of mathematical programming in general structural optimization systems is described. Some of the rst available general structural optimization systems, for instance OASIS, see Esping (1986), SAMCEF, see Fleury (1987) and Braibant & Morelle (1990), and CAOS, see Rasmussen (1989, 1990) and Rasmussen, Lund, Birker & Olho (1993) provide the user with a choice of a number of prede ned functions from which the optimization problem can be built, for instance, weight, de ection, stress, eigenfrequency, and compliance. Functions like these are made available by equipping the system with a module which can nd this information in the database of analysis results. Given the options above, the user may choose to formulate the problem as, for instance,
Minimize Subject to
maximum stress weight some upper limit lowest eigenfrequency some lower limit
or alternatively or any other combination of the functions. We see that the possible problem formulations of such a system are of very di erent types. Weigth and compliance are integral type functions, de ection or stress at a given point are local type functions, and maximum stress
Chapter 7. A General and Flexible Method of Problem De nition Minimize Subject to
123
weight maximum stress some upper limit de ection at a given point some upper limit compliance some upper limit
or minimum eigenfrequency are min/max or max/min type criteria which are inherently non-di erentiable. In addition to the mathematical diculties created by the di erent natures of these criteria, it is a programming diculty to make the system cope with any possible combination that the user may de ne. A very general and elegant solution is to make use of the so-called bound formulation which will be brie y described in Section 7.3. A system that provides a wide selection of basic functions, possibly more than the abovementioned, and enables the user to freely combine them into optimization problems is a very general tool for structural optimization. However, by developing and using such systems extensively for some years to solve industrial problems, we have learned that it is impossible for a system designer to foresee the necessary facilities for all types of problems that may arise in practical design cases. The universe of relevant design problems is too large to be covered by any practical nite set of criterion functions. Some straight-forward examples of unusual problem de nitions have been
cases where some mathematical combination of di erent stress types at dif-
ferent locations in the structure has been established as the design criterion through a long time of numerical and experimental investigations and perhaps practical use. Industries will typically rather abstain from the use of structural optimization than being forced into changing their well-tested design criteria just because of a limitation in the software system.
cases of multicriterion optimization in the form of using a weighted sum of di erent functions as one of the criteria of the problem.
cases of unusual material failure criteria in the form of expressions involving, for instance, di erent stress or strain types.
cases where manufacturing constraints must be included in the de nition of the optimization problem.
System developers have the possibility to taylor the system by continuously reprogramming it to t the problem at hand but this is not acceptable in the long run. Therefore we initiated a more fundamental solution of the problem of generality.
124
7.3. Bound Formulation
7.3 Bound Formulation In this section the bound formulation which is used as a basis for formulating and solving problems of mathematical programming in ODESSY is described. In order to keep an overview of a large system like ODESSY, it is absolutely necessary to maintain a rigorous modularity in the organization of the code and have the modules perform the necessary exchange of information through well-de ned interfaces. In relation to optimizers, this problem is well-known and has been addressed by Vanderplaats (1987) in connection with his ADS subroutine package. The optimization concept of ODESSY is based on the formation and solution of a sequence of explicit subproblems as opposed to systems that work with line-search oriented optimizers. From a theoretical point of view, the subproblem approach is known to be less stable than line-search algorithms and it is often dicult or impossible to prove convergence properties for the algorithm. However, from a system point of view, the subproblem approach is much easier to program and control because it is realized via a xed sequence of calculations independently of the iteration history. The subproblem approach works reliably in most cases and has gained a signi cant popularity among developers of structural optimization systems. In the formulation of problems of mathematical programming only rst order approximations are used, i.e., we do not require our analysis module to provide more than rst order sensitivities because higher order derivatives are computationally very costly to compute. This does not prevent the use of, for instance, a quasi-Newton approach in which second order approximations are gradually formed based on rst order information in several iterations. However, currently ODESSY only contains two optimizers, namely a sequential linear programming algorithm (SIMPLEX) and an implementation of the MMA method by Svanberg (1987), see also Fleury & Braibant (1986). The usual mathematical programming formulation of a structural optimization problem is as follows: Minimize g0(a)
a
Subject to
(7.1)
gj (a) Gj ; j = 1; : : : ; J ai ai ai; i = 1; : : : ; I where I is the number of so-called design variables, ai, and J is the number of constraints other than side constraints. Maximization problems are easily included in this form by simply minimizing g0(a). We notice that non-di erentiable functions occur very often in connection with practical structural optimization problems as the result of min/max or max/min criteria. The objective and constraint functions gj ; j = 0; : : : ; J , are speci ed by the user as a part of the problem formulation. These vector functions can be picked and combined freely
Chapter 7. A General and Flexible Method of Problem De nition
125
among all the analysis results that the system is able to evaluate as will be described in the next section. In the interest of simplicity, generality and, above all, ease of programming, a formulation that enables the system to handle the optimization problem in a uniform way regardless of the blend of local-, integral- and min/max-criteria is very much desired and can be obtained by use of the so-called bound formulation, see Bends e, Olho & Taylor (1983), Taylor & Bends e (1984), and Olho (1989). This technique very elegantly solves the problem of generality, and, assuming an adequate sensitivity analysis scheme, it also provides a simple solution to the non-di erentiability problem in connection with min/max and max/min problems. It is assumed that any function gj in the problem is actually a max over a set of functions (the formulation for max/min problems is completely equivalent), such that
gj (a) =
max gjk (a); j = 0; : : : ; J
k=1;:::;pj
(7.2)
The counters pj ; j = 0; : : : ; J , designate the number of functions among which gj is given as the max. In the case of, for instance, a max stress criterion, pj would be the number of nodal stresses among which the maximum is to be found. Ordinary scalar functions are just a special case signi ed by the corresponding pj = 1. This gives us the following form of 7.1: Minimize max g0k (a);
a
k=1;:::;p0
Subject to
max gjk (a) Gj ; j = 1; : : : ; J
(7.3)
k=1;:::;pj
ai ai ai ; i = 1; : : : ; I
The constraint functions gj = max(gjk ), j = 1; : : : ; J , do not cause di erentiability problems. They may simply be treated separately:
Minimize a Subject to
k=1;:::;p0
max g0k (a);
gjk (a) Gj ; k = 1; : : : ; pj ; j = 1; : : : ; J ai ai ai ; i = 1; : : : ; I
(7.4)
This leads to the idea of using a similar approach for the objective function g0 = max(g0k ). In order to make this possible, a new, arti cial design variable is introduced and a new, arti cial objective function f ( ). It is now possible to formulate an equivalent problem in which the previous non-di erentiable objective function plays the role of constraints:
Minimize
f ( )
126
7.3. Bound Formulation a; Subject to
g0k (a) f ( ); k = 1; : : : ; p0 ; gjk (a) Gj ; k = 1; : : : ; pj ; j = 1; : : : ; J ai ai ai; i = 1; : : : ; I
(7.5)
In this formulation, the only way of reducing the value of the arti cial objective function f ( ) while ful lling the constraints, is to simultaneously reduce all the functions comprised in the original objective, i.e., minimizing the maximum. The arti cial objective function f ( ) acts as a bound which suppresses the value of the original objective and gives this technique the name of \bound formulation". The speci c form of the arti cial objective function f ( ) has yet to be selected. If a linear method of mathematical programming is used, there is no reason not to select the simplest imaginable function of : f ( ) = (7.6) Next 7.6 is inserted in 7.5, and, in order to better cope with functions of varying nature and scale, we perform a normalization of the problem:
Minimize a; Subject to
g0k (a) 1; k = 1; : : : ; p ; 0 gjk (a) 1; k = 1; : : : ; p ; j = 1; : : : ; J j Gj ai ai ai; i = 1; : : : ; I
(7.7)
As previously mentioned, the counters pj ; j = 0; : : : ; J , designate the number of functions over which the max for criterion j is to be found. In the case of, for instance, minimization of maximum nodal stress, pj may often be many thousands, and the number of functions in the optimization becomes very large. In order to prevent this, an \active set strategy" can be used, which only includes the nodal stresses exceeding a certain fraction of the current maximum in the problem. Tableau 7.7 is valid regardless of the blend of vector functions gj ; j = 0; : : : ; J , and the numerical operations performed are therefore identical for any problem that the user could possibly de ne. Due to this standardization, the bound formulation has greatly simpli ed the programming of the optimization module. Having obtained a standard format for the mathematical program, a method for the user to de ne the problem and a database module for extracting vector functions gj ; j = 0; : : : ; J , are required.
Chapter 7. A General and Flexible Method of Problem De nition
127
7.4 Specifying the Problem Solving many real-life structural design optimization problems has indicated that the only way to cover all possible combinations of problem speci cations is to allow the user to specify the functions of the problem as mathematical relations between a set of basic data available in the system. Furthermore, the system must be designed such that it is relatively simple to expand the set of basic information as new analysis facilities are added to the system. The same is valid for the available mathematical expressions. The basic data of the system are available through a class of database operations that extract the required information from the ODESSY analysis result les. These functions are very similar to those used in conventional systems to extract the values and sensitivities of the objective and constraint functions. Examples of calls of the database functions could be: nstress(svm, mat=steel, ldc=1)
which returns the nodal von Mises stresses in material \steel" in load case 1, or weight( )
which returns the overall weight of the structure. The mathematical operations are available as the usual mathematical operators, for instance addition, subtraction, multiplication and division. In addition to these, a set of prede ned functions are available, for instance logarithm, trigonometric functions, maximum and minimum. We can imagine a criterion function speci ed as maxval [ 1.5[ nstress(svm, mat=steel) ] + 1.2[ nstress(st, mat=aluminium) ] ]
which would specify the maximum value of a vector containing a weighted sum of von Mises stresses in material \steel" and Tresca stresses in material \aluminium". Please notice the use of brackets as parantheses except in the case of arguments to database funtions. We shall return to this syntactic peculiarity later.
7.5 Database Operations The database operations are speci ed as function calls. Each function extracts a speci c type of information from the database, for instance nodal stresses, element strains, compliance or volume. The obtained vector functions can be divided into the three categories: nodal vector functions, element vector functions, and global vector functions. The database functions available in the ODESSY system are listed in the following and possible input speci cations to each function are also given.
128
7.5. Database Operations Table 7.1: Database operations for nodal vector functions with arguments.
Description
Name
Nodal stresses Nodal strains
nstress nstrain
Displacements
disp
Nodal forces
force
Nodal temperatures Nodal thermal forces Nodal coordinates Nodal radii Boundary radius of curvature
temp
Speci c Speci c Load Absolute Speci c Speci c Additional type material case value layer point options yes yes yes yes yes yes yes yes yes yes yes yes Absolute yes yes yes yes yes displacement yes yes yes yes yes Absolute force yes yes yes yes
tempforce
yes
yes
yes
yes
-
yes
x,y,z
-
-
-
yes
-
yes
rad
-
-
-
yes
-
yes
curvature
-
-
-
yes
-
no
Table 7.2: Database operations for element vector functions with arguments. Description Element stresses Element strains Element thicknesses
Name
Speci c Speci c Load Absolute Speci c Speci c Additional type material case value layer point options estress yes yes yes yes yes yes estrain
yes
yes
yes
yes
yes
yes
elthk
-
yes
-
-
-
-
The extraction of raw data by database operations is always the rst step in evaluating a user de ned criterion function gj . In order to be able to handle all combinations in a uni ed way, the system has been designed such that all database operations return the same data type, a vector of nodal values, VNV. The components of VNV's are nodal values (may also contain element or global values)
Chapter 7. A General and Flexible Method of Problem De nition
129
Table 7.3: Database operations for global vector functions with arguments. Description
Name
Volume Weight Compliance Mass moment of inertia Area moment of inertia Eigenfrequencies Eigenfrequencies incl. stress sti ening Buckling load factors Probability of failure
vol weight compliance massinertia
Speci c Load material case yes yes yes yes -
Additional options
areainertia
yes
-
eigenfreq
-
yes
Speci c number, include all eigenfrequencies above or below speci ed number
eigenfreqsti
-
yes
Speci c number, include all eigenfrequencies above or below speci ed number
buckload
-
yes
pof
yes
yes
Speci c number, include all buckling load factors above or below speci ed number
characterized by the following data structure (ODESSY is coded in ANSI C): struct ods nv f long nr; long mat; long ldc; double val; double vali; short sign;
g
/* Node or element number */ /* Material number */ /* Load case number */ /* Value */ /* Sensitivity value */ /* Sign of value */
The material number is part of this data structure because a given node may be on the interface between several materials and have a di erent value for each material. VNV's are simply arrays of the structure described above: struct ods vnv f long n nv; /* Number of components in the vector */ short type; /* Type of vector (nodal, element, or global) */ struct ods nv nv[ ]; /* The actual vector of the type struct ods nv */
g
Some of the database functions, for instance the volume calculation, return only a scalar
130
7.6. Operators and Functions
value. In this case there is no evident need for the data structures described above, but the subsequent operations are much simpler if data are represented in a uniform way.
7.6 Operators and Functions The usual way of writing mathematical expressions is by a mix of operators and functions, for instance log(3:2 x + y) where the two operators, * and +, are used together with a logarithm function. The usual mathematical operators and functions are available in ODESSY for operations between random combinations of VNV's and scalar numbers. Furthermore, some additional convenient operators, like concatenation of two VNV's, have been implemented. The currently available operators and functions are listed in tables 7.4 and 7.5. Table 7.4: Implemented operators. Operator: + / *
^ j
max min maxval minval maxvec minvec
\value" \value"
Description: Addition operators. Subtraction operator. Division operator. Multiplication operator. Power function. Operator which concatenates two vectors. Operator which speci es a maximization of the objective function. Operator which speci es a minimization of the objective function. Maximum value operator which nds the maximum value of a vector. Minimum value operator which nds the minimum value of a vector. Operator which compares two vectors and returns a vector consisting of maximum values for each node (or element). Operator which compares two vectors and returns a vector consisting of minimum values for each node (or element). \Top" operator which subtracts all values of a vector exceeding \value" multiplied by the maximum value of the vector. The scalar \value" must be between 0.0 and 1.0. \Bottom" operator which subtracts all values of a vector less than \value" multiplied by the minimum value of the vector. The scalar \value" must be larger than 1.0.
Each operator or function operates on the VNV's de ned in section 7.5 and is thus capable of evaluating either initial values or sensitivity values. For each operator and function, a set of evaluation rules is implemented, e.g., it is not allowed to multiply a VNV of nodal values with a VNV of element values.
Chapter 7. A General and Flexible Method of Problem De nition
131
Table 7.5: Implemented functions. Function: cos sin log
Description: Cosine. Sine. Logarithm base e.
7.7 Evaluation Sequence Each de nition of an objective or constraint function in the mathematical program is interpreted by a so-called parser. A parser is well-known as an important part of any computer programming language compiler. It breaks a mathematical expression into a sequence of simple operations and keeps track of input and output of each operation. The user can control the evaluation sequence by subdividing the expression using brackets. Brackets are used for this purpose to distinguish subdivisions from function arguments. In the absence of brackets, the parser breaks the expression down based on the usual matematical rules of precedence, such that a + b c is evaluated as a + [b c] rather than [a + b] c. The available operators are internally converted into functions, such that the expression results in a sequence of function calls each returning one VNV to be used in subsequent calls. Each function call involves either: 1. A database function de ning a VNV. 2. Call of an operation on one VNV. 3. Call of an operation between two VNV's. This decomposition of the evaluation sequence makes it easy to implement analytical expressions for derivatives of each VNV. Because a maximum of two VNV's and only one basic operation or function is involved in each function call, symbolic di erentiation is easily implemented in each step. To illustrate the evaluation sequence for di erent types of VNV's, we consider the example in Section 9.3 where the shape of a turbine disk is optimized. For this turbine disk the design optimization problem is de ned as where vM;i is von Mises reference stress at node i and Ti is the nodal temperature. The non-linear stress constraint is illustrated on Fig. 9.12 in Section 9.3. In ODESSY, this optimization problem can be de ned as min [ massinertia() ] maxval [ nstress(svm)/ 1.0E6 / [ minvec [ 550; [ 1484.5 - [ 1.5temp()] ] ] ] ] < 1.0 minval [ curvature() 5.0 ] > 0.0050 The stress constraint where the von Mises reference stress at every nodal point i is normalized with a temperature dependent non-linear function de ning the allowable stress
132
7.7. Evaluation Sequence
Minimize Subject to
the mass moment of inertia
8 > vM;i [MPa] > < 550 if Ti 623 K maximum normalized stress > vM;i [MPa] > : 1484 :5 1:5Ti if Ti > 623 K
1
minimum boundary radius of curvature 5 mm
illustrates the generality of this approach of de ning problems of mathematical programming. For these de nitions of objective and constraint functions, we get the following evaluation sequences:
Objective function: VNV1 = massinertia() VNV2 = min(VNV1)
Database function de ning VNV1 as the mass moment. Call of \min" operator with input VNV1 gives nal result.
Constraint function no. 1: VNV1 = 1.5 VNV2 = temp() VNV3 = mult(VNV1,VNV2 ) VNV4 = 1484.5 VNV5 = minus(VNV4,VNV3 ) VNV6 = 550 VNV7 = nstress(svm) VNV8 = 1.0E6 VNV9 = minvec(VNV6 ,VNV5 ) VNV10 = div(VNV7 ,VNV8 )
VNV1 is de ned as the digit 1.5. Database function de nes VNV2 as a vector containing nodal temperatures. Call of multiplication operator between VNV1 and VNV2 . VNV4 is de ned as the digit 1484.5. Call of subtraction operator between VNV4 and VNV3. VNV6 is de ned as the digit 550. Database function de nes VNV7 as a vector containing von Mises stresses for all nodes. VNV8 is de ned as the digit 1:0 106. Call of \minvec" operator between VNV6 and VNV5 returns a vector containing the smallest value of either VNV6 and VNV5 for each node, see Fig. 9.12. Call of division operator between VNV7 and VNV8 gives VNV10 containing von Mises stresses in [MPa].
Chapter 7. A General and Flexible Method of Problem De nition VNV11 = div(VNV10 ,VNV9 )
133
Call of division operator between VNV10 and VNV9 gives VNV11 containing normalized von Mises stresses. Call of \maxval" operator with input VNV11 gives nal result for constraint function no. 1 in VNV12 and de nes the maximum value of VNV12 to be less than the right hand side (= 1.0). VNV12 is just a copy of VNV11 .
VNV12 = maxval(VNV11)
Constraint function no. 2: VNV1 = 5.0 VNV2 = curvature() VNV3 = bottom(VNV2,VNV1 ) VNV4 = minval(VNV3)
VNV1 is de ned as the digit 5.0. Database function de nes VNV2 as a vector containing boundary radius of curvatures for all boundary nodes. Call of \bottom" operator between VNV2 and VNV1 gives VNV3 containing the boundary radii of curvature which are less than the smallest value multiplied by 5.0. Call of \minval" operator copies VNV3 to nal result for constraint function no. 2 in VNV4 and de nes the minimum value of VNV4 to be less than the right hand side (= 0.005).
Each VNV is evaluated for either initial values or sensitivity values, and we see that symbolic di erentiation is very easily incorporated in each function call. For instance, by usual di erentiation rules, the derivative of VNV10 for constraint function no. 1 is given as 7 VNV8 VNV10 = VNV7 VNV8 VNV 2 (VNV8) which is easily evaluated due to the structure of each VNV. 0
0
0
7.8 Using the Database Module This database module for evaluation of objective and constraint functions gj can be used in di erent ways. It is mainly used as a basic part of the iterative solution procedure for multicriterion optimum design which is realized through systematic sequences of redesign and reanalysis. This is illustrated on Fig. 7.1 where a ow diagram for structural optimization with ODESSY is shown. The user can improve the design by using this procedure for structural optimization. In each step of redesign, the available analysis modules in ODESSY compute current values of variables fm of the types listed in Tables 7.1, 7.2, and 7.3. These basic data are used to evaluate current values of the speci ed objective and constraint functions gj . The design sensitivity analysis is carried out for each design variable ai, resulting in derivatives rgj
134
7.8. Using the Database Module Start optimization
Preprocessor updates geometry, boundary conditions and loads
for specified number of iterations
fm
Structural analysis
Database module evaluates objective and constraint functions g j Preprocessor perturbs design variable ai and updates analysis model ∆
Design sensitivity analysis
for each design variable ai
fm
Database module evaluates derivatives g j of objective and constraint functions ai , g j , g j
∆
∆
Optimizer calculates improved values of design variables a i Converged?
no
yes Stop
Figure 7.1: Flow diagram for structural optimization using ODESSY. of objective and constraint functions. The bound formulation, see Eq. 7.7, is then used to set up a problem of mathematical programming which is solved by one of the implemented optimizers, resulting in improved values of the design variables. This iteration process continues until convergence is achieved or the speci ed number of iterations is reached. The database module can also be used in another way as it has been integrated in the module for postprocessing of the results. The user can, while using the postprocessor, make interactive function calls to the database module and have any VNV (of nodal or element values) plotted at the current geometry. Such a facility can be very helpful in the design phase in order to evaluate some well-tested design criteria or to establish new design criteria. Furthermore, it is much easier to check the validity of de nitions of objective and constraint functions by using this interactive graphical facility.
Chapter 7. A General and Flexible Method of Problem De nition
135
7.9 Conclusions In this chapter it has been demonstrated that it is not only possible but also relatively simple to create a structural optimization system that handles problem formulations comprising any mix of local, integral, min/max and max/min functions in a completely uniform way. The mathematical basis of this is the bound formulation. This, in combination with a parser capable of interpreting user de ned expressions, makes it possible to do analysis and sensitivity analysis of user de ned mathematical expressions and even performing the sensitivity analysis by exact and ecient symbolic di erentiation of the expressions. The approach described makes the system very easy to expand with new available mathematical expressions or with new basic information if new analysis facilities are implemented. The integration of the database module in the module for postprocessing has made it very easy to check the validity of de nitions of objective and constraint functions, and, furthermore, has equipped the system with a very useful tool for interactive, graphical visualization of user de ned mathematical expressions for design criteria. This development can be seen as one out of many necessary steps that will make structural optimization facilities applicable and accessible to real-life engineering designers. Problems involving mixed criteria like, for instance, thermo-elastic stresses are dicult to handle by the traditional \trial and error" methods because the consequences of a contemplated design change may be extremely dicult to foresee, and they call very much upon rational design techniques based on optimization.
136
7.9. Conclusions
Chapter Multiple Eigenvalues in Structural Design Problems
8
8.1 Introduction
to the diculties of solving structural optimum design probT lems with multipledevoted eigenvalues. In Section 4.9 expressions for design sensitivity analhis chapter is
ysis of both simple and multiple eigenvalues were derived and it was shown that multiple eigenvalues can only be expected to be directionally di erentiable. In Sections 6.6 and 6.7 these expressions for design sensitivity analysis were used successfully to compute changes of simple and multiple natural transverse vibration frequencies subject to changes of di erent design parameters of sti ener reinforced thin elastic plates. The question now is how to solve optimum design problems with multiple eigenvalues as traditional gradient based optimization algorithms cannot be applied due to the lack of usual Frechet di erentiability of multiple eigenvalues. The diculties of having multiple eigenvalues in structural design problems have been an active research area during the last 15 years and references to some important papers in this area are given in Section 8.2. The problem of solving optimization problems with multiple eigenvalues is then illustrated by a simple example involving 2x2 matrices depending on two design variables. Next the problem of optimization is formulated as maximization of the smallest (simple or multiple) eigenvalue subject to a constraint of given volume of material of the structure in Section 8.4. For such an optimization problem necessary optimality conditions are derived for arbitrary multiplicity of the smallest eigenvalue. The necessary optimality conditions express (I) linear dependence of a set of generalized gradient vectors of the multiple eigenvalue and the gradient vector of the constraint, and (II) positive semi-de niteness of a matrix of the coecients of the linear combination. The main advantage of these necessary optimality conditions is, when compared to those obtained by other researchers, that they do not contain variations of design variables. 137
138
8.2. Background
In Section 8.5 iterative numerical algorithms for eigenvalue optimization problems involving multiple eigenvalues are presented. The basic idea of the approaches described is to add constraints on the allowable design changes in case of multiple eigenvalues, whereby design sensitivity expressions for simple as well as multiple eigenvalues become identical. It is shown how the derived necessary optimality conditions can be applied for development of an iterative numerical method for optimization of structural eigenvalues of arbitrary multiplicity. Furthermore, an e ective mathematical programming approach for solving optimization problems is described. The mathematical programming approach is used to solve the problem of maximizing the lowest eigenvalue of a sti ener reinforced thin elastic plate in Section 8.6. Most of the results presented in this chapter originate from a joint work on multiple eigenvalues in structural optimization problems between Alexander P. Seyranian, Moscow State Lomonosov University, Niels Olho and myself. This joined work can be found in Seyranian, Lund & Olho (1994).
8.2 Background Multiple eigenvalues in the form of buckling loads and natural frequencies of vibration very often occur in complex structures that depend on many design parameters and have many degrees of freedom. For example, sti ener reinforced thin-walled plate and shell structures have a dense spectrum of eigenvalues, and multiple eigenvalues are found very often as illustrated in Sections 6.6 and 6.7. Also, symmetry of structural systems may lead to apperance of several linearly independent buckling modes and vibration modes with multiple eigenvalues. In 1977, Olho & Rasmussen discovered that the optimum buckling load of a clampedclamped column of given volume is bimodal. This optimization problem was rst considered by Lagrange, and its interesting history is presented in a recent paper by Cox (1992). Olho & Rasmussen (1977) showed that the bimodality of the optimum eigenvalue must be taken into account in the mathematical formulation of the problem in order to obtain the correct optimum solution. They rst demonstrated that an analytical solution obtained earlier by Tadjbakhsh & Keller (1962) under the tacit assumption of a simple buckling load is not optimal, then presented a bimodal formulation of the problem, solved it numerically, and obtained the correct optimum design. The optimum bimodal buckling load obtained was later con rmed to be correct to within a slight deviation of the sixth digit by analytical solutions obtained independently by Seyranian (1983, 1984) and Masur (1984). The discovery in 1977 of multiple optimum eigenvalues in structural optimization problems, and the necessity of applying a bi- or multimodal formulation in such cases, opened a new eld for theoretical investigations and development of methods of numerical analysis and solution. Prager & Prager (1979), Choi & Haug (1981), and Haug & Choi (1982) presented unimodal and bimodal optimum solutions for systems with few degrees of freedom con rming apperance of multiple eigenvalues in optimization problems. A wealth of references on mul-
Chapter 8. Multiple Eigenvalues in Structural Design Problems
139
timodal optimization problems and speci c results for columns, arches, plates and shells can be found in comprehensive surveys by Olho & Taylor (1983), Gajewski & Zyczkowski (1988), Zyczkowski (1989) and Gajewski (1990). A survey of other problems of optimum design with respect to structural eigenvalues was earlier published by Olho (1980), see also Olho (1981a, 1981b). One of the main problems related to multiple eigenvalues is their non-di erentiability in the common (Frechet) sense. This was revealed by Masur & Mroz (1979, 1980) and Haug & Rousselet (1980b). The non-di erentiability creates diculties in nding sensitivities of multiple eigenvalues with respect to design changes and derivation of necessary optimality conditions in optimization problems. Choi & Haug (1981) used a Lagrange multiplier method for bimodal problems and showed that this method which is very useful for di erentiable criteria and constraints may yield incorrect results. Haug & Rousselet (1980b) proved existence of directional derivatives of multiple eigenvalues and obtained explicit formulas for derivatives. Bratus & Seyranian (1983) and Seyranian (1987) presented sensitivity analysis of multiple eigenvalues based on a perturbation technique and derived necessary optimality conditions. The main advantage of these necessary optimality conditions is, when compared with those obtained by previous researchers, that they do not contain variations of design variables. Similar developments were presented by Masur (1984, 1985). It was with the use of these necessary optimality conditions Seyranian (1983, 1984) and Masur (1984) independently of each other obtained the analytical solution to the bimodal optimum clamped-clamped column problem mentioned above. Overton (1988) considered minimization of the maximum eigenvalue of a symmetric matrix. This problem is similar to the problem of maximizing the minimum eigenvalue. Derivation of necessary optimality conditions using the bound formulation of such problems had earlier been presented by Bends e, Olho & Taylor (1983) and Taylor & Bends e (1984). In a recent paper Cox & Overton (1992) presented new mathematical results for optimization problems of columns against buckling. They derived necessary optimality conditions using advanced nonsmooth optimization methods. Their results for optimum columns are in good agreement with the results obtained earlier by Olho & Rasmussen (1977), Seyranian (1983, 1984), and Masur (1984). Numerical algorithms for solution of structural optimization problems with multiple eigenvalues have been suggested and discussed by, among others, Olho & Rasmussen (1977), Choi, Haug & Lam (1982), Choi, Haug & Seong (1983), Olho & Plaut (1983), Bends e, Olho & Taylor (1983), Myslinski & Sokolowski (1985), Zhong & Cheng (1986), Plaut, Johnson & Olho (1986), Gajewski & Zyczkowski (1988), Overton (1988), and Cox & Overton (1992).
8.3 Illustrative Optimization Problem for Eigenvalues In order to illustrate the diculties in solving eigenvalue optimization problems with multiple eigenvalues, let us consider a simple illustrative example. The example involves 2x2
140
8.3. Illustrative Optimization Problem for Eigenvalues
matrices depending on two design variables x and y, and design sensitivities of multiple eigenvalues are calculated using the expressions given in Section 4.9. The example is taken from Seyranian, Lund & Olho (1994) where other simple illustrative examples can be seen. The example is described by the following K and M matrices
"
#
K = 1 +y x 1 y x ;
"
M = 10 01
#
(8.1)
The characteristic equation for this system is
2 2 + 1 x2 y2 = 0
(8.2)
and
q 1;2 = 1 x2 + y2 (8.3) The level curves = c, where c is a constant, for this function are described by the equation q c = 1 x2 + y2 (8.4) and
x2 + y2 = (c 1)2 This surface is a circular cone, see Fig. 8.1.
(8.5)
Figure 8.1: Circular cone surface for eigenvalue . Bimodality occurs at x = 0; y = 0, for which we have 1 = 2 = 1. It can be seen immediately from Eq. 8.3 and Fig. 8.1 that the eigenvalues are not di erentiable at the bimodal point in the usual (Frechet) sense. Indeed @1;2 = p x ; @1;2 = p y (8.6) @x @y x2 + y2 x2 + y2
Chapter 8. Multiple Eigenvalues in Structural Design Problems
141
As x ! 0 and y ! 0 the two right-hand side expressions become unde ned and have no limit. Furthermore, L'H^opital's Rule cannot help either because the derivatives of the denominators also tend to zero. Now we proceed to sensitivity analysis of the double eigenvalue ~ = 1 at x = 0; y = 0, using the results of Section 4.9.2 in Chapter 4. Taking the direction "e we have
p
x = "e1; y = "e2; e1 + e2 = 1 (8.7) For the sake of simplicity we can introduce the angle and write the directional vector e in the form e1 = cos ; e2 = sin (8.8) Let us determine the directional derivatives 1, 2 for the double eigenvalue ~ = 1. The orthonormalized eigenvectors corresponding to ~ are ! ! 1 0 1 = 0 ; 2 = 1 (8.9) Using the expressions in Eqs. 8.1 and 8.9 we obtain the vectors fsk according to Eq. 4.45 ! ! ! !! 1 0 1 0 1 f11T = 1 0 0 1 0 ; 1 0 1 0 10 = (1; 0) ! ! ! !! 1 0 0 0 1 f12T = 1 0 0 1 1 ; 1 0 1 0 01 = (0; 1) (8.10) ! ! ! !! 1 0 0 0 1 f22T = 0 1 0 1 1 ; 0 1 1 0 01 = ( 1; 0) Thus, Eq. 4.46 takes the form cos sin det sin (8.11) cos = 0 or 21;2 = sin2 + cos2 = 1 (8.12) So, 1 = 1 and 2 = 1 for any direction e = (cos ; sin ). Hence, the double eigenvalue ~ splits into 1;2 = 1 " for any direction "e. This means that the bimodal solution x = 0; y = 0 is the optimum solution to the problem Maximize min j ; j = 1; 2 (8.13) x; y This result, of course, can be seen immediately from Fig. 8.1.
8.4 Necessary Optimality Conditions for Eigenvalue Problems In this section necessary optimality conditions for eigenvalue optimization problems are described. The optimization problem considered concerns maximization of the lowest of the
142
8.4. Necessary Optimality Conditions for Eigenvalue Problems
eigenvalues j ; j = 1; : : : ; n, subject to a constant volume constraint and can be formulated as follows
Maximize min j ; j = 1; : : : ; n a1 ; : : : ; aI
Subject to
F (a1; : : : ; aI ) = 0
(8.14)
(8.15)
This formulation of the structural eigenvalue optimization problem is chosen as it is the most commonly used. In the following di erent situations of optimum fundamental eigenvalues are considered.
8.4.1 Simple Optimum Fundamental Eigenvalue First the situation where optimum is achieved at the simple lowest eigenvalue 1 with 1 < 2 3 : : : is studied. In this well-known case, due to Frechet di erentiability of simple eigenvalues, the necessary optimality condition implies linear dependence of the gradient vectors of 1 and F 1 0f0 = 0 (8.16) where ! ! ! @ K @ M @ K @ M T T 1 = 1 @a 1 @a 1 ; : : : ; 1 @a 1 @a 1 1 1 I I (8.17) ! @F ; : : : ; @F f0 = F = @a @aI 1
r
r
r
and 0 is a positive (Lagrangian) multiplier to be determined from Eqs. 8.15 and 8.16.
8.4.2 N -fold Optimum Fundamental Eigenvalue Let us consider the general case when in the optimization problem, Eqs. 8.14 and 8.15, the maximum is attained at an N -fold multiple lowest eigenvalue 1 = 2 = : : : = N < N +1 : : :. This is a non-di erentiable case, and we have to use directional derivatives as described in Section 4.9. Taking the vector of varied design variables in the form a + "e, kek = 1, we obtain the directional derivatives = j ; j = 1; : : : ; N , from Eq. 4.46, i.e.,
det fskT e sk = 0; s; k = 1; : : : ; N
(8.18)
where the generalized gradient vectors fsk are given by Eq. 4.45. Please note that the subscripts of the generalized gradient vector fsk refer to the modes from which it is calculated.
Chapter 8. Multiple Eigenvalues in Structural Design Problems
143
The direction e must ful l the volume constraint in Eq. 8.15, i.e.,
f0T e = 0
(8.19)
The necessary optimality conditions are derived for arbitrary multiplicity N of the lowest multiple eigenvalue in Appendix D and they give much insight in the diculties in maximizing a lowest multiple eigenvalue. In the sequel, a short summary of these conditions are given. In general, the optimum point is characterized by the fact that there must be no admissible direction e for which all = j ; j = 1; : : : ; N , are of the same sign, otherwise an improving direction e exists. In Appendix D it is proved that if the vectors f0 ; fsk , s; k = 1; : : : ; N , k s (the total number of these vectors is equal to (N + 1)N=2 + 1) are linearly independent, then there exists an improving direction e for which j > 0; j = 1; : : : ; N . It should be noted that the linear independence of the vectors is only possible if I (N + 1)N=2 + 1, where I is the dimension of the vector a of design variables. In this way the existence of an improving direction e can easily be checked. The necessary optimality conditions in case of an N -fold optimum fundamental eigenvalue can be stated as: If the vector of design variables a renders a lowest N -fold eigenvalue 1 = 2 = : : : = N a maximum, it is necessary that the vectors f0 ; fsk , s; k = 1; : : : ; N , k s, are linearly dependent N X
skfsk 0f0 = 0 (8.20) s;k=1
with the coecients 0 ; sk satisfying conditions of positive semi-de niteness of the symmetric matrix sk, s; k = 1; : : : ; N . The main advantage of these necessary optimality conditions is, when compared to those obtained by other researchers, that they do not contain variations of design variables. These necessary optimality conditions are proved in Appendix D both in case of a bimodal optimum eigenvalue and for arbitrary multiplicity N of the fundamental eigenvalue. The applicability of the necessary optimality conditions as basis for numerical iterative algorithms for solution of eigenvalue problems involving multiple eigenvalues is described in the following section.
8.5 Algorithms for Eigenvalue Problems with Multiple Eigenvalues This section is devoted to descriptions of algorithms that can be used to solve eigenvalue problems involving multiple eigenvalues. It will be illustrated how the necessary optimality conditions derived in Section 8.4 and Appendix D can be used as a basis for an iterative numerical method for solution of eigenvalue optimization problems, and furthermore, an e ective mathematical programming approach is described.
144
8.5. Algorithms for Eigenvalue Problems with Multiple Eigenvalues
The basic idea in our algorithms developed for solving the structural eigenvalue optimization problem de ned by Eqs. 8.14 and 8.15 have been that we would like to avoid the necessity of using directional derivatives. If directional derivatives should be implemented in the optimization algorithm, a line search method using directional derivatives may possibly be developed but such an algorithm will be computationally very costly. For summary, let us rewrite Eq. 4.47 which is used for computing the increments = j ; j = 1; : : : ; N , of the N -fold eigenvalue corresponding to the vector a of actual increments of the design variables:
det fskT a sk = 0; s; k = 1; : : : ; N
(8.21)
where the generalized gradient vectors fsk are de ned by Eq. 4.45. If the o -diagonal terms in the sensitivity matrix in Eqs. 8.21 are zero, then the traditional equations for determining increments of simple eigenvalues appear, i.e., traditional Frechet derivatives are valid. Such a situation can be obtained by adding the constraints
fskT a = 0; s; k = 1; : : : ; N; s 6= k
(8.22)
For an N -fold multiple eigenvalue, Eqs. 8.22 results in N (N 1)=2 constraints for the allowable direction a, and in case of only a few design variables ai ; i = 1; : : : ; I , the optimization problem may become too constrained. However, adding these constraints makes it possible to use traditional gradient based algorithms for solving the optimization problem where the increments j of an N -fold multiple eigenvalue can be computed as j = fjjT a; j = 1; : : : ; N
(8.23)
Furthermore, as demonstrated by the reinforced plate example in Section 6.7, it is necessary to decide correctly from the numerical results the multiplicity N of a multiple eigenvalue, otherwise the directional derivatives computed might be erroneous due to the solution of an incorrect subeigenvalue problem for determination of directional sensitivities, see Eq. 8.21 and the discussion in Section 6.7. This problem of determining correctly the multiplicity of an eigenvalue is now avoided by adding the constraints in Eq. 8.22 because Eq. 8.23 is valid for both simple and multiple eigenvalues.
Chapter 8. Multiple Eigenvalues in Structural Design Problems
145
8.5.1 Optimality Criteria Based Algorithm In the sequel it will be shown how the necessary optimality conditions derived in Section 8.4 can be used as basis for an optimization algorithm. Such an approach is also described in Seyranian, Lund & Olho (1994). At a given iteration stage, the design is associated with eigenvalues j ; j = 1; : : : ; n, and it is necessary to decide the multiplicity of the lowest eigenvalue. If the relative di erence between adjacent eigenvalues is less than , which is a small parameter assumed to be speci ed, these eigenvalues are considered to be multiple, and the multiplicity of the lowest eigenvalue determines the actual formulation for calculating the increments a of the design variables a.
Simple Eigenvalue If the lowest eigenvalue 1 is simple, the eigenvalue increment and the volume constraint can be expressed as 1 = T1 a = f11T a (8.24) f0T a = 0 (8.25) where f11 and f0 are de ned by Eqs. 4.45 and 8.17, respectively, and the volume constraint is assumed to be linear in the design variables. Guided by the results in Subsection 8.4.1, the vector of increments of the design variables is taken in the single modal form
r
a = k(f11 0f0 )
(8.26)
where k is a positive move limit type of scaling factor. At the optimum point, according to Eqs. 8.16 and 8.17, the vector a will tend towards the null vector. The a priori unknown constant 0 is determined by substituting Eq. 8.26 into Eq. 8.25 which gives T
0 = f0Tf11 (8.27)
f0 f0
and substitution of Eq. 8.27 into 8.24 yields the Cauchy-Bunyakowski inequality 1 = k
f11T f11
!
(f0T f11 )(f0T f11 ) 0 fT f 0 0
(8.28)
for the increment 1 of the eigenvalue. Thus, at each step of redesign the eigenvalue 1 increases while satisfying the volume constraint, Eq. 8.25, and this continues until f11 and f0 becomes linearly dependent, cf. Eqs. 8.16 and 8.17.
Bimodal Eigenvalue If the lowest eigenvalue is associated with a bimodal eigenvalue ~ = 1 = 2 or the two lowest eigenvalues 1 < 2 are very close such that 2 1 < , a bimodal formulation must be used to determine the increments a of the vector of design variables a.
146
8.5. Algorithms for Eigenvalue Problems with Multiple Eigenvalues
Guided by the results stated in Lemma 1 and Theorem 1 in Section D.1 in Appendix D, see also Eq. 8.20, the vector of increments a is taken as
a = k ( 11f11 + 2 12f12 + 22f22 0f0 )
(8.29)
where k again is a positive scaling factor. Obviously, the increments a for the iterative computational procedure can be chosen in many ways, but using the approach of adding the constraints given by Eq. 8.22 leads us to choose the following four simultaneous conditions as a basis for determining the coecients
11 , 2 12, 22, and 0 in the expression for a in Eq. 8.29 (where we disregard the scaling factor k): 1 = f11T a = 1 (8.30) 2 = f22T a = (1 + 1 2) (8.31) f12T a = 0 (8.32) f0T a = 0 (8.33) It is seen from Eqs. 8.30 and 8.31 that we specify the increments 1 and 2 with a view to diminish the (possible) di erence between 1 and 2 while going in a direction that increases the bimodal eigenvalue. Now, by substituting Eq. 8.29 into Eqs. 8.30-8.33, we obtain the following system of equations for determining the unknown coecients 11, 2 12, 22, and 0:
2 T T f22 f T f12 f T f0 3 8 f f
11 11 f11 11 11 11 7 > > 66 T T T < f22 f22 f22 f12 f22 f0 77 22 66 f12T f12 f12T f0 75 >> 2 12 4 symm f T f0 : 0 0
9 8 9 > > > 1 > > = < 1 + 1 2 > = = > > > 0 > > ; > : ; 0
(8.34)
Having solved Eq. 8.34 for the coecients 11, 2 12, 22, and 0, substitution of these coecients into Eq. 8.29 results in a new vector a of increments of design variables. It may be necessary to normalize these coecients in order to avoid numerical problems as described in Seyranian, Lund & Olho (1994). It should be noted that the determinant of the coecient matrix in Eq. 8.34 is nonnegative, and that it only vanishes if the vectors f11 , f12 , f22 , and f0 become linearly dependent, which is the necessary optimality condition for an optimum bimodal solution. This approach for a bimodal lowest eigenvalue is easily extended to higher multiplicity of the lowest eigenvalue, and during the optimization process, the optimization procedure switches between the di erent formulations, depending on the multiplicity of the lowest eigenvalue. The optimization procedure described in this section has been used successfully in Seyranian, Lund & Olho (1994) for solving optimization problems for maximum buckling load of stepped columns on an elastic foundation and my collegue Lars Krog has implemented this approach in ODESSY for solving topology design problems with eigenvalues. The major advantage of this approach is that the number of design variables, in most cases, is reduced. The number of design variables is I but using this approach, only four design
Chapter 8. Multiple Eigenvalues in Structural Design Problems
147
parameters 11, 2 12, 22, and 0 need to be determined in case of a bimodal eigenvalue. This approach may be very attractive in topology optimization where the number I of design variables can be very large.
8.5.2 Mathematical Programming Approach In this section it will be demonstrated that a mathematical programming approach for solving optimization problems can be used e ectively. The approach is based on the same ideas as described in Subsection 8.5.1, i.e., the constraints given by Eq. 8.22 are included such that the need for using directional derivatives is avoided, but in this mathematical programming formulation a set of the lowest eigenvalues j ; j = 1; : : : ; m, is included in each design step, ensuring better convergence properties than the algorithm based on necessary optimality conditions described in the preceding section where only the lowest eigenvalue is considered in each design step. The implementation of this mathematical programming approach has been made in cooperation with my collegue Lars Krog. The basic idea is to replace the original objective function in Eq. 8.14 with a linearized prediction of the objective function in the next iteration, i.e., the optimization problem, cf. Eqs. 8.14 and 8.15, is reformulated as
Maximize min j + j ; j = 1; : : : ; m
a1 ; : : : ; aI
Subject to
(8.35)
F (a1; : : : ; aI ) = 0
(8.36)
fskT a = 0; s 6= k
(8.37)
ai ai ai; i = 1; : : : ; I
(8.38)
The additional N (N 1)=2 constraints in Eq. 8.37 for each N -fold multiple eigenvalue make it possible to compute the linear increment of both simple and multiple eigenvalues j in the form j = fjj a (8.39) where the generalized gradient vector fjj is de ned by Eq. 4.45, i.e., the linear increment of eigenvalue j can be related directly to eigenvector j , although the eigenvalue is multiple. The optimization problem in Eqs. 8.35-8.38 is rewritten using the bound formulation described in Section 7.3, i.e.,
Maximize ai ;
(8.40)
148
8.6. Example: Maximization of Lowest Eigenvalue of Ribbed Plate Subject to
j + j ; j = 1; : : : ; m
(8.41)
F (a1; : : : ; aI ) = 0
(8.42)
fskT a = 0; s 6= k
(8.43)
ai ai ai; i = 1; : : : ; I
(8.44)
This mathematical programming problem is solved in each design iteration, and as the process converges to the optimum point, the vector a tends to the null vector. The necessary optimality conditions derived in Section 8.4 then can be applied to check that the optimum solution is obtained. The mathematical programming approach described in this section has proven to be very e ective as will be demonstrated in the next section where this iterative numerical method will be used for solution of an eigenvalue optimization problem involving multiple eigenvalues.
8.6 Example: Maximization of Lowest Eigenvalue of Ribbed Plate In order to illustrate the eciency of the mathematical programming approach for solving eigenvalue optimization problems with multiple eigenvalues as described in the preceding section, a reinforced plate example is studied. The aim of the optimization is to maximize the lowest eigenfrequency of the ribbed plate introduced in Section 6.6, taking a volume constraint into account. The plate is clamped at all edges and all dimensions can be seen in Fig. 8.2. The plate is made of steel with the following material properties Young's modulus = 210000 MPa Poisson's ratio = 0.3 Mass density = 7800 Kg/m3 The nite element model consists of 1156 4-node isoparametric Mindlin plate nite elements, the lowest 6 eigenfrequencies are calculated, and eigenfrequencies are considered to be identical if the relative di erence between the values is 10 4. The eigenmodes for the initial design are shown in Fig. 8.3. In order to improve the design of the plate, ve design variables are de ned. For simplicity, only 5 symmetric design variables are de ned, that is, the plate thickness, two di erent rib thicknesses, the distances a between the ribs, and the width b of the ribs as shown in Fig. 8.2.
Chapter 8. Multiple Eigenvalues in Structural Design Problems
149
L
b
L
a
b
plate
ribs
ribs
a b
b
: plate thickness = 0.005 m : rib thickness 1 = 0.05 m : rib thickness 2 = 0.05 m
a = 0.5 m b = 0.05 m L = 1.60 m
Figure 8.2: Initial design and design variables of square plate reinforced by ribs.
f1 = 92.1974 Hz.
f2 = 161.7083 Hz.
f3 = 161.7083 Hz.
f4 = 175.0327 Hz.
f5 = 176.0789 Hz. f6 = 177.0941 Hz. Figure 8.3: Eigenmodes for initial design of reinforced plate. The SIMPLEX algorithm is used as optimizer and 100 design iterations are performed using a move limit factor of 2%, i.e., the maximum change of a design variable in each design iteration is 2%. The optimized design is shown in Fig. 8.4. The side constraint for minimum rib width b becomes active in the optimized design as could be expected, see, e.g., the discussions about the \solid plate paradox" in Rozvany, Olho , Cheng & Taylor (1982), see also Olho (1974, 1975) and Cheng & Olho (1981). The optimization history is shown in Fig. 8.5 which demonstrates a stable convergence
8.6. Example: Maximization of Lowest Eigenvalue of Ribbed Plate : plate thickness = 0.006558 m : rib thickness 1 = 0.1637 m : rib thickness 2 = 0.1621 m
a
b
150
b
a = 0.4945 m b = 0.01 m
a
b
b
Figure 8.4: Optimized design. and comparisons between eigenfrequencies of the initial and the nal design are given in Table 8.1. The lowest eigenfrequency is increased by more than 106%. Eigenfrequency [Hz] 220 200 180 ω1 ω2 ω3 ω4 ω5 ω6
3
160 140 120 100 80
0
10
20
30
40
50
60
70
80
90
100
Iteration number
Figure 8.5: Optimization history. It is seen that the lowest eigenfrequency of the optimized design is distinct although the di erence between the rst and second eigenfrequency is less than 0.08%. During the optimization process the two lowest eigenfrequencies coalesce in several design iterations. The third and fourth eigenfrequency remain multiple in all design iterations, so the number of additional constraints given by Eq. 8.22 is one or two in each design iteration, depending on the multiplicity of the lowest eigenfrequency. The eigenmodes for the optimized design are shown in Fig. 8.6. This example illustrates that the mathematical programming approach described in Section 8.5 can be used e ectively for design problems involving multiple eigenvalues. However, let us slightly modify the example in order to obtain a multiple lowest eigenfrequency. Next four supporting springs with sti nesss k = 109 N/m are added to the structure as
Chapter 8. Multiple Eigenvalues in Structural Design Problems
151
Table 8.1: Result of design optimization for reinforced plate. Frequency: Initial design Final design f1 92.1974 190.3866 f2 161.7083 190.5347 f3 161.7083 191.9949 f4 175.0327 191.9949 f5 176.0789 193.2443 f6 177.0941 208.1040
f1 = 190.3866 Hz.
f2 = 190.5347 Hz.
f3 = 191.9949 Hz.
f4 = 191.9949 Hz.
f5 = 193.2443 Hz. f6 = 208.1040 Hz. Figure 8.6: Eigenmodes for optimized design of reinforced plate. illustrated in Fig. 8.7. Obviously, this increases the lowest eigenfrequency and results in a dense spectrum of eigenfrequencies as can be seen in Fig. 8.8 where the eigenmodes for the initial design are shown. Again, 100 design iterations are performed, the lowest eigenfrequency is increased by 29% and becomes double in the optimized design. None of the side constraints are active in the design iterations and the optimized design is shown in Fig. 8.9. The iteration history is illustrated in Fig. 8.10 and comparisons between initial and nal values of eigenfrequencies are given in Table 8.2. It is seen that the rst and second eigenfrequency and the fourth and fth eigenfrequency coalesce, and the third eigenfrequency is nearly equal to the lowest double eigenfrequency as the relative di erence is less than 2 10 4. This dense spectrum of lowest eigenfrequencies in the nal design results in two
152
8.6. Example: Maximization of Lowest Eigenvalue of Ribbed Plate
b
L
k k k b
L
a
k
a b
b
: plate thickness = 0.005 m : rib thickness 1 = 0.05 m : rib thickness 2 = 0.05 m
a = 0.5 m b = 0.05 m L = 1.60 m
Figure 8.7: Initial design and design variables of square plate reinforced by ribs and with four supporting springs.
f1 = 170.2444 Hz.
f2 = 175.0327 Hz.
f3 = 176.5525 Hz.
f4 = 176.5525 Hz.
f5 = 177.6583 Hz. f6 = 178.9780 Hz. Figure 8.8: Eigenmodes for initial design of reinforced plate with four springs. additional constraints given by Eq. 8.22 as eigenfrequencies are considered to be identical by the optimization algorithm if the relative di erence between the values is 10 4, that is, the lowest and the fourth eigenfrequency is considered double by the optimization algorithm.
Chapter 8. Multiple Eigenvalues in Structural Design Problems
153
a
b
: plate thickness = 0.007888 m : rib thickness 1 = 0.04766 m : rib thickness 2 = 0.07906 m
b
a = 0.5069 m b = 0.02082 m
a
b
b
Figure 8.9: Optimized design. The iteration history demonstrates a stable convergence process. Eigenfrequency [Hz] 240 230 220 ω1 ω2 ω3 ω4 ω5 ω6
3
210 200 190 180 170 0
10
20
30
40
50
60
70
80
90
100
Iteration number
Figure 8.10: Optimization history. The eigenmodes for the optimized design can be seen in Fig. 8.11. This example illustrates that the mathematical programming approach described in Section 8.5 is very ecient in case of multiple eigenvalues. Correct sensitivity information is used in each design step due to the additional constraints given by Eq. 8.22, which is the reason for the stable convergence.
154
8.6. Example: Maximization of Lowest Eigenvalue of Ribbed Plate
Table 8.2: Result of design optimization for reinforced plate with four springs. Frequency: Initial design Final design f1 170.2444 219.6628 f2 175.0327 219.6628 f3 176.5525 219.6980 f4 176.5525 219.9389 f5 177.6583 219.9389 f6 178.9780 231.4708
f1 = 219.6628 Hz.
f2 = 219.6628 Hz.
f3 = 219.6980 Hz.
f4 = 219.9389 Hz.
f5 = 219.9389 Hz. f6 = 231.4708 Hz. Figure 8.11: Eigenmodes for optimized design of reinforced plate with four springs.
Chapter
9
Examples of Interactive Engineering Design with ODESSY 9.1 Introduction
Ia fewof ODESSY can be used for interactive engineering design. For reasons of brevity, only representative examples will be presented here; a demonstration of all the facilities n this chapter it will be illustrated by means of examples how some of the facilities
of the system would require a large number of additional examples. In Section 9.2 it will be illustrated how design sensitivity display and what-if studies can be used to improve the design of a turbine disk. The use of design sensitivity display can greatly improve the design synthesis process as colour design sensitivity contours provide the designer information about which design variables are critical to some performance measure or the performance measures that are most e ected by changing a particular design variable. The use of design optimization based on mathematical programming is illustrated in Section 9.3 for improvement of the design of the turbine disk introduced in the preceding section. The objective is to reduce the mass moment of inertia in order to increase the acceleration capabilities of the disk, and at the same time, a quite complicated temperature dependent stress constraint and a manufacturing constraint on the minimum boundary radius of curvature are taken into account. The next section illustrates how the turbine disk can be designed using a ceramic material. For such design cases the probability of failure must be evaluated. A reliability evaluation based on a two-parameter Weibull distribution has been generally accepted in design of ceramics and has therefore been implemented in ODESSY. The low mass density of the ceramic material reduces the mass moment of inertia signi cantly, compared to the design made of steel, and the objective then is to reduce the probability of failure of the disk. Finally, the shape optimization of a shell structure in the form of the hood of a Mazda 323 automobile with the objective of maximizing the fundamental frequency of free vibrations 155
156
9.2. Design Sensitivity Display and What-If Studies of Turbine Disk
is shown.
9.2 Design Sensitivity Display and What-If Studies of Turbine Disk With expressions for design sensitivities of displacements, stresses, compliance, eigenvalues, etc., at hand, this section will brie y illustrate how design sensitivity analysis can be used to improve engineering designs. A turbine disk will be used as an example, and the relations between the parameterized design model and the nite element analysis model are also illustrated. Previously, when using the traditional design process, the designer was required to use intuition and trial and error procedures to nd ways of improving the design. Nowadays, by using a structural analysis program which has capabilities for design sensitivity analysis, the eciency of the design process can be highly improved. Through the use of, e.g., colour stress contour plots together with colour stress design sensitivity contour plots, the engineer easily identi es critical regions in which design improvements can be made. To illustrate this, let us consider the problem of the rotating turbine disk of Figs. 9.1 and 9.2.
Figure 9.1: Initial design of turbine disk. The disk has blades attached to its circumference and is driven by hot gas. Only the design of the cross section of the turbine disk is considered so an axisymmetric model can be used as illustrated in Fig. 9.2. The blades give rise to a xed centrifugal force which is modelled by a uniformly distributed load of value 310 MPa at the circumference of the
Chapter 9. Examples of Interactive Engineering Design with ODESSY 157 ω = 2094 rad/s
573 K
master node fixed boundary design boundary convection
0
R1
50
°
axis of revolution
20
723 K
16
310 MPa
10
20
convection 100
Figure 9.2: Design model of cross section of turbine disk. disk, and the gas exposes the circumference to a relatively high temperature of 723 K. The centre of the disk is attached to a relatively cold shaft of temperature 573 K, and forced convection of heat to the surroundings takes place in the region between the blades and the shaft. At these boundaries the temperature of the environment is speci ed to 723 K and the convection coecient is 0.0012 W/(mm2K). At maximum speed, the disk rotates at 2094 rad/s (= 20000 rev/min). The disk is made of steel with the mass density 7.75 Kg/mm3, Young's modulus 180000 MPa, Poisson's ratio 0.3, thermal expansion coecient 1.2 10 5 K 1, and thermal conductivity coecient 0.027 W/(Kmm). The design model consists of two design elements. There are two design boundaries, i.e., boundaries whose shapes are allowed to change. Each of these shapes are de ned by the positions of a number of master nodes, and this creates an evident connection between the design variables (the movements of the master nodes) and the shape of the geometry. In this example, the direction of the movement of each master node is constrained to follow some prede ned translation directions speci ed by the designer as shown in Fig. 9.3. Thus, in this example, the design variables are simply taken to be the sizes of the movements of the master nodes along the associated translation directions. In addition, the distribution of nite element nodes on the boundaries and the desired nite element type for each design element must be de ned. All necessary speci cations including loading conditions are assigned to the design model, and the preprocessor with facilities for automatic mesh generation automatically converts the design model into a nite element analysis model as shown in Fig. 9.4. The preprocessor has meshed the design elements with a mixture of 6 and 9 node isoparametric 2-D axisymmetric nite elements. All load speci cations are automatically converted into consistent nodal loads. The design model now has been converted into an analysis model, and by changing the
158
9.2. Design Sensitivity Display and What-If Studies of Turbine Disk master node fixed boundary design boundary vector defining allowable translation direction for master node
axis of revolution
Figure 9.3: Variable design model. ω = 2094 rad/s prescribed temperature = 573 K convection boundary conditions
consistent nodal loads
prescribed temperature = 723 K
axis of revolution convection boundary conditions
Figure 9.4: Finite element analysis model. values of the design variables, the geometry can be changed parametrically into other shapes. Now, as a starting point we would like to:
Decrease the maximum von Mises reference stress The convection boundary condition as well as the ow of heat from the hot circumference to the cold shaft clearly change with the design and give rise to a varying stress eld, and
Chapter 9. Examples of Interactive Engineering Design with ODESSY 159 so do the centrifugal force of the disk. This problem is therefore quite complex, because the stresses depend on the design, the temperatures, and the forces, which again depend on the design. The temperature eld and the von Mises stress eld in the initial disk are displayed in Figs. 9.5 and 9.6, respectively. In Fig. 9.7 the von Mises stresses are shown in the region near the lower boundary where the maximum value is found. ODESSY Postprocessor Name: wheel Date: Jul 19 1993 19:07 TEMPERATURES
723.00 698.00 673.00 648.00 623.00 598.00 573.00
Figure 9.5: Temperature eld in turbine disk. ODESSY Postprocessor Name: wheel Date: Jul 19 1993 19:07 STRESS LEVELS von Mises
5.711E+008 4.761E+008 3.811E+008 2.861E+008 1.911E+008 9.610E+007 1.095E+006
Figure 9.6: von Mises stress eld in turbine disk.
160
9.2. Design Sensitivity Display and What-If Studies of Turbine Disk ODESSY Postprocessor Name: wheel Date: Jun 7 1994 16:39 STRESS LEVELS von Mises 5.711E+008 4.761E+008 3.811E+008 2.861E+008 1.911E+008 9.610E+007 1.095E+006
Figure 9.7: Zoom of region with largest von Mises stress. As was already indicated in Fig. 9.3, the disk has been assigned 20 shape design variables which control the shape of the structural domain, and the geometry may be perturbed for each of these variables in order to display stress design sensitivities with colour contour plots. master node fixed boundary design boundary vector defining allowable translation direction for master node
axis of revolution
Figure 9.8: The two selected design variables for which sensitivities will be shown. For the two particular design variables indicated in Fig. 9.8, corresponding stress design sensitivity elds are shown in Figs. 9.9 and 9.10. Each stress design sensitivity eld is associated with translation of the master node in the direction indicated by the arrow.
Chapter 9. Examples of Interactive Engineering Design with ODESSY 161 ODESSY Postprocessor Name: wheel Date: Jun 7 1994 16:41 STRESS LEVELS von Mises 7.464E+010 5.000E+010 3.750E+010 2.500E+010 1.250E+010 0.000E+000 -9.270E+010
Figure 9.9: von Mises stress design sensitivity plot no. 1. ODESSY Postprocessor Name: wheel Date: Jul 20 1993 19:13 STRESS LEVELS von Mises
3.938E+010 5.000E+009 2.500E+009 0.000E+000 -2.500E+009 -5.000E+009 -6.428E+009
Figure 9.10: von Mises stress design sensitivity plot no. 2. Fig. 9.9 shows that most of the large von Mises stresses in this region will be increased if the master node is moved in the direction of the arrow. In other words, in order to decrease the large stresses in this region, the master node must be moved in the opposite direction of the arrow in Fig. 9.9. Fig. 9.10 shows that the indicated inward movement of the master node results in negative stress sensitivities in the highly stressed region at the lower boundary, i.e., a design change of this kind has a desirable decreasing e ect on the maximum von Mises stress. No doubt,
162
9.3. Shape Optimization of Turbine Disk
this is because such a design change will imply a decrease of the centrifugal forces in the rotating disk. When all 20 design perturbations have been performed and the corresponding design sensitivity elds determined, the designer can do a what-if study, where he decides on suitable changes of the values of the design variables according to the stress sensitivity information. The rst result of a what-if study of the disk example is shown in Fig. 9.11, where changes of all design variables have been guessed on the basis of the stress sensitivity information obtained, and without considering, e.g., a constraint on the volume of the disk. This new design implies a reduction of the maximum von Mises stress from 571 to 453 MPa. ODESSY Postprocessor Name: wheel Date: Jul 20 1993 20:58 STRESS LEVELS von Mises
4.525E+008 3.772E+008 3.018E+008 2.265E+008 1.511E+008 7.580E+007 4.524E+005
Figure 9.11: von Mises stresses in updated geometry obtained by what-if study. Based on this improved design, a new design sensitivity study can be performed, and the design of the disk presumeably can be further improved, if necessary. This leads us to another use of design sensitivity analysis, namely in engineering design optimization.
9.3 Shape Optimization of Turbine Disk The previous example illustrated that the designer can improve designs based on design sensitivity information by doing what-if studies. However, if the designer is confronted with a problem that involves more than just a few design variables and criteria to be taken into account, it becomes very dicult to survey and quantify the design sensitivity information so as to make decisions regarding values of the design variables that would satisfy design constraints and be any better than other alternative values. In such situations, facilities for engineering design optimization based on an iterative solution procedure as described in Section 7.8 can be used as will be illustrated on the turbine disk example introduced in the preceding section.
Chapter 9. Examples of Interactive Engineering Design with ODESSY 163 When designing turbine disks or other thermo-elastic problems involving high temperatures, it is very often necessary to consider the fact that the strength, in terms of yield stress, of most materials is strongly dependent upon the temperature. Metals are often assumed to have a constant strength up to a certain temperature and then a linear decay as shown in Fig. 9.12 for the material used for the turbine disk. It is noted that this stress constraint has a non-di erentiable behaviour. 600
Allowable von Mises stress σvM [Mpa]
500 400 300 200 100 0 273 323 373 423 473 523 573 623 673 723 773 823 Temperature [K]
Figure 9.12: Allowable von Mises stress. Furthermore, the manufacturing process used for the turbine disk requires the minimum boundary radius of curvature to be larger than 5 mm. The objective of the shape optimization is to increase the acceleration capabilities of the turbine disk, i.e., to minimize the mass moment of inertia, so the de nition of the structural optimization problem is
Minimize Subject to
the mass moment of inertia maximum normalized von Mises stress 1 minimum boundary radius of curvature 5 mm
This temperature dependent non-linear stress constraint is realized by normalizing the von Mises stress at the i'th node with the function shown in Fig. 9.12. It should be noted that the e ect of this non-linear stress constraint at a given material point changes during the optimization process as the temperature eld changes with design. The de nition of the optimization problem can be rewritten as This de nition of the structural optimization problem can be realized using the database module as described in detail in Section 7.7. The temperature dependent normalized stress constraint is shown for the initial structure in Fig. 9.13. The maximum value is 1:34 and corresponds to a violation of 34%. Performing a shape optimization of the turbine disk example using a SIMPLEX algorithm as optimizer leads to a reduction of the mass moment of inertia from 3:12 104 to 2:33 104 Kgm2. The stress constraint is now ful lled as shown in Fig. 9.14. Here, the very even
164
9.4. Design Optimization of Ceramic Components
Minimize Subject to
the mass moment of inertia
8 > vM;i [MPa] > < 550 if Ti 623 K maximum normalized stress > vM;i [MPa] > : 1484 :5 1:5Ti if Ti > 623 K
1
minimum boundary radius of curvature 5 mm ODESSY Postprocessor Name: wheel Date: Jun 7 1994 16:50 VNV LEVELS
1.340E+000 1.117E+000 8.938E-001 6.709E-001 4.480E-001 2.251E-001 2.182E-003
Vnv: maxval[ nstress(svm)/1.0E6 / [minvec [550; [1484.5 - [1.5*temp()] ] ] ] ] < 1.0
Figure 9.13: Temperature dependent stress constraint for initial design. distribution of the normalized stress criterion over the domain should be noted. The nal design can also be seen in Fig. 9.15. The nal minimum boundary radius of curvature is 13:5 mm, i.e. this constraint is not active in the nal design. During the optimization process, this manufacturing constraint became active in some iterations. This example illustrates the generality and exibility of the database module described in Chapter 7 for evaluating objective and constraint functions.
9.4 Design Optimization of Ceramic Components This section is devoted to problems concerning design optimization of ceramic components. Such design optimization problems are rarely discussed but design with ceramic materials calls for use of structural shape optimization as design with new materials very often cannot be based on design rules or engineering tradition. This will be demonstrated for
Chapter 9. Examples of Interactive Engineering Design with ODESSY 165 ODESSY Postprocessor Name: wheel Date: Jun 7 1994 16:51 VNV LEVELS
1.000E+000 8.335E-001 6.669E-001 5.004E-001 3.338E-001 1.673E-001 7.709E-004
Vnv: maxval[ nstress(svm)/1.0E6 / [minvec [550; [1484.5 - [1.5*temp()] ] ] ] ] < 1.0
Figure 9.14: Temperature dependent stress constraint for nal design.
Figure 9.15: Optimized design of turbine disk. the turbine disk introduced in the preceding sections. Discussions about multicriteria design optimization of ceramic components can be found in a very interesting paper by Koski & Silvennoinen (1990) contained in Eschenauer, Koski and Osyczka (1990) where many other interesting applications of multicriteria design optimization are discussed. In the last decades there has been an increasing use of ceramic materials in mechanical en-
166
9.4. Design Optimization of Ceramic Components
ginering applications where good wear resistance properties, high hardness, sucient hightemperature capability, high sti ness, and good corrosion resistance are needed. However, design with ceramic components is di erent from design with traditional ductile materials due to the brittle behaviour of ceramics. The use of ceramic materials for load carrying components involves two basic features that must be taken into account in the design phase. First, even at high temperature, the material has very low strain tolerance and practically exhibits no yielding. Thus, the material behaviour is linearly elastic up to the fracture point where an unstable crack growth suddenly takes place. Second, there is frequently large scatter in the strength data so probabilistic methods must be used. A reliability evaluation based on a twoparameter Weibull distribution has been generally accepted in design of ceramics, see, e.g., McLean & Hartsock (1989). Weibull developed a probabilistic failure criterion based only on tensile stresses in the component. Compressive failure is not considered in this criterion because brittle materials usually fail from tensile stresses due to their very high compressive strength. The probability of failure is computed for a ceramic component from its stress eld by using the weakest link theory based on the Weibull distribution, i.e., it is assumed that the weakest crack-stress combination will cause the overall damage, but the location of this critical point is not known. The mean fracture stress c is extremely sensitive to small defects in the material which may be caused by the manufacturing process or by environmental e ects during the everyday use, and as the ceramic material properties are variable due to the random distribution of aws in ceramics, there is a variation of strengths for various parts of a component. So, the probability of failure of a ceramic component is a function of the volume of material subjected to tensile stresses. Accordingly, the mean fracture stress c is associated with a certain reference volume Vc which usually encompasses that part of the test specimen where tensile stresses occur. The most popular method of generating material data is to use test specimens subjected to four point bending. The specimens are loaded to fracture and the mean fracture stress c is calculated for a corresponding reference volume Vc. Furthermore, the Weibull modulus m is determined. The Weibull modulus m can be interpreted as a measure of the narrowness of the strength distribution, i.e., the larger the value m, the smaller is the variation of the fracture stress of the material. For uniaxial stress, Weibull established the following function that describes the cumulative probability of failure Pf of a ceramic component: 1 m 1 m 1 Z m Pf = 1 exp (9.1) m! c Vc V dV where is the tensile stress at a given point. The term 1 1 Z 1 1 m (9.2) m ! = m + 1 = 0 t exp [ t] dt is the value of the gamma function at m1 + 1 which is easily evaluated. In order to expand Eq. 9.1 for three-dimensional stress states, the concept of integrating
Chapter 9. Examples of Interactive Engineering Design with ODESSY 167 the normal stress n around the portion of the unit radius sphere where the normal stress is positive is generally used, see, e.g., McLean & Hartsock (1989) and Fig. 9.16. σ3
dA = cos ϕ d ϕ d ψ σn ϕ
ψ
σ2
σ1
Figure 9.16: Geometric variables describing location on the unit sphere. Thus, the general equation for the probability of failure is 1 m 1 m 1 Z 2m + 1 Pf = 1 exp m! c Vc V 2
) # (9.3) i m cos2 1 cos2 + 2 sin2 + 3 sin2 cos d d dV
(Z 2 Z =2 h 0
0
The integration over the unit sphere can be carried out by numerical integration for any value of the Weibull modulus m, or analytically for discrete values of m. In the implementation in ODESSY, I have chosen the latter approach. When evaluating the volume integral in Eq. 9.3, the order of Gauss quadrature used for the numerical integration in general depends on m, but in the current implementation in ODESSY, the Gauss points used for evaluating the stresses are also used for computing the volume integral. This approximation might result in inaccurate results in case of large stress gradients. In the case of a three-dimensional stress state it might be computationally advantageous to neglect the mutual interrelationsship between the tensile principal stresses whereby the following expression for the probability of failure is obtained: 1 m 1 m 1 Z m m m Pf = 1 exp (9.4) m! c Vc V (1 + 2 + 3 ) dV This approximation might be acceptable if the second and third principal stresses are small percentages of the maximum principal stress. In case of other stress states, this approximation may lead to unacceptable errors. The principal stresses are only included in the volume integral if they are positive, i.e., i < 0 ) im = 0. Both Eqs. 9.3 and 9.4 have been implemented in ODESSY. To illustrate the use of this theory for determining the probability of failure of ceramic components, let us redesign the turbine disk introduced in the preceding sections using a ceramic material. Silicon nitride is chosen as material due to its strength properties at high temperatures and its low mass density compared to other ceramics, e.g., zirconia. The material properties are standard values given by a manufacturer (Kyocera Corporation). The
168
9.4. Design Optimization of Ceramic Components
chosen sintered silicon nitride has mass density 3.20 Kg/mm3, Young's modulus 304000 MPa, Poisson's ratio 0.27, thermal expansion coecient 3.0 10 6 K 1, and thermal conductivity coecient 0.029 W/(Kmm). The data used for computing the probability of failure are the mean fracture stress c = 785 MPa (at 723 K), the corresponding reference volume Vc = 13.5 mm3, and the Weibull modulus m = 13. It is assumed that the turbine blades have been redesigned in silicon nitride, resulting in smaller centrifugal forces. The uniformly distributed load of value 310 MPa corresponding to these centrifugal forces, see Fig. 9.2, is assumed to be reduced to 130 MPa due to the lower mass density of silicon nitride. All other boundary conditions and load speci cations are unchanged but one of the major reasons for using ceramic materials in gas turbines and turbo chargers is to allow higher working temperatures, whereby the eciency can be improved. The initial design is shown in Fig. 9.1 and the largest principal stresses are shown in Fig. 9.17 because the probability of failure highly depends on these stresses. ODESSY Postprocessor Name: wheelc Date: Apr 24 1994 18:23 STRESS LEVELS SIG1 2.397E+008 1.994E+008 1.590E+008 1.187E+008 7.833E+007 3.797E+007 -2.380E+006
Figure 9.17: Largest principal stresses for initial design of ceramic turbine disk. The probability of failure is computed by using Eq. 9.3 because the rst and second principal stresses are of the same magnitude. The probability of failure of the initial design is computed to 1.4710 3, i.e., 1 out of 680 turbine disks will fail. The value obtained using the simpli ed Eq. 9.4 di ers from the value obtained by Eq. 9.3 by a factor of 3 due to the stress state in the disk. Next, automatic shape optimization is performed. The mass moment of inertia for the initial design is 1.29104 Kgm2 which is 55% of the mass moment of inertia of the optimized design in Section 9.3. However, the objective of the optimization process is to decrease the probability of failure, so the mass moment of inertia is allowed to be increased. The constraint on the minimum boundary radius of curvature is maintained, so the de nition
Chapter 9. Examples of Interactive Engineering Design with ODESSY 169 of the structural optimization problem is
170
9.4. Design Optimization of Ceramic Components Minimize Subject to
probability of failure the mass moment of inertia 1.40104 Kgm2 minimum boundary radius of curvature 5 mm
Performing a shape optimization, the probability of failure is reduced from 1.4710 3 to 3.2210 6, i.e., 1 out of 310000 disks will fail. This probability of failure seems to be suciently reduced, but it is always very dicult to choose an acceptable reliability. The failure of a turbine disk may lead to a life-threatening situation so it is necessary to be absolutely certain about material data, loading conditions, and boundary conditions. The nal design can be seen in Fig. 9.18, and the largest principal stresses for the optimized design are shown in Fig. 9.19.
Figure 9.18: Optimized design of ceramic turbine disk. The constraints on the mass moment of inertia and the minimum boundary radius of curvature are both active for the nal design, i.e., the mass moment of inertia of the nal design is 40% smaller than that of the optimized design obtained in Section 9.3. The turbine disk designed and made of silicon nitride therefore has signi cantly improved acceleration capabilities.
Chapter 9. Examples of Interactive Engineering Design with ODESSY 171 ODESSY Postprocessor Name: wheelc Date: Apr 24 1994 18:19 STRESS LEVELS SIG1 1.343E+008 1.119E+008 8.945E+007 6.703E+007 4.461E+007 2.218E+007 -2.379E+005
Figure 9.19: Largest principal stresses for nal design of ceramic turbine disk.
9.5 Shape Optimization of an Automobile Hood This example involves shape optimization of a shell structure in the form of the hood of a Mazda 323 automobile with the objective of maximizing the fundamental frequency of free vibrations. In practical shape design optimization it is usually advantageous to start out with a relatively simple model in terms of geometry and nite element representation. The model is then re ned iteratively as the process converges towards a good solution and the designer acquires more knowledge about the nature of the problem. The nal result often will be the last in a long sequence of models. However, for reasons of brevity and the purpose of illustration, we shall start out this example with a fairly accurate geometric modeling of the original real-life structure and not attempt to generate improved models along the way. It is assumed that there is a constraint on the total amount of material such that the weight of the optimized structure does not exceed the original one. The hood is made from steel plates with thickness 1.0 mm, Young's modulus = 210 GPa, Poisson's ratio = 0.3 and the mass density = 7800 kg/m3. The overall dimensions of the hood are approximately 1.25 x 0.85 m and must remain unchanged.
9.5.1 Model The original geometry, which is visualized in Figs. 9.20 and 9.21, consists of two doubly curved shells joined by welding. One shell is the external surface of the hood, and the other shell, which is welded to the inner side of the hood, is a multi-connected set of sti eners. A total of 496 boundaries and 238 curved surface patches form the model. The shapes of the surfaces are controlled by their boundaries which in turn depend on a number of
172
9.5. Shape Optimization of an Automobile Hood
Engine hood Stiffening shell Joined area
Figure 9.20: The underside of the original geometry. The black area is the joined region (joined by welding), the dark gray is the sti ening shell, and the light gray is the upper side of the hood. B
C
B
Engine hood Stiffening shell Joined area
A
A
Figure 9.21: The original geometry with boundary conditions. master nodes as described in Chapter 2. Modi ers and design variables are de ned such that positions, heights and widths of the sti eners can be changed continuously along each sti ener. The outer surface of the engine hood is not changed during the design
Chapter 9. Examples of Interactive Engineering Design with ODESSY 173 optimization as we do not want to a ect the aerodynamical properties. Symmetry with respect to the vertical midplane in Fig. 9.21 is imposed on possible design changes, and the model then has a total of 32 independent design variables. The nite element model (see Figs. 9.21 and 9.22) is established using a mix of mapping and free meshing of curved surfaces. Fig. 9.22 shows a section through the two shells illustrating their interrelation.
Engine hood Stiffening shell Joined area
Figure 9.22: Section through the two shells. As seen in Fig. 9.21, the structure is supported by two hinges attached to the edge of the hood, and three support points on the opposite side. The hinges (A) x all degrees of freedom except for one rotation. The supports (B) x the out-of-plane translational degree of freedom while the support (C) xes the out-of-plane and vertical translational degrees of freedom. Isoparametric shell elements with 6 and 8 nodes are used for the analysis, and the model comprises a total of more than 20000 degrees of freedom. This modeling gives a rather crude description of the geometry, but a test of the convergence shows that it is adequate for the representation of the eigenmodes at hand.
9.5.2 Analysis The analysis of the original geometry using the Subspace iteration method yields the following three lowest eigenfrequencies:
f1 = 132 Hz f2 = 177 Hz f3 = 198 Hz We see that none of these eigenfrequencies are multiple. The lowest eigenfrequency corresponds to the eigenmode shown in Fig. 9.23.
174
9.5. Shape Optimization of an Automobile Hood
Frequency: 132 Hz
Figure 9.23: First eigenmode for initial geometry.
9.5.3 Result The nal geometry is shown in Figs. 9.24 and 9.25, and is obtained after 12 iterations using the SIMPLEX algorithm of ODESSY. The three lowest eigenfrequencies of the optimized design are found to be:
f1 = 163 Hz f2 = 182 Hz f3 = 216 Hz The minimum eigenfrequency of 163 Hz implies an increase 24% relative to that of the original structure, and the nal volume corresponds exactly to the original one. It is obvious that rather large geometric changes have taken place, and many of the design variables reach speci ed maximum (or minimum) allowable values which indicates that the topology of the initial geometry is not optimum for the case of maximizing the smallest eigenfrequency. Thus, the nal geometry may only be considered optimum within this topology and the limitations we have set for variations of the design variables in the initial mathematical formulation of the problem.
Chapter 9. Examples of Interactive Engineering Design with ODESSY 175
Engine hood Stiffening shell Joined area
Figure 9.24: Shape optimized geometry.
Engine hood Stiffening shell Joined area
Figure 9.25: Section through the two shells in optimized geometry.
176
9.5. Shape Optimization of an Automobile Hood
Frequency: 163 Hz
Figure 9.26: First eigenmode for optimized geometry.
Chapter Conclusions
10
M and optimization are covered in this Ph.D. project. These topics are related to the development of a variety of capabilities which constitute part of the backbone of the general any different topics in the eld of structural analysis, design sensitivity analysis
purpose computer aided engineering design system ODESSY, and a short summary and conclusions of the work presented are given in the following.
Analysis Capabilities for Structural Optimization The nite element method is used as the analysis tool and we have chosen to write our own code. This makes it possible to design the system exactly to desired purposes. 19 di erent isoparametric 2-D solids, 3-D solids, plate and shell nite elements have been implemented as described in Chapter 3. The nite element module has facilities for static stress analysis, natural frequency analysis, steady state thermal analysis, thermo-elastic analysis, eigenfrequency analysis with initial stress sti ening e ects due to mechanical or thermal loads, and linear buckling analysis with the possibility of including thermo-elastic e ects. The nite element library has reached a quite high level and has been programmed in a very structured way such that it is easy to implement other nite element types.
Design Sensitivity Analysis Facilities for design sensitivity analysis have been implemented for all available analysis modules, nite element types, and design variable types as described in Chapters 4 and 5. The design variables of the structural design problem can be either geometrical design variables like sizing or shape variables, material design variables like constitutive parameters of materials, support design variables like the position of supports for the structure, or loading design variables like the position of external loads applied to the structure. The method of design sensitivity analysis used is the direct approach which is computationally ecient and based on di erentiation of the state equations. Using the direct 177
178 approach to design sensitivity analysis, derivatives of various nite element matrices need to be determined. It is not possible, in ODESSY, to establish analytical relations between the derivatives of nite element matrices and the available types of generalized shape design variables due to the very general mesh generation and parameterization facilities implemented. As described in Section 5.7, the unstructured mesh generators that may be used to generate three-dimensional surface meshes include smoothing processes, and analytical relations between the position of surface nite element nodes and generalized shape design variables are therefore not available. The semi-analytical approach where derivatives of various nite element matrices and vectors are approximated by computationally inexpensive rst order nite di erences has therefore been chosen instead of the analytical approach. However, as discovered for static problems by Barthelemy & Haftka (1988), the semianalytical method of design sensitivity analysis is prone to large errors for certain types of problems involving shape design variables. The inaccuracy problems may occur for design sensitivities with respect to structural shape design variables in problems where the displacement eld is characterized by rigid body rotations which are large relative to actual deformations of the nite elements, i.e., for example in problems involving linearly elastic bending of long-span, beam-like structures, and of plate and shell structures. This error problem is entirely due to the nite di erence approximation involved in determining various element matrix derivatives. There has been developed a modi ed semi-analytical method that is based on \exact" numerical di erentiation of element matrices (\exact" up to round-o errors) by means of computationally inexpensive rst order nite di erences. The method of \exact" numerical di erentiation has been implemented for all analysis modules, nite element types, and design variable types as described in detail in Chapter 5. This method is computationally ecient, especially in problems involving many design variables as described in Section 5.9, because \exact" numerical derivatives of element matrices only need to be calculated once. Furthermore, the method can even be implemented in connection with existing nite element codes where di erent subroutines for computation of element sti ness matrices are only available as black-box routines. The new method of semi-analytical design sensitivity analysis completely eliminates the inaccuracy problems as has been demonstrated by several numerical examples of design sensitivity analysis in Chapter 6. The traditional semi-analytical method of design sensitivity analysis has been shown to give very accurate results for a classical llet example, i.e., in a case where the displacement eld is characterized by small rigid body rotations relative to actual deformations of the nite elements. For such design problems the traditional semi-analytical method is completely reliable. However, in cases where the displacement eld entails dominance of rigid body rotations relative to actual deformations of the nite elements, the traditional semi-analytical method has been shown to be prone to yield erroneous sensitivities. This has been demonstrated for static design sensitivity analysis of a long cantilever beam in Section 6.3 and of a thin clamped square plate in Section 6.5. Furthermore, the inaccuracy problem has also been demonstrated for thermo-elastic design sensitivity analysis of a two-material cantilever
Chapter 10. Conclusions
179
beam in Section 6.4. The inaccuracy problem can be expected in many thermo-elastic design problems as displacement elds with dominance of rigid body rotations often occur in such design problems. The design sensitivity analysis of simple as well as multiple eigenvalues of vibrating plates reinforced by ribs has been shown to yield accurate results in Sections 6.6 and 6.7. One of the major diculties in computing sensitivities of multiple eigenvalues is to decide correctly from numerical results the multiplicity of a given eigenvalue in cases where the spectrum of eigenvalues is dense. An incorrect estimation of the multiplicity of a repeated eigenvalue is shown to give erroneous sensitivities in Section 6.7 and this diculty associated with multiple eigenvalues, in addition to the lack of usual Frechet di erentiability, has to be taken into account when solving optimum design problems involving multiple eigenvalues. Finally, the inaccuracy problem associated with the traditional semi-analytical method is shown to occur also in dynamic design sensitivity analysis. This is illustrated by computation of design sensitivies of simple as well as multiple free vibration frequencies of a thin clamped square plate in Section 6.8. Furthermore, as shown in Section 6.9, if initial stress sti ening e ects are taken into account when computing eigenfrequencies of vibrating plates, the error problem is even worse because errors in stress sensitivities from the static design sensitivity analysis become accumulated in the dynamic design sensitivity analysis. The new approach to semi-analytical design sensitivity analysis based on \exact" numerical di erentiation of element matrices has been shown to yield accurate sensitivities for all studies made and must be regarded as a very reliable and useful tool in a general purpose computer aided engineering design system.
A General and Flexible Method of Problem De nition In Chapter 7 it has been demonstrated that it is not only possible but also relatively simple to develop a general database module that handles problem formulations comprising any mix of local, integral, min/max and max/min functions in a completely uniform way. The mathematical basis for this is the bound formulation. This, in combination with a parser capable of interpreting user de ned expressions, makes it possible to perform analyses and sensitivity analyses of user de ned mathematical expressions and even to carry out the sensitivity analysis by exact and ecient symbolic di erentiation of the expressions. The approach described makes it very easy to expand the system by new available mathematical expressions or by new basic information if new analysis facilities are to be implemented. The integration of the database module in the module for postprocessing has made it very easy to check the validity of de nitions of objective and constraint functions, and, furthermore, has equipped the system with a very useful tool for interactive, graphical visualization of user de ned mathematical expressions for design criteria.
180 Multiple Eigenvalues in Structural Design Problems In Chapter 8 the diculties of solving structural optimum design problems with multiple eigenvalues are described. The basic diculty is the lack of usual Frechet di erentiability, suct that traditional gradient based algorithms cannot be used in general, and directional derivatives must be employed instead. A very simple example has been used to illustrate the case of a bimodal eigenvalue. The necessary optimality conditions for an optimum solution to the problem of maximizing the lowest eigenvalue of a structure with a volume constraint are given, and they provide basic insight in the diculties connected with this type of optimization problems. Based on the experience gained by design sensitivity analysis studies and on the basis of deriving the necessary optimality conditions, di erent approaches to the development of iterative numerical algorithms for eigenvalue optimization problems are described. The basic idea of the approaches is to add constraints on the allowable design changes in cases of multiple eigenvalues, whereby design sensitivity expressions for simple as well as multiple eigenvalues become identical. The major advantage of using this approach is that the problem of determining the multiplicity of an eigenvalue correctly is no longer crucial when these additional constraints are considered. It has been shown that the necessary optimality conditions can be applied for development of an iterative numerical method for optimization of structural eigenvalues of arbitrary multiplicity. Thus, an e ective and elegant algorithm based on a mathematical programming approach has been developed. This mathematical programming approach has been used to solve the problem of maximizing the lowest eigenfrequency of a plate reinforced by ribs. If four supporting springs are added to the structure, the optimum solution has a bimodal lowest eigenvalue. The mathematical programming approach has been shown to work very well for problems involving multiple eigenvalues.
Interactive Engineering Design with ODESSY The di erent topics covered in this thesis have all been investigated with the aim of developing a general, exible, and reliable computer aided environement for interactive rational design, and several examples have been presented with a view to illustrate that ODESSY is a very e ective tool for interactive engineering design. The use of design sensitivity display and what-if studies for improving engineering designs has been illustrated via a turbine disk example. The use of design sensitivity display can greatly improve the design synthesis process as colour design sensitivity plots provide the designer with information concerning the in uence of design variables on performance measures. The use of automated design optimization has been illustrated on various design improvements of the turbine disk and on shape optimization of a hood of an automobile. Complicated design criteria can be taken into account, thereby making it easier for designers in industry to adopt and use ODESSY for design synthesis and optimization. The optimum design techniques implemented in ODESSY have proved to be invaluable tools in the development of new products. Not only do they speed up the development
process, but they often enable signi cant improvements of existing or even classical mechanical components. Optimization techniques are also most important in connection with design with new materials, e.g., ceramics or composites, where design rules and engineering tradition are often sparse or unavailable.
Further Work Although a very general design system has been developed, there are still many possibilities for extensions and improvement. The e orts pertaining to integration of ODESSY into commercial CAD systems, to develop facilities for automatic three-dimensional mesh generation, and other topics related to the pre- and postprocessing facilities will continue. Furthermore, the system should be expanded with respect to the classes of problems that can be handled in terms of implemented types of nite elements, analysis capabilities, and hence new types of structures and design objectives. Currently, only at shell elements generated from plate nite elements are available, so improved shell elements should be implemented. Furthermore, several other three-dimensional solid nite elements must be implemented when unstructured three-dimensional mesh facilities become available. Facilities for adaptive mesh generation can improve the reliability of the nite element model used during the redesign process and can therefore become a valuable tool for shape optimization. The extension into transient and non-linear types of analysis problems is a large task but such capabilities are very often necessary in order to solve practical real-life problems. Furthermore, expansions into other elds of analysis, for instance uid dynamics, acoustics, and magnetic eld theory are other possibilities. Inclusion of design criteria concerning fabrication cost can also be very useful. Expansions of analysis and design facilities into these areas will greatly improve the general applicability of the system.
181
182
References The references are listed alphabetically according to the last name of the author. Arora, J.S. (1989): Interactive Design Optimization of Structural Elements. In: Discretization Methods and Structural Optimization - Procedures and Applications (Eds. H.A. Eschenauer & G. Thierauf), pp. 10-16, Springer-Verlag, Berlin. Arora, J.S.; Haug, E.J. (1979): Methods of Design Sensitivity Analysis in Structural Optimization. AIAA Journal, Vol. 17, No. 9, pp. 970-974. Atrek, E. R.; Gallagher, H.; Ragsdell, K.M.; Zienkiewicz, O.C. (1984) (Eds.): New Directions in Optimum Structural Design, John Wiley & Sons, New York. Barthelemy, B.; Chon, C.T.; Haftka, R.T. (1988): Sensitivity Approximation of Static Structural Response. Finite Elements in Analysis and Design, Vol. 4, pp. 249-265. Barthelemy, B.; Haftka, R.T. (1988): Accuracy Analysis of the Semi-Analytical Method for Shape Sensitivity Analysis. AIAA Paper 88-2284, Proc. AIAA/ASME/ASCE/ASC 29th Structures, Structural Dynamics and Materials Conference, Part 1, pp. 562-581. Also Mechanics of Structures and Machines, Vol. 18, pp. 407-432 (1990). Bathe, K.-J. (1982): Finite Element Procedures in Engineering Analysis, Prentice-Hall, New Jersey. Bathe, K.-J.; Ramaswamy, S. (1980): An Accelerated Subspace Iteration Method. Computer Methods in Applied Mechanics and Engineering, Vol. 23, pp. 313-331. Bends e, M.P. (1989): Optimal Shape as a Material Distribution Problem. Structural Optimization, Vol. 1, No. 4, pp. 193-202. Bends e, M.P. (1994): Methods for the Optimization of Structural Topology, Shape and Material. Mathematical Institute, Technical University of Denmark, Lyngby, Denmark, 299 pp. Bends e, M.P.; Kikuchi, N. (1988): Generating Optimal Topologies in Structural Design Using a Homogenization Method. Computer Methods in Applied Mechanics and Engineering, Vol. 117, pp. 197-224. Bends e, M.P.; Mota Soares, C.A. (1993) (Eds.): Topology Design of Structures, Kluwer Academic Publishers, Dordrecht, The Netherlands. Bends e, M.P.; Olho , N.; Taylor, J.E. (1983): A Variational Formulation for Multicriteria Structural Optimization. Journal of Structural Mechanics, Vol. 11, pp. 523544. 183
Bends e, M.P.; Rodrigues, H.C. (1990): On Topology and Boundary Variations in
Shape Optimization. Control and Cybernetics, Vol. 19, No. 3-4, pp. 9-26. Bends e, M.P.; Rodrigues, H.C. (1991): Integrated Topology and Boundary Shape Optimization of 2-D Solids. Computer Methods in Applied Mechanics and Engineering, Vol. 87, pp. 15-34. Bends e, M.P.; Rodrigues, H.C.; Rasmussen, J. (1991): Topology and Boundary Shape Optimization As an Integrated Tool for Computer Aided Design. In: Engineering Optimization in Design Processes (Eds. H.A. Eschenauer, C. Mattheck & N. Olho ), pp. 27-34, Springer-Verlag, Berlin. Bennett, J.A.; Botkin, M.E. (1985): Structural Shape Optimization with Geometric Description and Adaptive Mesh Re nement. AIAA Journal, Vol. 23, No. 3, pp. 458-464. Bennett, J.A.; Botkin, M.E. (1986) (Eds.): The Optimum Shape. Automated Structural Design, Plenum Press, New York. Birker, T.; Lund, E. (1991): Development of a 3D Computer Aided Optimal Design System, Master Thesis. Institute of Mechanical Engineering, Aalborg University, Denmark. Botkin, M.E. (1988): Shape Optimization with Multiple Loading Conditions and Mesh Re nement. AIAA Paper 88-2299, Proc. AIAA/ASME/ASCE/AHS/ASC 29th Structures, Structural Dynamics and Materials Conf. (held in Williamsburg, Va., April 18-20), Vol. 2, pp. 689-695. Botkin, M.E. (1992): Three-Dimensional Shape Optimization Using Fully Automatic Mesh Generation. AIAA Journal, Vol. 30, No. 7, pp. 1932-1934. Botkin, M.E.; Bajorek, D.; Prasad, M. (1992): Feature-Based Structural Design of Solid Automotive Components. In: DE-Vol.42, Design Theory and Methodology -DTM '92 (Eds. D.L. Taylor & L.A. Stau er), pp. 57-64. Botkin, M.E.; Yang, R.J.; Bennett, J.A. (1986): Shape Optimization of Threedimensional Stamped and Solid Automotive Components. In: The Optimum Shape. Automated Structural Design (Eds. J.A. Bennett & M.E. Botkin), pp. 235-262, Plenum Press, New York. Braibant, V. (1986): Shape Sensitivity by Finite Elements. Journal of Structural Mechanics, Vol. 14, pp. 209-228. Braibant, V.; Fleury, C. (1984): Shape Optimal Design Using B-splines. Computer Methods in Applied Mechanics and Engineering, Vol. 44, pp. 247-267. Braibant, V.; Morelle, P. (1990): Shape Optimal Design and Free Mesh Generation. Structural Optimization, Vol. 2, pp. 223-231. Bratus, A.S.; Seyranian, A.P. (1983): Bimodal Solutions in Eigenvalue Optimization Problems. Prikl.Matem.Mekhan., Vol. 47, No. 4, pp. 546-554; Also in: Applied Mathematics in Mechanics, Vol. 47, pp. 451-457. Bremicker, M.; Chirehdast, M.; Kikuchi, N.; Papalambros, P. (1991): Integrated Topology and Shape Optimization in Structural Design. Mechanics of Structures and Machines, Vol. 19, pp. 551-587. 184
Cea, J. (1981): Problems of Shape Optimal Design. In: Optimization of Distributed
Parameter Structures (Eds. E.J. Haug & J. Cea), Vol. 2, pp. 1005-1048, Sijtho & Nordho , The Netherlands. Cheng, G.; Gu, Y.; Wang, X (1991): Improvement of Semi-Analytic Sensitivity Analysis and MCADS. In: Engineering Optimization in Design Processes (Eds. H.A. Eschenauer, C. Mattheck & N. Olho ), pp. 221-223, Springer-Verlag, Berlin. Cheng, G.; Gu, Y.; Zhou, Y. (1989): Accuracy of Semi-Analytical Sensitivity Analysis. Finite Elements in Analysis and Design, Vol. 6, pp. 113-128. Cheng, G.; Liu, Y. (1987): A New Computational Scheme for Sensitivity Analysis. Engineering Optimization, Vol. 12, pp. 219-235. Cheng, G.; Olho , N. (1981): An Investigation Concerning Optimal Design of Solid Elastic Plates. International Journal of Solids and Structures, Vol. 17, pp. 305-323. Cheng, G.; Olho , N. (1991): New Method of Error Analysis and Detection in SemiAnalytical Sensitivity Analysis. Report No. 36, Institute of Mechanical Engineering, Aalborg University, Denmark, 34 pp. Cheng, G.; Olho , N. (1993): Rigid Body Motion Test Against Error in SemiAnalytical Sensitivity Analysis. Computers & Structures, Vol. 46, No. 3, pp. 515-527. Choi, K.K. (1985): Shape Design Sensitivity Analysis of Displacement and Stress Constraints. Journal of Structural Mechanics, Vol. 13, pp. 27-41. Choi, K.K.; Chang, K.-H. (1991): Shape Design Sensitivity Analysis and WhatIf Workstation For Elastic Solids. Technical Report R-105, Center for Simulation and Design Optimization and Department of Mechanical Engineering, The University of Iowa, 33 pp. Choi, K.K.; Haug, E.J. (1981a): Optimization of Structures with Repeated Eigenvalues. In: Optimization of Distributed Parameter Structures (Eds. E.J. Haug & J. Cea), Vol. 1, pp. 219-277, Sijtho & Nordho , The Netherlands. Choi, K.K.; Haug, E.J. (1981b): Repeated Eigenvalues in Mechanical Optimization Problems. Contemporary Mathematics, Vol. 4, pp. 61-86. Choi, K.K.; Haug, E.J. (1983): Shape Design Sensitivity Analysis of Elastic Structures. Journal of Structural Mechanics, Vol. 11, No. 2, pp. 231-269. Choi, K.K.; Haug, E.J.; Lam, H.L. (1982): A Numerical Method For Distributed Parameter Structural Optimization Problems with Repeated Eigenvalues. Journal of Structural Mechanics, Vol. 10, No. 2, pp. 191-207. Choi, K.K.; Haug, E.J.; Seong, H.G. (1983): An Iterative Method For Finite Dimensional Structural Optimization Problems with Repeated Eigenvalues. International Journal for Numerical Methods in Engineering, Vol. 19, pp. 93-112. Choi, K.K.; Santos, J.L.T.; Frederick, M.C. (1985): Implementation of Design Sensitivity Analysis with Existing Finite Element Codes. Center for Computer Aided Design and Department of Mechanical Engineering, The University of Iowa, 9 pp. Choi, K.K.; Seong, H.W. (1986): A Domain Method for Shape Design of Built-up Structures. Computer Methods in Applied Mechanics and Engineering, Vol. 57, pp. 1-15.
185
Choi, K.K.; Twu, S.-L. (1988): On Equivalence of Continuum and Discrete Methods
of Shape Sensitivity Analysis. AIAA Journal, Vol. 27, No. 10, pp. 1418-1424 Clarke, F. (1990): Optimization and Nonsmooth Analysis, Classics in Applied Mathematics 5. Society for Industrial and Applied Mathematics, Philadelphia. Cook, R.D.; Malkus, D.S.; Plesha, M.E. (1989): Concepts and Applications of Finite Element Analysis, 3rd Edition, Wiley & Sons, New York. Courant, R.; Hilbert, D. (1953): Methods of Mathematical Physics, Vol. 1, Interscience Publishers, New York. Cox, S.J. (1992): The Shape of the Ideal Column. The Mathematical Intelligencer, Vol. 14, pp. 6-24. Cox, S.J.; Overton, M.L. (1992): On the Optimal Design of Columns against Buckling. SIAM Journal on Mathematical Analysis, Vol. 23, No. 2, pp. 287-325. Dems, K.; Haftka, R.T. (1989): Two Approaches to Sensitivity Analysis for Shape Variation of Structures. Mechanics of Structures and Machines, Vol. 16, No. 4, pp. 501-522. Dems, K.; Mroz, Z. (1983): Variational Approach by Means of Adjoint Systems to Structural Optimization and Sensitivity Analysis - I Variation of Material Parameters within Fixed Domain. International Journal of Solids and Structures, Vol. 19, pp. 677692. Dems, K.; Mroz, Z. (1984): Variational Approach by Means of Adjoint Systems to Structural Optimization and Sensitivity Analysis - II Structure Shape Variation. International Journal of Solids and Structures, Vol. 20, pp. 527-552. Dems, K.; Mroz, Z. (1993): On Shape Sensitivity Approaches in the Numerical Analysis of Structures. Structural Optimization, Vol. 6, pp. 86-93. Dhatt, G.; Touzot, G. (1984): The Finite Element Method Displayed, Wiley & Sons, New York. Diaz, A.R.; Kikuchi, N. (1992): Solution to Shape and Topology Eigenvalue Optimization Problems using a Homogenization Method. International Journal for Numerical Methods in Engineering, Vol. 35, pp. 1487-1502. Diaz, A.R.; Kikuchi, N.; Papalambros, P.; Taylor, J.E. (1983): Design of an Optimal Grid for Finite Element Methods. Journal of Structural Mechanics, Vol. 11, pp. 215-230. Ding, Y. (1986): Shape Optimization of Structures: a Literature Survey. Computers & Structures, Vol. 24, No. 6, pp. 985-1004. El-Sayed, M.E.M.; Zumwalt, K.W. (1991): Ecient Design Sensitivity Derivatives for Multi-Load Case Structures as an Integrated Part of Finite Element Analysis. Computers & Structures, Vol. 40, No. 6, pp. 1461-1467. Eschenauer, H.A. (1986): Numerical and Experimental Investigations of Structural Optimization of Engineering Designs, Siegen: Bonn + Fries, Druckerei und Verlag. Eschenauer, H.A.; Koski, J.; Osyczka, A. (1990) (Eds.): Multicriteria Design Optimization, Springer-Verlag, Berlin. 186
Eschenauer, H.A.; Mattheck, C.; Olho , N. (1991) (Eds.): Engineering Optimization in Design Processes, Springer-Verlag, Berlin.
Eschenauer, H.A.; Olho , N. (1983) (Eds.): Optimization Methods in Structural Design, B.I.-Wissenschaftsverlag, Mannheim.
Eschenauer, H.A.; Thierauf, G. (1989) (Eds.): Discretization Methods and Structural
Optimization - Procedures and Applications, Springer-Verlag, Berlin. Esping, B.J.D. (1983): Minimum Weight Design of Membrane Structures, Ph.D. Thesis, Dept. Aeronautical Structures and Materials, The Royal Institute of Technology, Stockholm, Report 83-1. Esping, B.J.D. (1984): Minimum Weight Design of Membrane Structures Using Eight Node Isoparametric Elements and Numerical Derivatives. Computers & Structures, Vol. 19, No. 4, pp. 591-604. Esping, B.J.D. (1986): The OASIS Structural Optimization System. Computers & Structures, Vol. 23, No. 3, pp. 365-377. Fenyes, P.A.; Lust, R.V. (1991): Error Analysis of Semi-Analytical Displacement Derivatives for Shape and Sizing Variables. AIAA Journal, Vol. 29, pp. 271-279. Fleury, C. (1987): Computer Aided Optimal Design of Elastic Structures. In: Computer Aided Optimal Design: Structural and Mechanical Systems (Ed. C.A. Mota Soares), pp. 831-900, Springer-Verlag, Berlin. Fleury, C. (1989): CONLIN: An Ecient Dual Optimizer Based On Convex Approximation Concepts. Structural Optimization, Vol. 1, pp. 81-89. Fleury, C.; Braibant, V. (1986): Structural Optimization: A New Dual Method Using Mixed Variables. International Journal for Numerical Methods in Engineering, Vol. 23, 409-428. Fleury, C.; Braibant, V. (1987): Structural Synthesis: Towards a Reliable CAD-tool. In: Computer Aided Optimal Design: Structural and Mechanical Systems (Ed. C.A. Mota Soares), pp. 137-149, Springer-Verlag, Berlin. Gajewski, A. (1990): Multimodal Optimal Design of Structural Elements (A Survey). Mechanika Teoretyczna i Stosowana, Vol. 28, pp. 75-92. Gajewski, A.; Zyczkowski, M. (1988): Optimal Structural Design Under Stability Constraints. Kluwer Academic Publishers, Dordrecht-Boston-London. George, P.L. (1988): Modulef: Generation Automatique de Maillages (in French). Collection Didactique, edite par Institut National de Recherche en Informatique et en Automatique, pp. 75-80. ISBN 2-7261-0522-X, ISSN 0299-0733. Gilmore, B.J.; Hoeltzel, D.A.; Azarm, S.; Eschenauer, H.A. (1993) (Eds.): Advances in Design Automation. Proc. 1993 ASME Design Technical Conferences - 19th Design Automation Conference, Albuquerque, New Mexico, The American Society of Mechanical Engineers, New York. Gu, Y.; Cheng, G. (1990): Structural Shape Optimization Integrated with CAD Environment. Structural Optimization, Vol. 2, pp. 23-28.
187
Haber, R.B. (1987): A New Variational Approach to Structural Shape Sensitivity Anal-
ysis. In: Computer Aided Optimal Design: Structural and Mechanical Systems (Ed. C.A. Mota Soares), pp. 573-587, Springer-Verlag, Berlin. Haftka, R.T.; Adelman, H.M. (1989): Recent Developments in Structural Sensitivity Analysis. Structural Optimization, Vol. 1, pp. 137 - 151. Haftka, R.T.; Grandhi, R.V. (1986): Structural Shape Optimization - A Survey. Computer Methods in Applied Mechanics and Engineering, Vol. 57, pp. 91-106. Haug, E.J. (1981): A Review of Distributed Parameter Structural Optimization Literature. In: Optimization of Distributed Parameter Structures (Eds. E.J. Haug & J. Cea), Vol. 1, pp. 3-74, Sijtho & Nordho , The Netherlands. Haug, E.J. (1993) (Ed.): Concurrent Engineering: Tools and Technologies for Mechanical System Design, Springer-Verlag, Berlin. Haug, E.J.; Cea, J. (1981) (Eds.): Optimization of Distributed Parameter Structures, Vol. 1: pp. 1-842, Vol. 2: pp. 843-1609, Sijtho & Nordho , The Netherlands. Haug, E.J.; Choi, K.K. (1982): Systematic Occurence of Repeated Eigenvalues in Structural Optimization. Journal of Optimization Theory and Applications, Vol. 38, No. 2, pp. 251-274. Haug, E.J.; Choi, K.K. (1986): Material Derivative Methods for Shape Design Sensitivity Analysis. In: The Optimum Shape. Automated Structural Design (Eds. J.A. Bennett & M.E. Botkin), pp. 29-60, Plenum Press, New York. Haug, E.J.; Choi, K.K.; Komkov, V. (1986): Design Sensitivity Analysis of Structural Systems, Academic Press, New York. Haug, E.J.; Rousselet, B. (1980a): Design Sensitivity Analysis in Structural Mechanics. I: Static Response Variation. Journal of Structural Mechanics, Vol. 8, No. 1, pp. 17-41. Haug, E.J.; Rousselet, B. (1980b): Design Sensitivity Analysis in Structural Mechanics. II: Eigenvalue Variations. Journal of Structural Mechanics, Vol. 8, No. 2, pp. 161-186. Herskovits, J. (1993): Proc. Structural Optimization '93 - The World Congress on Optimal Design of Structural Systems, Vol. 1 & 2, Federal University of Rio de Janeiro, Brazil. Hughes, T.J.R. (1987): The Finite Element Method, Prentice-Hall, New Jersey. Hughes, T.J.R.; Cohen, M. (1978): The \Heterosis" Finite Element for Plate Bending. Computers & Structures, Vol. 9, pp. 445-450. Hughes, T.J.R.; Cohen, M.; Haroun, M. (1978): Reduced and Selective Integration Techniques in the Finite Element Analysis of Plates. Nuclear Engineering and Design, Vol. 46, pp. 203-222. Hornlein, H.R.E.M. (1987): Take-o in Optimum Structural Design. In: Computer Aided Optimal Design. Structural and Mechanical Systems (Ed. C.A. Mota Soares), NATO ASI Series F: Computer and System Sciences, Vol. 27, pp. 901-919, Springer-Verlag, Berlin. 188
Kibsgaard, S. (1991): Analysis and Optimization of FE-discretized Structures, Ph.D.
Thesis. Special Report No. 6, Institute of Mechanical Engineering, Aalborg University, Denmark. Kibsgaard, S.; Olho , N.; Rasmussen, J. (1989): Concept of an Optimization System. In: Proc. OPTI89 - First Int. Conf. on Computer Aided Optimum Design of Structures: Applications (Eds. C.A. Brebbia & S. Hernandez), pp. 79-88, Computational Mechanics Publications, Springer-Verlag, Berlin. Kikuchi, N.; Chung, K.Y.; Torigaki, T.; Taylor, J.E. (1986): Adaptive Finite Element Methods for Shape Optimization of Linearly Elastic Structures. In: The Optimum Shape. Automated Structural Design (Eds. J.A. Bennett & M.E. Botkin), pp. 139-170, Plenum Press, New York. Koski, J.; Silvennoinen; R. (1990): Multicriteria Design of Ceramic Components. In: Multicriteria Design Optimization (Eds. H.A. Eschenauer, J. Koski & A. Osyczka), pp. 447-463, Springer-Verlag, Berlin. Kristensen, E.S.; Madsen, N.F. (1976): On the Optimum Shape of Fillets in Plates Subjected to Multiple In-plane Loading Cases. International Journal for Numerical Methods in Engineering, Vol. 10, pp. 1007-1019. Ladefoged, T. (1988): Triangular Ring Element with Analytic Expressions for Sti ness and Mass Matrix. Computer Methods in Applied Mechanics and Engineering, Vol. 67, pp. 171-187. Lancaster, P. (1964): On Eigenvalues of Matrices Dependent on a Parameter. Numerische Mathematik, Vol. 6, pp. 377-387. Lund, E. (1993): Design Sensitivity Analysis of Finite Element Discretized Structures. Report No. 55, Institute of Mechanical Engineering, Aalborg University, Denmark, 31 pp. An earlier version of the paper can be found in: Shape Design Sensitivity Analysis of Finite Element Discretized Structures. In: Lecture Notes for COMETT course on Computer Aided Optimum Design of Structures, August 1993, Aalborg. Lund, E.; Birker, T.; Rasmussen, J. (1991): CAOS Computer Aided Optimization System Ver. 2.1. User's Manual and Reference. Special Report No. 9, Institute of Mechanical Engineering, Aalborg University, Denmark, 161 pp. Lund, E.; Olho , N. (1993a): Reliable and Ecient Finite Element Based Design Sensitivity Analysis of Eigenvalues. In: Proc. Structural Optimization '93 - The World Congress on Optimal Design of Structural Systems (Ed. J. Herskovits), Vol. 2, pp. 197204, Federal University of Rio de Janeiro, Brazil. Lund, E.; Olho , N. (1993b): Shape Design Sensitivity Analysis of Eigenvalues Using \Exact" Numerical Di erentiation of Finite Element Matrices. Report No. 54, Institute of Mechanical Engineering, Aalborg University, Denmark, 19 pp. To appear in Structural Optimization, Vol. 8 (1994). Lund, E.; Rasmussen, J. (1994): A General and Flexible Method of Problem De nition in Structural Optimization Systems. Report No. 58, Institute of Mechanical Engineering, Aalborg University, Denmark, 18 pp. To appear in Structural Optimization, Vol. 8 (1994)). 189
Masmoudi, M.; Broudiscou, C.; Guillaume, P. (1993): Automatic Di erentiation and Shape Optimization. In: Proc. Structural Optimization '93 - The World Congress on Optimal Design of Structural Systems (Ed. J. Herskovits), Vol. 2, pp. 189-196, Federal University of Rio de Janeiro, Brazil. Masur, E.F. (1984): Optimal Structural Design under Multiple Eigenvalue Constraints. International Journal of Solids and Structures, Vol. 20, pp. 211-231. Masur, E.F. (1985): Some Additional Comments on Optimal Structural Design under Multiple Eigenvalue Constraints. International Journal of Solids and Structures, Vol. 21, pp. 117-120. Masur, E.F.; Mroz, Z. (1979): Non-stationary Optimality Conditions in Structural Design. International Journal of Solids and Structures, Vol. 15, pp. 503-512. Masur, E.F.; Mroz, Z. (1980): Singular Solutions in Structural Optimization Problems. In: Variational Methods in the Mechanics of Solids (Ed. S. Nemat-Nasser), pp. 337-343, Evanston, Illinois, Pergamon Press, New York. McLean, A.F.; Hartsock, D.L. (1989): Design with Structural Ceramics. In: Structural Mechanics (Ed. J.B. Wachtman), pp. 27-97, Academic Press, London. Mills-Curran, W.C. (1988): Calculation of Eigenvector Derivatives for Structures with Repeated Eigenvalues. AIAA Journal, Vol. 26, No. 7, pp. 867-871. Mlejnek, H.P. (1992): Accuracy of Semi-Analytical Sensitivities and Its Improvement by the \Natural Method". Structural Optimization, Vol. 4, pp. 128-131. Morelle, P.; Braibant, V. (1991): Multi-Model Optimization of Large Scale Structures. Samtech, Liege, Belgium. Morris, A.J. (1982) (Ed.): Foundations of Structural Optimization: A Uni ed Approach, Chichester, England. Mota Soares, C.A. (1987) (Ed.): Computer Aided Optimal Design: Structural and Mechanical Systems, Springer-Verlag, Berlin. Mota Soares, C.A.; Rodrigues, H.C.; Choi, K.K. (1984): Shape Optimal Structural Design Using Boundary Elements and Minimum Compliance Techniques. Journal of Mechanisms, Transmissions and Automation in Design, Vol. 106, pp. 518-523. Mouritsen, O. . (1992): Introduktion til Elementmetoden (In Danish), Aalborg University, Denmark, 63 pp. Myslinski, A.; Sokolowski, J. (1985): Nondi erentiable Optimization Problems For Elliptic Systems. SIAM Journal on Control and Optimization, Vol. 23, No. 4, pp. 632-648. Olho , N. (1974): Optimal Design of Vibrating Rectangular Plates. International Journal of Solids and Structures, Vol. 10, pp. 93-109. Olho , N. (1975): On Singularities, Local Optima, and Formation of Sti eners in Optimal Design of Plates. In: IUTAM Symposium, Warsaw/Poland 1973 (Eds. A. Sawczuk & Z. Mroz), pp. 82-103, Springer-Verlag, Berlin. Olho , N. (1980): Optimal Design with Respect to Structural Eigenvalues. In: Proc. XVth International IUTAM Congress on Theoretical and Applied Mechanics, Toronto 1980 (Eds. F.P.J. Rimrott & B. Tabarrok), pp. 133-149, Nord-Holland, The Netherlands. 190
Olho , N. (1981a): Optimization of Columns against Buckling. In: Optimization of
Distributed Parameter Structures (Eds. E.J. Haug & J. Cea), Vol. 1, pp. 152-176, Sijtho & Nordho , The Netherlands. Olho , N. (1981b): Optimization of Transversely Vibrating Beams and Rotating Shafts. In: Optimization of Distributed Parameter Structures (Eds. E.J. Haug & J. Cea), Vol. 1, pp. 177-199, Sijtho & Nordho , The Netherlands. Olho , N. (1989): Multicriterion Structural Optimization Via Bound Formulation and Mathematical Programming. Structural Optimization, Vol. 1, pp. 11-17. Olho , N.; Bends e, M.P.; Rasmussen, J. (1991): On CAD-integrated Structural Topology and Design Optimization. Computer Methods in Applied Mechanics and Engineering, Vol. 89, pp. 259-279. Olho , N., Krog, L.; Thomsen, J. (1993): Bi-Material Topology Optimization. In: Proc. Structural Optimization '93 - The World Congress on Optimal Design of Structural Systems (Ed. J. Herskovits), Vol. 1, pp. 327-334, Federal University of Rio de Janeiro, Brazil. Olho , N., Lund, E. (1994): Finite Element Based Engineering Design Sensitivity Analysis and Optimization. In: Advances in Structural Optimization (Ed. J. Herskovits), Kluwer (to appear). Olho , N.; Lund, E.; Rasmussen, J. (1992): Concurrent Engineering Design Optimization In a CAD Environment. Special Report No. 16, Institute of Mechanical Engineering, Aalborg University, Denmark, 64 pp; Also in: Concurrent Engineering: Tools and Technologies for Mechanical System Design (Ed. E.J. Haug), pp. 523-586, Springer-Verlag, Berlin (1993). Olho , N.; Plaut, R.H. (1983): Bimodal Optimization of Vibrating Shallow Arches. International Journal of Solids and Structures, Vol. 19, No. 6, pp. 553-570. Olho , N.; Rasmussen, J. (1991a): Study of Inaccuracy in Semi-Analytical Sensitivity Analysis - A Model Problem. Structural Optimization, Vol. 3, pp. 203-213. Olho , N.; Rasmussen, J. (1991b): Method of Error Elimination for a Class of SemiAnalytical Sensitivity Analysis Problems. In: Engineering Optimization in Design Processes (Eds. H.A. Eschenauer, C. Mattheck & N. Olho ), pp. 193-200, Springer-Verlag, Berlin. Olho , N.; Rasmussen, J.; Lund, E. (1992): A Method of \Exact" Numerical Di erentiation for Error Elimination in Finite Element Based Semi-Analytical Shape Sensitivity Analysis. Special Report No. 10, Institute of Mechanical Engineering, Aalborg University, Denmark, 79 pp. Also Mechanics of Structures and Machines, Vol. 21, pp. 1-66 (1993). Olho , N.; Rasmussen, S.H. (1977): On Single and Bimodal Optimum Buckling Loads of Clamped Columns. International Journal of Solids and Structures, Vol. 13, pp. 605-614. Olho , N.; Taylor, J.E. (1983): On Structural Optimization. Journal of Applied Mechanics, Vol. 50, pp. 1139-1151.
191
Overton, M.L. (1988): On Minimizing the Maximum Eigenvalue of a Symmetric Matrix.
SIAM Journal on Matrix Analysis and Applications, Vol. 9, No. 2, pp. 256-268. Pedersen, P. (1973): Some Properties of Linear Strain Triangles and Optimal Finite Element Models. International Journal of Numerical Methods in Engineering, Vol. 7, pp. 415-429. Pedersen, P. (1981): Design with Several Eigenvalue Constraints by Finite Elements and Linear Programming. Journal of Structural Mechanics, Vol. 10, No. 3, pp. 243-271. Pedersen, P. (1983): A Uni ed Approach to Optimal Design. In: Optimization Methods in Structural Design (Eds. H.A. Eschenauer & N. Olho ), pp. 182-187, B.I.-Wissenschaftsverlag, Mannheim. Pedersen, P. (1988): Design for Minimum Stress Concentration - Some Practical Aspects. In: Structural Optimization (Eds. G.I.N. Rozvany & B.L. Karihaloo), pp. 225-232, Kluwer, Dordrecht, The Netherlands. Pedersen, P. (1989): On Optimal Orientation of Orthotropic Materials. Structural Optimization, Vol. 1, pp. 101-106. Pedersen, P. (1990): Bounds on Elastic Energy in Solids of Orthotropic Materials. Structural Optimization, Vol. 2, pp. 55-63. Pedersen, P. (1991): On Thickness and Orientational Design with Orthotropic Materials. Structural Optimization, Vol. 3, pp. 69-78. Pedersen, P. (1993) (Ed.): Optimal Design with Advanced Materials, Elsevier, The Netherlands. Pedersen, P.; Cheng, G.; Rasmussen, J. (1989): On Accuracy Problems for SemiAnalytical Sensitivity Analysis. Mechanics of Structures and Machines, Vol. 17, No. 3, pp. 373-384. Pedersen, P.; Laursen, C.L. (1983): Design for Minimum Stress Concentration by Finite Elements and Linear Programming. Journal of Structural Mechanics, Vol. 10, No. 4, pp. 375-391. Plaut, R.H.; Johnson, L.W.; Olho , N. (1986): Bimodal Optimization of Compressed Columns on Elastic Foundations. Journal of Applied Mechanics, Vol. 53, pp. 130-134. Prager, S.; Prager, W. (1979): A Note on Optimal Design of Columns. Int. J. Mech. Sci., Vol. 21, pp. 249-251. Rasmussen, J. (1989): Development of the Interactive Structural Shape Optimization System CAOS (in Danish), Ph.D. Thesis, Institute of Mechanical Engineering, Aalborg University, Denmark, Special Report No. 1a. Rasmussen, J. (1990): The Structural Optimization System CAOS. Structural Optimization, Vol. 2, pp. 109-115. Rasmussen, J. (1991): Shape Optimization and CAD. International Journal of Systems Automation: Research and Applications (SARA), Vol. 89, pp. 259-279. Rasmussen, J.; Lund, E.; Birker, T. (1992): Collection of Examples, CAOS Optimization System, 3rd edition. Special Report No. 1c, Institute of Mechanical Engineering, 192
Aalborg University, Denmark, 119 pp.
Rasmussen, J.; Lund, E.; Birker, T.; Olho , N. (1993): The CAOS System.
International Series of Numerical Mathematics, Vol. 110, pp. 75-96, Birkhuser Verlag, Basel. Rasmussen, J.; Lund, E.; Olho , N. (1992): Structural Optimization by the Finite Element Method. In: Lecture Notes for European School CIMI/COMETT on Computer Aided Design of Structures, Journees Universitaires, 13-17 July 1992, Nice, France. Rasmussen, J.; Lund, E.; Olho , N. (1993a): Parametric Modeling and Analysis for Optimum Design. In: Proc. Structural Optimization '93 - The World Congress on Optimal Design of Structural Systems (Ed. J. Herskovits), Vol. 2, pp. 407-414, Federal University of Rio de Janeiro, Brazil. Rasmussen, J.; Lund, E.; Olho , N. (1993b): Integration of Parametric Modeling and Structural Analysis for Optimum Design. In: Advances in Design Automation. Proc. 1993 ASME Design Technical Conferences - 19th Design Automation Conference, Albuquerque, New Mexico (Eds. B.J. Gilmore, D.A. Hoeltzel, S. Azarm & H.A. Eschenauer), Vol. 1, pp. 637-648, The American Society of Mechanical Engineers, New York. Rasmussen, J.; Lund, E.; Olho , N.; Birker, T. (1993): Structural Topology and Shape Optimization with the CAOS System. In: Lecture Notes for COMETT course on Computer Aided Optimal Design, February 1993, Eindhoven. Rasmussen, J.; Olho , N.; Lund, E. (1992): Foundations for Optimum Design System Development. In: Lecture Notes for Advanced TEMPUS course on Numerical Methods in Computer Optimal Design, 11-15 maj 1992, Zakopane, Poland. Rasmussen, J.; Thomsen, J.; Olho , N. (1993): Topology and Boundary Variation Design Methods in a CAD System. In: Topology Design of Structures (Eds. M.P. Bends e & C.A. Mota Soares), Kluwer Academic Publishers, Dordrecht, The Netherlands. Ringertz, U. (1992): Optimal Design and Imperfection Sensitivity of Nonlinear Shell Structures. FFA TN 1992-30, The Aeronautical Research Institute of Sweden, 33 pp. Rodrigues, H.C. (1988): Shape Optimal Design of Elastic Bodies Using a Mixed Variational Formulation. Computer Methods in Applied Mechanics and Engineering, Vol. 69, pp. 29-44. Rosengren, R. (1989): Shape Optimization of a Saab 9000 Wishbone. SAAB-SCANIA, Sweden. Rozvany, G.I.N. (1989): Structural Design via Optimality Criteria - The Prager Approach to Structural Optimization, Kluwer, Dordrecht, The Netherlands. Rozvany, G.I.N. (1993) (Ed.): Optimization of Large Structural Systems, Vol. 1: pp. 1-622, Vol. 2: 623-1198, Kluwer, Dordrecht, The Netherlands. Rozvany, G.I.N.; Olho , N.; Cheng, G.; Taylor, J.E. (1982): On the Solid Plate Paradox in Structural Optimization. Journal of Structural Mechanics, Vol. 10, No. 1, pp. 1-32. Rozvany, G.I.N.; Karihaloo, B.L. (1988) (Eds.): Structural Optimization, Kluwer, Dordrecht, The Netherlands.
193
Santos, J.L.T.; Choi, K.K. (1989): Integrated Computational Considerations for Large
Scale Structural Design Sensitivity Analysis and Optimization. In: Discretization Methods and Structural Optimization - Procedures and Applications (Eds. H.A. Eschenauer & G. Thierauf), pp. 299-307, Springer-Verlag, Berlin. Seyranian, A.P. (1983): A Solution of a Problem of Lagrange. Dokl. Akad. Nauk SSSR, Vol. 271, pp. 337-340; Also Sov. Phys. Dokl., Vol. 28, pp. 550-551. Seyranian, A.P. (1984): On a Problem of Lagrange. Mekhanika Tverdogo Tela, No. 2, pp. 101-111; Also Mechanics of Solids, Vol. 19, No. 2, pp. 100-111. Seyranian, A.P. (1987): Multiple Eigenvalues in Optimization Problems. Prikl. Mat. Mekh., Vol. 51, pp. 349-352; Also Applied Mathematics in Mechanics, Vol. 51, pp. 272275. Seyranian, A.P.; Lund, E.; Olho , N. (1994): Multiple Eigenvalues in Structural Optimization Problems. Special Report No. 21, Institute of Mechanical Engineering, Aalborg University, Denmark, 51 pp. (To appear in Structural Optimization). Schmit, L.A. (1982): Structural Synthesis - Its Genesis and Development. AIAA Journal, Vol. 20, pp. 992-1000. Sobieszanski-Sobieski, J.; Rogers, J.L. (1984): A Programming System for Research and Applications in Structural Optimization. In: New directions in optimum structural design (Eds. E. Atrek, R.H. Gallagher, K.M. Ragsdell & O.C. Zienkiewicz), pp. 563-585, John Wiley & Sons, New York. Stanton, E.L. (1986): Geometric Modeling for Structural and Material Shape Optimization. In: The Optimum Shape. Automated Structural Design (Eds. J.A. Bennett & M.E. Botkin), pp. 365-383, Plenum Press, New York. Svanberg, K. (1987): The Method of Moving Asymptotes - A New Method For Structural Optimization. International Journal for Numerical Methods in Engineering, Vol. 24, pp. 359-373. Tadjbakhsh, I.; Keller, J.B. (1962): Strongest Columns and Isoperimetric Inequalities For Eigenvalues. Journal of Applied Mechanics, Vol. 29, pp. 159-164. Taylor, J.E.; Bends e, M.P. (1984): An Interpretation of Min-Max Structural Design Problems Including a Method for Relaxing Constraints. International Journal of Solids and Structures, Vol. 20, pp. 301-314. Thomsen, J. (1992): Optimering af anisotrope materialers egenskaber og konstruktioners topologi (in Danish), Ph.D. Thesis. Special Report No. 14. Institute of Mechanical Engineering, Aalborg University, Denmark. Vanderplaats, G.N. (1987): ADS - A FORTRAN program for Automated Design Synthesis - version 2.01. Engineering Design Optimization, INC. Wang, S.-Y.; Sun, Y.; Gallagher, R.H. (1985): Sensitivity Analysis in Shape Optimization of Continuum Structures. Computers & Structures, Vol. 20, No. 5, pp. 855-867. Wexler, A.S. (1987): Automatic Evaluation of Derivatives. Appl. Math. Comp., Vol. 24, pp. 19-46. 194
Wittrick, W.H. (1962): Rates of Change of Eigenvalues, With Reference to Buckling
and Vibration Problems. Journal of the Royal Aeronautical Society, Vol. 66, pp. 590-591. Yang, R.J.; Botkin, M.E. (1986): The Relationship Between the Variational Approach and the Implicit Di erentiation Approach to Shape Design Sensitivities. In: The Optimum Shape. Automated Structural Design (Eds. J.A. Bennett & M.E. Botkin), pp. 61-77, Plenum Press, New York. Yang, R.-J.; Choi, K.K. (1985): Accuracy of Finite Element Based Shape Design Sensitivity Analysis. Journal of Structural Mechanics, Vol. 13, No. 2, pp. 223-239. Yao, T.-M.; Choi, K.K. (1989): 3-D Shape Optimal Design and Automatic Finite Element Regridding. International Journal for Numerical Methods in Engineering, Vol. 28, pp. 369-384. Zhong, W.; Cheng, G. (1986): Second-order Sensitivity Analysis of Multimodal Eigenvalues and Related Optimization Techniques. Journal of Structural Mechanics, Vol. 14, No. 4, pp. 421-436. Zienkiewicz, O.C.; Campbell, J.S. (1973): Shape Optimization and Sequential Linear Programming. In: Optimum Structural Design, Theory and Applications (Eds. R.H. Gallagher & O.C.Zienkiewicz), Wiley and Sons, London, pp. 109-126. Zienkiewicz, O.C.; Taylor, R.L. (1989): The Finite Element Method. Volume 1 & 2, 4th Edition, McGraw-Hill, London. Zolesio, J.-P. (1981): The Material Derivative (or Speed) Method for Shape Optimization. In: Optimization of Distributed Parameter Structures (Eds. E.J. Haug & J. Cea), Vol. 2, pp. 1089-1151, Sijtho & Nordho , The Netherlands. Zyczkowski, M. (1989) (ed.): Structural Optimization Under Stability and Vibration Constraints. Springer, Wien - New York.
195
196
APPENDIX
197
198
Appendix
A
3D Solid Isoparametric Finite Elements
T an 8-node and a 20-node element. The 20-node element can have curved element sides as illustrated in Fig. A.1.
wo 3D isoparametric nite elements have been implemented in ODESSY, that is,
z,w
ζ
η ξ
ζ
η ξ
x,u
y,v
Figure A.1: Domain, node numbering, and nodal degrees of freedom of 8- and 20-node isoparametric nite elements.
A.1 Shape Functions for 3D Solid Isoparametric Finite Elements. Within the isoparametric formulation of a nite element with an arbitrary number n of nodal points, the same set of shape functions Ni = Ni(; ; ); i = 1; : : : ; n (A.1) 199
200
A.2. Element Sti ness Matrix
is used for interpolation of global x, y, and z coordinates from nodal values xi , yi, and zi and of displacements functions u, v, and w from nodal values ui, vi, and wi, i.e.,
x = u(x; y; z) =
n X i=1
n X i=1
Nixi ; y =
n X i=1
Niui; v(x; y; z) =
Niyi; z =
n X i=1
n X i=1
Nizi
Nivi; w(x; y; z) =
Shape functions for the 8-node element are given in Table A.1.
(A.2) n X i=1
Niwi
(A.3)
Table A.1: Shape functions for 8-node 3D solid isoparametric element.
N1 = 18 (1 )(1 )(1 ) N3 = 81 (1 + )(1 + )(1 ) N5 = 81 (1 )(1 )(1 + ) N7 = 81 (1 + )(1 + )(1 + )
N2 = 81 (1 + )(1 )(1 ) N4 = 81 (1 )(1 + )(1 ) N6 = 18 (1 + )(1 )(1 + ) N8 = 81 (1 )(1 + )(1 + )
and shape functions for the 20-node element are given in Table A.2.
A.2 Element Sti ness Matrix The element sti ness matrix k is given by
k=
Z
BT E B jJj d
(A.4)
Here, is the domain of the nite element described in curvilinear, non-dimensional -- coordinates for the element, see Fig. A.1, and jJj is the determinant of the Jacobian matrix J which at each point de nes the transformation of di erentials d , d, and d into dx, dy, and dz. Like J, the strain-displacement matrix B depends on coordinates of the nodal points, whereas the constitutive matrix E depends only on the constitutive parameters of the assumed linearly elastic material which, in the implementation in ODESSY, can be either isotropic or anisotropic. The expressions for the Jacobian J and for the straindisplacement matrix B are given in the following. In terms of the vector di of nodal degrees of freedom
di = fui vi wigT ; i = 1; : : : ; n
(A.5)
the element nodal vector d containing nodal displacements is
d = fdT1 dT2 : : : dTi : : : dTn gT and the strain vector function " is "(x; y; z) = f"x "y "z xy yz xz gT
(A.6) (A.7)
Appendix A. 3D Solid Isoparametric Finite Elements
201
Table A.2: Shape functions for 20-node 3D solid isoparametric element.
N9 = 14 (1 2)(1 )(1 ) N10 = 41 (1 + )(1 2)(1 ) N11 = 14 (1 2)(1 + )(1 ) N12 = 14 (1 )(1 2)(1 ) N13 = 14 (1 2)(1 )(1 + ) N14 = 14 (1 + )(1 2)(1 + ) N15 = 14 (1 2)(1 + )(1 + ) N16 = 41 (1 )(1 2)(1 + ) N17 = 14 (1 )(1 )(1 2) N18 = 41 (1 + )(1 )(1 2) N19 = 14 (1 + )(1 + )(1 2) N20 = 41 (1 )(1 + )(1 2) N1 = 18 (1 )(1 )(1 ) 21 (N17 + N12 + N9 ) N2 = 18 (1 + )(1 )(1 ) 12 (N18 + N10 + N9 ) N3 = 81 (1 + )(1 + )(1 ) 12 (N19 + N11 + N10 ) N4 = 18 (1 )(1 + )(1 ) 12 (N20 + N12 + N11) N5 = 18 (1 )(1 )(1 + ) 12 (N17 + N16 + N13) N6 = 18 (1 + )(1 )(1 + ) 12 (N18 + N14 + N13 ) N7 = 18 (1 + )(1 + )(1 + ) 12 (N19 + N15 + N14 ) N8 = 18 (1 )(1 + )(1 + ) 12 (N20 + N16 + N15 ) with their mutual relationship de ned by
"=Bd
(A.8)
The strain-displacement matrix B is determined by operating on the shape functions Ni, and it is found that B = [ b1 b2 : : : bi : : : bn ] (A.9) where the submatrix bi , which is associated with the nodal point i of the nite element, has the form 2 3 N 0 0 i;x 66 7 66 0 Ni;y 0 777 bi = 6666 N0 N0 N0i;z 7777 ; i = 1; : : : ; n (A.10) 66 i;y i;x 77 4 0 Ni;z Ni;y 5 Ni;z 0 Ni;x Here, the derivatives of the shape functions Ni with respect to x, y, and z are given by
8 9 2 9 38 > > > N ; ; ; N < i;x = 6 x x x 7 < i; > = 64 ;y ;y ;y 75 Ni; = = N i;y > > > : Ni;z > ; ;z ;z ;z : Ni; ;
8 9 > < Ni; > = ; i = 1; : : : ; n N i; > : Ni; > ;
(A.11)
202
A.3. Consistent Nodal Load Vector
where the matrix is the inverse of the Jacobian
=J
1
(A.12)
2 3 x; y; z; n 66 77 X J = 4 x; y; z; 5 = i=1 x; y; z;
2 3 N x N y N z i; i i; i i; i 66 77 (A.13) N x N y N z i; i i; i i; i 4 5 Ni; xi Ni; yi Ni; zi Note that J is expressed in terms of the derivatives of Ni; i = 1; : : : ; n, with respect to the curvilinear element coordinates , , and , and of the coordinates (xi ; yi; zi), i = 1; : : : ; n, of each of the n nodal points of the nite element.
A.3 Consistent Nodal Load Vector The consistent element load vector f is given by
f=
Z
T F jJj d + N B
Z
T F jJj d! N S !
(A.14)
where is the domain of the nite element in its local coordinate system, FB represents body forces, ! the surface described in curvilinear, non-dimensional , , or coordinates for the element at which surface forces FS are applied, and N contains shape functions Ni. In the surface integral, N and jJj are evaluated on !. If initial thermally induced strains have to be taken into account, the consistent nodal force vector f th due to thermally induced strains is calculated as
f th
=
Z
BT E "th jJj d
(A.15)
where "th is an element vector containing thermally induced strains, i.e.,
n o n o "th = "thx "thy "thz xyth yzth xzth T = T^ T^ T^ 0 0 0 T ; T^ = T T0
(A.16)
where is a matrix containing thermal expansion coecients, T is the temperature at the given point, and T0 is the temperature at which the structure is free of thermally induced strains (typically 20C). These thermally induced strains are based on Gauss points values for obtaining the highest accuracy.
A.4 Consistent and Lumped Mass Matrices The consistent element mass matrix m is given by
m=
Z
% NT N jJj d
(A.17)
Here, is the domain of the nite element in its local coordinate system, see Fig. A.1, % the mass density, N contains shape functions Ni, and jJj is the determinant of the Jacobian matrix J.
Appendix A. 3D Solid Isoparametric Finite Elements
203
Lumped mass matrices are also available in ODESSY. These are generated using the popular HRZ lumping scheme, see Cook, Malkus and Plesha (1989). The idea of this method is to use only the diagonal terms of the consistent mass matrix, but to scale them in such a way that the total mass of the element is preserved. The procedural steps are as follows: 1. Compute only the diagonal coecients mii of the consistent mass matrix. 2. Compute the total mass of the element, m. 3. Compute a number s by adding the diagonal coecients mii associated with translational d.o.f. (but not rotational d.o.f., if any) that are mutually parallel and in the same direction. The number of these translational d.o.f. is denoted by x. 4. Scale all the diagonal coecients by multiplying them by the ratio xm=s, thus preserving the total mass of the element. This HRZ lumping procedure is used for all the elements in ODESSY when generating lumped mass matrices.
A.5 Element Initial Stress Sti ness Matrix In the derivation of element initial stress sti ness matrices it is convenient to reorder nodal degrees of freedom by introducing the element displacement vector d, where translational d.o.f. are reordered so that rst all x-direction d.o.f. are given, then y, and then z as follows
d = fu1 u2 : : : ui : : : un v1 v2 : : : vi : : : vn w1 w2 : : : wi : : : wngT (A.18) Relating d.o.f. to the reordered element vector d the element initial stress sti ness matrix k for the 3D isoparametric nite elements is given by k =
Z
GT S G jJj d
(A.19)
Here, is the domain of the nite element described in curvilinear, non-dimensional -- coordinates for the element, see Fig. A.1, G a matrix obtained by appropriate di erentiation of shape functions Ni, S a matrix of initial stresses, and jJj is the determinant of the Jacobian matrix J. The matrix G is given by 2 3
66 g 0 0 77 G=4 0 g 0 5
0 0 g
(A.20)
204
A.6. Thermal Element \Sti ness Matrix"
where each submatrix g is given by
2 66 Ni;x g = 4 Ni;y Ni;z
The stress matrix S is given by
3 77 5 ; i = 1; : : : ; n
(A.21)
2 3 s 0 0 S = 664 0 s 0 775
(A.22)
0 0 s
and each submatrix s is de ned as
2 3 x xy xz s = 664 xy y yz 775 xz yz z
(A.23)
Here x, xy , etc., are stresses found by an initial static stress analysis.
A.6 Thermal Element \Sti ness Matrix" The thermal element \sti ness matrix" consists of contributions from the heat conduction matrix kth given by Z kth = BthT Bth jJj d
(A.24) Here, is the domain of the nite element in its local coordinate system, see Fig. A.1, Bth a matrix obtained by appropriate di erentiation of shape functions Ni, the thermal conductivity matrix, and jJj is the determinant of the Jacobian matrix J. If the material is isotropic, can simply be replaced by the scalar , the conductivity coecient. The matrix Bth is given by
h
Bth = bth1 bth2 : : : bthi : : : bthn
i
(A.25)
where the submatrix bthi, which is associated with the nodal point i of the nite element, has the form 2 3 N i;x bthi = 664 Ni;y 775 ; i = 1; : : : ; n (A.26) Ni;z In case of boundary conditions in terms of convection heat transfer, the thermal \sti ness matrix" receives additional contributions given by the element matrix h
h=
Z
!2
NT h N jJj d!
(A.27)
Here, !2 is the surface of the nite element described in curvilinear, non-dimensional , , or coordinates for the element, for which the convection boundary condition is applied. N contains shape functions Ni, h is the convection coecient speci ed, and jJj is the determinant of the Jacobian matrix J for the surface !2 .
Appendix A. 3D Solid Isoparametric Finite Elements
205
A.7 Consistent Thermal Nodal Flux Vector The consistent thermal nodal ux vector q is given by
q=
Z
!1
NT qS jJj d! +
Z
!2
NT h Te jJj d!
(A.28)
where the rst term derives from speci ed ux at the surface !1 and the latter term from a speci ed convection boundary condition at surface !2. The surfaces !1, !2 are described in curvilinear, non-dimensional , , or coordinates for the element. The scalar qS is prescribed ux normal to the surface !1, N contains shape functions Ni that are evaluated on the surface !, jJj the determinant of the Jacobian matrix for the surface !, h the convection coecient speci ed, and Te is the environmental temperature speci ed for the convection boundary condition.
A.8 Gauss Quadrature The Gauss integration rules used for the 3D isoparametric nite elements are given in Table A.3.
206
A.8. Gauss Quadrature Table A.3: Gauss quadrature used for 3D solid isoparametric nite elements. Integration rule: Element type k m k kth "; 8-node 8-point 8-point 8-point 8-point 8-point 20-node 14-point 14-point 8-point 14-point 8-point
Appendix
B
2D Solid Isoparametric Finite Elements solid isoparametric nite elements have been implemented in F ODESSY. These isoparametric nite elements are formulated in a uni ed way for ive different 2D
both plane stress, plane strain, and axisymmetric situations. The elements have 3-, 4-, 6-, 8-, and 9-nodes, respectively, and are shown in Fig. B.1. The 6-, 8-, and 9-node elements can have straight as well as curved boundaries. η
(
ξ1 ξ2 ξ 3
)
axis of rotational symmetry
ξ
y,v (z,w)
η
η ξ2
ξ1 ξ3
ξ
ξ
x,u (r,u)
Figure B.1: Domain, node numbering, and nodal degrees of freedom of 3-, 4-, 6-, 8-, and 9node 2D isoparametric nite elements. Local, non-dimensional coordinates , and area coordinates 1, 2, 3 are shown for the quadrilateral and triangular elements, respectively. Text in parantheses refer to standard notations for problems with rotational symmetry.
207
208
B.1. Shape Functions for 2D Solid Isoparametric Finite Elements.
B.1 Shape Functions for 2D Solid Isoparametric Finite Elements. Within the isoparametric formulation of a nite element with an arbitrary number n of nodal points, the same set of shape functions
Ni = Ni(; ); i = 1; : : : ; n
(B.1)
is used for interpolation of global x, y coordinates from nodal values xi, yi and of displacements functions u, v from nodal values ui, vi, i.e.,
x = u(x; y) =
n X
i=1 n X i=1
Nixi ; y =
n X i=1
Niui; v(x; y) =
Ni yi n X i=1
Ni vi
(B.2) (B.3)
Shape functions for the 4-, 8-, and 9-node elements are given in Tables B.1, B.2, and B.3, respectively. Table B.1: Shape functions for 4-node 2D solid isoparametric element.
N1 = 41 (1 )(1 ) N2 = 14 (1 + )(1 ) N3 = 14 (1 + )(1 + ) N4 = 41 (1 )(1 + )
Table B.2: Shape functions for 8-node 2D solid isoparametric element.
N5 = 21 (1 2)(1 ) N6 = 21 (1 + )(1 2 ) N7 = 12 (1 2)(1 + ) N8 = 12 (1 )(1 2) N1 = 41 (1 )(1 ) 12 (N5 + N8 ) N2 = 14 (1 + )(1 ) 12 (N5 + N6 ) N3 = 14 (1 + )(1 + ) 12 (N6 + N7 ) N4 = 14 (1 )(1 + ) 12 (N7 + N8 ) In case of triangular isoparametric elements, the interpolation functions are de ned conveniently in terms of non-dimensional area coordinates 1, 2, and 3 within the element as shown in Fig. B.1. Only two of the three dimensionless area coordinates are mutually independent, due to the area constraint relation
1 + 2 + 3 = 1
(B.4)
Appendix B. 2D Solid Isoparametric Finite Elements
209
Table B.3: Shape functions for 9-node 2D solid isoparametric element.
N9 = (1 2)(1 2) N5 = 12 (1 2)(1 ) 12 N9 N6 = 21 (1 + )(1 2 ) N7 = 12 (1 2)(1 + ) 12 N9 N8 = 12 (1 )(1 2) N1 = 14 (1 )(1 ) 21 (N5 + N8) 14 N9 N2 = 14 (1 + )(1 ) 12 (N5 + N6) 14 N9 N3 = 14 (1 + )(1 + ) 12 (N6 + N7 ) 14 N9 N4 = 14 (1 )(1 + ) 12 (N7 + N8) 14 N9
1N 2 9 1N 2 9
If 1 and 2 are selected as independent coordinates the following relations are obtained
1 = ;
2 = ;
3 = 1
(B.5)
Derivatives with respect to and , with the constraint in Eq. B.4 taken into account, can be found as @ = @ @ ; @ = @ @ (B.6) @ @1 @3 @ @2 @3 The element matrices for the triangular elements can be formulated similarly to the quadrilateral elements by using Eqs. B.4 - B.6. The shape functions for the 3- and 6-node elements are given in Tables B.4 and B.5. Table B.4: Shape functions for 3-node 2D solid isoparametric element.
N1 = 1 N2 = 2 N3 = 3
Table B.5: Shape functions for 6-node 2D solid isoparametric element.
N1 = 1(2 1 1) N2 = 2(2 2 1) N3 = 3(2 3 1) N4 = 4 1 2 N5 = 4 2 3 N3 = 4 1 3
210
B.2. Element Sti ness Matrix
B.2 Element Sti ness Matrix The element sti ness matrix k is given by
k=
Z
T E B jJj d! B !
(B.7)
Here, ! is the domain of the nite element in its local coordinate system, see Fig. B.1, and jJj is the determinant of the Jacobian matrix J which at each point de nes the transformation of di erentials d and d into dx and dy. Like J, the strain-displacement matrix B depends on coordinates of the nodal points, whereas the constitutive matrix E depends only on the constitutive parameters of the assumed linearly elastic material which can be either isotropic or anisotropic in the implementation in ODESSY. For the plane stress and strain situations, d! = t d d, where t is the thickness of the nite element, and for rotationel symmetry we have d! = 2 r d d, where r is the radius. The axis of rotational symmetry is assumed to be parallel with the y-axis as shown in Fig. B.1. The expressions for the Jacobian J and for the strain-displacement matrix B can be found in a uniform way for plane stress, plane strain and axisymmetric nite elements, if the strain vector " is de ned in the following way
"(x; y) = f"x "y xy "z gT
(B.8)
In order to have the relations between standard notations for axisymmetric problems and the notations used here, it should be noted that r = x, z = y, w = v, "r = "x, "z = "y ,
rz = xy , and " = "z as indicated on Fig. B.1. The element nodal vector d containing nodal displacements is given by
d = fdT1 dT2 : : : dTi : : : dTn gT
(B.9)
where the vector di of nodal degrees of freedom is
di = fui vigT ; i = 1; : : : ; n
(B.10)
The relation between the strain vector " and the displacement vector d is
"=Bd
(B.11)
The strain-displacement matrix B can be found to have the following form
B = [ b1 b2 : : : bi : : : bn ]
(B.12)
where the submatrix bi , which is associated with the nodal point i of the nite element, has the form 3 2 N 0 i;x 7 66 66 0 Ni;y 777 (B.13) bi = 66 Ni;y Ni;x 77 ; i = 1; : : : ; n 5 4 Ni 0 r
Appendix B. 2D Solid Isoparametric Finite Elements
211
The fourth row in bi is used only in case of an axisymmetric problem. The derivatives of the shape functions with respect to x and y are given by
(
) " #( ) ( ) Ni;x = ;x ;x Ni; = Ni; ; i = 1; : : : ; n Ni;y ;y ;y Ni; Ni;
where the matrix is the inverse of the Jacobian
=J
1
"
# X n " N x N y # x; y; i; i i; i J = x; y; = N x N i; i i; yi i=1
(B.14) (B.15) (B.16)
B.3 Consistent Nodal Load Vector The consistent element load vector f is given by
f=
Z
Z
NT FB jJj d! + NT FS jJj d !
(B.17)
where ! is the domain of the nite element in its local coordinate system, FB represents body forces, the boundary described in curvilinear, non-dimensional or coordinates for the element at which boundary forces FS are applied, and N contains shape functions Ni. In the boundary integral, N and jJj are evaluated on . For the plane stress and strain situations, d = t d , where t is the thickness of the nite element, and for rotationel symmetry we have d = 2 r d , where r is the radius. If initial thermally induced strains have to be taken into account, the consistent nodal force vector f th due to thermally induced strains is calculated as
f th
=
Z
T E "th jJj d! B !
(B.18)
where "th is an element vector containing thermally induced strains, i.e.,
n o n o "th = "thx "thy xyth "thz T = T^ T^ 0 T^ T ; T^ = T T0
(B.19)
where is a matrix containing thermal expansion coecients, T is the temperature at the given point, and T0 is the temperature at which the structure is free of thermally induced strains. These thermal strains are based on Gauss points values for obtaining the highest accuracy.
B.4 Consistent and Lumped Mass Matrices The consistent element mass matrix m is given by
m=
Z
!
% NT N jJj d!
(B.20)
212
B.5. Element Initial Stress Sti ness Matrix
Here, ! is the domain of the nite element described in curvilinear, non-dimensional coordinates for the element, see Fig. B.1, % the mass density, N contains shape functions Ni , and jJj is the determinant of the Jacobian matrix J. Lumped mass matrices for these elements are also available. These are generated using the HRZ lumping scheme as described in Appendix A.
B.5 Element Initial Stress Sti ness Matrix In the derivation of element initial stress sti ness matrices it is convenient to reorder nodal degrees of freedom by introducing the element displacement vector d, where translational d.o.f. are reordered so that rst all x-direction d.o.f. are given, and then y as follows
d = fu1 u2 : : : ui : : : un v1 v2 : : : vi : : : vngT (B.21) Relating d.o.f. to the reordered element vector d , the element initial stress sti ness matrix k for the 2D isoparametric nite elements is given by Z k = ! GT S G jJj d! (B.22) Here, ! is the domain of the nite element in its local coordinate system, see Fig. B.1, G a matrix obtained by appropriate di erentiation of shape functions Ni, S a matrix of initial stresses, and jJj is the determinant of the Jacobian matrix J. The matrix G is given by " # g 0 G= 0 g (B.23) where each submatrix g is given by
"
# N i;x g = N ; i = 1; : : : ; n i;y
The stress matrix S is given by where each submatrix s is de ned as
"
S = 0s 0s
#
"
# x xy s= xy y Here x, xy , etc., are stresses found by an initial static stress analysis.
(B.24) (B.25) (B.26)
B.6 Thermal Element \Sti ness Matrix" The thermal element \sti ness matrix" consists of contributions from the heat conduction matrix kth given by Z kth = BthT Bth jJj d! (B.27) !
Appendix B. 2D Solid Isoparametric Finite Elements
213
Here, ! is the domain of the nite element described in curvilinear, non-dimensional coordinates for the element, see Fig. B.1, Bth a matrix obtained by appropriate di erentiation of shape functions Ni, the thermal conductivity matrix, and jJj is the determinant of the Jacobian matrix J. If the material is isotropic, can simply be replaced by the scalar , the conductivity coecient. The matrix Bth is given by h i Bth = bth1 bth2 : : : bthi : : : bthn (B.28) where the submatrix bthi, which is associated with the nodal point i of the nite element, has the form " # N i;x th bi = N ; i = 1; : : : ; n (B.29) i;y
In case of boundary conditions in terms of convection heat transfer, the thermal \sti ness matrix" receives additional contributions given by the element matrix h
h=
Z
2
NT h N jJj d
(B.30)
Here, 2 is the boundary of the nite element described in curvilinear, non-dimensional or coordinates for the element, for which the convection boundary condition is applied. N contains shape functions Ni, h is the convection coecient speci ed, and jJj is the determinant of the Jacobian matrix J for the boundary 2.
B.7 Consistent Thermal Nodal Flux Vector The consistent thermal nodal ux vector q is given by
q=
Z
1
NT
qS jJj d +
Z
2
NT h Te jJj d
(B.31)
where the rst term derives from speci ed ux at the boundary 1 and the latter term from a speci ed convection boundary condition at boundary 2. The boundaries 1, 2 are described in curvilinear, non-dimensional or coordinates for the element. The scalar qS is prescribed ux normal to the boundary 1 , N contains shape functions Ni that are evaluated on the boundary , jJj the determinant of the Jacobian matrix J for the boundary , h the convection coecient speci ed, and Te is the environmental temperature speci ed for the convection boundary condition.
B.8 Gauss Quadrature The Gauss integration rules used for the 2D isoparametric nite elements are given in Table B.6. It should be noted that the 4-point integration rule can be employed everywhere in the numerical integration for the 9-node Lagrange element with nearly as good results as the computationally more expensive 9-point integration.
214
B.8. Gauss Quadrature
Table B.6: Gauss quadrature used for 2D solid isoparametric nite elements. Integration rule: Element type 3-node 4-node 6-node 8-node 9-node
k
1-point 4-point 3-point 4-point 9-point
m
1-point 4-point 3-point 4-point 9-point
k
1-point 1-point 3-point 4-point 4-point
kth
1-point 4-point 3-point 4-point 9-point
";
1-point 1-point 3-point 4-point 4-point
Appendix
C
Isoparametric Mindlin Plate and Shell Finite Elements Mindlin plate nite elements have been implemented S in ODESSY, and all these plate elements can also be used as at shell elements. The ix different isoparametric
element matrices are rst described for Mindlin plate elements with in-plane membrane capability, where the elements are formed by combining a plane membrane element, i.e., the 2D solid isoparametric elements, with a standard Mindlin plate bending element. In Section C.10 it is shown how these elements are transformed into at shell elements. This is an easy way to formulate shell elements and the elements pass patch tests and do not exhibit strain under rigid body motion. However, although membrane-bending coupling is present throughout an actual curved shell, it is absent in individual at shell elements and this might lead to inaccurate results, especially if a small number of elements are used to model a curved surface. This must be taken into account when generating nite element models using these at shell elements. The elements have 3-, 4-, 6-, 8-, and 9-nodes, respectively, and are shown in Fig. C.1. The 6-, 8-, and 9-node elements can have straight as well as curved boundaries in the element plane. Both a 9-node \Lagrange" and a 9-node \heterosis" Mindlin plate nite element have been implemented, see Section C.9. It is important to notice that all element matrices for the Mindlin elements are given in terms of standard right-hand-rule rotations x , y as illustrated in Figs. C.1 and C.2. When the Mindlin plate theory is derived it is normally done using the rotations 1 , 2 which are de ned as 1 = y (C.1) = 2
x
The use of rotations 1 , 2 greatly simplify the algebra when developing the theory but as the aim of this description of the implemented Mindlin plate elements is to give an overview of the element matrices, standard right-hand-rule rotations x, y are used everywhere. Furthermore, in the transformation of these at plate elements to at shell elements with 215
216 η ξ ξ2 1 ξ3
ξ
y
η ξ2
y
η
ξ1 ξ3
ξ
vi
ξ
θyi
i z
wi
x
z
θxi
ui
x
Figure C.1: Domain, node numbering, and nodal degrees of freedom of 3-, 4-, 6-, 8-, and 9-node isoparametric Mindlin plate nite elements. Local, non-dimensional coordinates , and area coordinates 1, 2, 3 are shown for the quadrilateral and triangular elements, respectively. z
y x θ2
θx
θy
θ1
Figure C.2: Sign convections for rotations. arbitrary orientation it is necessary to use right-hand-rule rotations x , y . The Mindlin plate theory is based on the four following main assumptions: 1. The domain is of the following special form:
= (x; y; z) 2 = (C.13) > x;y > 2;y > > > > > > > > > > > 1;y + 2;x > y;y + x;x > > > > > > > > > w; w; 2 y x y > > > > > > > : 1 w;x ; : y w;x > ; then the submatrix bi , which is associated with the nodal point i of the nite element, has the form 2 3 N 0 0 0 0 i;x 66 0 N 7 0 0 0 77 i;y 66 66 Ni;y Ni;x 0 0 0 777 6 0 0 Ni;x 777 ; i = 1; : : : ; n bi = 666 00 00 (C.14) 0 Ni;y 0 77 66 66 0 0 0 Ni;x Ni;y 777 64 0 0 Ni;y Ni 0 75 0 0 Ni;x 0 Ni
It is seen that the rst three rows in bi originate from the membrane part, the next three from the bending terms and the last two rows from the shear part. The submatrix bi is divided into these three parts, and the evaluation of the sti ness matrix k is carried out for each part, possibly using di erent orders of Gauss quadrature as will be described in Section C.8. The derivatives of the shape functions Ni with respect to x and y must be calculated when evaluating the submatrix bi , and these are given by
(
) " #( ) ( ) Ni;x = ;x ;x Ni; = Ni; ; i = 1; : : : ; n Ni;y ;y ;y Ni; Ni; where the matrix is the inverse of the Jacobian " # X n " N x N y # x; y; i; i i; i J = x; y; = i=1 Ni; xi Ni; yi
(C.15)
(C.16)
C.3 Consistent Nodal Load Vector The consistent element load vector f is given by
f=
Z
!
NT
FB jJj d! +
Z
!
NT
Z
FS jJj d! + NT FS jJj t d
(C.17)
where ! is the domain of the nite element in its local coordinate system, t the thickness of the element, FB represents body forces, the boundary described in curvilinear, nondimensional or coordinates for the element, FS represents surface forces, and N contains
220
C.4. Consistent and Lumped Mass Matrices
shape functions Ni. In the surface and boundary integrals, N and jJj are evaluated on ! and , respectively. If initial thermally induced strains have to be taken into account, the consistent nodal force vector f th due to thermally induced strains is calculated as
f th =
Z
!
BT E "th jJj d!
where "th is an element vector containing thermally induced strains, i.e., n o n o "th = "thx "thy xyth yzth xzth T = T^ T^ 0 0 0 T ; T^ = T T0
(C.18) (C.19)
where is a matrix containing thermal expansion coecients, T is the temperature at the given point, and T0 is the temperature at which the structure is free of thermally induced strains. These thermally strains are based on Gauss points values for obtaining the highest accuracy.
C.4 Consistent and Lumped Mass Matrices The consistent element mass matrix m is given by
m=
Z
!
% NT N jJj d!
(C.20)
Here, ! is the domain of the nite element in its local coordinate system, % the mass density, N contains shape functions Ni , and jJj is the determinant of the Jacobian matrix J. Lumped mass matrices are generated using the HRZ lumping scheme as described in Appendix A.
C.5 Element Initial Stress Sti ness Matrix In the derivation of element initial stress sti ness matrices it is convenient to reorder and omit some nodal degrees of freedom by introducing the reordered, condensed element displacement vector d that only contains translational degrees of freedom. These translational d.o.f. are reordered so that rst all x-direction d.o.f. are given, then y, and then z as follows
d = fu1 u2 : : : ui : : : un v1 v2 : : : vi : : : vn w1 w2 : : : wi : : : wngT (C.21) Relating d.o.f. to the reordered, condensed element vector d , the element initial stress sti ness matrix k for an isoparametric Mindlin plate nite element is given by Z k = ! GT S G jJj d! (C.22)
where ! is the domain of the nite element described in curvilinear, non-dimensional coordinates for the element, G a matrix obtained by appropriate di erentiation of shape
Appendix C. Isoparametric Mindlin Plate and Shell Finite Elements
221
functions Ni , S a matrix of initial membrane stresses, and jJj is the determinant of the Jacobian matrix J. The matrix G is given by 2 3
66 g 0 0 77 G=4 0 g 0 5
where each submatrix g is given by
(C.23)
0 0 g
"
# N i;x g = N ; i = 1; : : : ; n i;y
The stress matrix S is given by
(C.24)
2 3 s 0 0 S = 664 0 s 0 775
(C.25)
0 0 s
where each submatrix s is de ned as
"
# x xy s= (C.26) xy y Here x, xy , etc., are membrane stresses in the plate found by an initial static stress analysis.
C.6 Thermal Element \Sti ness Matrix" The thermal element sti ness matrix consists of contributions from the heat conduction matrix kth given by Z kth = BthT Bth jJj d! (C.27) !
Here, ! is the domain of the nite element in its local coordinate system, Bth a matrix obtained by appropriate di erentiation of shape functions Ni, the thermal conductivity matrix, and jJj is the determinant of the Jacobian matrix J. If the material is isotropic, can simply be replaced by the scalar , the conductivity coecient. The matrix Bth is given by
h
Bth = bth1 bth2 : : : bthi : : : bthn
i
(C.28)
where the submatrix bthi, which is associated with the nodal point i of the nite element, has the form " # N i;x th bi = N ; i = 1; : : : ; n (C.29) i;y
In case of boundary conditions in terms of convection heat transfer, the thermal \sti ness matrix" receives additional contributions given by the element matrix h
h=
Z
!2
NT
h N jJj d! +
Z
2
NT h N jJj t d
(C.30)
222
C.7. Consistent Thermal Nodal Flux Vector
The convection boundary condition is applied to either the surface !2 or the boundary 2 of the nite element, both described in local coordinates. N contains shape functions Ni, t is the thickness, h the convection coecient speci ed, and jJj is the determinant of the Jacobian matrix J for the surface !2 or the boundary 2 .
C.7 Consistent Thermal Nodal Flux Vector The consistent thermal nodal ux vector q is given by
q =
Z
qS jJj d! +
Z
T q jJj t d N S !Z Z + NT h Te jJj d! + NT h Te jJj t d ! 1
NT 2
1
2
(C.31)
where the rst two terms derive from speci ed ux at either surface !1 or boundary 1 and the two latter terms from speci ed convection boundary conditions at surface !2 or boundary 2 . The surfaces !1 , !2 and boundaries 1 , 2 are described in curvilinear, non-dimensional coordinates for the element and t is the thickness of the element. The scalar qS is prescribed ux normal to the surface !1 or the boundary 1 , N contains shape functions Ni that are evaluated on the surface ! or the boundary , jJj the determinant of the Jacobian matrix J for the surface ! or the boundary , h the convection coecient speci ed, and Te is the environmental temperature speci ed for the convection boundary condition.
C.8 Gauss Quadrature Before the Gauss integration rules used for element matrices of the Mindlin plate elements are described, a few general remarks about these elements are given. The main advantage of isoparametric Mindlin plate elements is the simple theory as only C 0 continuity is required of displacement and rotation variables. The basis is a \thick" plate theory in which transverse shear strains are accounted for, but the elements can also be used for \thin" plates. However, all the implemented elements will su er from shear \locking" in the thin plate limit, i.e., when t ! 0. In order to understand this locking behaviour the sti ness matrix k can be regarded as being composed of a membrane sti ness matrix km , a bending sti ness matrix kb, and a shear sti ness matrix ks. The strain-displacement matrix B is similarly divided into these three parts, i.e., B = Bm + Bb + Bs. Here, Bm is obtained by using the rst three rows in B in Eq. C.12 associated with in-plane membrane strains "x, "y , and xy and setting all other rows in B to zero. Similarly, Bb is associated with in-plane bending strains "x, "y , and xy and is obtained by using row 4, 5, and 6 of B. The matrix Bs is associated with transverse shear strains yz and xz , and is obtained by using the 2 last rows in B. In this way the element sti ness matrix can be written as
k =
Z
Z
Z
T D B jJj d! + BT D B jJj d! + BT D B jJj d! B m M m b M b s M s |! {z } |! {z } |! {z } km kb ks
(C.32)
Appendix C. Isoparametric Mindlin Plate and Shell Finite Elements
223
When Gauss quadrature is used for the numerical integration of ks, it brings two constraints to a Mindlin plate element for each integration point, one associated with yz and the other with xz as described by, e.g., Cook, Malkus & Plesha (1989). Too many integration points may therefore lead to shear locking for a thin plate. This locking phenomenon can be avoided by adopting a reduced or selective integration rule to generate k as described by Hughes, Cohen & Haroun (1978), and their guidelines for using selective integration rules have therefore been followed. Alternatively, one could rede ne the transverse shear interpolation as described by Hughes (1987). The Gauss quadrature used is given in Table C.1. Table C.1: Gauss quadrature used for isoparametric Mindlin plate nite elements. Integration rule: Element type: 3-node 4-node 6-node 8-node 9-node
km
1-point 4-point 3-point 4-point 9-point
kb
1-point 4-point 3-point 9-point 9-point
ks
1-point 1-point 3-point 4-point 4-point
m
1-point 4-point 3-point 4-point 4-point
k
1-point 4-point 3-point 4-point 4-point
kth
";
1-point 1-point 4-point 1-,4-,1-point 3-point 3-point 4-point 4-point 4-point 4-point
The use of selective integration, however, leads to the unpleasant property that the elements can have zero-energy modes, i.e., mechanisms. This is avoided when full integration rules are used but then shear locking appears. Using the selective integration rules given in Table C.1, the 4-node bilinear element has two possible zero-energy modes, where only one of them is communicable between adjacent elements. The 8-node serendipity element has one mechanism which is not communicable between adjacent elements, and the 9-node Lagrange element has one mechanism. The two triangular elements implemented have no mechanisms but su er from shear locking. The 9-node Lagrange element, in general, gives the most accurate results, see Hughes, Cohen & Haroun (1978) and Cook, Malkus & Plesha (1989), but it is not a \foolproof" element as it has a spurious zero-energy mode. However, an improved 9-node element called the \heterosis" element has been implemented in ODESSY, and the theory for this element is given in the following.
C.9 The \Heterosis" Plate Element The \heterosis" Mindlin plate element was developed by Hughes & Cohen (1978). The element is a 9-node element which employs serendipity shape functions for the transverse displacement w, and Lagrange shape functions for displacements u, v and rotations x, y .
224
C.10. Generating Flat Shell Elements
That is, the transverse displacement w is omitted for the 9-th node which is situated in the center of the element, see Fig. C.1. The zero-energy mode observed for the 9-node Lagrange element is thereby eliminated and the \heterosis" element has no mechanisms. Selective integration as described for the 9-node element in Table C.1 is used for this element. Using the notations Ni8 and Ni for shape functions for the 8-node serendipity element and 9-node Lagrange element, respectively, the submatrix bi, see Eq. C.14, for the Heterosis nite element is given by 2 3 N 0 0 0 0 i;x 66 0 N 7 0 0 0 77 i;y 66 66 Ni;y Ni;x 0 0 0 777 6 0 0 Ni;x 777 ; i = 1; : : : ; 8 bi = 666 00 00 (C.33) 0 Ni;y 0 77 66 66 0 0 0 Ni;x Ni;y 777 64 0 0 Ni;y8 Ni 0 75 8 0 0 Ni;x 0 Ni and 2 3 N 0 0 0 9 ;x 66 0 N 7 0 0 77 9;y 66 66 N9;y N9;x 0 0 777 6 7 b9 = 666 00 00 N0 N09;x 777 (C.34) 9;y 66 77 66 0 0 N9;x N9;y 77 64 0 0 N9 0 75 0 0 0 N9 When the consistent nodal load vector f is calculated, see Eq. C.17, the serendipity shape functions Ni8 must be used for interpolating loads in the z-direction. The \heterosis" element has proven to be a very good plate element which can be used for both thin and thick plates, and it has no mechanisms.
C.10 Generating Flat Shell Elements The six di erent Mindlin plate nite elements can also be used as shell elements with 6 d.o.f. per node. The six Mindlin plate elements described above have no \drilling freedoms" z included in the formulation, and an element rotation z is thus not measured and gives no contribution to the strain energy stored in the element. For analysis of a at or folded plate structure the rotation z is not necessary, and furthermore, for a slightly curved shell structure the sti ness corresponding to a z degree of freedom is small. However, for a strongly curved shell structure, where the curvature can be measured as the angle between the element planes of two adjacent elements, this contribution to the sti ness of the element may be large. I have chosen to use the approximation of transforming the Mindlin plate elements into general at shell elements, but in order to avoid an ill-conditioned or even singular sti -
Appendix C. Isoparametric Mindlin Plate and Shell Finite Elements
225
ness matrix, the at shell elements are given a small sti ness for \drilling rotations" z according to Bathe (1982). The element sti ness matrix is expanded from 5 to 6 d.o.f., and the sti ness coecients in the diagonal of the element sti ness matrix corresponding to rotations z are set to 1=1000 of the smallest diagonal element in the original sti ness matrix. These added sti nesses must be large enough to enable the accurate solution of the nite element equilibrium equations and small enough not to a ect the system response signi cantly. This expanded element matrix k is then transformed from the element plane to the global coordinate system using the standard transformation
k = TT k T
(C.35)
where T is the transformation matrix between the local and global element degrees of freedom. All other element vectors and matrices for these at shell elements are calculated in a somewhat similar way; they are established in their local coordinate system and then transformed to the global system. In this way the computer code for the plate elements can be used in all operations for the shell elements. These at shell elements pass patch tests and do not exhibit strain under rigid body motion. However, although membrane-bending coupling is present throughout an actual curved shell, it is absent in individual at shell elements and this might lead to inaccurate results, especially if a small number of elements are used to model a curved surface. This must be taken into account when generating nite element models using these at shell elements.
226
C.10. Generating Flat Shell Elements
Appendix
D
Necessary Optimality Conditions for Eigenvalue Problems
In this appendix necessary optimality conditions for eigenvalue optimization problems are derived in case of a multiple optimum eigenvalue. These derivations were mainly performed by Alexander P. Seyranian in our joint work on problems involving multiple eigenvalues, and they can also be found in Seyranian, Lund & Olho (1994). In case of a simple optimum fundamental eigenvalue, the necessary optimality conditions are given in Section 8.4.1. The optimization problem considered concerns maximization of the lowest of the eigenvalues j ; j = 1; : : : ; n, subject to a constant volume constraint as de ned by Eqs. 8.14 and 8.15, i.e.,
Maximize min j ; j = 1; : : : ; n a1 ; : : : ; aI
Subject to
F (a1; : : : ; aI ) = 0
(D.1)
(D.2)
The necessary optimality conditions rst are derived for a double optimum fundamental eigenvalue and then for any multiplicity N of the lowest eigenvalue and they give much insight in the diculties in maximizing a lowest multiple eigenvalue.
D.1 Double Optimum Fundamental Eigenvalue Let us consider the case when the optimum is achieved at the double lowest eigenvalue 1 = 2 , where 1 = 2 < 3 : : :. This is a non-di erentiable case, and we have to use directional derivatives as described in Section 4.9. 227
228
D.1. Double Optimum Fundamental Eigenvalue
Taking the vector of varied design variables in the form a + "e, kek = 1, according to Eq. 4.46 we obtain the directional derivatives 1 and 2 from
T T e f e f 11 12 det f T e f T e = 0 12 22 This is a quadratic equation in . Solving it we obtain for any direction e q T e + f T e (f T e f T e)2 + 4 (f T e)2 f 11 22 12 1;2 = 11 22 2
(D.3)
(D.4)
The necessary optimality condition for a maximum is min(1 ; 2) 0
(D.5)
for any direction e satisfying the condition f0T e = 0. From Eq. D.3 we see that if we take the direction as e, then both 1 and 2 will change their signs to the opposite ones. This means that if for some direction e both derivatives 1 , 2 are negative then the design point is not a maximum, since a change in sign of the direction e leads to 1 > 0; 2 > 0, i.e., a better design. This means that the necessary optimality condition in the bimodal case is
12 0
(D.6)
for any admissible direction e, i.e., direction that satis es the condition
f0T e = 0
(D.7)
The optimality condition in Eq. D.6 is fundamentally di erent from that of the di erentiable case due to its non-linear nature. The condition was rst formulated by Masur & Mroz (1979, 1980). Using Eqs. D.3 and D.4 we can express the necessary optimality condition of Eq. D.6 in the form (f11T e)(f22T e) (f12T e)2 0 (D.8) for any arbitrary direction e satisfying the condition in Eq. D.7.
D.1.1 Lemma 1: Existence of Improving Direction in Bimodal Case Let us formulate the Lemma for existence of an improving direction e: Lemma 1:
If the vectors f11 ; f12 ; f22; f0 are linearly independent then there exists an improving direction e for which 1 > 0; 2 > 0.
Appendix D. Necessary Optimality Conditions for Eigenvalue Problems 229
D.1.2 Proof of Lemma 1 To prove Lemma 1 consider the following system of linear algebraic equations of the variables e1; : : : ; eI
f11T e f12T e f22T e f0T e
= = = =
10 > 0 0 20 > 0 0
(D.9)
If the vectors f11 ; f12; f22 ; f0 are linearly independent, then a solution e to the system in e 0 0 Eq. D.9 exists for arbitrary values of 1 and 2 . The vector e~ = kek is then an improving direction since from Eqs. D.3 and D.9 we have 0 1 = f11T e~ = ke1k > 0 0 = f T e~ = 2 > 0 2
22
(D.10)
kek
which proves Lemma 1.
D.1.3 Theorem 1: Necessary Optimality Conditions for Bimodal Case Now let us formulate the necessary optimality conditions for the bimodal case Theorem 1:
If the vector of design variables a constitutes the solution of the optimization problem, Eqs. D.1 and D.2, with the double eigenvalue 1 = 2 < 3 : : :, then the vectors f11 ; f12; f22 ; f0 are linearly dependent
11f11 + 2 12f12 + 22 f22 0f0 = 0
(D.11)
with the coecients sk satisfying the inequality
11 22 122
(D.12)
i.e., satisfying conditions of positive semi-de niteness of the symmetric matrix sk ; s; k = 1; 2. Here it is assumed that the rank of the matrix consisting of the vectors f11 ; f12 ; f22 ; f0 is equal to 3. Note that the linear independence of the four vectors mentioned above is possible only when the dimension I of the vector of design variables a is greater than 3.
230
D.2. N -fold Optimum Fundamental Eigenvalue
D.1.4 Proof of Theorem 1 At the optimum point, linear dependence of the vectors fsk , f0 in Eq. D.11 is a consequence of Lemma 1, and to prove Eq. D.12 we express, for example, f22 from Eq. D.11 f22 =
11 f11 2
12 f12 +
0 f0 (D.13) 22 22 22 and substitute this expression into Eq. D.8. Using Eq. D.7 we get
11 f T e2 + 2 12 f T e f T e + f T e2 0 (D.14) 12
22 11
22 12 11 This quadratic form of f11T e and f12T e is positive semi-de nite only if its coecients satisfy the inequality in Eq. D.12. Lemma 1 and Theorem 1 were formulated and proved for the rst time by Bratus & Seyranian (1983).
D.2 N -fold Optimum Fundamental Eigenvalue Consider next the general case when in the optimization problem, Eqs. D.1 and D.2, the maximum is attained at an N -fold multiple lowest eigenvalue 1 = 2 = : : : = N < N +1 : : :. In this case for any admissible direction e, i.e., direction satisfying the condition in Eq. D.7, we nd directional derivatives j ; j = 1; : : : ; N , from Eq. 4.46, i.e.,
det fskT e sk = 0; s; k = 1; : : : ; N
(D.15)
If the maximum is attained then there must be no admissible direction e for which all = j ; j = 1; : : : ; N , are of the same sign. This is an obvious generalization of the necessary optimality condition in Eq. D.6. The Lemma for existence of improving directions and the Theorem for necessary optimality conditions in the general case when maximum is attained at an N -fold multiple lowest eigenvalue are given in the following.
D.2.1 Lemma 2: Existence of Improving Direction in the General Case The Lemma for existence of an improving direction e can be formulated as Lemma 2:
If the vectors f0; fsk , s; k = 1; : : : ; N , k s (the total number of these vectors is equal to (N + 1)N=2 + 1) are linearly independent, then there exists an improving direction e for which j > 0; j = 1; : : : ; N .
Note that the linear independence of the vectors is only possible if I (N + 1)N=2 + 1, where I is the dimension of the vector a of design variables.
Appendix D. Necessary Optimality Conditions for Eigenvalue Problems 231
D.2.2 Proof of Lemma 2 To prove Lemma 2 we consider the system of linear equations in e1 ; : : : ; eI
fskT e = sk s0 ; f0T e = 0
s; k = 1; : : : ; N; k s
(D.16)
where s0 are given positive constants. If the vectors f0 ; fsk are linearly independent, a solution to Eq. D.16 exists for any s0 , in particular when s0 > 0. Suppose the vector e is a solution to the system in Eq. D.16 and let us normalize this vector as e~ = keek . Then we obtain from Eqs. D.16 and D.15 0 j = fjjT e~ = kejk > 0; j = 1; : : : ; N (D.17) which implies existence of an improving direction. This proves Lemma 2, see also Seyranian (1987). When I < (N + 1)N=2 + 1, the vectors fsk ; f0 are always linearly dependent and hence an improving direction may not exist.
D.2.3 Theorem 2: Necessary Optimality Conditions for the General Case Let us formulate the theorem for the necessary optimality conditions Theorem 2:
If the vector of design variables a renders a lowest N -fold eigenvalue 1 = 2 = : : : = N a maximum, it is necessary that the vectors f0 ; fsk , s; k = 1; : : : ; N , k s, are linearly dependent N X s;k=1
sk fsk 0f0 = 0
(D.18)
with the coecients 0; sk satisfying conditions of positive semi-de niteness of the symmetric matrix sk , s; k = 1; : : : ; N . Note that due to the symmetry sk fsk = ksfks we have N X s;k=1
sk fsk =
N X s=1
ssfss + 2
N X s;k=1
sk fsk
(D.19)
s>k
Nevertheless, we prefer the form of Eq. D.18 due to its convenience.
D.2.4 Proof of Theorem 2 Linear dependence of the vectors fsk ; f0 at the optimum point is an obvious consequence of Lemma 2 and we just need to prove the necessity of positive semi-de niteness of the matrix
232
D.2. N -fold Optimum Fundamental Eigenvalue
sk . To this end we choose a new basis of eigenvectors ~ 1; : : : ; ~ N for which the matrix
sk is diagonal, and show that if an optimum is attained then all ~ss 0, s = 1; : : : ; N . Let us transform the eigenvectors N X ~ s = gks k ; s = 1; : : : ; N (D.20) k=1
Here ~ s are transformed eigenvectors satisfying the orthonormality condition in Eq. 4.26, and gks is the transformation matrix. Using Eq. D.20 in Eq. 4.26 we obtain
~ Ti M ~ j
N X
= = = = =
s=1 N X s;k=1 N X s;k=1 N X s=1 ij ;
!
gsi Ts M
N X k=1
gkj k
!
gsigkj Ts M k gsigkj sk
(D.21)
gsigsj i; j = 1; : : : ; N
In matrix form this equation is equivalent to
gT g = I and gT = g
1
(D.22)
where I is the unit matrix. The last equation means that the transformation matrix g is an orthogonal matrix. Now let us express vectors k from Eq. D.20 by ~ s. Due to Eq. D.22 we have N X s = gsk ~ k ; s = 1; : : : ; N (D.23) Using the notation
k=1
! @ K @ M @ K @ M K ~ M = @a ~ @a ; : : : ; @a ~ @a 1 1 I I and Eq. 4.45, we obtain N N X X
ksfks =
ks Tk K ~ M s k;s=1 k;s=1 ! ! N N N X X X T =
ks gkt ~ t K ~ M gsm ~ m m =1 k;s=1 0 t=1 1 N N X X @ gkt ksgsmA ~ Tt K ~ M ~ m = t;m=1 k;s=1 N X =
~tm~ftm
r
r
r
t;m=1
r r
r
r
r
(D.24)
(D.25)
Appendix D. Necessary Optimality Conditions for Eigenvalue Problems 233 So, in the new basis the matrix ks takes the form
~tm = In matrix form we have
N X k;s=1
gkt ksgsm
(D.26)
~ = gT g = g 1 g
(D.27) This means that there exists a basis in which the matrix ~ is diagonal. Then the optimality condition in Eq. D.18 takes the form N X
~ss~fss 0f0 = 0
s=1
(D.28)
To show that the condition ~ss > 0, s = 1; : : : ; N , is the necessary condition for optimality, let us consider an admissible direction e, i.e., direction satisfying the condition Eq. D.7. Multiplying Eq. D.28 by e we obtain N X
~ss~fssT e = 0
s=1
(D.29)
Suppose that in Eqs. D.28 the j -th coecient ~jj 6= 0. Let us take the admissible direction e such that ~fskT e = 0; s; k = 1; : : : ; N; k > s ~fttT e = t0 ; t = 1; : : : ; N; t 6= j (D.30) T f0 e = 0 where t0 are arbitrary positive constants. Such a direction e exists if we assume that the rank of the matrix consisting of the vectors f0 , ~fsk ,e s; k = 1; : : : ; N , k s, is equal to (N + 1)N=2 I . Normalizing this vector we get e~ = kek . Then according to Eq. D.15 for this direction e~ we obtain and
0 s = ~fssT e~ = kesk > 0;
s = 1; : : : ; N; s 6= j (D.31)
j = ~fjjT e~
So we can nd j from Eq. D.29 taking the direction e~
j = ~fjjT e~ = =
N 1 X ~T
~jj s=1 ~ssfsse~ s6=j N ~ ! 1 X ss 0 kek s=1 ~jj s s6=j
Here we have used Eq. D.31.
(D.32)
234
D.2. N -fold Optimum Fundamental Eigenvalue
If maximum of the lowest N -fold eigenvalue is achieved, then for any admissible direction e the sensitivities k , k = 1; : : : ; N , must not be of the same sign. Since we have chosen e~ such that all s, s = 1; : : : ; N , s 6= j , are positive, then j must be less than or equal to zero, i.e., j 0. Using Eq. D.32 we get N ~ ! X ss 0 ~jj s 0 s=1
(D.33)
s6=j
for arbitrary choice of the positive constants s0 , s = 1; : : : ; N , s 6= j . The inequality in Eq. D.33 can be satis ed only if
~ss 0; s = 1; : : : ; N; s 6= j
~jj
(D.34)
since, otherwise, the constants s0 can be chosen such that the inequality in Eq. D.34 is violated. This means that all ~ss, s = 1; : : : ; N , must be of the same sign. Without loss of generality, all ~ss can be regarded as non-negative quantities. So, we have proved Lemma 2, i.e., that
~ss 0; s = 1; : : : ; N; (D.35) which implies positive semi-de niteness of the matrix of coecients sk , s; k = 1; : : : ; N , and this constitutes the necessary optimality condition in the general case of an N -fold multiple lowest optimum eigenvalue. Similar results for minimizing the maximum eigenvalue were obtained by Overton (1988). Recently, Cox & Overton (1992), derived the necessary optimality conditions for discrete and distributed eigenvalue problems using Clarke's generalized gradient, see Clarke (1990). They also considered lower and upper bounds on design variables ai ai ai . These results approve optimality conditions suggested by Olho & Rasmussen (1977), and also used by many others, see, e.g., Gajewski & Zyczkowski (1988).
