Computational structural analysis and finite element methods

Computational

Structural Analysis

and Finite Element

Methods

A. Kaveh

Computational Structural Analysis

and Finite Element Methods

.

A. Kaveh

Computational Structural

Analysis and Finite Element

Methods

A. Kaveh

Centre of Excellence for Fundamental Studies in

Structural Engineering

School of Civil Engineering

Iran University of Science and Technology

Tehran

Iran

ISBN 978-3-319-02963-4 ISBN 978-3-319-02964-1 (eBook)

DOI 10.1007/978-3-319-02964-1

Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013956541

© Springer International Publishing Switzerland 2014

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part

of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,

recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or

information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar

methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts

in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being

entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication

of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the

Publisher’s location, in its current version, and permission for use must always be obtained from

Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center.

Violations are liable to prosecution under the respective Copyright Law.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this

publication does not imply, even in the absence of a specific statement, that such names are exempt

from the relevant protective laws and regulations and therefore free for general use.

While the advice and information in this book are believed to be true and accurate at the date of

publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for

any errors or omissions that may be made. The publisher makes no warranty, express or implied, with

respect to the material contained herein.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Recent advances in structural technology require greater accuracy, efficiency and

speed in the analysis of structural systems. It is therefore not surprising that new

methods have been developed for the analysis of structures with complex config￾urations and large number of elements.

The requirement of accuracy in analysis has been brought about by the need for

demonstrating structural safety. Consequently, accurate methods of analysis had to

be developed, since conventional methods, although perfectly satisfactory when

used on simple structures, have been found inadequate when applied to complex

and large-scale structures. Another reason why higher speed is required results from

the need to have optimal design, where analysis is repeated hundred or even

thousands of times.

This book can be considered as an application of discrete mathematics rather

than the more usual calculus-based methods of analysis of structures and finite

element methods. The subject of graph theory has become important in science and

engineering through its strong links with matrix algebra and computer science.

At first glance, it seems extraordinary that such abstract material should have quite

practical applications. However, as the author makes clear, the early relationship

between graph theory and skeletal structures and finite element models is now

obvious: the structure of the mathematics is well suited to the structure of the

physical problem. In fact, could there be any other way of dealing with this

structural problem? The engineer studying these applications of structural analysis

has either to apply the computer programs as a black box, or to become involved in

graph theory, matrix algebra and sparse matrix technology. This book is addressed

to those scientists and engineers, and their students, who wish to understand the

theory.

The methods of analysis in this book employ matrix algebra and graph theory,

which are ideally suited for modern computational mechanics. Although this text

deals primarily with analysis of structural engineering systems, it should be

recognised that these methods are also applicable to other types of systems such

as hydraulic and electrical networks.

v

The author has been involved in various developments and applications of graph

theory in the last four decades. The present book contains part of this research

suitable for various aspects of matrix structural analysis and finite element methods,

with particular attention to the finite element force method.

In Chap. 1, the most important concepts and theorems of structures and theory of

graphs are briefly presented. Chapter 2 contains different efficient approaches for

determining the degree of static indeterminacy of structures and provides systematic

methods for studying the connectivity properties of structural models. In this chapter,

force method of analysis for skeletal structures is described mostly based on the

author’s algorithms. Chapter 3 provides simple and efficient methods for construction

of stiffness matrices. These methods are especially suitable for the formation of well￾conditioned stiffness matrices. In Chaps. 4 and 5, banded, variable banded and frontal

methods are investigated. Efficient methods are presented for both node and element

ordering. Many new graphs are introduced for transforming the connectivity proper￾ties of finite element models onto graph models. Chapters 6 and 7 include powerful

graph theory and algebraic graph theory methods for the force method of finite

element meshes of low order and high order, respectively. These new methods use

different graphs of the models and algebraic approaches. In Chap. 8, several

partitioning algorithms are developed for solution of multi-member systems, which

can be categorized as graph theory methods and algebraic graph theory approaches.

In Chap. 9, an efficient method is presented for the analysis of near-regular structures

which are obtained by addition or removal of some members to regular structural

models. In Chap. 10, energy formulation based on the force method is derived and a

new optimization algorithm called SCSS is applied to the analysis procedure. Then,

using the SCSS and prescribed stress ratios, structures are analyzed and designed. In

all the chapters, many examples are included to make the text easier to be understood.

I would like to take this opportunity to acknowledge a deep sense of gratitude to

a number of colleagues and friends who in different ways have helped in the

preparation of this book. Mr. J. C. de C. Henderson, formerly of Imperial College

of Science and Technology, first introduced me to the subject with most stimulating

discussions on various aspects of topology and combinatorial mathematics. Profes￾sor F. Ziegler and Prof. Ch. Bucher encouraged and supported me to write this

book. My special thanks are due to Mrs. Silvia Schilgerius, the senior editor of the

Applied Sciences of Springer, for her constructive comments, editing and unfailing

kindness in the course of the preparation of this book. My sincere appreciation is

extended to our Springer colleagues Ms. Beate Siek and Ms. G. Ramya Prakash.

I would like to thank my former Ph.D. and M.Sc. students, Dr. H. Rahami,

Dr. M. S. Massoudi, Dr. K. Koohestani, Dr. P. Sharafi, Mr. M. J. Tolou Kian,

Dr. A. Mokhtar-zadeh, Mr. G. R. Roosta, Ms. E. Ebrahimi, Mr. M. Ardalan, and

Mr. B. Ahmadi for using our joint papers and for their help in various stages of

writing this book. I would like to thank the publishers who permitted some of our

papers to be utilized in the preparation of this book, consisting of Springer-Verlag,

John Wiley and Sons, and Elsevier.

My warmest gratitude is due to my family and in particular my wife, Mrs.

Leopoldine Kaveh, for her continued support in the course of preparing this book.

vi Preface

Every effort has been made to render the book error free. However, the author

would appreciate any remaining errors being brought to his attention through his

email-address: [email protected].

Tehran A. Kaveh

December 2013

Preface vii

.

Contents

1 Basic Definitions and Concepts of Structural Mechanics and Theory

of Graphs ............................................ 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Definitions . . ................................. 1

1.1.2 Structural Analysis and Design . . . . . . . . . . . . . . . . . . . . 4

1.2 General Concepts of Structural Analysis ................... 5

1.2.1 Main Steps of Structural Analysis .................. 5

1.2.2 Member Forces and Displacements . ................ 6

1.2.3 Member Flexibility and Stiffness Matrices ............ 7

1.3 Important Structural Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.2 Castigliano’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.3 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.4 Contragradient Principle . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.5 Reciprocal Work Theorem . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Basic Concepts and Definitions of Graph Theory . . . . . . . . . . . . . 18

1.4.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.2 Definition of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.3 Adjacency and Incidence . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.4 Graph Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.5 Walks, Trails and Paths . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4.6 Cycles and Cutsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4.7 Trees, Spanning Trees and Shortest Route Trees . . . . . . . . 23

1.4.8 Different Types of Graphs . . . . . . . . . . . . . . . . . . . . . . . . 23

1.5 Vector Spaces Associated with a Graph . . . . . . . . . . . . . . . . . . . . 25

1.5.1 Cycle Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.5.2 Cutset Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.5.3 Orthogonality Property . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.5.4 Fundamental Cycle Bases . . . . . . . . . . . . . . . . . . . . . . . . 27

1.5.5 Fundamental Cutset Bases . . . . . . . . . . . . . . . . . . . . . . . . 27

ix

1.6 Matrices Associated with a Graph . . . . . . . . . . . . . . . . . . . . . . . . 28

1.6.1 Matrix Representation of a Graph . . . . . . . . . . . . . . . . . . 29

1.6.2 Cycle Bases Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.6.3 Special Patterns for Fundamental Cycle Bases . . . . . . . . . 33

1.6.4 Cutset Bases Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.6.5 Special Patterns for Fundamental Cutset Bases . . . . . . . . . 34

1.7 Directed Graphs and Their Matrices . . . . . . . . . . . . . . . . . . . . . . 35

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Optimal Force Method: Analysis of Skeletal Structures . . . . . . . . . 39

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2 Static Indeterminacy of Structures . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2.1 Mathematical Model of a Skeletal Structure . . . . . . . . . . . 41

2.2.2 Expansion Process for Determining the Degree

of Static Indeterminacy . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 Formulation of the Force Method . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3.1 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3.2 Member Flexibility Matrices . . . . . . . . . . . . . . . . . . . . . . 49

2.3.3 Explicit Method for Imposing Compatibility . . . . . . . . . . . 52

2.3.4 Implicit Approach for Imposing Compatibility . . . . . . . . . 53

2.3.5 Structural Flexibility Matrices . . . . . . . . . . . . . . . . . . . . . 55

2.3.6 Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3.7 Optimal Force Method . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.4 Force Method for the Analysis of Frame Structures . . . . . . . . . . . 60

2.4.1 Minimal and Optimal Cycle Bases . . . . . . . . . . . . . . . . . . 61

2.4.2 Selection of Minimal and Subminimal Cycle Bases . . . . . . 62

2.4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.4.4 Optimal and Suboptimal Cycle Bases . . . . . . . . . . . . . . . . 69

2.4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.4.6 An Improved Turn Back Method for the Formation

of Cycle Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.4.8 Formation of B0 and B1 Matrices . . . . . . . . . . . . . . . . . . . 78

2.5 Generalized Cycle Bases of a Graph . . . . . . . . . . . . . . . . . . . . . . 82

2.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.5.2 Minimal and Optimal Generalized Cycle Bases . . . . . . . . . 85

2.6 Force Method for the Analysis of Pin-Jointed Planar Trusses . . . . 86

2.6.1 Associate Graphs for Selection of a Suboptimal GCB . . . . 86

2.6.2 Minimal GCB of a Graph . . . . . . . . . . . . . . . . . . . . . . . . 89

2.6.3 Selection of a Subminimal GCB: Practical Methods . . . . . 89

2.7 Algebraic Force Methods of Analysis . . . . . . . . . . . . . . . . . . . . . 91

2.7.1 Algebraic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

x Contents

3 Optimal Displacement Method of Structural Analysis . . . . . . . . . . 101

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.2.1 Coordinate Systems Transformation . . . . . . . . . . . . . . . . . 102

3.2.2 Element Stiffness Matrix Using Unit

Displacement Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.2.3 Element Stiffness Matrix Using Castigliano’s Theorem . . . 109

3.2.4 The Stiffness Matrix of a Structure . . . . . . . . . . . . . . . . . . 111

3.2.5 Stiffness Matrix of a Structure;

an Algorithmic Approach . . . . . . . . . . . . . . . . . . . . . . . . 116

3.3 Transformation of Stiffness Matrices . . . . . . . . . . . . . . . . . . . . . . 118

3.3.1 Stiffness Matrix of a Bar Element . . . . . . . . . . . . . . . . . . 118

3.3.2 Stiffness Matrix of a Beam Element . . . . . . . . . . . . . . . . . 120

3.4 Displacement Method of Analysis . . . . . . . . . . . . . . . . . . . . . . . . 122

3.4.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.4.2 General Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.5 Stiffness Matrix of a Finite Element . . . . . . . . . . . . . . . . . . . . . . 128

3.5.1 Stiffness Matrix of a Triangular Element . . . . . . . . . . . . . 129

3.6 Computational Aspects of the Matrix Displacement Method . . . . . 132

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4 Ordering for Optimal Patterns of Structural Matrices: Graph

Theory Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.2 Bandwidth Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

4.4 A Shortest Route Tree and Its Properties . . . . . . . . . . . . . . . . . 142

4.5 Nodal Ordering for Bandwidth Reduction . . . . . . . . . . . . . . . . 142

4.5.1 A Good Starting Node . . . . . . . . . . . . . . . . . . . . . . . . 143

4.5.2 Primary Nodal Decomposition . . . . . . . . . . . . . . . . . . 145

4.5.3 Transversal P of an SRT . . . . . . . . . . . . . . . . . . . . . . 146

4.5.4 Nodal Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.5.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.6 Finite Element Nodal Ordering for Bandwidth Optimisation . . . 147

4.6.1 Element Clique Graph Method (ECGM) . . . . . . . . . . 149

4.6.2 Skeleton Graph Method (SkGM) . . . . . . . . . . . . . . . . 149

4.6.3 Element Star Graph Method (EStGM) . . . . . . . . . . . . 150

4.6.4 Element Wheel Graph Method (EWGM) . . . . . . . . . . 151

4.6.5 Partially Triangulated Graph Method (PTGM) . . . . . . 152

4.6.6 Triangulated Graph Method (TGM) . . . . . . . . . . . . . 153

4.6.7 Natural Associate Graph Method (NAGM) . . . . . . . . 153

4.6.8 Incidence Graph Method (IGM) . . . . . . . . . . . . . . . . 155

4.6.9 Representative Graph Method (RGM) . . . . . . . . . . . . 156

4.6.10 Computational Results . . . . . . . . . . . . . . . . . . . . . . . 157

4.6.11 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Contents xi

4.7 Finite Element Nodal Ordering for Profile Optimisation . . . . . . 160

4.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

4.7.2 Graph Nodal Numbering for Profile Reduction . . . . . 162

4.7.3 Nodal Ordering with Element

Clique Graph (NOECG) . . . . . . . . . . . . . . . . . . . . . . 164

4.7.4 Nodal Ordering with Skeleton Graph (NOSG) . . . . . . 165

4.7.5 Nodal Ordering with Element Star Graph (NOESG) . . . 166

4.7.6 Nodal Ordering with Element Wheel Graph

(NOEWG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.7.7 Nodal Ordering with Partially Triangulated Graph

(NOPTG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

4.7.8 Nodal Ordering with Triangulated Graph (NOTG) . . . 167

4.7.9 Nodal Ordering with Natural Associate Graph

(NONAG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

4.7.10 Nodal Ordering with Incidence Graph (NOIG) . . . . . 168

4.7.11 Nodal Ordering with Representative Graph (NORG) . . . 168

4.7.12 Nodal Ordering with Element Clique Representative

Graph (NOECRG) . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4.7.13 Computational Results . . . . . . . . . . . . . . . . . . . . . . . 170

4.7.14 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4.8 Element Ordering for Frontwidth Reduction . . . . . . . . . . . . . . . 171

4.9 Element Ordering for Bandwidth Optimisation of Flexibility

Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.9.1 An Associate Graph . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.9.2 Distance Number of an Element . . . . . . . . . . . . . . . . . 175

4.9.3 Element Ordering Algorithms . . . . . . . . . . . . . . . . . . . 175

4.10 Bandwidth Reduction for Rectangular Matrices . . . . . . . . . . . . 177

4.10.1 Definitions . . . .................................. 177

4.10.2 Algorithms .................................... 178

4.10.3 Examples ..................................... 179

4.10.4 Bandwidth Reduction of Finite Element Models . . . . . . . . . 181

4.11 Graph-Theoretical Interpretation of Gaussian Elimination . . . . . 182

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5 Ordering for Optimal Patterns of Structural Matrices:

Algebraic Graph Theory and Meta-heuristic Based Methods . . . . . 187

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

5.2 Adjacency Matrix of a Graph for Nodal Ordering . . . . . . . . . . . . 187

5.2.1 Basic Concepts and Definitions . . . . . . . . . . . . . . . . . . . . 187

5.2.2 A Good Starting Node . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5.2.3 Primary Nodal Decomposition . . . . . . . . . . . . . . . . . . . . . 190

5.2.4 Transversal P of an SRT . . . . . . . . . . . . . . . . . . . . . . . . . 191

5.2.5 Nodal Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

5.2.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

xii Contents

5.3 Laplacian Matrix of a Graph for Nodal Ordering . . . . . . . . . . . . . 192

5.3.1 Basic Concepts and Definitions . . . . . . . . . . . . . . . . . . . . 192

5.3.2 Nodal Numbering Algorithm . . . . . . . . . . . . . . . . . . . . . . 196

5.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

5.4 A Hybrid Method for Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . 196

5.4.1 Development of the Method . . . . . . . . . . . . . . . . . . . . . . . 197

5.4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5.4.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

5.5 Ordering via Charged System Search Algorithm . . . . . . . . . . . . . 203

5.5.1 Charged System Search . . . . . . . . . . . . . . . . . . . . . . . . . . 203

5.5.2 The CSS Algorithm for Nodal Ordering . . . . . . . . . . . . . . 208

5.5.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

6 Optimal Force Method for FEMs: Low Order Elements . . . . . . . . 215

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

6.2 Force Method for Finite Element Models: Rectangular and

Triangular Plane Stress and Plane Strain Elements . . . . . . . . . . . . 215

6.2.1 Member Flexibility Matrices . . . . . . . . . . . . . . . . . . . . . . 216

6.2.2 Graphs Associated with FEMs . . . . . . . . . . . . . . . . . . . . . 220

6.2.3 Pattern Corresponding to the Self Stress Systems . . . . . . . 221

6.2.4 Selection of Optimal γ-Cycles Corresponding

to Type II Self Stress Systems . . . . . . . . . . . . . . . . . . . . . 224

6.2.5 Selection of Optimal Lists . . . . . . . . . . . . . . . . . . . . . . . . 225

6.2.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

6.3 Finite Element Analysis Force Method: Triangular and Rectangular

Plate Bending Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

6.3.1 Graphs Associated with Finite Element Models . . . . . . . . 233

6.3.2 Subgraphs Corresponding to Self-Equilibrating Systems . . . 233

6.3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

6.4 Force Method for Three Dimensional Finite Element Analysis . . . 244

6.4.1 Graphs Associated with Finite Element Model . . . . . . . . . 244

6.4.2 The Pattern Corresponding to the Self Stress Systems . . . . 245

6.4.3 Relationship Between γ(S) and b1(A(S)) . . . . . . . . . . . . . 248

6.4.4 Selection of Optimal γ-Cycles Corresponding

to Type II Self Stress Systems . . . . . . . . . . . . . . . . . . . . . 251

6.4.5 Selection of Optimal Lists . . . . . . . . . . . . . . . . . . . . . . . . 252

6.4.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

6.5 Efficient Finite Element Analysis Using Graph-Theoretical Force

Method: Brick Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

6.5.1 Definition of the Independent Element Forces . . . . . . . . . . 258

6.5.2 Flexibility Matrix of an Element . . . . . . . . . . . . . . . . . . . 259

6.5.3 Graphs Associated with Finite Element Model . . . . . . . . . 261

6.5.4 Topological Interpretation of Static Indeterminacy . . . . . . 263

Contents xiii

6.5.5 Models Including Internal Node . . . . . . . . . . . . . . . . . . . . 270

6.5.6 Selection of an Optimal List Corresponding to Minimal

Self-Equilibrating Stress Systems . . . . . . . . . . . . . . . . . . . 271

6.5.7 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

7 Optimal Force Method for FEMS: Higher Order Elements . . . . . . 281

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

7.2 Finite Element Analysis of Models Comprised of Higher Order

Triangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

7.2.1 Definition of the Element Force System . . . . . . . . . . . . . . 282

7.2.2 Flexibility Matrix of the Element . . . . . . . . . . . . . . . . . . . 282

7.2.3 Graphs Associated with Finite Element Model . . . . . . . . . 282

7.2.4 Topological Interpretation of Static Indeterminacies . . . . . 284

7.2.5 Models Including Opening . . . . . . . . . . . . . . . . . . . . . . . . 287

7.2.6 Selection of an Optimal List Corresponding to Minimal

Self-Equilibrating Stress Systems . . . . . . . . . . . . . . . . . . . 290

7.2.7 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

7.3 Finite Element Analysis of Models Comprised of Higher Order

Rectangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

7.3.1 Definition of Element Force System . . . . . . . . . . . . . . . . . 298

7.3.2 Flexibility Matrix of the Element . . . . . . . . . . . . . . . . . . . 300

7.3.3 Graphs Associated with Finite Element Model . . . . . . . . . 301

7.3.4 Topological Interpretation of Static Indeterminacies . . . . . 303

7.3.5 Selection of Generators for SESs of Type II and Type III . . . 307

7.3.6 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

7.3.7 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

7.4 Efficient Finite Element Analysis Using Graph-Theoretical

Force Method: Hexa-Hedron Elements . . . . . . . . . . . . . . . . . . . . 316

7.4.1 Independent Element Forces and Flexibility Matrix

of Hexahedron Elements . . . . . . . . . . . . . . . . . . . . . . . . . 317

7.4.2 Graphs Associated with Finite Element Models . . . . . . . . 321

7.4.3 Negative Incidence Number . . . . . . . . . . . . . . . . . . . . . . . 325

7.4.4 Pattern Corresponding to Self-Equilibrating Systems . . . . . 325

7.4.5 Selection of Generators for SESs of Type II and

Type III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

7.4.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

8 Decomposition for Parallel Computing: Graph Theory Methods. . . . 341

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

8.2 Earlier Works on Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

8.2.1 Nested Dissection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

8.2.2 A Modified Level-Tree Separator Algorithm . . . . . . . . . . . 342

8.3 Substructuring for Parallel Analysis of Skeletal Structures . . . . . . 343

8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

8.3.2 Substructuring Displacement Method . . . . . . . . . . . . . . . . 344

xiv Contents