Efficient methods leading to highly sparse and banded structural matrices
Application of graph theory for efficient analysis of skeletal structures
Many worked examples and exercises will help the reader to appreciate the theory
Graph theory gained initial prominence in science and engineering through its strong links with matrix algebra and computer science. Moreover, the structure of the mathematics is well suited to that of engineering problems in analysis and design. The methods of analysis in this book employ matrix algebra, graph theory and meta-heuristic algorithms, which are ideally suited for modern computational mechanics. Efficient methods are presented that lead to highly sparse and banded structural matrices. The main features of the book include: application of graph theory for efficient analysis; extension of the force method to finite element analysis; application of meta-heuristic algorithms to ordering and decomposition (sparse matrix technology); efficient use of symmetry and regularity in the force method; and simultaneous analysis and design of structures.
Content Level » Research
Keywords » Application of Graph Theory for Efficient Analysis - Finite Element Analysis - Meta-heuristic Algorithms
Related subjects » Computational Intelligence and Complexity - Computational Science & Engineering
Author(s): Kaveh A.
Publisher: Springer
Year: 2014
Language: English
Pages: 445
Tags: Математика;Вычислительная математика;Метод конечных элементов;
Preface......Page 6
Contents......Page 10
1.1.1 Definitions......Page 18
1.1.2 Structural Analysis and Design......Page 21
1.2.1 Main Steps of Structural Analysis......Page 22
1.2.2 Member Forces and Displacements......Page 23
1.2.3 Member Flexibility and Stiffness Matrices......Page 24
1.3.1 Work and Energy......Page 28
1.3.3 Principle of Virtual Work......Page 30
1.3.4 Contragradient Principle......Page 33
1.3.5 Reciprocal Work Theorem......Page 34
1.4 Basic Concepts and Definitions of Graph Theory......Page 35
1.4.2 Definition of a Graph......Page 36
1.4.4 Graph Operations......Page 37
1.4.5 Walks, Trails and Paths......Page 38
1.4.6 Cycles and Cutsets......Page 39
1.4.8 Different Types of Graphs......Page 40
1.5 Vector Spaces Associated with a Graph......Page 42
1.5.3 Orthogonality Property......Page 43
1.5.5 Fundamental Cutset Bases......Page 44
1.6 Matrices Associated with a Graph......Page 45
1.6.1 Matrix Representation of a Graph......Page 46
1.6.2 Cycle Bases Matrices......Page 49
1.6.3 Special Patterns for Fundamental Cycle Bases......Page 50
1.6.5 Special Patterns for Fundamental Cutset Bases......Page 51
1.7 Directed Graphs and Their Matrices......Page 52
References......Page 54
2.1 Introduction......Page 55
2.2 Static Indeterminacy of Structures......Page 56
2.2.1 Mathematical Model of a Skeletal Structure......Page 57
2.2.2 Expansion Process for Determining the Degree of Static Indeterminacy......Page 58
2.2.2.2 A Unifying Function......Page 59
2.2.2.4 An Intersection Theorem......Page 60
2.2.2.5 A Method for Determining the DSI of Structures......Page 61
2.3.1 Equilibrium Equations......Page 62
2.3.2 Member Flexibility Matrices......Page 65
2.3.3 Explicit Method for Imposing Compatibility......Page 68
2.3.4 Implicit Approach for Imposing Compatibility......Page 69
2.3.6 Computational Procedure......Page 71
2.4 Force Method for the Analysis of Frame Structures......Page 76
2.4.1 Minimal and Optimal Cycle Bases......Page 77
2.4.2 Selection of Minimal and Subminimal Cycle Bases......Page 78
2.4.3 Examples......Page 83
2.4.4.1 Suboptimal Cycle Bases; A Direct Approach......Page 85
2.4.4.2 Suboptimal Cycle Bases; an Indirect Approach......Page 87
2.4.5 Examples......Page 88
2.4.6 An Improved Turn Back Method for the Formation of Cycle Bases......Page 91
2.4.7 Examples......Page 92
2.4.8 Formation of B0 and B1 Matrices......Page 94
2.5 Generalized Cycle Bases of a Graph......Page 98
2.5.1 Definitions......Page 99
2.5.2 Minimal and Optimal Generalized Cycle Bases......Page 101
2.6.1 Associate Graphs for Selection of a Suboptimal GCB......Page 102
2.6.3 Selection of a Subminimal GCB: Practical Methods......Page 105
2.7.1 Algebraic Methods......Page 107
References......Page 114
3.2 Formulation......Page 116
3.2.1 Coordinate Systems Transformation......Page 117
3.2.2 Element Stiffness Matrix Using Unit Displacement Method......Page 120
3.2.3 Element Stiffness Matrix Using Castigliano´s Theorem......Page 124
3.2.4 The Stiffness Matrix of a Structure......Page 126
3.2.5 Stiffness Matrix of a Structure; an Algorithmic Approach......Page 131
3.3.1 Stiffness Matrix of a Bar Element......Page 133
3.3.2 Stiffness Matrix of a Beam Element......Page 135
3.4 Displacement Method of Analysis......Page 137
3.4.1 Boundary Conditions......Page 139
3.4.2 General Loading......Page 140
3.5 Stiffness Matrix of a Finite Element......Page 143
3.5.1 Stiffness Matrix of a Triangular Element......Page 144
3.6 Computational Aspects of the Matrix Displacement Method......Page 147
References......Page 150
4.1 Introduction......Page 151
4.2 Bandwidth Optimisation......Page 152
4.3 Preliminaries......Page 154
4.5 Nodal Ordering for Bandwidth Reduction......Page 156
4.5.1 A Good Starting Node......Page 157
4.5.2 Primary Nodal Decomposition......Page 159
4.5.4 Nodal Ordering......Page 160
4.6 Finite Element Nodal Ordering for Bandwidth Optimisation......Page 161
4.6.2 Skeleton Graph Method (SkGM)......Page 163
4.6.3 Element Star Graph Method (EStGM)......Page 164
4.6.4 Element Wheel Graph Method (EWGM)......Page 165
4.6.5 Partially Triangulated Graph Method (PTGM)......Page 166
4.6.7 Natural Associate Graph Method (NAGM)......Page 167
4.6.8 Incidence Graph Method (IGM)......Page 169
4.6.9 Representative Graph Method (RGM)......Page 170
4.6.10 Computational Results......Page 171
4.6.11 Discussions......Page 172
4.7.1 Introduction......Page 174
4.7.2 Graph Nodal Numbering for Profile Reduction......Page 176
4.7.3 Nodal Ordering with Element Clique Graph (NOECG)......Page 178
4.7.4 Nodal Ordering with Skeleton Graph (NOSG)......Page 179
4.7.6 Nodal Ordering with Element Wheel Graph (NOEWG)......Page 180
4.7.8 Nodal Ordering with Triangulated Graph (NOTG)......Page 181
4.7.11 Nodal Ordering with Representative Graph (NORG)......Page 182
4.7.14 Discussions......Page 184
4.8 Element Ordering for Frontwidth Reduction......Page 185
4.9.1 An Associate Graph......Page 188
4.9.3 Element Ordering Algorithms......Page 189
4.10.1 Definitions......Page 191
4.10.2 Algorithms......Page 192
4.10.3 Examples......Page 193
4.10.4 Bandwidth Reduction of Finite Element Models......Page 195
4.11 Graph-Theoretical Interpretation of Gaussian Elimination......Page 196
References......Page 199
5.2.1 Basic Concepts and Definitions......Page 201
5.2.3 Primary Nodal Decomposition......Page 204
5.2.5 Nodal Ordering......Page 205
5.3.1 Basic Concepts and Definitions......Page 206
5.4 A Hybrid Method for Ordering......Page 210
5.4.1 Development of the Method......Page 211
5.4.2 Numerical Results......Page 212
5.4.3 Discussions......Page 213
5.5.1.1 Background Definitions......Page 217
5.5.1.2 Newtonian Mechanics Laws......Page 218
5.5.1.3 The Rules of the Charged System Search......Page 219
5.5.2 The CSS Algorithm for Nodal Ordering......Page 222
5.5.3 Numerical Examples......Page 225
References......Page 227
6.2 Force Method for Finite Element Models: Rectangular and Triangular Plane Stress and Plane Strain Elements......Page 229
6.2.1 Member Flexibility Matrices......Page 230
6.2.2 Graphs Associated with FEMs......Page 234
6.2.3 Pattern Corresponding to the Self Stress Systems......Page 235
6.2.3.3 Type III Self Stress System......Page 237
6.2.4 Selection of Optimal gamma-Cycles Corresponding to Type II Self Stress Systems......Page 238
6.2.5 Selection of Optimal Lists......Page 239
6.2.6 Numerical Examples......Page 241
6.3 Finite Element Analysis Force Method: Triangular and Rectangular Plate Bending Elements......Page 244
6.3.2.1 Definitions of Independent Elements Forces......Page 247
6.3.2.3 Self-Equilibrating Systems of Type II......Page 250
6.3.2.4 Self-Equilibrating Systems of Type III......Page 251
6.3.2.5 Self-Equilibrating Systems of Type IV......Page 252
6.3.3 Numerical Examples......Page 254
6.4.1 Graphs Associated with Finite Element Model......Page 258
6.4.2 The Pattern Corresponding to the Self Stress Systems......Page 259
6.4.2.2 Type II Self Stress Systems......Page 260
6.4.2.4 Type II Minimal Cycles......Page 261
6.4.3.1 Number of Nodes of the Associate Graph......Page 262
6.4.3.2 The Number of Members of the Associate Graph......Page 263
6.4.4 Selection of Optimal gamma-Cycles Corresponding to Type II Self Stress Systems......Page 265
6.4.5 Selection of Optimal Lists......Page 266
6.4.6 Numerical Examples......Page 268
6.5 Efficient Finite Element Analysis Using Graph-Theoretical Force Method: Brick Element......Page 271
6.5.1 Definition of the Independent Element Forces......Page 272
6.5.2 Flexibility Matrix of an Element......Page 273
6.5.3.2 Natural Associate Graph......Page 275
6.5.4.2 Pattern of Type I Self-Equilibrating Systems......Page 277
6.5.4.3 Relationship Between gamma(S) and NAG(FEM)......Page 278
6.5.4.5 Pattern of Type III Self-Equilibrating Systems......Page 281
6.5.4.6 Type I Minimal Cycles......Page 282
6.5.4.7 Type II Minimal Cycles......Page 283
6.5.5 Models Including Internal Node......Page 284
6.5.6 Selection of an Optimal List Corresponding to Minimal Self-Equilibrating Stress Systems......Page 285
6.5.7 Numerical Examples......Page 286
References......Page 293
7.2 Finite Element Analysis of Models Comprised of Higher Order Triangular Elements......Page 295
7.2.3 Graphs Associated with Finite Element Model......Page 296
7.2.3.1 Natural Associate Graph......Page 297
7.2.4.1 Degree of Static Indeterminacy of the FEM......Page 298
7.2.4.4 Models Excluding Openings......Page 299
7.2.5.1 Pattern of Type II Self-Equilibrating Systems......Page 301
7.2.5.5 Self-Equilibrating Systems Corresponding to the External Indeterminacies......Page 303
7.2.6 Selection of an Optimal List Corresponding to Minimal Self-Equilibrating Stress Systems......Page 304
7.2.7 Numerical Examples......Page 305
7.3 Finite Element Analysis of Models Comprised of Higher Order Rectangular Elements......Page 311
7.3.1 Definition of Element Force System......Page 312
7.3.2 Flexibility Matrix of the Element......Page 314
7.3.3 Graphs Associated with Finite Element Model......Page 315
7.3.3.2 An Interface Graph......Page 316
7.3.4.1 Degree of Static Indeterminacy of the FEM......Page 317
7.3.4.3 Type II Self-Equilibrating Systems......Page 318
Type II Minimal Cycles of NAG(S)......Page 320
7.3.5 Selection of Generators for SESs of Type II and Type III......Page 321
7.3.6 Algorithm......Page 322
7.3.7 Numerical Examples......Page 323
7.4 Efficient Finite Element Analysis Using Graph-Theoretical Force Method: Hexa-Hedron Elements......Page 330
7.4.1 Independent Element Forces and Flexibility Matrix of Hexahedron Elements......Page 331
7.4.2.1 An Interface Graph......Page 335
7.4.2.2 Natural Associate Graph......Page 337
7.4.4 Pattern Corresponding to Self-Equilibrating Systems......Page 339
7.4.4.1 Type I Self-Equilibrating Systems......Page 340
7.4.4.2 Type II Self-Equilibrating Systems......Page 341
7.4.4.3 Type III Self-Equilibrating Systems......Page 343
7.4.5 Selection of Generators for SESs of Type II and Type III......Page 345
7.4.6 Numerical Examples......Page 348
References......Page 352
8.1 Introduction......Page 354
8.2.2 A Modified Level-Tree Separator Algorithm......Page 355
8.3.1 Introduction......Page 356
8.3.2 Substructuring Displacement Method......Page 357
8.3.3.2 Iterative Methods......Page 359
8.3.3.3 Hybrid Methods......Page 360
8.3.5 Examples......Page 361
8.3.6 Simplified Algorithm for Substructuring......Page 363
8.4 Domain Decomposition for Finite Element Analysis......Page 365
8.4.1 Introduction......Page 366
8.4.2 A Graph Based Method for Subdomaining......Page 367
8.4.4 Computational Results of the Graph Based Method......Page 369
8.4.5 Discussions on the Graph Based Method......Page 372
8.4.6 Engineering Based Method for Subdomaining......Page 373
8.4.7 Genre Structure Algorithm......Page 374
8.4.8 Example......Page 377
8.4.10 Discussions......Page 380
8.5.1 Algorithm for the Force Method Substructuring......Page 383
8.5.2 Examples......Page 386
References......Page 389
9.2.1 Boolean Operation on Graphs......Page 390
9.2.2 Cartesian Product of Two Graphs......Page 391
9.2.3 Strong Cartesian Product of Two Graphs......Page 393
9.2.4 Direct Product of Two Graphs......Page 394
9.3 Analysis of Near-Regular Structures Using Force Method......Page 396
9.3.1 Formulation of the Flexibility Matrix......Page 398
9.3.2 A Simple Method for the Formation of the Matrix AT......Page 401
9.4 Analysis of Regular Structures with Excessive Members......Page 402
9.4.2 Investigation of a Simple Example......Page 403
9.5.1 Investigation of an Illustrative Simple Example......Page 406
9.6 Practical Examples......Page 409
References......Page 419
10.1 Introduction......Page 420
10.2 Supervised Charged System Search Algorithm......Page 421
10.3 Analysis by Force Method and Charged System Search......Page 422
10.4 Procedure of Structural Design Using Force Method and the CSS......Page 427
10.4.1 Pre-selected Stress Ratio......Page 428
10.4.1.1 Fully Stress Design (FSD) for Statically Indeterminate Structures......Page 430
10.5 Minimum Weight......Page 433
References......Page 445
