Title Computational structural analysis and finite element methods / Ali Kaveh

Published New York : Springer, 2013 ©2014

Table of 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
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
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
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
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
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
8.3.3.	Methods of Substructuring					346
8.3.4.	Main Algorithm for Substructuring					348
8.3.5.	Examples					348
8.3.6.	Simplified Algorithm for Substructuring					350
8.3.7.	Greedy Type Algorithm					352
8.4.	Domain Decomposition for Finite Element Analysis					352
8.4.1.	Introduction					353
8.4.2.	A Graph Based Method for Subdomaining					354
8.4.3.	Renumbering of Decomposed Finite Element Models					356
8.4.4.	Computational Results of the Graph Based Method					356
8.4.5.	Discussions on the Graph Based Method					359
8.4.6.	Engineering Based Method for Subdomaining					360
8.4.7.	Genre Structure Algorithm					361
8.4.8.	Example					364
8.4.9.	Computational Results of the Engineering Based Method					367
8.4.10.	Discussions					367
8.5.	Substructuring: Force Method					370
8.5.1.	Algorithm for the Force Method Substructuring					370
8.5.2.	Examples					373
	References					376
9.	Analysis of Regular Structures Using Graph Products					377
9.1.	Introduction					377
9.2.	Definitions of Different Graph Products					377
9.2.1.	Boolean Operation on Graphs					377
9.2.2.	Cartesian Product of Two Graphs					378
9.2.3.	Strong Cartesian Product of Two Graphs					380
9.2.4.	Direct Product of Two Graphs					381
9.3.	Analysis of Near-Regular Structures Using Force Method					383
9.3.1.	Formulation of the Flexibility Matrix					385
9.3.2.	A Simple Method for the Formation of the Matrix AT					388
9.4.	Analysis of Regular Structures with Excessive Members					389
9.4.1.	Summary of the Algorithm					390
9.4.2.	Investigation of a Simple Example					390
9.5.	Analysis of Regular Structures with Some Missing Members					393
9.5.1.	Investigation of an Illustrative Simple Example					393
9.6.	Practical Examples					396
	References					406
10.	Simultaneous Analysis, Design and Optimization of Structures Using Force Method and Supervised Charged System Search					407
10.1.	Introduction					407
10.2.	Supervised Charged System Search Algorithm					408
10.3.	Analysis by Force Method and Charged System Search					409
10.4.	Procedure of Structural Design Using Force Method and the CSS					414
10.4.1.	Pre-selected Stress Ratio					415
10.5.	Minimum Weight					420
	References					432

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Description 1 online resource (445 pages) Contents From the Contents: Basic concepts of structural analysis and graph theory -- Optimal force method of structural analysis -- Optimal displacement method of structural analysis -- Ordering for optimal patterns of structural matrices: graph theory methods Summary 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 Notes Online resource; title from PDF title page (ebrary, viewed January 13, 2014) Subject Graphic statics. Structural analysis (Engineering) -- Computer programs. graphic statics. TECHNOLOGY & ENGINEERING -- Engineering (General) TECHNOLOGY & ENGINEERING -- Reference. Ingénierie. Graphic statics Structural analysis (Engineering) -- Computer programs Form Electronic book ISBN 9783319029641 3319029649 3319029630 9783319029634

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