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E-raamat: Optimal Analysis of Structures by Concepts of Symmetry and Regularity

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  • Ilmumisaeg: 16-May-2013
  • Kirjastus: Springer Verlag GmbH
  • Keel: eng
  • ISBN-13: 9783709115657
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Optimal Analysis of Structures by Concepts of Symmetry and RegularitySuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 16-May-2013
  • Kirjastus: Springer Verlag GmbH
  • Keel: eng
  • ISBN-13: 9783709115657
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Optimal analysis is defined as an analysis that creates and uses sparse, well-structured and well-conditioned matrices. The focus is on efficient methods for eigensolution of matrices involved in static, dynamic and stability analyses of symmetric and regular structures, or those general structures containing such components. Powerful tools are also developed for configuration processing, which is an important issue in the analysis and design of space structures and finite element models. Different mathematical concepts are combined to make the optimal analysis of structures feasible. Canonical forms from matrix algebra, product graphs from graph theory and symmetry groups from group theory are some of the concepts involved in the variety of efficient methods and algorithms presented. The algorithms elucidated in this book enable analysts to handle large-scale structural systems by lowering their computational cost, thus fulfilling the requirement for faster analysis and design of future complex systems. The value of the presented methods becomes all the more evident in cases where the analysis needs to be repeated hundreds or even thousands of times, as for the optimal design of structures by different metaheuristic algorithms. The book is of interest to anyone engaged in computer-aided analysis and design and software developers in this field. Though the methods are demonstrated mainly through skeletal structures, continuum models have also been added to show the generality of the methods. The concepts presented are not only applicable to different types of structures but can also be used for the analysis of other systems such as hydraulic and electrical networks.

This book presents algorithms that will enable analysts to handle large-scale structural systems by lowering their computational cost, thus fulfilling the requirement for faster analysis and design of future complex systems.
1 Introduction to Symmetry and Regularity 1(14)
1.1 Symmetric Structures
1(4)
1.1.1 Definition of Symmetry
1(2)
1.1.2 History of the Developments of Symmetry in Structural Engineering
3(2)
1.2 Regular Structures
5(2)
1.2.1 Repetitive and Cyclic Structures
5(1)
1.2.2 Definition of Regularity
6(1)
1.3 Examples of Symmetric and Regular Structural Models
7(3)
1.4 Optimal Analysis of Structures
10(1)
References
11(4)
2 Introduction to Graph Theory and Algebraic Graph Theory 15(22)
2.1 Introduction
15(1)
2.2 Basic Concepts and Definitions of Graph Theory
16(6)
2.2.1 Definition of a Graph
16(1)
2.2.2 Adjacency and Incidence
17(1)
2.2.3 Graph Operations
17(1)
2.2.4 Walks, Trails and Paths
18(1)
2.2.5 Cycles and Cutsets
19(1)
2.2.6 Trees, Spanning Trees and Shortest Route Trees
19(1)
2.2.7 Directed Graphs
20(1)
2.2.8 Different Types of Graphs
21(1)
2.3 Vector Spaces Associated with a Graph
22(2)
2.3.1 Cycle Space
22(1)
2.3.2 Cutset Space
23(1)
2.3.3 Cycle Bases Matrices
23(1)
2.3.4 Cutset Bases Matrices
24(1)
2.4 Graphs Associated with Matrices
24(1)
2.5 Planar Graphs: Euler's Polyhedron Formula
25(2)
2.5.1 Planar Graphs
26(1)
2.6 Definitions from Algebraic Graph Theory
27(5)
2.6.1 Incidence, Adjacency and Laplacian Matrices of a Graph
27(1)
2.6.2 Incidence and Adjacency Matrices of a Directed Graph
28(1)
2.6.3 Adjacency and Laplacian Matrices of a Weighted Graph
29(1)
2.6.4 Eigenvalues and Eigenvectors of an Adjacency Matrix
30(1)
2.6.5 Eigenvalues and Eigenvectors of a Laplacian Matrix
31(1)
2.6.6 Additional Properties of a Laplacian Matrix
31(1)
2.7 Matrix Representation of a Graph in Computer
32(2)
2.8 Historical Problem of Graph Theory
34(1)
References
35(2)
3 Graph Products and Configuration Processing 37(32)
3.1 Introduction
37(1)
3.2 Definitions of Different Graph Products
38(7)
3.2.1 Boolean Operation on Graphs
38(1)
3.2.2 Cartesian Product of Two Graphs
38(2)
3.2.3 Strong Cartesian Product of Two Graphs
40(1)
3.2.4 Direct Product of Two Graphs
41(2)
3.2.5 Lexicographic Product of Two Graphs
43(2)
3.3 Directed Graph Products
45(5)
3.3.1 Type I Directed Graph Products
46(1)
3.3.2 Type II Directed Graph Products
47(1)
3.3.3 Type III Directed Graph Products
48(1)
3.3.4 Type IV Directed Graph Products
49(1)
3.4 Weighted Triangular and Circular Graph Products for Configuration Processing
50(3)
3.4.1 Extension of Classic Graph Products
50(1)
3.4.2 Formulation of Weighted Strong Cartesian Product
51(1)
3.4.3 Formulation of Weighted Direct New Product
52(1)
3.4.4 Weighted Cartesian Direct Graph Products
52(1)
3.5 Definition of Weighted Triangular Graph Products
53(3)
3.5.1 Weights Assigned to Nodes of the Generators and Product Graphs
54(1)
3.5.2 Weighted Triangular Strong Cartesian Graph Product
55(1)
3.5.3 Weighted Triangular Semistrong Cartesian Graph Product
55(1)
3.6 Definition of a Weighted Circular Graph Product
56(4)
3.6.1 Weighted Circular Cartesian Graph Products
57(1)
3.6.2 Weighted Circular Strong Cartesian Graph Product
57(1)
3.6.3 Weighted Circular Direct Graph Product
58(2)
3.6.4 Weighted Circular Cartesian Direct Graph Product
60(1)
3.7 Weighted Cut-Out in Graph Products
60(3)
3.7.1 Weighted Cut-Outs in Cartesian Graph Product Models
61(1)
3.7.2 Weighted Cut-Out Cartesian Direct Graph Product
61(1)
3.7.3 Weighted Cut-Out Strong Cartesian Graph Product
62(1)
3.7.4 Weighted Cut-Out Semistrong Cartesian Graph Product
62(1)
3.8 Covered Graph Products
63(4)
3.8.1 Covered Cut-Out Cartesian Graph Product
64(1)
3.8.2 Covered Cut-Out Strong Cartesian Graph Product
65(1)
3.8.3 Weighted Covered Cut-Out Strong Cartesian Graph Product
66(1)
3.8.4 Weighted Covered Cut-Out Semistrong Cartesian Graph Product
66(1)
References
67(2)
4 Canonical Forms, Basic Definitions and Properties 69(46)
4.1 Introduction
69(1)
4.2 Decomposition of Matrices to Special Forms
69(12)
4.2.1 Canonical Form I
70(1)
4.2.2 Canonical Form II
70(2)
4.2.3 Canonical Form III
72(2)
4.2.4 Transformation of Form III into Form II
74(2)
4.2.5 Form IV Symmetry
76(2)
4.2.6 Method for the Formation of el and e2 Matrices
78(3)
4.3 Generalization of Form IV to Higher-Order Matrices
81(2)
4.4 Special Pattern Form IV Matrices
83(2)
4.5 Eig[ M] Operator
85(1)
4.6 Laplacian Matrices for Different Forms
86(11)
4.6.1 Symmetry and Laplacian of Graphs
86(2)
4.6.2 Factorisation of Symmetric Graphs
88(4)
4.6.3 Form III as an Augmented Form II
92(4)
4.6.4 Mixed Models
96(1)
4.7 Graph Representation of Form IV Symmetry
97(4)
4.7.1 Graph Representation
97(1)
4.7.2 Examples
98(3)
4.8 Generalised Form III Matrix
101(1)
4.9 Block Diagonalization of Compound Matrices
102(5)
4.10 Matrices as the Sum of Three Kronecker Products
107(1)
4.11 The Commutating Condition
108(1)
4.12 A Block Tri-diagonal Matrix with Corner Blocks and Its Block Diagonalisation
109(4)
References
113(2)
5 Canonical Forms for Combinatorial Optimisation, Nodal Ordering and Graph Partitioning 115(16)
5.1 Introduction
115(1)
5.2 Preliminary Definitions
115(1)
5.3 Algebraic Graph Theory for Ordering and Partitioning
116(1)
5.4 Eigenvalue Problems and Similarity Transformation
117(1)
5.5 A Special Canonical Form and Its Block Diagonalisation
117(2)
5.6 Adjacency and Laplacian Matrices for Models of Different Topologies
119(4)
5.6.1 Configuration of Type 1
119(1)
5.6.2 Configurations of Type 2, Type 3 and Type 4
120(3)
5.7 Examples from Structural Models
123(5)
References
128(3)
6 Graph Products for Ordering and Domain Decomposition 131(22)
6.1 Introduction
131(1)
6.2 Graph Models of Finite Element Meshes
132(1)
6.3 Eigenvalues of Graph Matrices for Cartesian Product
132(4)
6.3.1 Properties of Kronecker Product
132(1)
6.3.2 Theorem
133(1)
6.3.3 Eigenvalues of Graph Matrices for Cycle and Path Graphs
134(1)
6.3.4 Example
135(1)
6.4 Spectral Method for Bisection
136(1)
6.4.1 Computing λ2 for Laplacian of Regular Models
136(1)
6.4.2 Algorithm
136(1)
6.5 Numerical Results
137(3)
6.6 Spectral Method for Nodal Ordering
140(1)
6.7 Spectral Method for Different Product Graphs: An Approximate Method
141(8)
6.7.1 Main Theorem
143(1)
6.7.2 Eigensolution in Cartesian Product of Two Graphs
144(1)
6.7.3 Eigensolution in Direct Product of Two Graphs
145(1)
6.7.4 Eigensolution in Strong Cartesian Product of Two Graphs
145(1)
6.7.5 Examples
146(3)
6.8 Numerical Examples
149(2)
References
151(2)
7 Canonical Forms Applied to Structural Mechanics 153(112)
7.1 Introduction
153(1)
7.2 Vibrating Cores for a Mass-Spring Vibrating System
154(11)
7.2.1 The Graph Model of a Mass-Spring System
156(1)
7.2.2 Vibrating Systems with Form II Symmetry
157(2)
7.2.3 Vibrating Systems with Form III Symmetry
159(2)
7.2.4 Generalized Form III and Vibrating System
161(4)
7.2.5 Discussion
165(1)
7.3 Buckling Load of Symmetric Frames
165(17)
7.3.1 Buckling Load for Symmetric Frames with Odd Number of Spans per Storey
165(10)
7.3.2 Buckling Load for Symmetric Frames with an Even Number of Spans per Storey
175(6)
7.3.3 Discussion
181(1)
7.4 Eigenfrequencies of Symmetric Planar Frame
182(13)
7.4.1 Eigenfrequencies of Planar Symmetric Frames with Odd Number of Spans
182(8)
7.4.2 Decomposition of Symmetric Planar Frames with Even Number of Spans
190(4)
7.4.3 Discussion
194(1)
7.5 Eigenfrequencies of Symmetric Planar Trusses via Weighted Graph Symmetry and New Canonical Forms
195(22)
7.5.1 Modified Symmetry Forms
195(5)
7.5.2 Numerical Results
200(16)
7.5.3 Discussion
216(1)
7.6 General Canonical Forms for Analytical Solution of Problems in Structural Mechanics
217(13)
7.6.1 Definitions
217(1)
7.6.2 Decomposition of a Tri-diagonal Matrix
218(3)
7.6.3 A New Form for Efficient Solution of Eigenproblem
221(5)
7.6.4 Canonical Penta-diagonal Form
226(4)
7.7 Numerical Examples for the Matrices as the Sum of Three Kronecker Products
230(6)
7.8 Symmetric Finite Element Formulation Using Canonical Forms: Truss and Frame Elements
236(13)
7.8.1 Sign Convention
236(1)
7.8.2 Truss Element
237(6)
7.8.3 Beam Element
243(5)
7.8.4 Discussion
248(1)
7.9 Eigensolution of Rotationally Repetitive Space Structures
249(14)
7.9.1 Basic Formulation of the Used Stiffness Matrix
249(2)
7.9.2 A Canonical Form Associated with Rotationally Repetitive Structures
251(1)
7.9.3 Eigensolution for Finding Buckling Load of Structure with the BTMCB Form
252(3)
7.9.4 Eigensolution for Free Vibration of Structural Systems with the BTMCB Form
255(1)
7.9.5 Reducing Computational Efforts by Substructuring the System
256(2)
7.9.6 Numerical Examples
258(4)
7.9.7 Concluding Remarks
262(1)
References
263(2)
8 Graph Products Applied to the Analysis of Regular Structures 265(50)
8.1 Introduction
265(1)
8.2 Analysis of Repetitive Structures
266(15)
8.2.1 Eigenvectors for Sum of the Kronecker Products
266(2)
8.2.2 Solution of Linear Equations via Eigenvalues and Eigenvectors
268(1)
8.2.3 Kronecker Product of a Path and a Cycle
269(2)
8.2.4 An Illustrative Example
271(2)
8.2.5 Algorithm for the Analysis
273(1)
8.2.6 Numerical Examples
274(7)
8.3 Static and Modal Analyses of Structures with Different Repeated Patterns
281(6)
8.3.1 Static Analysis of Structures with Repeated Patterns
282(5)
8.4 Free Vibration Analysis of Irregular Structure Comprising of Regular Parts
287(12)
8.4.1 Illustrative Examples
288(9)
8.4.2 Discussion
297(2)
8.5 Block Circulant Matrices and Applications in Free Vibration Analysis of Cyclically Repetitive Structures
299(8)
8.5.1 Some Basic Definitions and Concepts of Block Circulant Matrices
299(1)
8.5.2 Some Properties of Permutation Matrices
300(2)
8.5.3 Some Properties of Block Circulant Matrices
302(3)
8.5.4 The Complete Study of a Simple Example
305(2)
8.6 Complementary Examples
307(6)
References
313(2)
9 Graph Products Applied to the Locally Modified Regular Structures Using Direct Methods 315(26)
9.1 Introduction
315(1)
9.2 Analysis of Non-regular Graphs Using the Results of Regular Models via an Iterative Method
315(14)
9.2.1 Main Method
316(3)
9.2.2 Numerical Examples
319(9)
9.2.3 Discussion
328(1)
9.3 Application of Kronecker Product to the Analysis of Modified Regular Structures
329(10)
9.3.1 Inversion of Block Matrices
329(2)
9.3.2 Proposed Method
331(5)
9.3.3 Numerical Examples
336(2)
9.3.4 Concluding Remarks
338(1)
References
339(2)
10 Graph Products Applied to the Regular and Locally Modified Regular Structures Using Iterative Methods 341(60)
10.1 Introduction
341(1)
10.2 Eigensolution of Symmetric and Regular Structures Using Canonical Forms
341(22)
10.2.1 Canonical Form II
343(1)
10.2.2 Canonical Form III
344(3)
10.2.3 Nested Form II
347(1)
10.2.4 Nested Form III
348(2)
10.2.5 Generalised Form II
350(3)
10.2.6 Block Circulant Form
353(6)
10.2.7 Augmented Block Circulant (ABC) Form
359(4)
10.3 Eigensolution of Locally Modified Regular Structures Using Iterative Methods
363(22)
10.3.1 Eigensolution of Locally Modified Regular Structures Using Shifted Inverse Iteration Method
364(9)
10.3.2 Approximate Eigensolution of Locally Modified Regular Structures Using a Substructuring Technique
373(12)
10.4 Substructure Representation for Efficient Eigensolution of Regular Structures
385(13)
10.4.1 Substructure Representation of TRS
387(2)
10.4.2 Modal Truncation
389(1)
10.4.3 Reduced Eigenproblem
390(1)
10.4.4 Evaluation of the Residual Flexibility Matrix
391(1)
10.4.5 Numerical Experiments
391(7)
References
398(3)
11 Group Theory and Applications in Structural Mechanics 401(32)
11.1 Introduction
401(1)
11.2 Basic Concepts of Symmetry Groups and Representation Theory
402(6)
11.2.1 Definition of a Group
402(1)
11.2.2 Classes of a Group
402(1)
11.2.3 Symmetry and Symmetry Operations
403(1)
11.2.4 Symmetry Group
404(1)
11.2.5 Representation Theory
404(4)
11.3 Stability Analysis of Hyper Symmetric Skeletal Structures Using Group Theory
408(8)
11.3.1 A Review of the Present Method Through a Simple Example
408(7)
11.3.2 More Complicated Forms of Symmetry
415(1)
11.4 Finding the Factors of a Symmetric Column Element
416(2)
11.4.1 Hyper Symmetry
418(1)
11.5 Symmetric Frames Having Numerous Symmetry Operators
418(14)
11.5.1 Frames with Symmetrical Factors
427(4)
11.5.2 Discussions
431(1)
References
432(1)
12 Graph-Group Method for the Analysis of Symmetric-Regular Structures 433(26)
12.1 Introduction
433(1)
12.2 Symmetry Groups of Graph Products
433(4)
12.3 Symmetry Analysis of Product Graphs
437(12)
12.4 Application in Analysis of Prestressed Cable Nets
449(9)
12.5 Discussion
458(1)
References
458(1)
Index 459
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