Topological Transformations for Efficient Structural Analysis

Preface
Contents
1 The Main Objective of the Present Book
References
2 Introduction to Graph Theory and Algebraic Graph Theory
2.1 Basic Concepts and Definitions of Graph Theory
2.1.1 Definition of a Graph
2.1.2 Adjacency and Incidence
2.1.3 Graph Operations
2.1.4 Walks, Trails and Paths
2.1.5 Cycles and Cutsets
2.1.6 Trees, Spanning Trees and Shortest Route Trees
2.2 Different Types of Graphs
2.3 Vector Spaces of a Graph
2.3.1 Cycle Space
2.3.2 Cutset Space
2.3.3 Cycle Bases Matrices
2.3.4 Cutset Bases Matrices
2.4 Planar Graphs; Polyhedron Formula of Euler
2.4.1 Planar Graphs
2.5 Maximal Matching in Bipartite Graphs
2.5.1 Theorems on Matching
2.5.2 Maximum Matching
2.6 Historical Problem of Graph Theory
2.7 Definitions from Algebraic Graph Theory
2.7.1 The Adjacency Matrix
2.7.2 The Incidence Matrix
2.7.3 Incidence Matrix of an Oriented Graph
2.7.4 Some Properties of Symmetric Matrices
2.7.5 The Laplacian Matrix
2.8 Graphs Associated with Matrices
References
3 Embedding Graphs on Lower Dimensional Spaces
3.1 Introduction
3.2 Graph Drawing for Calculating the DSI of Space Structures
3.3 Ordering for Constructing Well Structured Sparse Matrices: Graph Theory Methods
3.4 Bandwidth Optimization
3.4.1 Preliminaries
3.4.2 A Shortest Route Tree and Its Properties
3.4.3 Nodal Ordering for Bandwidth Reduction
3.5 Finite Element Nodal Ordering
3.5.1 Element Clique Graph Method
3.5.2 Skeleton Graph Method
3.5.3 Element Star Graph Method
3.5.4 Element Wheel Graph Method
3.5.5 Partially Triangulated Graph Method
3.5.6 Triangulated Graph Method
3.5.7 Natural Associate Graph Method (NAGM)
3.5.8 Incidence Graph Method (INGM)
3.5.9 Representative Graph Method (REGM)
3.5.10 Complete Representative Graph Method (REGM)
3.6 Graph Models for Meshless Discretization
3.6.1 Strongly Connected Associate Graph (SCAG)
3.6.2 Partially Connected Associate Graph (PCAG)
3.6.3 Weakly Connected Associate Graph (WCAG)
3.6.4 Associate Bipartite Graph (ABG)
References
4 Embedding Graphs on Higher Dimensional Spaces
4.1 Introduction
4.2 Force Method of Structural Analysis
4.2.1 Equilibrium Equations
4.2.2 Compatibility Equations
4.3 Embeddings on Higher Dimensional Spaces
4.3.1 Definitions from Topology and Algebraic Topology
4.3.2 Orientable 2-Manifolds
4.3.3 Simplicial Complexes
4.3.4 CW-Complexes
4.3.5 Collapsible and Contractible Complexes
4.3.6 Homology Group
4.4 Cycle Bases Selection: Topological Methods
4.4.1 A 2-Dimensional Polyhedron Embedding
4.4.2 Admissible Embeddings
4.4.3 Modified Manifold Embedding
4.4.4 Embedding S on a Union of Disks
4.5 Graph-Theoretical Force Method
References
5 Embedding Graphs on Spaces of Identical Dimensions
5.1 Introduction
5.2 Element Ordering for Frontwidth Reduction; A Line Graph
5.3 Element and Nodal Ordering; A K-Total Graph
5.3.1 Definitions
5.3.2 Algorithm for Bandwidth Reduction of Rectangular Matrices
5.3.3 Entire Graph
5.4 Generalized Cycle Bases; Interchange Graph
5.5 Cycle and Generalized Cycle Basis Ordering
5.6 The Rigidity of Planar Trusses
5.6.1 Simple, Compound and Complex Trusses
5.6.2 The Rigidity of Grid-Shaped Planar Trusses
5.6.3 Grids with Diagonal Rods
5.6.4 Grid with Diagonal Rods and/or Cables
5.6.5 Henneberg Sequence for Examining the Rigidity of Trusses
5.6.6 Complete Matching and Rigidity of Trusses
5.7 Duality of Cycle Bases and Cut Set Bases; Dual Graph
5.7.1 A Planar Truss and Its Maxwell Diagram as Dual Graphs
5.7.2 Flow Graph and Potential Graph, and Their Applications
5.7.3 Potential Graph and Its Application
5.8 Other Applications
References
6 Structural Configuration Generation
6.1 Introduction
6.2 Algebraic Representation of a Graph in Integer Coordinate System
6.3 Representations of Operations on Graphs
6.3.1 Addition of Two Subgraphs
6.3.2 Subtraction of Two Subgraphs
6.4 Some Functions for Configuration Processing
6.4.1 Translation Functions
6.4.2 Rotation Functions
6.4.3 Reflection Functions
6.4.4 Projection Functions
6.5 Geometry of Structures
References
7 Symmetry Using Linear Algebra and Graph Theory
7.1 Introduction
7.2 Basic Definitions
7.2.1 Basic Concepts from the Theory of Graphs
7.2.2 Basic Definitions from Linear Algebra
7.2.3 Definitions from Algebraic Graph Theory
7.3 Canonical Forms of a Matrix
7.3.1 Form I
7.3.2 Form II
7.3.3 Form III
7.3.4 Form IV
7.4 A Canonical Form Associated with Rotationally Repetitive Structures
7.5 Different Kinds of Symmetry
7.5.1 Form I Symmetry
7.5.2 Form II Symmetry
7.5.3 Form III Symmetry
7.6 The Form Associated with Rotationally Repetitive Structures
7.7 The Relations Between the Canonical Forms
7.7.1 The Relation Between the Form I and Form II
7.7.2 The Relation Between the Form II and Form III
7.7.3 The Relation Between the Form IV and Form III
7.8 The Relation Between the Canonical Form II and the Form Associated with Rotationally Repetitive Structures
7.9 Examples
7.10 Concluding Remarks
References
8 Complementary Space of Graphs
8.1 Introduction
8.2 Theorems for Graph Partitioning
8.3 Largest Eigenvector of the Complementary Laplacian Matrix
8.4 Numerical Results
8.5 Concluding Remarks
References
9 Miscellaneous Applications Graph Problems Using Meta-Heuristic Algorithms
9.1 Introduction
9.2 Optimal Domain Decomposition Using Colliding Bodies Optimization and k-median Method
9.2.1 The Formulation of the CBO Algorithm
9.2.2 Mathematical Formulation of the k-median Problem
9.2.3 Numerical Examples
9.2.4 Results and Discussion on Examples
9.2.5 Concluding Remarks
9.3 Simulated Annealing Algorithm for Selecting Suboptimal Cycle Basis of a Graph
9.3.1 Definitions from Theory of Graphs
9.3.2 Simulated Annealing Algorithm
9.3.3 Simulated Annealing Algorithm for the Formation of a Suboptimal Cycle Basis
9.3.4 Generating the Initial Cycle Basis
9.3.5 Generating a Neighbor Solution
9.3.6 Reannealing (Restarting)
9.3.7 Algorithm Parameters
9.3.8 Examples
9.3.9 Conclusions
9.4 Further Miscellaneous Applications
9.4.1 Size Reduction Transformation
References