main-content

This book proposes and validates a number of methods and shortcuts for frugal engineers, which will allow them to significantly reduce the computational costs for analysis and reanalysis and, as a result, for structural design processes. The need for accuracy and speed in analyzing structural systems with ever-tighter design tolerances and larger numbers of elements has been relentlessly driving forward research into methods that are capable of analyzing structures at a reasonable computational cost. The methods presented are of particular value in situations where the analysis needs to be repeated hundreds or even thousands of times, as is the case with the optimal design of structures using different metaheuristic algorithms. Featuring methods that are not only applicable to skeletal structures, but by extension also to continuum models, this book will appeal to researchers and engineers involved in the computer-aided analysis and design of structures, and to software developers in this field. It also serves as a complement to previous books on the optimal analysis of large-scale structures utilizing concepts of symmetry and regularity. Further, its novel application of graph-theoretical methods is of interest to mathematicians.

### Chapter 1. Definitions from Graph Theory and Graph Products

Abstract
Graph theory is a branch of mathematics that has found many applications in engineering and science, such as chemical, electrical, civil and mechanical engineering, architecture, management and control, communication, operational research, sparse matrix technology, combinatorial optimisation, and computer science. Therefore many books have been published on applied graph theory. In this chapter some basic definitions and concepts of graph theory and graph products are presented.
Ali Kaveh, Hossein Rahami, Iman Shojaei

### Chapter 2. Basic Concepts and Definitions of Symmetry and Regularity

Abstract
Concepts and definitions for different types of symmetry and regularity are already published in a book by Kaveh, “Optimal Analysis of Structures by Concepts of Symmetry and Regularity”. In this chapter definitions and concepts relevant to the contents of the present book are provided to make it self-content. In this chapter different types of graph products, method of block diagonaliztion, techniques of calculating the eigenvalues and eigenvectors, decomposition of adjacency and Laplacian, and circulant matrices are presented.
Ali Kaveh, Hossein Rahami, Iman Shojaei

### Chapter 3. Static Analysis of Near-Regular Skeletal Structures: Additional Members

Abstract
A structure with decomposable stiffness matrix is hereafter called a regular structure. In addition to regular structures, there is another group of structures for which the stiffness matrix cannot be directly decomposed. These structures might be solvable (i.e., eigensolution or inversion) using the relationships in Chap. 2, through manipulation of either the stiffness matrix or the structure. However, such manipulations have to be compensated for, which might be computationally demanding. A structure that can be solved with minimum manipulations is hereafter called a near-regular structure. In general, there are two types of near-regular structures solution of which requires different algorithms and techniques. In this chapter near regular structures with additional members are discussed using displacement and force methods and methods are developed for the analysis and eigensolution of these structures.
Ali Kaveh, Hossein Rahami, Iman Shojaei

### Chapter 4. Static Analysis of Near-Regular Skeletal Structures: Additional Nodes

Abstract
In Chap. 3, methods for calculation of eigenpairs and inverse of matrices of near-regular structures with additional members were developed. There is another group of near-regular structures wherein the number of degrees of freedom for the near-regular structure is larger than that of the regular structure. Therefore, these structure can be decomposed to regular and irregular sub-structures to get advantages of swift solution of regular structures and parallel computations. This chapter is devoted to the analysis of near-regular structures with additional nodes using displacement and force methods as well as eigensolution of these structures.
Ali Kaveh, Hossein Rahami, Iman Shojaei

### Chapter 5. Static Analysis of Nearly Regular Continuous Domains

Abstract
Regarding the style and elegance of new designs as well as growing applications of prefabrication in structural and mechanical systems, many systems hold regular and near-regular geometrical patterns. Because of natural repetition of elements and nodes in numerical methods and regularity of meshes, regular and near regular patterns are also observed in many numerical solution resulting in decomposable stiffness matrices. In this chapter, finite element and mesh free solutions of regular and near-regular structural and mechanical systems are developed and discussed.
Ali Kaveh, Hossein Rahami, Iman Shojaei

### Chapter 6. Dynamic Analysis of Near-Regular Structures

Abstract
In previous chapters methods for eigensolution and static analysis of regular and near-regular skeletal and continuum structures were presented. In this chapter methods are developed for efficient dynamic analysis of structures. The proposed methods are based on converting initial-value problems into boundary-value problems and taking advantages of the properties of the latter problems in efficient solution of the former ones. Finite difference formulation of boundary-value problems leads to matrices with repetitive tri-diagonal and block tri-diagonal patterns whose eigensolution and matrix inversion are obtained using the relationships in Chap. 2.
Ali Kaveh, Hossein Rahami, Iman Shojaei

### Chapter 7. Swift Analysis of Linear and Non-linear Structures and Applications Using Reanalysis

Abstract
In this chapter efficient methods are presented for reducing the computational complexity of analysis for structures in the process of size and geometry optimization. These methods result in simultaneous swift analysis and optimal design of structures. A swift analysis for optimal size is performed using a modified solution for structures with changed members and the optimal analysis is performed via a modified solution for structures with changed supports and nodes. The methods are further generalized to non-linear analysis within optimization procedures and to mechanical/biomechanical systems other than structures.
Ali Kaveh, Hossein Rahami, Iman Shojaei

### Chapter 8. Global Near-Regular Mechanical Systems

Abstract
There is another group of near-regular mechanical systems that have global deviations from their corresponding regular system (hereafter called global near-regular mechanical systems). In such systems a large number of degrees of freedom are affected by irregularities. Despite extensive research on mechanical systems with local irregularities, the literature on global near-regular mechanical systems is scant. In this chapter, methods are developed for static and dynamic analysis/reanalysis as well as eigensolution (vibration analysis) of such systems using Kronecker products and matrix manipulations.
Ali Kaveh, Hossein Rahami, Iman Shojaei

### Chapter 9. Mappings for Transformation of Near-Regular Domains to Regular Domains

Abstract
In this chapter a numerical method is presented for efficient solution of many differential equations with arbitrary domains using geometrical transformation and Kronecker product rules. Initially, a mesh free formulation for rectangular domains is developed and a full decomposition of matrix equations is achieved using Kronecker product rules. The solution of a governing equation on an arbitrary domain is sought through a geometrical transformation from the rectangular domain into the original domain using conformal mapping. The efficiency of the proposed method is examined using various engineering examples.
Ali Kaveh, Hossein Rahami, Iman Shojaei

### Chapter 10. Numerical Solution for System of Linear Equations Using Tridiagonal Matrix

Abstract
In this Chapter methods are developed for numerical solution of system of linear equations through taking advantages of the properties of repetitive tridiagonal matrices. A system of linear equations is usually obtained in the final step of many science and engineering problems such as problems involving partial differential equations. In the proposed solutions, the problem is first solved for repetitive tridiagonal matrices and a closed-from relationship is obtained. This relationship is then utilized for the solution of a general matrix through converting the matrix into a repetitive tridiagonal matrix and the remaining matrix.
Ali Kaveh, Hossein Rahami, Iman Shojaei