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2014 | Buch

Computational Structural Analysis and Finite Element Methods

verfasst von: A. Kaveh

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Basic Definitions and Concepts of Structural Mechanics and Theory of Graphs

Abstract

This chapter consists of two parts. In the first part, basic definitions, concepts and theorems of structural mechanics are presented. These theorems are employed in the following chapters and are very important for their understanding. For determination of the distribution of internal forces and displacements, under prescribed external loading, a solution to the basic equations of the theory of structures should be obtained, satisfying the boundary conditions. In the matrix methods of structural analysis, one must also use these basic equations. In order to provide a ready reference for the development of the general theory of matrix structural analysis, the most important basic theorems are introduced in this chapter, and illustrated through simple examples.

In the second part, basic concepts and definitions of graph theory are presented. Since some of the readers may be unfamiliar with the theory of graphs, simple examples are included to make it easier to understand the presented concepts.

A. Kaveh

Chapter 2. Optimal Force Method: Analysis of Skeletal Structures

Abstract

This chapter starts with presenting simple and general methods for calculating the degree of static indeterminacy of different types of skeletal structures, such as rigid-jointed planar and space frames, pin-jointed planar trusses and ball-jointed space trusses. Then the progress made in the force method of structural analysis in recent years is presented. Efficient methods are developed for the formation of highly sparse flexibility matrices for different types of skeletal structures.

A. Kaveh

Chapter 3. Optimal Displacement Method of Structural Analysis

Abstract

In this chapter, the principles introduced in Chap. 1 are used for the formulation of the general displacement method of structural analysis. Computational aspects are discussed and many worked examples are included to illustrate the concepts and principles being used. In order to show the generality of the methods introduced for the formation of the element stiffness matrices, the stiffness matrix of a simple finite element is also derived.

A. Kaveh

Chapter 4. Ordering for Optimal Patterns of Structural Matrices: Graph Theory Methods

Abstract

In this chapter, methods are presented for ordering to form special patterns for sparse structural matrices. Such transformation reduces the storage and the number of operations required for the solution, and leads to more accurate results. Graph theory methods are presented for different approaches to reordering equations to preserve their sparsity, leading to predefined patterns. Alternative, objective functions are considered and heuristic algorithms are presented to achieve these objectives. Three main methods for the solution of structural equations require the optimisation of bandwidth, profile and frontwidth, especially for those encountered in finite element analysis. Methods are presented for reducing the bandwidth of the flexibility matrices. Bandwidth optimisation of rectangular matrices is presented for its use in the formation of sparse flexibility matrices.

A. Kaveh

Chapter 5. Ordering for Optimal Patterns of Structural Matrices: Algebraic Graph Theory and Meta-heuristic Based Methods

Abstract

There are different matrices associated with a graph, such as incidence matrix, the adjacency matrix and the Laplacian matrix. One of the aims of algebraic graph theory is to determine how properties of graphs are reflected in algebraic properties of these matrices. The eigenvalues and eigenvectors of these matrices provide valuable tools for combinatorial optimisation and in particular for ordering of sparse symmetric matrices such as the stiffness and flexibility matrices of the structures. Here, algebraic graph-theoretical methods and metaheuristic-based algorithms are provided for nodal ordering for bandwidth and profile reduction.

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Chapter 6. Optimal Force Method for FEMs: Low Order Elements

Abstract

In this chapter force method finite element models comprising of low order elements are presented.

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Chapter 7. Optimal Force Method for FEMS: Higher Order Elements

Abstract

In this chapter force method for the analysis of finite element models comprising of higher order elements are studied.

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Chapter 8. Decomposition for Parallel Computing: Graph Theory Methods

Abstract

In the last decade, parallel processing has come to be widely used in the analysis of large-scale structures. This chapter is devoted to the optimal decomposition of structural models using graph theory approaches. First, efficient graph theory methods are presented for the optimal decomposition of space structures. The subdomaining approaches are then presented for partitioning of finite element models. A substructuring technique for the force method of structural analysis is discussed.

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Chapter 9. Analysis of Regular Structures Using Graph Products

Abstract

In this chapter, an efficient method is presented for the analysis of non-regular structures which are obtained by addition or removal of some members to regular structural models. Here a near-regular structure is divided into two sets, namely “the regular part of the structure” and “the excessive members”. Regular part refers to the structure for which the inverse of the stiffness matrix can be obtained by the previously developed simplified methods, and excessive members refer to those which cause the non-regularity of the regular structure.

A. Kaveh

Chapter 10. Simultaneous Analysis, Design and Optimization of Structures Using Force Method and Supervised Charged System Search

Abstract

Developing methods with higher computation efficiency is a crucial subject in advanced engineering problems of multi-physics nature. For instance, analyzing structures with larger number of members requires larger memory size and longer computation time. In addition, this costly computation has to be repeated many times, typically over 10,000 times, because the cross section size of the members is not determined in the early stages of designing such structures. Therefore, reducing the size of structural matrices and eliminating the unduly repetitions in the design and analysis procedures can lead to a considerable reduction in the computation efficiency. In this chapter, this goal is achieved utilizing meta-heuristics algorithms which minimize the energy function indirectly. Besides, design procedure and minimizing the weight of the structure is added to the analysis procedure. One of the most reliable meta-heuristic methods recently developed is Charged System Search (CSS), that is used in here. In this chapter, supervisor agents are considered to increase the exploration ability of the CSS algorithm. This method is called supervised CSS abbreviated as SCSS.

A. Kaveh

Titel: Computational Structural Analysis and Finite Element Methods
verfasst von: A. Kaveh
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-02964-1
Print ISBN: 978-3-319-02963-4
DOI: https://doi.org/10.1007/978-3-319-02964-1

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