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2024 | Buch

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Finite Element Methods

Parallel-Sparse Statics and Eigen-Solutions

verfasst von: Duc Thai Nguyen

Verlag: Springer Nature Switzerland

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

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Über dieses Buch

This new edition includes three new chapters, 7 through 9, that have very broad, practical applications in engineering and science. In addition, the author’s latest research results incorporated into the new textbook demonstrates better performance than the popular METIS software for partitioning graphs, partitioning finite element meshes, and producing fill-reducing orderings for sparse matrices. The new Chapter 8, and its pre-requisite, Chapter 7, present a state-of-the-art algorithm for computing the shortest paths for real-life (large-scale) transportation networks with minimum computational time. This approach has not yet appeared in any existing textbooks and it could open the doors for other transportation engineering applications. Chapter 9 vastly expands the scope of the previous edition by including sensitivity (gradient) computation and MATLAB’s built-in function “fmincon” for obtaining the optimum (or best) solution for general engineering problems.

Inhaltsverzeichnis

Frontmatter

Chapter 1. A Review of Basic Finite Element Procedures

Abstract

A thorough review of all basic (fundamental) components of the finite element (FE) procedures is presented in this chapter, which includes the “weak formulation,” how to identify the geometric versus natural boundary conditions, flowcharts for “statics and dynamics” finite element analysis (FEA), one−/two−/three-dimensional finite element equations, assembling process, how to incorporate boundary conditions into FE sparse solvers, and Gauss quadrature rules for integration.

Duc Thai Nguyen

Chapter 2. Simple MPI/FORTRAN/MATLAB Application

Abstract

A thorough explanation of the massage passing interface (MPI) FORTRAN and MATLAB programming (with examples) in both sequential and parallel computing environments is presented in this chapter. Powerful “vector unrolling techniques” for efficient matrix times matrix and Cholesky solver operations are also discussed in this chapter.

Duc Thai Nguyen

Chapter 3. Direct Sparse Equation Solvers

Abstract

A detailed explanation (including examples) of sparse storage schemes for storing large sparse (stiffness) matrix, sparse symbolic factorization, sparse numerical factorization, sparse forward and backward solvers, different reordering algorithms (to minimize nonzero fills-in terms occurred during the symbolic factorization phase), super-nodes and unrolling strategies to improve computational efficiencies of Cholesky (and unsymmetrical) solvers, and alternative approach for handling indefinite system of linear equations are all presented in this chapter.

Duc Thai Nguyen

Chapter 4. Sparse Assembly Process

Abstract

A detailed explanation (including examples) of the sparse (symbolic and numerical) assembling process to obtain finite element “system” stiffness equations for both symmetrical and unsymmetrical stiffness equations is explained in this chapter. How to incorporate Dirichlet boundary conditions into the finite element sparse assembling process is also presented in this chapter.

Duc Thai Nguyen

Chapter 5. Generalized Eigen-Solvers

Abstract

Detailed explanation (including examples) of solving the standard and generalized eigenproblems is provided in this chapter. Popular subspace iterations and Lanczos algorithms (including major subtopics, such as inverse and forward iterations, shifted eigenproblems, Sturm sequence check, unsymmetrical eigensolvers, balanced matrix, reduction to Hessenberg form, QR factorization, and modified Gram-Schmidt re-orthogonalization) are presented in this chapter.

Duc Thai Nguyen

Chapter 6. Finite Element Domain Decomposition Procedures

Abstract

A detailed explanation (including examples) of different finite element algorithms based on “divide-and-conquer” strategies (such as classical substructuring method, FETI-D, FETI-DP methods) is presented in this chapter. Major subtopics including the generalized/pseudo-inverse, preconditioned conjugate gradient (PCG) algorithm and mixed direct-iterative solvers are also discussed in this chapter.

Duc Thai Nguyen

Chapter 7. Heuristic Partitioning Algorithm for General Purpose Transportation Networks and Finite Element Meshes

Abstract

Recently developed heuristic algorithms for efficient domain partitioning (DP) of large-scale finite element applications are thoroughly discussed in this chapter. Examples are provided to clarify the steps involved in DP. Numerical performance (to reduce/minimize the system boundary nodes among adjacent subdomains) of the developed DP algorithm is also compared to popular METIS software. Real-life transportation networks and finite element meshes are used to validate the developed DP algorithms.

Duc Thai Nguyen

Chapter 8. Parallel Domain Partitioning Shortest Path Algorithms

Abstract

New Dijkstra shortest path (SP) algorithm that takes full advantage of DP algorithms developed in Chap. 7 is presented in this chapter. A small-scale numerical example of “10-node” transportation network is used to explain detailed steps to get the correct solution for Dijkstra SP algorithm by integrating its subdomains’ individual solutions. This chapter is ended with the solutions of several real-life (large-scale) transportation networks using the newly developed “parallel Dijkstra SP algorithm” with finite element based domain partitioning (DP) procedures.

Duc Thai Nguyen

Chapter 9. Sensitivity Analysis and Optimization with Partitioned Subdomains

Abstract

Finite element-based design sensitivity analysis (DSA) and optimal design within a general framework of domain partitioning (DP) formulation are thoroughly explained in this chapter. Three different methods, such as the adjoint method, the direct method, and the finite difference methods, for computing “sensitivity (or derivative)” information are presented in this chapter. Both “generic” and “structural analysis” examples are used in this chapter to clarify detailed computational issues involved in these three methods. This chapter is ended with an explanation of incorporating the computed derivative information into MATLAB built-in function “FMINCON” to iteratively achieve the “optimal, or near-optimal” structural (size, shape, and topology) optimization solutions.

Duc Thai Nguyen

Backmatter

Titel: Finite Element Methods
verfasst von: Duc Thai Nguyen
Verlag: Springer Nature Switzerland
Electronic ISBN: 978-3-031-48788-0
Print ISBN: 978-3-031-48787-3
DOI: https://doi.org/10.1007/978-3-031-48788-0

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