Finite Element Method

Name: Finite Element Method
Brand: Wiley-ISTE
Price: 238.99 EUR
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Gouri Dhatt Emmanuel Lefrançois Gilbert Touzot(Autor*in)

Wiley-ISTE (Verlag)

Erschienen am 27. Dezember 2012

624 Seiten

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978-1-118-56970-2 (ISBN)

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Inhalt

Cover
Finite Element Method
Title Page
Copyright Page
Table of Contents
Introduction
0.1 The finite element method
0.1.1 General remarks
0.1.2 Historical evolution of the method
0.1.3 State of the art
0.2 Object and organization of the book
0.2.1 Teaching the finite element method
0.2.2 Objectives of the book
0.2.3 Organization of the book
0.3 Numerical modeling approach
0.3.1 General aspects
0.3.2 Physical model
0.3.3 Mathematical model
0.3.4 Numerical model
0.3.5 Computer model
Bibliography
Conference proceedings
Monographs
Periodicals
Chapter 1. Approximations with finite elements
1.0 Introduction
1.1 General remarks
1.1.1 Nodal approximation
1.1.2 Approximations with finite elements
1.2 Geometrical definition of the elements
1.2.1 Geometrical nodes
1.2.2 Rules for the partition of a domain into elements
1.2.3 Shapes of some classical elements
1.2.4 Reference elements
1.2.5 Shapes of some classical reference elements
1.2.6 Node and element definition tables
1.3 Approximation based on a reference element
1.3.1 Expression of the approximate function u(x)
1.3.2 Properties of approximate function u(x)
1.4 Construction of functions N (? ) and N (? )
1.4.1 General method of construction
1.4.2 Algebraic properties of functions N and N
1.6 Computation of functions N, their derivatives and the Jacobian matrix
1.6.1 General remarks
1.6.2 Explicit forms for N
1.5 Transformation of derivation operators
1.5.1 General remarks
1.5.2 First derivatives
1.5.3 Second derivatives
1.5.4 Singularity of the Jacobian matrix
1.7 Approximation errors on an element
1.7.1 Notions of approximation errors
1.7.2 Error evaluation technique
1.7.3 Improving the precision of approximation
1.8 Example of application: rainfall problem
Bibliography
Chapter 2. Various types of elements
2.0 Introduction
2.1 List of the elements presented in this chapter
2.2 One-dimensional elements
2.2.1 Linear element (two nodes, C0)
2.2.2 High-precision Lagrangian elements: (continuity C0)
2.2.3 High-precision Hermite elements
2.2.4 General elements
2.3 Triangular elements (two dimensions)
2.3.1 Systems of coordinates
2.3.2 Linear element (triangle, three nodes, C0)
2.3.3 High-precision Lagrangian elements (continuity C0)
2.3.4 High-precision Hermite elements
2.4 Quadrilateral elements (two dimensions)
2.4.1 Systems of coordinates
2.4.2 Bilinear element (quadrilateral, 4 nodes, C0)
2.4.3 High-precision Lagrangian elements
2.4.4 High-precision Hermite element
2.5 Tetrahedral elements (three dimensions)
2.5.1 Systems of coordinates
2.5.2 Linear element (tetrahedron, four nodes, C0)
2.5.3 High-precision Lagrangian elements (continuity C0)
2.5.4 High-precision Hermite elements
2.6 Hexahedric elements (three dimensions)
2.6.1 Trilinear element (hexahedron, eight nodes, C0)
2.6.2 High-precision Lagrangian elements (continuity C0)
2.6.3 High-precision Hermite elements
2.7 Prismatic elements (three dimensions)
2.7.1 Element with six nodes (prism, six nodes, C0)
2.7.2 Element with 15 nodes (prism, 15 nodes, C0)
2.8 Pyramidal element (three dimensions)
2.8.1 Element with five nodes
2.9 Other elements
2.9.1 Approximation of vectorial values
2.9.2 Modifications of the elements
2.9.3 Elements with a variable number of nodes
2.9.4 Superparametric elements
2.9.5 Infinite elements
Bibliography
Chapter 3. Integral formulation
3.0 Introduction
3.1 Classification of physical systems
3.1.1 Discrete and continuous systems
3.1.2 Equilibrium, eigenvalue and propagation problems
3.2 Weighted residual method
3.2.1 Residuals
3.2.2 Integral forms
3.3 Integral transformations
3.3.1 Integration by parts
3.3.2 Weak integral form
3.3.3 Construction of additional integral forms
3.4 Functionals
3.4.1 First variation
3.4.2 Functional associated with an integral form
3.4.3 Stationarity principle
3.4.4 Lagrange multipliers and additional functionals
3.5 Discretization of integral forms
3.5.1 Discretization of W
3.5.2 Approximation of the functions u
3.5.3 Choice of the weighting functions ?
3.5.4 Discretization of a functional (Ritz method)
3.5.5 Properties of the systems of equations
3.6 List of PDEs and weak expressions
3.6.1 Scalar field problems
3.6.2 Solid mechanics
3.6.3 Fluid mechanics
Bibliography
Chapter 4. Matrix presentation of the finite element method
4.0 Introduction
4.1 The finite element method
4.1.1 Finite element approach
4.1.2 Conditions for convergence of the solution
4.1.3 Patch test
4.2 Discretized elementary integral forms We
4.2.1 Matrix expression of We
4.2.2 Case of a nonlinear operator L
4.2.3 Integral form We on the reference element
4.2.4 A few classic forms of We and of elementary matrices
4.4 Assembly of the global discretized form W
4.4.1 Assembly by expansion of the elementary matrices
4.4.2 Assembly in structural mechanics
4.3 Techniques for calculating elementary matrices
4.3.1 Explicit calculation for a triangular element (Poisson's equation)
4.3.2 Explicit calculation for a quadrangular element (Poisson's equation)
4.3.3 Organization of the calculation of the elementary matrices by numerical integration
4.3.4 Calculation of the elementary matrices: linear problems
4.5 Technique of assembly
4.5.1 Stages of assembly
4.5.2 Rules of assembly
4.5.3 Example of a subprogram for assembly
4.5.4 Construction of the localization table LOCE
4.6 Properties of global matrices
4.6.1 Band structure
4.6.2 Symmetry
4.6.3 Storage methods
4.7 Global system of equations
4.7.1 Expression of the system of equations
4.7.2 Introduction of the boundary conditions
4.7.3 Reactions
4.7.4 Transformation of variables
4.7.5 Linear relations between variables
4.8 Example of application: Poisson's equation
4.9 Some concepts about convergence, stability and error calculation
4.9.1 Notations
4.9.2 Properties of the exact solution
4.9.3 Properties of the solution obtained by the finite element method
4.9.4 Stability and locking
4.9.5 One-dimensional exact finite elements
Bibliography
Chapter 5. Numerical Methods
5.0 Introduction
5.1 Numerical integration
5.1.1 Introduction
5.1.2 One-dimensional numerical integration
5.1.3 Two-dimensional numerical integration
5.1.4 Numerical integration in three dimensions
5.1.5 Precision of integration
5.1.6 Choice of number of integration points
5.1.7 Numerical integration codes
5.2 Solving systems of linear equations
5.2.1 Introduction
5.2.2 Gaussian elimination method
5.2.3 Decomposition
5.2.4 Adaptation of algorithm (5.44) to the case of a matrix stored by the skyline method
5.3 Solution of nonlinear systems
5.3.1 Introduction
5.3.2 Substitution method
5.3.3 Newton-Raphson method
5.3.4 Incremental (or step-by-step) method
5.3.5 Changing of independent variables
5.3.6 Solution strategy
5.3.7 Convergence of an iterative method
5.4 Resolution of unsteady systems
5.4.1 Introduction
5.4.2 Direct integration methods for first-order systems
5.4.3 Modal superposition method for first-order systems
5.4.4 Methods for direct integration of second-order systems
5.4.5 Modal superposition method for second-order systems
5.5 Methods for calculating the eigenvalues and eigenvectors
5.5.1 Introduction
5.5.2 Recap of some properties of eigenvalue problems
5.5.3 Methods for calculating the eigenvalues
Bibliography
Chapter 6. Programming technique
6.0 Introduction
6.1 Functional blocks of a finite element program
6.2 Description of a typical problem
6.3 General programs
6.3.1 Possibilities of general programs
6.3.2 Modularity
6.4 Description of the finite element code
6.4.1 Introduction
6.4.2 General organization
6.4.3 Description of tables and variables
6.5 Library of elementary finite element method programs
6.5.1 Functional blocks
6.5.2 List of thermal elements
6.5.3 List of elastic elements
6.5.4 List of elements for fluid mechanics
6.6 Examples of application
6.6.1 Heat transfer problems
6.6.2 Planar elastic problems
6.6.3 Fluid flow problems
Appendix. Sparse solver
7.0 Introduction
7.1 Methodology of the sparse solver
7.1.1 Assembly of matrices in sparse form: row-by-row format
7.1.2 Permutation using the "minimum degree" algorithm
7.1.3 Modified column-column storage format
7.1.4 Symbolic factorization
7.1.5 Numerical factorization
7.1.6 Solution of the system by descent/ascent
7.2 Numerical examples
Bibliography
Index

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