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Finite Element Analysis
An updated and comprehensive review of the theoretical foundation of the finite element method
The revised and updated second edition of Finite Element Analysis: Method, Verification, and Validation offers a comprehensive review of the theoretical foundations of the finite element method and highlights the fundamentals of solution verification, validation, and uncertainty quantification. Written by noted experts on the topic, the book covers the theoretical fundamentals as well as the algorithmic structure of the finite element method. The text contains numerous examples and helpful exercises that clearly illustrate the techniques and procedures needed for accurate estimation of the quantities of interest. In addition, the authors describe the technical requirements for the formulation and application of design rules.
Designed as an accessible resource, the book has a companion website that contains a solutions manual, PowerPoint slides for instructors, and a link to finite element software. This important text:
* Offers a comprehensive review of the theoretical foundations of the finite element method
* Puts the focus on the fundamentals of solution verification, validation, and uncertainty quantification
* Presents the techniques and procedures of quality assurance in numerical solutions of mathematical problems
* Contains numerous examples and exercises
Written for students in mechanical and civil engineering, analysts seeking professional certification, and applied mathematicians, Finite Element Analysis: Method, Verification, and Validation, Second Edition includes the tools, concepts, techniques, and procedures that help with an understanding of finite element analysis.
List of contents
1 Introduction to FEM 3
1.1 An introductory problem 6
1.2 Generalized formulation 9
1.2.1 The exact solution 9
1.2.2 The principle of minimum potential energy 14
1.3 Approximate solutions 16
1.3.1 The standard polynomial space 17
1.3.2 Finite element spaces in one dimension 20
1.3.3 Computation of the coefficient matrices 22
1.3.4 Computation of the right hand side vector 26
1.3.5 Assembly 27
1.3.6 Condensation 30
1.3.7 Enforcement of Dirichlet boundary conditions 30
1.4 Post-solution operations 33
1.4.1 Computation of the quantities of interest 33
1.5 Estimation of error in energy norm 37
1.5.1 Regularity 38
1.5.2 A priori estimation of the rate of convergence 38
1.5.3 A posteriori estimation of error 40
1.5.4 Error in the extracted QoI 46
1.6 The choice of discretization in 1D 47
1.6.1 The exact solution lies in Hk(I), k . 1 > p 47
1.6.2 The exact solution lies in Hk(I), k . 1 <= p 49
1.7 Eigenvalue problems 52
1.8 Other finite element methods 57
1.8.1 The mixed method 59
1.8.2 Nitsche's method 60
2 Boundary value problems 63
2.1 Notation 63
2.2 The scalar elliptic boundary value problem 65
2.2.1 Generalized formulation 66
2.2.2 Continuity 68
2.3 Heat conduction 68
2.3.1 The differential equation 70
2.3.2 Boundary and initial conditions 71
2.3.3 Boundary conditions of convenience 73
2.3.4 Dimensional reduction 75
2.4 Linear elasticity - strong form 82
2.4.1 The Navier equations 86
2.4.2 Boundary and initial conditions 86
2.4.3 Symmetry, antisymmetry and periodicity 88
2.4.4 Dimensional reduction in linear elasticity 89
2.4.5 Incompressible elastic materials 93
2.5 Stokes flow 95
2.6 Elasticity - generalized formulation 96
2.6.1 The principle of minimum potential energy 98
2.6.2 The RMS measure of stress 100
2.6.3 The principle of virtual work 101
2.6.4 Uniqueness 102
2.7 Residual stresses 106
2.8 Chapter summary 108
3 Implementation 111
3.1 Standard elements in two dimensions 111
3.2 Standard polynomial spaces 111
3.2.1 Trunk spaces 111
3.2.2 Product spaces 112
3.3 Shape functions 112
3.3.1 Lagrange shape functions 113
3.3.2 Hierarchic shape functions 115
3.4 Mapping functions in two dimensions 118
3.4.1 Isoparametric mapping 118
3.4.2 Mapping by the blending function method 121
3.4.3 Mapping algorithms for high order elements 123
3.5 Finite element spaces in two dimensions 125
3.6 Essential boundary conditions 125
3.7 Elements in three dimensions 126
3.7.1 Mapping functions in three-dimensions 127
3.8 Integration and differentiation 129
3.8.1 Volume and area integrals 129
3.8.2 Surface and contour integrals 131
3.8.3 Differentiation 131
3.9 Stiffness matrices and load vectors 132
3.9.1 Stiffness matrices 133
3.9.2 Load vectors 134
3.10 Post-solution operations 135
3.11 Computation of the solution and its first derivatives 135
3.12 Nodal forces 137
3.12.1 Nodal forces in the h-version 137
3.12.2 Nodal forces in the p-version 140
3.12.3 Nodal forces and stress resultants 141
3.13 Chapter summary 142
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About the author
Barna Szabó is Senior Professor in the Department of Mechanical Engineering and Materials Science at Washington University in St. Louis, USA. He is also co-founder and chairman of Engineering Software Research and Development, Inc.
Ivo Babu?ka is Professor Emeritus of The University of Texas at Austin, USA, Professor of Aerospace Engineering and Engineering Mechanics, Professor of Mathematics, and Senior Research Scientist of the Oden Institute of Computational Engineering and Sciences.
Summary
Finite Element Analysis
An updated and comprehensive review of the theoretical foundation of the finite element method
The revised and updated second edition of Finite Element Analysis: Method, Verification, and Validation offers a comprehensive review of the theoretical foundations of the finite element method and highlights the fundamentals of solution verification, validation, and uncertainty quantification. Written by noted experts on the topic, the book covers the theoretical fundamentals as well as the algorithmic structure of the finite element method. The text contains numerous examples and helpful exercises that clearly illustrate the techniques and procedures needed for accurate estimation of the quantities of interest. In addition, the authors describe the technical requirements for the formulation and application of design rules.
Designed as an accessible resource, the book has a companion website that contains a solutions manual, PowerPoint slides for instructors, and a link to finite element software. This important text:
* Offers a comprehensive review of the theoretical foundations of the finite element method
* Puts the focus on the fundamentals of solution verification, validation, and uncertainty quantification
* Presents the techniques and procedures of quality assurance in numerical solutions of mathematical problems
* Contains numerous examples and exercises
Written for students in mechanical and civil engineering, analysts seeking professional certification, and applied mathematicians, Finite Element Analysis: Method, Verification, and Validation, Second Edition includes the tools, concepts, techniques, and procedures that help with an understanding of finite element analysis.
