Finite Element Analysis: Method, Verification and Validation

✍ Scribed by Barna Szabó

• Publisher: Wiley
• Year: 2021
• Tongue: English
• Leaves: 387
• Series: Wiley Series in Computational Mechanics
• Edition: 2
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover
Title Page
Copyright
Contents
Preface to the second edition
Preface to the first edition
Preface
About the companion website
Chapter 1 Introduction to the finite element method
1.1 An introductory problem
1.2 Generalized formulation
1.2.1 The exact solution
1.2.2 The principle of minimum potential energy
1.3 Approximate solutions
1.3.1 The standard polynomial space
1.3.2 Finite element spaces in one dimension
1.3.3 Computation of the coefficient matrices
1.3.4 Computation of the right hand side vector
1.3.5 Assembly
1.3.6 Condensation
1.3.7 Enforcement of Dirichlet boundary conditions
1.4 Post‐solution operations
1.4.1 Computation of the quantities of interest
1.5 Estimation of error in energy norm
1.5.1 Regularity
1.5.2 A priori estimation of the rate of convergence
1.5.3 A posteriori estimation of error
1.5.4 Error in the extracted QoI
1.6 The choice of discretization in 1D
1.6.1 The exact solution lies in Hk(I), k−1<p
1.6.2 The exact solution lies in Hk(I), k−1≤p
1.7 Eigenvalue problems
1.8 Other finite element methods
1.8.1 The mixed method
1.8.2 Nitsche's method
Chapter 2 Boundary value problems
2.1 Notation
2.2 The scalar elliptic boundary value problem
2.2.1 Generalized formulation
2.2.2 Continuity
2.3 Heat conduction
2.3.1 The differential equation
2.3.2 Boundary and initial conditions
2.3.3 Boundary conditions of convenience
2.3.4 Dimensional reduction
2.4 Equations of linear elasticity – strong form
2.4.1 The Navier equations
2.4.2 Boundary and initial conditions
2.4.3 Symmetry, antisymmetry and periodicity
2.4.4 Dimensional reduction in linear elasticity
2.4.5 Incompressible elastic materials
2.5 Stokes flow
2.6 Generalized formulation of problems of linear elasticity
2.6.1 The principle of minimum potential energy
2.6.3 The principle of virtual work
2.6.4 Uniqueness
2.7 Residual stresses
2.8 Chapter summary
Chapter 3 Implementation
3.1 Standard elements in two dimensions
3.2 Standard polynomial spaces
3.2.1 Trunk spaces
3.2.2 Product spaces
3.3 Shape functions
3.3.1 Lagrange shape functions
3.3.2 Hierarchic shape functions
3.4 Mapping functions in two dimensions
3.4.1 Isoparametric mapping
3.4.2 Mapping by the blending function method
3.4.3 Mapping algorithms for high order elements
3.5 Finite element spaces in two dimensions
3.6 Essential boundary conditions
3.7 Elements in three dimensions
3.7.1 Mapping functions in three dimensions
3.8 Integration and differentiation
3.8.1 Volume and area integrals
3.8.2 Surface and contour integrals
3.8.3 Differentiation
3.9 Stiffness matrices and load vectors
3.9.1 Stiffness matrices
3.9.2 Load vectors
3.10 Post‐solution operations
3.11 Computation of the solution and its first derivatives
3.12 Nodal forces
3.12.1 Nodal forces in the h‐version
3.12.2 Nodal forces in the p‐version
3.12.3 Nodal forces and stress resultants
3.13 Chapter summary
Chapter 4 Pre‐ and postprocessing procedures and verification
4.1 Regularity in two and three dimensions
4.2 The Laplace equation in two dimensions
4.2.1 2D model problem, uEX∈Hk(Ω),k−1<p
4.2.2 2D model problem, uEX∈Hk(Ω),k−1≤p
4.2.3 Computation of the flux vector in a given point
4.2.4 Computation of the flux intensity factors
4.2.5 Material interfaces
4.3 The Laplace equation in three dimensions
4.4 Planar elasticity
4.4.1 Problems of elasticity on an L‐shaped domain
4.4.2 Crack tip singularities in 2D
4.4.3 Forcing functions acting on boundaries
4.5 Robustness
4.6 Solution verification
Chapter 5 Simulation
5.1 Development of a very useful mathematical model
5.1.1 The Bernoulli‐Euler beam model
5.1.2 Historical notes on the Bernoulli‐Euler beam model
5.2 Finite element modeling and numerical simulation
5.2.1 Numerical simulation
5.2.2 Finite element modeling
5.2.3 Calibration versus tuning
5.2.4 Simulation governance
5.2.5 Milestones in numerical simulation
5.2.6 Example: The Girkmann problem
5.2.7 Example: Fastened structural connection
5.2.8 Finite element model
5.2.9 Example: Coil spring with displacement boundary conditions
5.2.10 Example: Coil spring segment
Chapter 6 Calibration, validation and ranking
6.1 Fatigue data
6.1.1 Equivalent stress
6.1.2 Statistical models
6.1.3 The effect of notches
6.1.4 Formulation of predictors of fatigue life
6.2 The predictors of Peterson and Neuber
6.2.1 The effect of notches – calibration
6.2.2 The effect of notches – validation
6.2.3 Updated calibration
6.2.4 The fatigue limit
6.2.5 Discussion
6.3 The predictor Gα
6.3.1 Calibration of β(V,α)
6.3.2 Ranking
6.3.3 Comparison of Gα with Peterson′s revised predictor
6.4 Biaxial test data
6.4.1 Axial, torsional and combined in‐phase loading
6.4.2 The domain of calibration
6.4.3 Out‐of‐phase biaxial loading
6.5 Management of model development
6.5.1 Obstacles to progress
Chapter 7 Beams, plates and shells
7.1 Beams
7.1.1 The Timoshenko beam
7.1.2 The Bernoulli‐Euler beam
7.2 Plates
7.2.1 The Reissner‐Mindlin plate
7.2.2 The Kirchhoff plate
7.2.3 The transverse variation of displacements
7.3 Shells
7.3.1 Hierarchic thin solid models
7.4 Chapter summary
Chapter 8 Aspects of multiscale models
8.1 Unidirectional fiber‐reinforced laminae
8.1.1 Determination of material constants
8.1.2 The coefficients of thermal expansion
8.1.3 Examples
8.1.4 Localization
8.1.5 Prediction of failure in composite materials
8.1.6 Uncertainties
8.2 Discussion
Chapter 9 Non‐linear models
9.1 Heat conduction
9.1.1 Radiation
9.1.2 Nonlinear material properties
9.2 Solid mechanics
9.2.1 Large strain and rotation
9.2.2 Structural stability and stress stiffening
9.2.3 Plasticity
9.2.4 Mechanical contact
9.3 Chapter summary
Appendix A Definitions
A.1 Normed linear spaces, linear functionals and bilinear forms
A.1.1 Normed linear spaces
A.1.2 Linear forms
A.1.3 Bilinear forms
A.2 Convergence in the space X
A.2.1 The space of continuous functions
A.2.2 The space Lp(Ω)
A.2.3 Sobolev space of order 1
A.2.4 Sobolev spaces of fractional index
A.3 The Schwarz inequality for integrals
Appendix B Proof of h‐convergence
Appendix C Convergence in 3D: Empirical results
Appendix D Legendre polynomials
D.1 Shape functions based on Legendre polynomials
Appendix E Numerical quadrature
E.1 Gaussian quadrature
E.2 Gauss‐Lobatto quadrature
Appendix F Polynomial mapping functions
F.1 Interpolation on surfaces
F.1.1 Interpolation on the standard quadrilateral element
F.1.2 Interpolation on the standard triangle
Appendix G Corner singularities in two‐dimensional elasticity
G.1 The Airy stress function
G.2 Stress‐free edges
G.2.1 Symmetric eigenfunctions
G.2.2 Antisymmetric eigenfunctions
G.2.3 The L‐shaped domain
G.2.4 Corner points
Appendix H Computation of stress intensity factors
H.1 Singularities at crack tips
H.2 The contour integral method
H.3 The energy release rate
H.3.1 Symmetric (Mode I) loading
H.3.2 Antisymmetric (Mode II) loading
H.3.3 Combined (Mode I and Mode II) loading
H.3.4 Computation by the stiffness derivative method
Appendix I Fundamentals of data analysis
I.1 Statistical foundations
I.2 Test data
I.3 Statistical models
I.4 Ranking
I.5 Confidence intervals
Appendix J Estimation of fastener forces in structural connections
Appendix K Useful algorithms in solid mechanics
K.1 The traction vector
K.2 Transformation of vectors
K.3 Transformation of stresses
K.4 Principal stresses
K.5 The von Mises stress
K.6 Statically equivalent forces and moments
K.6.1 Technical formulas for stress
Bibliography
Index
EULA