Оглавление
Barna Szabó. Finite Element Analysis
Table of Contents
List of Tables
List of Illustrations
Guide
Pages
WILEY SERIES IN COMPUTATIONAL MECHANICS
Finite Element Analysis. Method, Verification and Validation
Preface to the second edition
Preface to the first edition
Notes
Preface
About the companion website
1 Introduction to the finite element method
1.1 An introductory problem
The choice of basis functions
Summary of the main points
1.2 Generalized formulation
1.2.1 The exact solution
Summary of the main points
1.2.2 The principle of minimum potential energy
1.3 Approximate solutions
1.3.1 The standard polynomial space
Lagrange shape functions
Legendre shape functions
1.3.2 Finite element spaces in one dimension
1.3.3 Computation of the coefficient matrices
Computation of the stiffness matrix
Computation of the Gram matrix
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
Computation of uFE(x0)
Direct computation of
Indirect computation of in node points
Nodal forces
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
Error estimation based on extrapolation
Examples
1.5.4 Error in the extracted QoI
1.6 The choice of discretization in 1D
1.6.1 The exact solution lies in ,
1.6.2 The exact solution lies in ,
1.7 Eigenvalue problems
1.8 Other finite element methods
1.8.1 The mixed method
1.8.2 Nitsche's method
Stabilization
Numerical example
Notes
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
Numerical treatment of periodic functions
2.3.4 Dimensional reduction
Planar problems
Axisymmetric models
Heat conduction in a bar
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
Planar elastostatic models: Notation
Plane strain
Plane stress
The Navier equations
Axisymmetric elastostatic models
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
Isotropic elasticity
2.6.2 The RMS measure of stress
2.6.3 The principle of virtual work
2.6.4 Uniqueness
2.7 Residual stresses
2.8 Chapter summary
Notes
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
Quadrilateral elements
Triangular elements
3.3.2 Hierarchic shape functions
Quadrilateral elements
Triangular elements
3.4 Mapping functions in two dimensions
3.4.1 Isoparametric mapping
Isoparametric mapping for quadrilateral elements
Isoparametric mapping for triangular elements
3.4.2 Mapping by the blending function method
3.4.3 Mapping algorithms for high order elements
Rigid body rotations
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
Volume forces
Surface tractions
Thermal loading
Summary of the main points
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
Notes
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,
4.2.2 2D model problem,
Dirichlet boundary condition
4.2.3 Computation of the flux vector in a given point
4.2.4 Computation of the flux intensity factors
Path‐independent integral
Orthogonality
Extraction of
4.2.5 Material interfaces
The Steklov method
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
Computation of stress intensity factors
4.4.3 Forcing functions acting on boundaries
Concentrated force
Step function
4.5 Robustness
4.6 Solution verification
Solution
Discussion
Notes
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
Calibration
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
Model 1: Strength of materials
Model 2: The fasteners are modeled by linear springs
Model 3: The fasteners are modeled by nonlinear springs
Model 4: The three‐dimensional contact problem
Discussion
5.2.8 Finite element model
Equilibrium of nodal forces
Discussion
5.2.9 Example: Coil spring with displacement boundary conditions
Solution of the linear model
Solution of the nonlinear model
Discussion
5.2.10 Example: Coil spring segment
Solution
Discussion
Notes
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
Edge notched specimen with inch
Edge notched specimen with r = 0.0035 inch
Conclusion
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
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
Extension A
Extension B
Ranking
Predictive performance
Selection of the prior
Inferential statistics
Validation
The updated domain of calibration
The number of experiments
6.5 Management of model development
6.5.1 Obstacles to progress
Notes
7 Beams, plates and shells
7.1 Beams
7.1.1 The Timoshenko beam
Shear correction
Numerical solution
Shear locking in Timoshenko beams
7.1.2 The Bernoulli‐Euler beam
Numerical solution
7.2 Plates
7.2.1 The Reissner‐Mindlin plate
Shear correction for plate models
7.2.2 The Kirchhoff plate
Enforcement of continuity
7.2.3 The transverse variation of displacements
Case A: The material properties are independent of
Case B: The material properties are symmetric functions of
7.3 Shells
The Naghdi shell model
The Novozhilov‐Koiter shell model
7.3.1 Hierarchic thin solid models
7.4 Chapter summary
Notes
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
Hexagonal pattern
Square pattern
Comparison
8.1.4 Localization
8.1.5 Prediction of failure in composite materials
Example
8.1.6 Uncertainties
8.2 Discussion
Notes
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
Notation
Assumptions
Incremental stress‐strain relationship
The deformation theory of plasticity
9.2.4 Mechanical contact
Gap elements in two dimensions
Outline of the algorithm
9.3 Chapter summary
Notes
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
A.2.1 The space of continuous functions
A.2.2 The space
A.2.3 Sobolev space of order 1
A.2.4 Sobolev spaces of fractional index
A.3 The Schwarz inequality for integrals
Notes
Appendix B Proof of h‐convergence
Appendix C Convergence in 3D: Empirical results
Input data
Reference solution
Discussion
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
Note
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
Complex eigenvalues
G.2.4 Corner points
Notes
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
Note
Appendix I Fundamentals of data analysis
I.1 Statistical foundations
The product rule
Bayes' theorem
Marginalization
I.2 Test data
I.3 Statistical models
The bilinear model
The fatigue limit model
The random fatigue limit model
I.4 Ranking
The Bayes factor
I.5 Confidence intervals
Notes
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
Normal traction
Shearing tractions
Notes
Bibliography
Index. a
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