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E-raamat: Finite Element Analysis - Method, Verification and Validation, Second Edition: Method, Verification and Validation 2nd Edition [Wiley Online]

(Washington University, St. Louis, Missouri), (University of Maryland)
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Finite Element Analysis - Method, Verification and Validation, Second Edition: Method, Verification and Validation 2nd EditionSuurem pilt
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Raamatu kodulehekülg: https://onlinelibrary.wiley.com/doi/book/10.1002/9781119426479
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.
Preface to the second edition xi
Preface to the first edition xiii
Preface xv
About the companion website xvii
1 Introduction To The Finite Element Method
1(50)
1.1 An introductory problem
3(3)
1.2 Generalized formulation
6(6)
1.2.1 The exact solution
6(5)
1.2.2 The principle of minimum potential energy
11(1)
1.3 Approximate solutions
12(14)
1.3.1 The standard polynomial space
13(3)
1.3.2 Finite element spaces in one dimension
16(1)
1.3.3 Computation of the coefficient matrices
17(3)
1.3.4 Computation of the right hand side vector
20(1)
1.3.5 Assembly
21(3)
1.3.6 Condensation
24(1)
1.3.7 Enforcement of Dirichlet boundary conditions
24(2)
1.4 Post-solution operations
26(4)
1.4.1 Computation of the quantities of interest
26(4)
1.5 Estimation of error in energy norm
30(8)
1.5.1 Regularity
30(1)
1.5.2 A priori estimation of the rate of convergence
31(1)
1.5.3 A posteriori estimation of error
32(4)
1.5.4 Error in the extracted QoI
36(2)
1.6 The choice of discretization in ID
38(4)
1.6.1 The exact solution lies in Hk(I), k -- 1 > p
38(1)
1.6.2 The exact solution lies in Hk(I), k -- 1 < p
39(3)
1.7 Eigenvalue problems
42(4)
1.8 Other finite element methods
46(5)
1.8.1 The mixed method
47(1)
1.8.2 Nitsche's method
48(3)
2 Boundary Value Problems
51(40)
2.1 Notation
51(2)
2.2 The scalar elliptic boundary value problem
53(2)
2.2.1 Generalized formulation
53(2)
2.2.2 Continuity
55(1)
2.3 Heat conduction
55(12)
2.3.1 The differential equation
57(1)
2.3.2 Boundary and initial conditions
58(1)
2.3.3 Boundary conditions of convenience
59(2)
2.3.4 Dimensional reduction
61(6)
2.4 Equations of linear elasticity - strong form
67(11)
2.4.1 The Navier equations
70(1)
2.4.2 Boundary and initial conditions
71(1)
2.4.3 Symmetry, antisymmetry and periodicity
72(1)
2.4.4 Dimensional reduction in linear elasticity
73(3)
2.4.5 Incompressible elastic materials
76(2)
2.5 Stokes flow
78(1)
2.6 Generalized formulation of problems of linear elasticity
78(9)
2.6.1 The principle of minimum potential energy
80(2)
2.6.2 The RMS measure of stress
82(1)
2.6.3 The principle of virtual work
83(1)
2.6.4 Uniqueness
84(3)
2.7 Residual stresses
87(2)
2.8
Chapter summary
89(2)
3 Implementation
91(28)
3.1 Standard elements in two dimensions
91(1)
3.2 Standard polynomial spaces
91(2)
3.2.1 Trunk spaces
91(1)
3.2.2 Product spaces
92(1)
3.3 Shape functions
93(4)
3.3.1 Lagrange shape functions
93(2)
3.3.2 Hierarchic shape functions
95(2)
3.4 Mapping functions in two dimensions
97(5)
3.4.1 Isoparametric mapping
97(2)
3.4.2 Mapping by the blending function method
99(2)
3.4.3 Mapping algorithms for high order elements
101(1)
3.5 Finite element spaces in two dimensions
102(1)
3.6 Essential boundary conditions
103(1)
3.7 Elements in three dimensions
103(3)
3.7.1 Mapping functions in three dimensions
105(1)
3.8 Integration and differentiation
106(3)
3.8.1 Volume and area integrals
106(1)
3.8.2 Surface and contour integrals
107(1)
3.8.3 Differentiation
108(1)
3.9 Stiffness matrices and load vectors
109(2)
3.9.1 Stiffness matrices
109(1)
3.9.2 Load vectors
110(1)
3.10 Post-solution operations
111(1)
3.11 Computation of the solution and its first derivatives
111(2)
3.12 Nodal forces
113(4)
3.12.1 Nodal forces in the h-version
113(2)
3.12.2 Nodal forces in the p-version
115(2)
3.12.3 Nodal forces and stress resultants
117(1)
3.13
Chapter summary
117(2)
4 Pre-And Postprocessing Procedures And Verification
119(36)
4.1 Regularity in two and three dimensions
119(1)
4.2 The Laplace equation in two dimensions
120(13)
4.2.1 2D model problem, uEX Hk(Ω), k -- 1 > p
121(2)
4.2.2 2D model problem, uEX Hk(Ω), k -- 1 < p
123(3)
4.2.3 Computation of the flux vector in a given point
126(2)
4.2.4 Computation of the flux intensity factors
128(3)
4.2.5 Material interfaces
131(2)
4.3 The Laplace equation in three dimensions
133(4)
4.4 Planar elasticity
137(6)
4.4.1 Problems of elasticity on an L-shaped domain
137(2)
4.4.2 Crack tip singularities in 2D
139(3)
4.4.3 Forcing functions acting on boundaries
142(1)
4.5 Robustness
143(5)
4.6 Solution verification
148(7)
5 Simulation
155(32)
5.1 Development of a very useful mathematical model
156(3)
5.1.1 The Bernoulli-Euler beam model
156(2)
5.1.2 Historical notes on the Bernoulli-Euler beam model
158(1)
5.2 Finite element modeling and numerical simulation
159(28)
5.2.1 Numerical simulation
159(1)
5.2.2 Finite element modeling
160(3)
5.2.3 Calibration versus tuning
163(1)
5.2.4 Simulation governance
164(1)
5.2.5 Milestones in numerical simulation
165(2)
5.2.6 Example: The Girkmann problem
167(3)
5.2.7 Example: Fastened structural connection
170(6)
5.2.8 Finite element model
176(4)
5.2.9 Example: Coil spring with displacement boundary conditions
180(4)
5.2.10 Example: Coil spring segment
184(3)
6 Calibration, Validation And Ranking
187(36)
6.1 Fatigue data
187(4)
6.1.1 Equivalent stress
188(1)
6.1.2 Statistical models
189(1)
6.1.3 The effect of notches
190(1)
6.1.4 Formulation of predictors of fatigue life
190(1)
6.2 The predictors of Peterson and Neuber
191(11)
6.2.1 The effect of notches -- calibration
193(2)
6.2.2 The effect of notches -- validation
195(2)
6.2.3 Updated calibration
197(2)
6.2.4 The fatigue limit
199(2)
6.2.5 Discussion
201(1)
6.3 The predictor Gα
202(3)
6.3.1 Calibration of β(V, α)
203(1)
6.3.2 Ranking
204(1)
6.3.3 Comparison of Gα with Peterson's revised predictor
205(1)
6.4 Biaxial test data
205(13)
6.4.1 Axial, torsional and combined in-phase loading
206(2)
6.4.2 The domain of calibration
208(2)
6.4.3 Out-of-phase biaxial loading
210(8)
6.5 Management of model development
218(5)
6.5.1 Obstacles to progress
220(3)
7 Beams, Plates And Shells
223(32)
7.1 Beams
223(11)
7.1.1 The Timoshenko beam
225(4)
7.1.2 The Bernoulli-Euler beam
229(5)
7.2 Plates
234(13)
7.2.1 The Reissner-Mindlin plate
236(4)
7.2.2 The Kirchhoff plate
240(3)
7.2.3 The transverse variation of displacements
243(4)
7.3 Shells
247(7)
7.3.1 Hierarchic thin solid models
249(5)
7.4
Chapter summary
254(1)
8 Aspects Of Multiscale Models
255(10)
8.1 Unidirectional fiber-reinforced laminae
255(9)
8.1.1 Determination of material constants
257(1)
8.1.2 The coefficients of thermal expansion
258(1)
8.1.3 Examples
258(3)
8.1.4 Localization
261(1)
8.1.5 Prediction of failure in composite materials
262(1)
8.1.6 Uncertainties
263(1)
8.2 Discussion
264(1)
9 Non-Linear Models
265(86)
9.1 Heat conduction
265(1)
9.1.1 Radiation
265(1)
9.1.2 Nonlinear material properties
266(1)
9.2 Solid mechanics
266(21)
9.2.1 Large strain and rotation
266(4)
9.2.2 Structural stability and stress stiffening
270(5)
9.2.3 Plasticity
275(6)
9.2.4 Mechanical contact
281(6)
9.3
Chapter summary
287(2)
Appendix A Definitions
289(1)
A.1 Normed linear spaces, linear functionals and bilinear forms
289(2)
A.1.1 Normed linear spaces
290(1)
A.1.2 Linear forms
290(1)
A.1.3 Bilinear forms
290(1)
A.2 Convergence in the space X
291(2)
A.2.1 The space of continuous functions
291(1)
A.2.2 The space LP(Ω)
291(1)
A.2.3 Sobolev space of order 1
291(1)
A.2.4 Sobolev spaces of fractional index
292(1)
A.3 The Schwarz inequality for integrals
293(2)
Appendix B Proof of h-convergence
295(2)
Appendix C Convergence in 3D: Empirical results
297(4)
Appendix D Legendre polynomials
301(1)
D.1 Shape functions based on Legendre polynomials
302(1)
Appendix E Numerical quadrature
303(1)
E.1 Gaussian quadrature
303(1)
E.2 Gauss-Lobatto quadrature
304(3)
Appendix F Polynomial mapping functions
307(1)
F.1 Interpolation on surfaces
308(3)
F.1.1 Interpolation on the standard quadrilateral element
309(1)
F.1.2 Interpolation on the standard triangle
309(2)
Appendix G Corner singularities in two-dimensional elasticity
311(1)
G.1 The Airy stress function
311(1)
G.2 Stress-free edges
312(7)
G.2.1 Symmetric eigenfunctions
313(2)
G.2.2 Antisymmetric eigenfunctions
315(1)
G.2.3 The L-shaped domain
315(2)
G.2.4 Corner points
317(2)
Appendix H Computation of stress intensity factors
319(1)
H.1 Singularities at crack tips
319(1)
H.2 The contour integral method
320(1)
H.3 The energy release rate
321(4)
H.3.1 Symmetric (Mode I) loading
322(1)
H.3.2 Antisymmetric (Mode II) loading
323(1)
H.3.3 Combined (Mode I and Mode II) loading
323(1)
H.3.4 Computation by the stiffness derivative method
323(2)
Appendix I Fundamentals of data analysis
325(1)
I.1 Statistical foundations
325(1)
I.2 Test data
326(2)
I.3 Statistical models
328(7)
I.4 Ranking
335(1)
I.5 Confidence intervals
335(2)
Appendix J Estimation of fastener forces in structural connections
337(4)
Appendix K Useful algorithms in solid mechanics
341(1)
K.1 The traction vector
341(1)
K.2 Transformation of vectors
342(1)
K.3 Transformation of stresses
343(1)
K.4 Principal stresses
344(1)
K.5 The von Mises stress
344(1)
K.6 Statically equivalent forces and moments
345(6)
K.6.1 Technical formulas for stress
348(3)
Bibliography 351(6)
Index 357
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 Babuška 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.
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