Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics,9780521011952

Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics

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Edition: 2nd
ISBN13: 9780521011952
ISBN10: 0521011957
Format: Paperback
Pub. Date: 2001-04-23
Publisher(s): Cambridge University Press
Availability: This title is currently not available.
List Price: $47.25

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Summary

This is a thoroughly revised version of the successful first edition. In addition to up-dating the existing text, the author has added new material that will prove useful for research or application of the finite element method. The most important application of finite elements is the numerical solution of elliptic partial differential equations. The author gives a thorough coverage of this subject and includes aspects such as saddle point problems which require a more in-depth mathematical treatment. This is a book for graduate students who do not necessarily have any particular background in differential equations, but require an introduction to finite element methods. Specifically, the chapter on finite elements in solid mechanics provides a bridge between mathematics and engineering.

Table of Contents

Preface to the Second English Edition x
Preface to the First English Edition xi
Preface to the German Edition xii
Notation xiv
Introduction
1(26)
Examples and Classification of PDE's
2(10)
Examples
2(6)
Classification of PDE's
8(1)
Well-posed problems
9(1)
Problems
10(2)
The Maximum Principle
12(4)
Examples
13(1)
Corollaries
14(1)
Problem
15(1)
Finite Difference Methods
16(6)
Discretization
16(3)
Discrete maximum principle
19(3)
A Convergence Theory for Difference Methods
22(5)
Consistency
22(1)
Local and global error
22(2)
Limits of the convergence theory
24(2)
Problems
26(1)
Conforming Finite Elements
27(78)
Sobolev Spaces
28(6)
Introduction to Sobolev spaces
29(1)
Friedrichs' inequality
30(1)
Possible singularities of H1 functions
31(1)
Compact imbeddings
32(1)
Problems
33(1)
Variational Formulation of Elliptic Boundary-Value Problems of Second Order
34(10)
Variational formulation
35(1)
Reduction to homogeneous boundary conditions
36(2)
Existence of solutions
38(4)
Inhomogeneous boundary conditions
42(1)
Problems
42(2)
The Neumann Boundary-Value Problem. A Trace Theorem
44(9)
Ellipticity in H1
44(1)
Boundary-value problems with natural boundary conditions
45(1)
Neumann boundary conditions
46(1)
Mixed boundary conditions
47(1)
Proof of the trace theorem
48(2)
Practical consequences of the trace theorem
50(2)
Problems
52(1)
The Ritz--Galerkin Method and Some Finite Elements
53(7)
Model problem
56(2)
Problems
58(2)
Some Standard Finite Elements
60(16)
Requirements on the meshes
61(1)
Significance of the differentiability properties
62(2)
Triangular elements with complete polynomials
64(3)
Remarks on C1 elements
67(1)
Bilinear elements
68(1)
Quadratic rectangular elements
69(1)
Affine families
70(4)
Choice of an element
74(1)
Problems
74(2)
Approximation Properties
76(13)
The Bramble-Hilbert lemma
77(1)
Triangular elements with complete polynomials
78(3)
Bilinear quadrilateral elements
81(2)
Inverse estimates
83(1)
Clement's interpolation
84(1)
Appendix: On the optimality of the estimates
85(2)
Problems
87(2)
Error Bounds for Elliptic Problems of Second Order
89(8)
Remarks on regularity
89(1)
Error bounds in the energy norm
90(1)
L2 estimates
91(2)
A simple L∞ estimate
93(1)
The L2-projector
94(1)
Problems
95(2)
Computational Considerations
97(8)
Assembling the stiffness matrix
97(2)
Static condensation
99(1)
Complexity of setting up the matrix
100(1)
Effect on the choice of a grid
100(1)
Local mesh refinement
100(2)
Refinements of partitions of 3-dimensional domains
102(1)
Problems
102(3)
Nonconforming and Other Methods
105(72)
Abstract Lemmas and a Simple Boundary Approximation
106(11)
Generalizations of Cea's lemma
106(2)
Duality methods
108(1)
The Crouzeix-Raviart element
109(3)
A Simple approximation to curved boundaries
112(2)
Modifications of the duality argument
114(2)
Problems
116(1)
Isoparametric Elements
117(5)
Isoparametric triangular elements
117(2)
Isoparametric quadrilateral elements
119(2)
Problems
121(1)
Further Tools from Functional Analysis
122(7)
Negative norms
122(2)
Adjoint operators
124(1)
An abstract existence theorem
124(2)
An abstract convergence theorem
126(1)
Proof of Theorem 3.4
127(1)
Problems
128(1)
Saddle Point Problems
129(14)
Saddle points and minima
129(1)
The inf-sup condition
130(4)
Mixed finite element methods
134(2)
Fortin interpolation
136(1)
Saddle point problems with penalty term
137(4)
Problems
141(2)
Mixed Methods for the Poission Equation
143(11)
The Poisson equation as a mixed problem
143(3)
The Raviart-Thomas element
146(1)
Interpolation by Raviart-Thomas elements
147(2)
Implementation and postprocessing
149(1)
Mesh-dependent norms for the Raviart-Thomas element
150(1)
The softening behaviour of mixed methods
151(2)
Problems
153(1)
The Stokes Equation
154(5)
Variational formulation
155(1)
The inf-sup condition
156(1)
Remarks on the Brezzi condition
157(1)
Nearly incompressible flows
158(1)
Problems
158(1)
Finite Elements for the Stokes Problem
159(10)
An instable element
159(5)
The Taylor-Hood element
164(1)
The MINI element
165(2)
The divergence-free nonconforming P1 element
167(1)
Problems
168(1)
A Posteriori Error Estimates
169(8)
Residual estimators
171(2)
Lower estimates
173(2)
Other estimators
175(1)
Local mesh refinement
176(1)
The Conjugate Gradient Method
177(39)
Classical Iterative Methods for Solving Linear Systems
178(9)
Stationary linear processes
178(2)
The Jacobi and Gauss-Seidel methods
180(3)
The model problem
183(1)
Overrelaxation
184(2)
Problems
186(1)
Gradient Methods
187(5)
The general gradient method
187(1)
Gradient methods and quadratic functions
188(2)
Convergence behavior in the case of large condition numbers
190(1)
Problems
191(1)
Conjugate Gradient and the Minimal Residual Method
192(9)
The CG algorithm
194(2)
Analysis of the CG method as an optimal method
196(2)
The minimal residual method
198(1)
Indefinite and unsymmetric matrices
199(1)
Problems
200(1)
Preconditioning
201(11)
Preconditioning by SSOR
204(1)
Preconditioning by ILU
205(2)
Remarks on parallelization
207(1)
Nonlinear problems
208(1)
Problems
209(3)
Saddle Point Problems
212(4)
The Uzawa algorithm and its variants
212(2)
An alternative
214(1)
Problems
215(1)
Multigrid Methods
216(53)
Multigrid Methods for Variational Problems
217(11)
Smoothing properties of classical iterative methods
217(1)
The multigrid idea
218(1)
The algorithm
219(4)
Transfer between grids
223(3)
Problems
226(2)
Convergence of Multigrid Methods
228(11)
Discrete norms
229(2)
Connection with the Sobolev norm
231(2)
Approximation property
233(2)
Convergence proof for the two-grid method
235(1)
An alternative short proof
236(1)
Some variants
236(1)
Problems
237(2)
Convergence for Several Levels
239(7)
A recurrence formula for the W-cycle
239(1)
An improvement for the energy norm
240(2)
The convergence proof for the V-cycle
242(3)
Problems
245(1)
Nested Iteration
246(6)
Computation of starting values
246(2)
Complexity
248(1)
Multigrid methods with a small number of levels
249(1)
The CASCADE algorithm
250(1)
Problems
251(1)
Multigrid Analysis via Space Decomposition
252(11)
Schwarz' alternating method
253(2)
Algebraic description of space decomposition algorithms
255(1)
Assumptions
256(1)
Direct consequences
257(1)
Convergence of multiplicative methods
258(2)
Verification of A1
260(1)
Local mesh refinements
261(1)
Problems
262(1)
Nonlinear Problems
263(6)
The multigrid Newton method
264(1)
The nonlinear multigrid method
265(2)
Starting values
267(1)
Problems
268(1)
Finite Elements in Solid Mechanics
269(68)
Introduction to Elasticity Theory
270(11)
Kinematics
270(2)
The equilibrium equations
272(2)
The Piola transform
274(1)
Constitutive Equations
275(4)
Linear material laws
279(1)
Problem
280(1)
Hyperelastic Materials
281(3)
Problems
283(1)
Linear Elasticity Theory
284(20)
The variational problem
284(4)
The displacement formulation
288(3)
The mixed method of Hellinger and Reissner
291(2)
The mixed method of Hu and Washizu
293(2)
Nearly incompressible material
295(4)
Locking
299(3)
Locking of the Timoshenko beam
302(1)
Problems
303(1)
Membranes
304(8)
Plane stress states
304(1)
Plane strain states
305(1)
Membrane elements
305(1)
The PEERS element
306(4)
Problems
310(2)
Beams and Plates: The Kirchhoff Plate
312(12)
The hypotheses
312(3)
Note on beam models
315(1)
Mixed methods for the Kirchoff plate
315(2)
DKT elements
317(6)
Problems
323(1)
The Mindlin-Reissner Plate
324(13)
The Helmholtz decomposition
325(2)
The mixed formulation with the Helmholtz decomposition
327(1)
MITC elements
328(4)
The model without a Helmholtz decomposition
332(3)
Problems
335(2)
References 337(10)
Some Additional Books on Finite Elements 347(1)
Index 348

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