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

ISBN-10: 0521705185

ISBN-13: 9780521705189

Edition: 3rd 2007 (Revised)

Authors: Dietrich Braess

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This definitive introduction to finite element methods has been thoroughly updated for a third edition which features important new material for both research and application of the finite element method. The discussion of saddle-point problems is a highlight of the book and has been elaborated to include many more nonstandard applications. The chapter on applications in elasticity now contains a complete discussion of locking phenomena. The numerical solution of elliptic partial differential equations is an important application of finite elements and the author discusses this subject comprehensively. These equations are treated as variational problems for which the Sobolev spaces are the right framework. Graduate students who do not necessarily have any particular background in differential equations, but require an introduction to finite element methods will find this text invaluable. Specifically, the chapter on finite elements in solid mechanics provides a bridge between mathematics and engineering.
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Book details

List price: $77.00
Edition: 3rd
Copyright year: 2007
Publisher: Cambridge University Press
Publication date: 4/12/2007
Binding: Paperback
Pages: 384
Size: 6.25" wide x 9.00" long x 0.75" tall
Weight: 0.440
Language: English

Preface to the Third English Edition page
Preface to the First English Edition
Preface to the German Edition
Examples and Classification of PDE's
Classification of PDE's
Well-posed Problems
The Maximum Principle
Finite Difference Methods
Discrete maximum principle
A Convergence Theory for Difference Methods
Local and global error
Limits of the convergence theory
Conforming Finite Elements
Sobolev Spaces
Introduction to Sobolev spaces
Friedrichs' inequality
Possible singularities of H1 functions
Compact imbeddings
Variational Formulation of Elliptic Boundary-Value Problems of Second Order
Variational formulation
Reduction to homogeneous boundary conditions
Existence of solutions
Inhomogeneous boundary conditions
The Neumann Boundary-Value Problem. A Trace Theorem
Ellipticity in H1
Boundary-value problems with natural boundary conditions
Neumann boundary conditions
Mixed boundary conditions
Proof of the trace theorem
Practical consequences of the trace theorem
The Ritz-Galerkin Method and Some Finite Elements
Model Problem
Some Standard Finite Elements
Requirements on the meshes
Significance of the differentiability properties
Triangular elements with complete polynomials
Remarks on C1 elements
Bilinear elements
Quadratic rectangular elements
Affine families
Choice of an element
Approximation Properties
The Bramble-Hilbert lemma
Triangular elements with complete polynomials
Bilinear quadrilateral elements
Inverse estimates
Cl�ment's interpolation
Appendix: On the optimality of the estimates
Error Bounds for Elliptic Problems of Second Order
Remarks on regularity
Error bounds in the energy norm
L2 estimates
A simple L∞ estimate
The L2-projector
Computational Considerations
Assembling the stiffness matrix
Static condensation
Complexity of setting up the matrix
Effect on the choice of a grid
Local mesh refinement
Implementation of the Neumann boundary-value Problem
Nonconforming and Other Methods
Abstract Lemmas and a Simple Boundary Approximation
Generalizations of C�a's lemma
Duality methods
The Crouzeix-Raviart element
A Simple approximation to curved boundaries
Modifications of the duality argument
Isoparametric Elements
Isoparametric triangular elements
Isoparametric quadrilateral elements
Further Tools from Functional Analysis
Negative norms
Adjoint operators
An abstract existence theorem
An abstract convergence theorem
Proof of Theorem 3.4
Saddle Point Problems
Saddle points and minima
The inf-sup condition
Mixed finite element methods
Fortin interpolation
Saddle point problems with penalty term
Typical applications
Mixed Methods for the Poisson Equation
The Poisson equation as a mixed problem
The Raviart - Thomas element
Interpolation by Raviart-Thomas elements
Implementation and postprocessing
Mesh-dependent norms for the Raviart-Thomas element
The Softening behaviour of mixed methods
The Stokes Equation
Variational formulation
The inf-sup condition
Nearly incompressible flows
Finite Elements for the Stokes Problems
An instable element
The Taylor-Hood element
The MINI element
The divergence-free nonconforming P1 element
A Posteriori Error Estimates
Residual estimators
Lower estimates
Remark on other estimators
Local mesh refinement and convergence
A Posteriori Error Estimates via the Hypercircle Method
The Conjugate Gradient Method
Classical Iterative Methods for Solving Linear Systems
Stationary linear processes
The Jacobi and Gauss-Seidel methods
The model problem
Gradient Methods
The general gradient method
Gradient methods and quadratic functions
Convergence behavior in the case of large condition numbers
Conjugate Gradient and the Minimal Residual Method
The CG algorithm
Analysis of the CG method as an optimal method
The minimal residual method
Indefinite and unsymmetric matrices
Preconditioning by SSOR
Preconditioning by ILU
Remarks on parallelization
Nonlinear Problems
Saddle Point Problems
The Uzawa algorithm and its variants
An alternative
Multigrid Methods
Multigrid Methods for Variational Problems
Smoothing properties of classical iterative methods
The multigrid idea
The algorithm
Transfer between grids
Convergence of Multigrid Methods
Discrete norms
Connection with the Sobolev norm
Approximation property
Convergence proof for the two-grid method
An alternative short proof
Some variants
Convergence for Several Levels
A recurrence formula for the W-cycle
An improvement for the energy norm
The convergence proof for the V-cycle
Nested Iteration
Computation of starting values
Multigrid methods with a small number of levels
The CASCADE algorithm
Multigrid Analysis via Space Decomposition
Schwarz alternating method
Direct consequences
Convergence of multiplicative methods
Verification of A1
Local mesh refinements
Nonlinear Problems
The multigrid-Newton method
The nonlinear multigrid method
Starting values
Finite Elements in Solid Mechanics
Introduction to Elasticity Theory
The equilibrium equations
The Piola transform
Constitutive Equations
Linear material laws
Hyperelastic Materials
Linear Elasticity Theory
The variational problem
The displacement formulation
The mixed method of Hellinger and Reissner
The mixed method of Hu and Washizu
Nearly incompressible material
Locking of the Timoshenko beam and typical remedies
Plane stress states
Plane strain states
Membrane elements
The PEERS element
Beams and Plates: The Kirchhoff Plate
The hypotheses
Note on beam models
Mixed methods for the Kirchoff plate
DKT elements
The Mindlin-Reissner Plate
The Helmholtz decomposition
The mixed formulation with the Helmholtz decomposition
MITC elements
The Model without a Helmboltz decomposition
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