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Numerical methods for mixed finite element problems : applications to incompressible materials and contact problems / Jean Deteix, Thierno Diop and Michel Fortin.
Author
Deteix, Jean
[Browse]
Format
Book
Language
English
Published/Created
Cham, Switzerland : Springer, [2022]
©2022
Description
1 online resource (119 pages)
Availability
Available Online
Springer Nature - Springer Lecture Notes in Mathematics eBooks
Springer Nature - Springer Mathematics and Statistics eBooks 2022 English International
Details
Subject(s)
Finite element method
[Browse]
Author
Diop, Thierno (Mathematician)
[Browse]
Fortin, Michel
[Browse]
Series
Lecture Notes in Mathematics
[More in this series]
Lecture Notes in Mathematics ; v.2318
[More in this series]
Bibliographic references
Includes bibliographical references and index.
Source of description
Description based on print version record.
Contents
Intro
Contents
1 Introduction
2 Mixed Problems
2.1 Some Reminders About Mixed Problems
2.1.1 The Saddle Point Formulation
2.1.2 Existence of a Solution
2.1.3 Dual Problem
2.1.4 A More General Case: A Regular Perturbation
2.1.5 The Case
2.2 The Discrete Problem
2.2.1 Error Estimates
2.2.2 The Matricial Form of the Discrete Problem
2.2.3 The Discrete Dual Problem: The Schur Complement
2.3 Augmented Lagrangian
2.3.1 Augmented or Regularised Lagrangians
2.3.2 Discrete Augmented Lagrangian in Matrix Form
2.3.3 Augmented Lagrangian and the Condition Number of the Dual Problem
2.3.4 Augmented Lagrangian: An Iterated Penalty
3 Iterative Solvers for Mixed Problems
3.1 Classical Iterative Methods
3.1.1 Some General Points
Linear Algebra and Optimisation
Norms
Krylov Subspace
Preconditioning
3.1.2 The Preconditioned Conjugate Gradient Method
3.1.3 Constrained Problems: Projected Gradient and Variants
Equality Constraints: The Projected Gradient Method
Inequality Constraints
Positivity Constraints
Convex Constraints
3.1.4 Hierarchical Basis and Multigrid Preconditioning
3.1.5 Conjugate Residuals, Minres, Gmres and the Generalised Conjugate Residual Algorithm
The Generalised Conjugate Residual Method
The Left Preconditioning
The Right Preconditioning
The Gram-Schmidt Algorithm
GCR for Mixed Problems
3.2 Preconditioners for the Mixed Problem
3.2.1 Factorisation of the System
Solving Using the Factorisation
3.2.2 Approximate Solvers for the Schur Complement and the Uzawa Algorithm
The Uzawa Algorithm
3.2.3 The General Preconditioned Algorithm
3.2.4 Augmented Lagrangian as a Perturbed Problem
4 Numerical Results: Cases Where Q= Q
4.1 Mixed Laplacian Problem
4.1.1 Formulation of the Problem.
4.1.2 Discrete Problem and Classic Numerical Methods
The Augmented Lagrangian Formulation
4.1.3 A Numerical Example
4.2 Application to Incompressible Elasticity
4.2.1 Nearly Incompressible Linear Elasticity
4.2.2 Neo-Hookean and Mooney-Rivlin Materials
Mixed Formulation for Mooney-Rivlin Materials
4.2.3 Numerical Results for the Linear Elasticity Problem
4.2.4 The Mixed-GMP-GCR Method
Approximate Solver in u
4.2.5 The Test Case
Number of Iterations and Mesh Size
Comparison of the Preconditioners of Sect.3.2
Effect of the Solver in u
4.2.6 Large Deformation Problems
Neo-Hookean Material
Mooney-Rivlin Material
4.3 Navier-Stokes Equations
4.3.1 A Direct Iteration: Regularising the Problem
4.3.2 A Toy Problem
5 Contact Problems: A Case Where Q≠Q
5.1 Imposing Dirichlet's Condition Through a Multiplier
5.1.1 and Its Dual
5.1.2 A Steklov-Poincaré operator
Using This as a Solver
5.1.3 Discrete Problems
The Matrix Form and the Discrete Schur Complement
5.1.4 A Discrete Steklov-Poincaré Operator
5.1.5 Computational Issues, Approximate Scalar Product
Simplified Forms of the ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper S script upper P Subscript h) /StPNE pdfmark [/StBMC pdfmarkSPhps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Operator and Preconditioning
5.1.6 The Formulation
The Choice of h
5.1.7 A Toy Model for the Contact Problem
The Discrete Formulation
The Active Set Strategy
5.2 Sliding Contact
5.2.1 The Discrete Contact Problem
Contact Status
5.2.2 The Algorithm for Sliding Contact
A Newton Method
5.2.3 A Numerical Example of Contact Problem
6 Solving Problems with More Than One Constraint
6.1 A Model Problem
6.2 Interlaced Method
6.3 Preconditioners Based on Factorisation.
6.3.1 The Sequential Method
6.4 An Alternating Procedure
7 Conclusion
Bibliography
Index.
Show 96 more Contents items
ISBN
3-031-12616-5
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Numerical methods for mixed finite element problems : applications to incompressible materials and contact problems / Jean Deteix, Thierno Diop, Michel Fortin.
id
99126222017206421