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)

Details

Subject(s)
Author
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.
ISBN
3-031-12616-5
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