Numerical methods for mixed finite element problems : applications to incompressible materials and contact problems / / Jean Deteix, Thierno Diop and Michel Fortin

• Record Nr.: UNINA9910595041303321
• Autore: Deteix Jean
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 3-031-12616-5
• Descrizione fisica: 1 online resource (119 pages)
• Collana: Lecture Notes in Mathematics ; ; v.2318
• Disciplina: 620.00151535
• Soggetti: Finite element method; Mètode dels elements finits; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: 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 -- The Active Set Strategy -- 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.