Numerical methods for mixed finite element problems: applications to incompressible materials and contact problems
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Cham, Switzerland
Springer
[2022]
|
| Schriftenreihe: | Lecture notes in mathematics
Volume 2318 |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033904723&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Abstract: | This book focuses on iterative solvers and preconditioners for mixed finite element methods. It provides an overview of some of the state-of-the-art solvers for discrete systems with constraints such as those which arise from mixed formulations. Starting by recalling the basic theory of mixed finite element methods, the book goes on to discuss the augmented Lagrangian method and gives a summary of the standard iterative methods, describing their usage for mixed methods. Here, preconditioners are built from an approximate factorisation of the mixed system. A first set of applications is considered for incompressible elasticity problems and flow problems, including non-linear models. An account of the mixed formulation for Dirichlet’s boundary conditions is then given before turning to contact problems, where contact between incompressible bodies leads to problems with two constraints. This book is aimed at graduate students and researchers in the field of numerical methods and scientific computing. |
| Umfang: | vi, 114 Seiten Diagramme |
| ISBN: | 9783031126154 |
Internformat
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| 100 | 1 | |a Deteix, Jean |0 (DE-588)1271053993 |4 aut | |
| 245 | 1 | 0 | |a Numerical methods for mixed finite element problems |b applications to incompressible materials and contact problems |c Jean Deteix ; Thierno Diop ; Michel Fortin |
| 264 | 1 | |a Cham, Switzerland |b Springer |c [2022] | |
| 300 | |a vi, 114 Seiten |b Diagramme | ||
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| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v Volume 2318 | |
| 520 | 3 | |a This book focuses on iterative solvers and preconditioners for mixed finite element methods. It provides an overview of some of the state-of-the-art solvers for discrete systems with constraints such as those which arise from mixed formulations. Starting by recalling the basic theory of mixed finite element methods, the book goes on to discuss the augmented Lagrangian method and gives a summary of the standard iterative methods, describing their usage for mixed methods. Here, preconditioners are built from an approximate factorisation of the mixed system. A first set of applications is considered for incompressible elasticity problems and flow problems, including non-linear models. An account of the mixed formulation for Dirichlet’s boundary conditions is then given before turning to contact problems, where contact between incompressible bodies leads to problems with two constraints. This book is aimed at graduate students and researchers in the field of numerical methods and scientific computing. | |
| 700 | 1 | |a Diop, Thierno |0 (DE-588)141943181 |4 aut | |
| 700 | 1 | |a Fortin, Michel |d 1945- |0 (DE-588)105353325X |4 aut | |
| 710 | 2 | |a Springer Nature Switzerland AG |0 (DE-588)1211528561 |4 pbl | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |a Deteix, Jean |t Numerical Methods for Mixed Finite Element Problems |b 1st ed. 2022. |d Cham : Springer International Publishing, 2022 |h 1 Online-Ressource(VI, 116 p. 29 illus., 22 illus. in color.) |z 9783031126161 |w (DE-604)BV048495910 |
| 830 | 0 | |a Lecture notes in mathematics |v Volume 2318 |w (DE-604)BV000676446 |9 2318 | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-033904723 | |
Datensatz im Suchindex
| _version_ | 1819376432079437824 |
|---|---|
| adam_text | Contents
1 Introduction 0c cceccececeeceeseacsaueueceassaesssessesereresess 1
2 Mixed Problems 0 0c ccccecececssasenseuscecsaesesscceuecens 3
2 1 Some Reminders About Mixed Problems ccccecceeceuee 3
211 The Saddle Point Formulation c cc 4
212 Existence of a Solution cccccccaeecescuecesseecenees 4
213 Dual Problem ccccccesssceescuscevecsauecueseneeses 5
214A More General Case: A Regular Perturbation 6
215 The Case b(v, q) = (BU, Q)Q cccceccceccecsesceneeeeneeseenes 6
2 2 The Discrete Problem 0cccccseccneceecetevesseuesseseuseeseus 7
221 Error Estimates 200 ccccccceececceecusuecusceceeneseeness 8
222 The Matricial Form of the Discrete Problem 4+ 9
223 The Discrete Dual Problem: The Schur Complement 11
2 3 Augmented Lagrangian ccceccecneceeceeseseesenssnereeeeees 13
231 Augmented or Regularised Lagrangians - m 13
232 Discrete Augmented Lagrangian in Matrix Form 15
233 Augmented Lagrangian and the Condition Number
of the Dual Problem cersrssneeasenenenennenenener 15
234 Augmented Lagrangian: An Iterated Penalty 17
3 Iterative Solvers for Mixed Problems ee 19
3 1 Classical Iterative Methods een 19
311 Some General Points nennen nenn 20
312 The Preconditioned Conjugate Gradient Method - 22
313 Constrained Problems: Projected Gradient and
Variants ueeeaeeeneeeeenersennennennsnns nennen een ent 24
314 Hierarchical Basis and Multigrid Preconditioning --- 26
315 Conjugate Residuals, Minres, Gmres and the
Generalised Conjugate Residual Algorithm +++++++- a
3 2 Preconditioners for the Mixed Problem een
321 Factorisation of the System enter 3
vi Contents
322 Approximate Solvers for the Schur Complement
and the Uzawa Algorithm seeeesuesesnesnsernennenennnn 35
323 The General Preconditioned Algorithm 00 08 38
324 Augmented Lagrangian as a Perturbed Problem 4]
4 Numerical Results: Cases Where Q = ©’ nnceennennnneenn 43
4 1 Mixed Laplacian Problem -:0:sceseeeeeeereee eee e eee eeenneees 43
411 Formulation of the Problem ceccccceeeecceeenerenes 43
412 Discrete Problem and Classic Numerical Methods 45
413A Numerical Example seresereseesesenenneenennennenen 47
4 2 Application to Incompressible Blasticity cscesceeseeeeevenes 48
421 Nearly Incompressible Linear Elasticity 000- 49
422 Neo-Hookean and Mooney-Rivlin Materials 52
423 Numerical Results for the Linear Elasticity Problem 55
424 The Mixed-GMP-GCR Method :e cscs eseeeeeeeees 56
425 The Test Case 2o2sseeeenaunussenneneesnennnennenneenennnn 58
426 Large Deformation Problems -ss0rerenreeenen ern 64
4 3 Navier-Stokes Equations c cece ese cce eee ee enone ee seeeaeseeanees 68
431A Direct Iteration: Regularising the Problem 71
432A Toy Problem nesesseessensensennneensenensneanenen nennen 72
5 Contact Problems: A Case Where Q £ 0° ccnennensnenenennen ern 75
5 1 Imposing Dirichlet’s Condition Through a Multiplier - 75
511 Ayl (De) and Its Dual AH VAT) nennen: 7
512A Steklov-Poincaré operator cccccceuceeneeeeceeeeees 78
513 Discrete Problems uceeenenncneenenennneneenenenenenee nenn 719
514A Discrete Steklov-Poincaré Operator 00cceeeeeceeeees 80
515 Computational Issues, Approximate Scalar Product 81
516 The L?(Tc) Formulation cc0cccccscesesseeceseceeseeees 84
517A Toy Model for the Contact Problem nennen: 84
5 2 Sliding Contact ccceenneee nennen 87
521 The Discrete Contact Problem cccccccccseeeceeeceeeees 89
522 The Algorithm for Sliding Contact cccccceeeeeeeees 92
523A Numerical Example of Contact Problem 2 - 93
6 Solving Problems with More Than One Constraint nnn 97
61A Model Problem ccccccccccscccccscccccececcececceceeceets 97
6 2 Interlaced Method cccneenannnnnneaannnnnnnnne nn eceeseeues 98
6 3 Preconditioners Based on Factorisation cneaeeeeeeeeenneneaeenee 100
631 The Sequential Method uuceaeeaananannannnnnnnnnnnnnnnn 101
6 4 An Alternating Procedure vessesesseseacuesstvacstcccescsevevseseeeven 102
7 Comelusion nn 105
Bibliography 222 107
Index
|
| any_adam_object | 1 |
| author | Deteix, Jean Diop, Thierno Fortin, Michel 1945- |
| author_GND | (DE-588)1271053993 (DE-588)141943181 (DE-588)105353325X |
| author_facet | Deteix, Jean Diop, Thierno Fortin, Michel 1945- |
| author_role | aut aut aut |
| author_sort | Deteix, Jean |
| author_variant | j d jd t d td m f mf |
| building | Verbundindex |
| bvnumber | BV048527939 |
| classification_rvk | SI 850 |
| ctrlnum | (OCoLC)1347779928 (DE-599)KXP1819014363 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV048527939 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T19:47:35Z |
| institution | BVB |
| institution_GND | (DE-588)1211528561 |
| isbn | 9783031126154 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-033904723 |
| oclc_num | 1347779928 |
| open_access_boolean | |
| owner | DE-188 DE-824 DE-83 |
| owner_facet | DE-188 DE-824 DE-83 |
| physical | vi, 114 Seiten Diagramme |
| publishDate | 2022 |
| publishDateSearch | 2022 |
| publishDateSort | 2022 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spellingShingle | Deteix, Jean Diop, Thierno Fortin, Michel 1945- Numerical methods for mixed finite element problems applications to incompressible materials and contact problems Lecture notes in mathematics |
| title | Numerical methods for mixed finite element problems applications to incompressible materials and contact problems |
| title_auth | Numerical methods for mixed finite element problems applications to incompressible materials and contact problems |
| title_exact_search | Numerical methods for mixed finite element problems applications to incompressible materials and contact problems |
| title_full | Numerical methods for mixed finite element problems applications to incompressible materials and contact problems Jean Deteix ; Thierno Diop ; Michel Fortin |
| title_fullStr | Numerical methods for mixed finite element problems applications to incompressible materials and contact problems Jean Deteix ; Thierno Diop ; Michel Fortin |
| title_full_unstemmed | Numerical methods for mixed finite element problems applications to incompressible materials and contact problems Jean Deteix ; Thierno Diop ; Michel Fortin |
| title_short | Numerical methods for mixed finite element problems |
| title_sort | numerical methods for mixed finite element problems applications to incompressible materials and contact problems |
| title_sub | applications to incompressible materials and contact problems |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033904723&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
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