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Nonconforming boundary elements and finite elements for interface and contact problems with friction [Elektronische Ressource] : hp-version for mortar, penalty and Nitsche's methods / von Alexey Chernov

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Nonconforming boundary elementsand finite elements for interfaceand contact problems with friction –hp-version for mortar,penalty and Nitsche’s methodsVon der Fakulta¨t fu¨r Mathematik und Physikder Universitat Hannover¨zur Erlangung des GradesDoktor der NaturwissenschaftenDr. rer. nat.genehmigte DissertationvonDipl.-Math. Alexey Chernovgeboren am 13.08.1981 in Moskau2006Referent: Prof. Dr. E. P. Stephan, Universita¨t HannoverKorreferent: Prof. Dr. P. Hansbo, Chalmers University of Technology, Go¨teborgKorreferent: Prof. Dr. J. Gwinner, Universitat der Bundeswehr, Munchen¨ ¨Tag der Promotion: 27.06.2006AbstractTheobjectiveofthisthesisistheconstruction,analysisandimplementationofhighorderFE,BEandFE/BEcoupling methods forinterface andfrictionalcontact problems withnonmatching discretizations, which have a wide industrial application.A new hp-Nitsche’s FE/BE coupling method for interface problems is designed andanalysed. The method is proven to be consistent and stable, independent of the dis-cretization parameters. A priori error analysis shows that the method is optimal in hand is suboptimal in p on quasiuniform meshes.The question of unique solvability is addressed for the one-body contact problem withTresca’s friction. Constructing a chain of equivalent formulations the frictional contactproblem is approximated with a sequence of frictionless contact problems (Uzawa algo-rithm). Conditions for the convergence of the algorithm are obtained.

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Nonconforming boundary elements
and finite elements for interface
and contact problems with friction –
hp-version for mortar,
penalty and Nitsche’s methods
Von der Fakulta¨t fu¨r Mathematik und Physik
der Universitat Hannover¨
zur Erlangung des Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
von
Dipl.-Math. Alexey Chernov
geboren am 13.08.1981 in Moskau
2006Referent: Prof. Dr. E. P. Stephan, Universita¨t Hannover
Korreferent: Prof. Dr. P. Hansbo, Chalmers University of Technology, Go¨teborg
Korreferent: Prof. Dr. J. Gwinner, Universitat der Bundeswehr, Munchen¨ ¨
Tag der Promotion: 27.06.2006Abstract
Theobjectiveofthisthesisistheconstruction,analysisandimplementationofhighorder
FE,BEandFE/BEcoupling methods forinterface andfrictionalcontact problems with
nonmatching discretizations, which have a wide industrial application.
A new hp-Nitsche’s FE/BE coupling method for interface problems is designed and
analysed. The method is proven to be consistent and stable, independent of the dis-
cretization parameters. A priori error analysis shows that the method is optimal in h
and is suboptimal in p on quasiuniform meshes.
The question of unique solvability is addressed for the one-body contact problem with
Tresca’s friction. Constructing a chain of equivalent formulations the frictional contact
problem is approximated with a sequence of frictionless contact problems (Uzawa algo-
rithm). Conditions for the convergence of the algorithm are obtained. An hp-penalty
BEMforone-bodyfrictionlesscontact isdeveloped. Astheapriorierroranalysisshows,
1−ǫthe penalty parameter ε must be chosen proportional to (h/p) , for optimal conver-n
gence rate (in the energy norm) of the discrete penalty solution to the solution of the
original variationalinequality formulation. A residual based a posteriori error estimator
for the h-version of penalty FE and BE for one-body contact with Tresca friction is
investigated. The error estimator, motivated so far with heuristical arguments and only
for FEM, is shown to be reliable and efficient for both FEM and BEM.
The two-body elastoplastic contact problem with Coulomb’s law of friction is solved
with the FE/BE coupling and pure BE methods. The incremental loading procedure
with Newton iterations on each loading step is used. Linearization of the frictional con-
tact and plasticity terms as well as a description of the solution procedure are given in
detail. Theresidualaposteriorierrorestimate, obtainedforone-bodyfrictionalcontact,
is generalized to this two-body frictional contact problem. A novel hp-mortar method
for two-body contact with Tresca’s friction is designed and analysed for a variational
inequality formulation. The contact constraints are imposed on the discrete global set
of Gauss-Lobatto points. The nonmatched meshes are connected in terms of the hp-
1/4mortar projection. The a priori error analysis shows the convergence rate O((h/p) )
in the energy norm under additional assumptions on the discretization parameters. A
Dirichlet-to-NeumannalgorithmandanUzawa algorithmareused tosolve theproblem.
A heuristically motivated error indicator is used to perform an hp automatic refine-
ment procedure. The h-version of the constructed method is extended onto two-body
elastoplastic frictional contact problems and is compared to the results provided by the
penalty method.
The theoretical results are supported by numerical benchmark computations.
Key words. frictional contact, interface problems, finite elements, boundary elements,
FE/BEcoupling,hp-methods,apriorierror,aposteriorierror,mortar,penalty,Nitsche’s
method
3Zusammenfassung
DasZieldieser DissertationistdieKonstruktion,AnalyseundImplementierung vonFE,
BE und FE/BE Kopplungsverfahren fur Interface- und Reibungskontaktprobleme mit¨
unpassenden Diskretisierungen.
Eine neue hp-Nitsche FE/BE Kopplungsmethode fur Interface-Probleme wird konstru-¨
iert und analysiert. Es wird bewiesen, dass das Verfahren konsistent und stabil ist,
unabha¨ngig von den Diskretisierungsparametern. Die durchgefu¨hrte a priori Fehlerana-
lyse zeigt, dass die Methode optimal in h und suboptimal in p auf den quasiuniformen
Gittern ist.
DieFragedereindeutigenLosbarkeitwirdfurdasEin-Korper-KontaktproblemmitTres-¨ ¨ ¨
ca Reibung untersucht. Die Losung des Ausgangsproblems mit Reibung wird durch eine¨
FolgevonreibungslosenProblemenapproximiert(UzawaAlgorithmus).DieKonvergenz-
bedingungen fur den Algorithmus werden hergeleitet. Eine hp-Penalty BE Methode fur¨ ¨
das Ein-Ko¨rper-Kontaktproblem wird entwickelt. Wie die a priori Fehleranalyse zeigt,
1−ǫmuß der Penalty-Parameter ε proportional zu (h/p) , gew¨ahlt werden, um die opti-n
male Konvergenzordnung (in der Energienorm) der diskreten Penalty-Lo¨sung gegen die
exakte L¨osung der variationellen Ungleichung zu erreichen. Als na¨chstes wird ein resi-
dueller a posteriori Fehlerscha¨tzer fu¨r die h-Versionen von FEM und BEM untersucht.
FurdenFehlerschatzer, derbisher nurmitheuristischen Argumentenmotiviert undaus-¨ ¨
schließlich fur FEM benutzt wurde, wird bewiesen, dass er zuverlassig und effizient ist.¨ ¨
Ferner werden die Zwei-Ko¨rper-Kontaktprobleme mit Coulomb’scher Reibung fu¨r die
h-Versionen von FE/BE und reinem BE Verfahren betrachtet. Die inkrementelle Last-
aufbringung mit dem Newton-Verfahren in jedem Iterationsschritt wird eingesetzt. Die
Linearisierung der Reibungskontaktterme und der Plastizita¨tsterme sowie die Beschrei-
bung der L¨osungsprozedur werden detailliert angegeben. Die residuelle a posteriori Feh-
lerabschatzung, die im Falle des Ein-Korper-Reibungskontaktproblems gewonnen wur-¨ ¨
de, wird auf ein Zwei-Korper-Reibungskontaktproblem verallgemeinert. Eine neue hp-¨
Mortar Methode fur das Zwei-Korper-Kontaktproblem mit Tresca Reibung wird kon-¨ ¨
struiert und die variationelle Ungleichung analysiert. Die Kontaktbedingungen sind auf
der diskreten globalen Menge der Gauß-Lobatto Knoten definiert. Nichtpassende Git-
ter sind durch die hp-Mortarprojektion verbunden. Die a priori Fehleranalyse zeigt die
1/4KonvergenzordnungO((h/p) )inderEnergienormunterzusa¨tzlichen Bedingungenfu¨r
die Diskretisierungsparametern. Dirichlet-zu-Neumann Verfahren und Uzawa Verfahren
werden als Lo¨sungsprozedur benutzt. Ein Fehlerindikator wird heuristisch begru¨ndet
und in einer automatischen hp Gitterverfeinerungsprozedur eingesetzt. Die h-Version
des obigen Verfahrens wird auf Zwei-Korper elastoplastische Kontaktprobleme mit Rei-¨
bung generalisiert und mit den Ergebnissen des Penalty-Verfahrens verglichen.
Die theoretische Ergebnisse werden durch die Benchmark-Rechnungen unterstu¨tzt.
Schlagworte: Reibungskontakt, Interface-Probleme, finite Elemente, Randelemente,
FE/BE Kopplung, hp-Methoden, a priori, a posteriori, Mortar, Penalty, Nitsche Ver-
fahren
4Acknowledgements
It is a great pleasure for me to thank my advisor, Prof. Dr. Ernst P. Stephan, for
inspiring me to work in this area. I am very grateful to him for his intensive guidance
of my work, for helpful discussions and important remarks. Also I would like to thank
PD Dr. Matthias Maischak for his support and help in my theoretical investigations
as well as in the programming. His software package maiprogs became a basis for
implementation of numerical experiments, presented in the thesis.
Iamverygratefultomyco-advisorandmyco-refereeProf. Dr. PeterHansbo(Chalmers
University ofTechnology, Gothenburg, Sweden) forhishospitalityandnumerous discus-
sions during my visit in Gothenburg. I would like to thank my co-referee Prof. Dr.
Joachim Gwinner (Universitat der Bundeswehr, Munchen, Germany) for his support¨ ¨
and readiness to examine this thesis in a very short period of time. I would also like to
thank Prof. Dr. Patrick Hild (Laboratoire de Math´ematiques, Universit´e de Franche-
Comt´e, Besanc¸on, France) for his comments and sending me several important papers.
Furthermore, I would like to give my thanks to the members of our working group
in the Institute of Applied Mathematics, University of Hanover, for the friendly and
stimulating atmosphere. My special thanks go to Dipl.-Math. Sergey Geyn for his
cooperation, particularly for performing several numerical examples from Section 4.2.7.
Also I am very grateful to all members of the DFG Graduiertenkolleg 615 for their
collaboration and interest in my work.
Finally, I would like to thank wholeheartedly my family and give a very special thanks
to my wife Natalia for her strong support and patience during my work on this thesis.
This thesis was supported by the DFG Graduiertenkolleg 615 ”Interaction of Modeling,
Computation Methods and Software Concepts for Scientific-Technological Problems”.
Hanover, May 2006 Alexey Chernov
56Contents
Introduction 13
1 Foundations 19
1.1 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2 Boundary integral operators for elliptic problems . . . . . . . . . . . . . 22
1.3 Symmetricboundaryelementformulationformixedboundaryvalueprob-
lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Discretization of the Steklov-Poincar´e operator . . . . . . . . . . . . . . . 26
2 Nonconforming methods for interface problems 29
2.1 The model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 hp-Nitsche’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Continuity and coercivity of a (,) . . . . . . . . . . . . . . . . . 33h
2.2.2 Interpolation in the|||||| -norm . . . . . . . . . . . . . . . . . . 35h
2.2.3 A priori error analysis . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.4 Algebraic formulation. . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Contact between a body and a rigid obstacle 51
3.1 Boundary weak formulations for contact problems with Tresca’s law of
friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.1 Boundary integral variational inequality . . . . . . . . . . . . . . 53
3.1.2 Existence and uniqueness of the weak solution . . . . . . . . . . . 56
3.1.3 Saddle point formulation - Uzawa algorithm . . . . . . . . . . . . 59
3.2 Penalty hp-BEM for one-body contact problem . . . . . . . . . . . . . . 65
3.2.1 Variationalinequality, Lagrangemultiplier, and penalty formulation 65
3.2.2 Inf-sup condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.3 Consistency error in the penalty approximation . . . . . . . . . . 70
3.2.4 A priori error analysis . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.5 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3 Residual FE and BE a posteriori error estimates for contact with friction 84
3.3.1 Regularization of the frictional contact problem . . . . . . . . . . 84
3.3.2 Residual a posteriori error estimates for finite elements . . . . . . 87
3.3.3 Residual a posteriori error estimates for boundary elements . . . . 91
7Contents
3.3.4 Mesh refinement strategy for the h-version . . . . . . . . . . . . . 99
3.3.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4 Nonconforming methods for two-body contact problems with friction 107
4.1 Classical and weak formulation for two-body contact problems with friction108
4.2 h-version of the penalty method . . . . . . . . . . . . . . . . . . . . . . . 112
4.2.1 Constitutive relations for contact with friction . . . . . . . . . . . 112
4.2.2 Constitutive relations for plasticity: J flow theory with isotropic2
/ kinematic hardening . . . . . . . . . . . . . . . . . . . . . . . . 114
4.2.3 FE/BE coupling for elastoplastic contact problems with friction . 115
4.2.4 Pure BEM for elastoplastic contact problems with friction . . . . 120
4.2.5 Linearization of the contact terms . . . . . . . . . . . . . . . . . . 124
4.2.6 Linearization of the plasticity terms in the FE domain – Return
mapping for plasticity . . . . . . . . . . . . . . . . . . . . . . . . 128
4.2.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.2.8 Numerical examples for adaptive mesh refinement . . . . . . . . . 137
4.3 hp-mortar BEM for variational inequality . . . . . . . . . . . . . . . . . . 141
4.3.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.3.2 A priori error analysis . . . . . . . . . . . . . . . . . . . . . . . . 144
4.3.3 Dirichlet-to-Neumann algorithm . . . . . . . . . . . . . . . . . . . 156
4.3.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.3.5 Uzawa algorithm and hp-adaptive error control . . . . . . . . . . 161
4.4 Mortar and penalty methods for elastoplastic contact problems . . . . . . 165
Bibliography 169
8List of Figures
2.1 Geometry and the numerical solution: smooth case . . . . . . . . . . . . 41
2.2 h-version with p =p = 2 vs. p-version with h = 1/5 h = 1/6 . . . . . 431 2 1 2
2.3 Geometry and the numerical solution: singular case . . . . . . . . . . . . 44
2.4 h-version with p =p = 1 vs. p-version with h = 1/6 h = 1/8 . . . . . 461 2 1 2
Nit2.5 h-version with λ =1: smooth solution . . . . . . . . . . . . . . . . . . 47
Nit2.6 h-version with λ =5: smooth solution . . . . . . . . . . . . . . . . . . 47
Nit2.7 h-version with λ =10: smooth solution . . . . . . . . . . . . . . . . . 48
Nit2.8 h-version with λ =20: smooth solution . . . . . . . . . . . . . . . . . 48
Nit2.9 h- and p-version for λ = 1: singular solution . . . . . . . . . . . . . . . 49
Nit2.10 h- and p-version for λ = 5: singular solution . . . . . . . . . . . . . . . 49
Nit2.11 h- and p-version for λ = 10: singular solution . . . . . . . . . . . . . . 50
Nit2.12 h- and p-version for λ = 20: singular solution . . . . . . . . . . . . . . 50
3.1 Error behaviour for varied penalty parameter ε=C h . . . . . . . . . . . 83ε
3.2 Contact geometry of Example 1 . . . . . . . . . . . . . . . . . . . . . . . 100
3.3 Initial meshT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101h
3.4 Sequence of the adaptively generated meshes and deformed geometries
4(value of the displacement is multiplied by 10 ) . . . . . . . . . . . . . . 102
3.5 x- and y-components of the displacement inside the body and deforma-
thtion of the auxiliary FE-grid after 6 refinement step, obtained with the
representation formula (1.2) . . . . . . . . . . . . . . . . . . . . . . . . . 102