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Dual-Dual Formulations for Frictional Contact Problems in Mechanics

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Dual-Dual Formulations for Frictional Contact Problems in Mechanics View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Institutionelles Repositorium der Leibniz Universität... Dual-dual formulations for frictional contact problems in mechanics Von der Fakultät für Mathematik und Physik der Gottfried Wilhelm Leibniz Universität Hannover zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von Dipl. Math. Michael Andres geboren am 28. 12. 1980 in Siemianowice (Polen) 2011 Referent: Prof. Dr. Ernst P. Stephan, Leibniz Universität Hannover Korreferent: Prof. Dr. Gabriel N. Gatica, Universidad de Concepción, Chile Korreferent: PD Dr. Matthias Maischak, Brunel University, Uxbridge, UK Tag der Promotion: 17. 12. 2010 ii To my Mum. Abstract This thesis deals with unilateral contact problems with Coulomb friction. The main focus of this work lies on the derivation of the dual-dual formulation for a frictional contact problem. First, we regard the complementary energy minimization problem and apply Fenchel’s duality theory. The result is a saddle point formulation of dual type involving Lagrange multipliers for the governing equation, the symmetry of the stress tensor as well as the boundary conditions on the Neumann boundary and the contact boundary, respectively. For the saddle point problem an equivalent variational inequality problem is presented. Both formulations include a nondiffer- entiable functional arising from the frictional boundary condition. Therefore, we introduce an additional dual Lagrange multiplier denoting the friction force. This procedure yields a dual-dual formulation of a two-fold saddle point structure. For the corresponding variational inequality problem we show the unique solvability. Two different inf-sup conditions are introduced that allow an a priori error analysis of the dual-dual variational inequality problem. To solve the problem numerically we use the Mixed Finite Element Method. We pro- pose appropriate finite element spaces satisfying the discrete version of the inf-sup conditions. A modified nested Uzawa algorithm is used to solve the corresponding discrete system. We prove its convergence based on the discrete inf-sup condi- tions. Furthermore, we present a reliable a posteriorierror estimator based on a Helmholtz decomposition. Numerical experiments are performed to underline the theoretical results. Keywords. Mixed Finite Element Method, dual formulation, saddle point problem, variational inequality, inf-sup condition, error estimator iv Zusammenfassung In dieser Arbeit betrachten wir Ein-Körper-Kontaktproblememit Coulomb Reibung. Das Ziel dieser Arbeit ist eine Herleitung der dual-dualen Formulierung dieser Rei- bungskontakprobleme. Ausgehend vom Minimierungsproblem der komplemen- tären Energie leiten wir mit Hilfe der Fenchel’schen Dualitätstheorie ein äquiva- lentes Sattelpunktsproblem her. Dieses beinhaltet Lagrange Multiplikatoren für die zugehörige Differentialgleichung, die Symmetrie des Spannungstensors sowie die Randbedingungen auf dem Neumannrand und dem Kontaktrand. Für das Sat- telpunktsproblem geben wir ein äquivalentes Variationsungleichungsproblem an. Beide Formulierungen enthalten ein nichtdifferenzierbares Reibungsfunktional. Da- her führen wir mit der Reibungskraft einen weiteren Lagrange Multiplikator ein und erhalten somit eine dual-duale Formulierung des Problems, welches eine zweifache Sattelpunktsstruktur aufweist. Für das zugehörige Variationsungleichungsprob- lem zeigen wir die eindeutige Lösbarkeit. Basierend auf zwei Inf-Sup-Bedingungen können wir eine a priori Analysis des dual-dualen Variationsungleichungsproblems durchführen. Wir verwenden die Gemischte Finite Elemente Methode zur numerischen Approx- imation der Lösung des Problems. Hierzu führen wir geeignete Finite Element Räume ein, welche die diskrete Version der Inf-Sup-Bedingungen erfüllen. Um das diskrete System zu lösen, benutzen wir einen modifizierten, geschachtelten Uzawa Algorithmus. Wir beweisen seine Konvergenz mit Hilfe der diskreten Inf-Sup- Bedingungen. Darüberhinaus leiten wir einen, auf einer Helmholtz-Zerlegung basierenden, a pos- teriori Fehlerschätzer her. Zudem stellen wir numerische Experimente vor, die unsere theoretischen Ergebnisse bestätigen. Schlagwörter. Gemischte Finite Elemente Methode, duale Formulierung, Sattel- punktproblem, Variationsungleichung, Inf-Sup-Bedingung, Fehlerschätzer v Acknowledgements It is a pleasure for me to thank my advisor, Prof. Dr. Ernst P. Stephan, for giving me the opportunity to work in his group for more than four years. I am very grateful that he suggested the topic of my thesis and for many lively discussions which helped me finishing this work. I also would like to thank all members of the working group “Numerical Analysis”. First of all I thank my friend Leo Nesemann for sharing the office and helping me to overcome many difficulties during the long time. Catalina Domínguez, Elke Ostermann, Florian Leydecker and Ricardo Prato also helped me having a good time at the institute. Furthermore, I like to thank my co-advisor PD Dr. Matthias Maischak who examined my thesis and supported me continuously, especially during my stays at the Brunel University. I thank my second examiner, Prof. Dr. Gabriel N. Gatica, for his readiness to examine my thesis. Finally, I would like to thank wholeheartedly my family, especially my beloved. They encouraged me every time when I got doubts. This work was made possible by the German Research Foundation (DFG) who supported me within the priority program SPP 1180 Prediction and Manipulation of Interactions between Structure and Process under grant STE 573/2-3. vi Contents 1. Introduction 1 2. Basic foundations 7 2.1. Notations................................... 7 2.2. Some properties of the Sobolev space H(div, Ω)............. 9 2.3. Convexanalysis............................... 10 3. Dual-dual formulation for a contact problem with friction in 2D 13 3.1. Dual-dualformulationin2D. 18 3.2. Mixedfiniteelements.. ......................... 58 3.3. NumericalAlgorithms . ......................... 76 3.4. NumericalExperiments . 88 4. Adaptive methods for the dual-dual contact problem with friction 105 4.1. Aposteriorierrorestimate . 105 4.2. Numerical experimentusing adaptive algorithms . 120 A. Numerical methods for process-oriented structures in metal chipping 127 vii 1. Introduction Contact problems in mechanics are investigated for more than hundred years. In 1881 Hertz [58] analyzed the so-called Hertz contact problem where he derives analytical representations for the shape of the contact area between two elastic spheres and determines the resulting contact pressure. The need for a more precise analysis and simulation in manufacturing and other fields of engineering science came along with better approximations of real life problems in terms of mathematical models and their numerical resolution. The mathematical theory of elasticity, see Sokolnikoff [79], is governed by Lamé’s equation. Combined with Hooke’s material law we obtain problems of linear elas- ticity. If the stresses occurring in a body subject to extern and intern forces exceed a certain value, e.g. the yield stress, the theory of linear elasticity is no longer valid and more complex nonlinear models have to be used, e.g. models for plasticity. For an introduction to elasto-plasticity we refer to Neˇcas and Hlaváˇcek [69]. The theory of plasticity is described in Han and Reddy [57] and an overview of other inelastic models can be found in Simo and Hughes [78]. The second field where nonlinearities appear are contact boundary conditions. In the simplest setting some linear elastic body is coming into contact with a rigid fric- tionless foundation. This is the so-called Signorini problem, see Kikuchi and Oden [63]. As the material points of the body must not pervade the rigid foundation they satisfy a nonpenetration condition, which is an inequality condition. Therefore, the theory of variational inequalities is closely connected to contact problems. In most cases contact problems are formulated in terms of variational inequality problems. Duvaut and Lions [31], Glowinski et al. [53], Kikuchi and Oden [63] and Neˇcas et al. [60] give an elaborate review of this topic. An abstract introduction to variational inequalities can be found in Kinderlehrer and Stampacchia [64] and Glowinski [52]. In the latter work the author distinguishes between variational inequalities of the first and of the second kind. Variational inequalities of the first kind are restricted to a convex set. For example the displacement field solving the Signorini problem is restricted to all vector fields that do not violate the nonpenetration condition. Vari- ational inequalities of the second kind are formulated on a whole space but involve a nonlinear convex functional, for example if the rigid foundation in the Signorini problem causes friction. In this case the tangential part of the stress on the contact boundary depends on the normal part. The common model of this phenomenon is the Coulomb law of friction, where the absolute value of the tangential stress cannot exceed a multiple of the absolute value of the normal stress, denoted as the 1 1. Introduction friction force. A simpler model is the law of Tresca friction. Here, the friction force is assumed to be given. There are several other models known in literature. Oden and Martins [74] give an extensive overview. Johansson and Klarbring [62] consider additionally the influ- ence of temperature and state a thermoelastic frictional contact
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