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Mortar Methods for Computational Contact Mechanics and General Interface Problems

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Mortar Methods for Computational Contact Mechanics and General Interface Problems TECHNISCHE UNIVERSITAT¨ MUNCHEN¨ Lehrstuhl fur¨ Numerische Mechanik Mortar Methods for Computational Contact Mechanics and General Interface Problems Alexander Popp Vollstandiger¨ Abdruck der von der Fakultat¨ fur¨ Maschinenwesen der Technischen Universitat¨ Munchen¨ zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr.-Ing. Klaus Drechsler Prufer¨ der Dissertation: 1. Univ.-Prof. Dr.-Ing. Wolfgang A. Wall 2. Univ.-Prof. Dr.-Ing. Michael W. Gee 3. Prof. Tod A. Laursen, Ph.D. Khalifa University, Abu Dhabi / UAE Die Dissertation wurde am 28. Juni 2012 bei der Technischen Universitat¨ Munchen¨ eingereicht und durch die Fakultat¨ fur¨ Maschinenwesen am 17. September 2012 angenommen. Abstract There exists a broad range of applications ranging from machine parts and tribology to crash- worthiness assessment and biomedical engineering, which all demand very general and powerful simulation tools for computational contact mechanics. Contact interaction of deformable bodies and its associated effects, for example frictional sliding, introduce significant additional com- plexities into the typical numerical modeling procedures of solid mechanics. This is mainly due to the inherent nonlinearity and lack of smoothness of the underlying physical principles. Sev- eral challenges have to be met on the way towards a truly general purpose simulation tool for computational contact mechanics, with the two most important aspects being given by robust space and time discretization as well as accurate and efficient constraint enforcement. In this thesis, the applicability of mortar finite element methods for problems of computational contact mechanics is explored. In particular, a novel mortar approach for contact interaction in the fully nonlinear realm (i.e. including finite deformations and nonlinear material behav- ior) is presented, which draws its superior effectiveness from a sound mathematical foundation based on findings in the fields of domain decomposition and non-conforming discretization. Ad- vantageous properties of mortar finite element methods include numerical stability with easily constructible discrete Lagrange multiplier spaces and preservation of the optimal spatial conver- gence rates of conforming discretizations, though with the added benefit of being able to deal with arbitrary non-matching interface meshes. Within the present work, mortar methods are first investigated for mesh tying in nonlinear solid mechanics before considering the actual unilateral contact case. In this regard, an emphasis is put on the so-called dual Lagrange multiplier approach, where the discrete coupling variables are de- fined based on a biorthogonality relationship with the primary unknowns (i.e. displacements). As compared with standard Lagrange multiplier techniques, a localization of the interface coupling conditions is achieved, and thus the dual Lagrange multiplier approach significantly facilitates the resulting algorithms without impinging upon the optimality of mortar methods. The compre- hensive framework presented here is the first one including feasible dual Lagrange multipliers for all types of first-order and second-order mortar finite elements. The main contribution of this thesis, however, lies in the consistent extension of a mortar method with dual Lagrange multiplier interpolation to nonlinear unilateral contact scenarios. The first fully consistent linearization of the dual mortar approach in the context of implicit time integration is presented, from which very efficient nonlinear solution methods can be derived. Moreover, the inequality nature of contact constraints is accommodated with an enhancement of so-called primal-dual active set strategies or semi-smooth Newton methods, thus combining the search for the active contact constraints and all other sources of nonlinearities within one single iteration scheme. As a result, the accuracy of Lagrange multiplier methods and the robustness of mortar-based discretization are paired with an arguably unrivaled numerical efficiency drawn from the dual Lagrange multiplier concept and tailored nonlinear solution methods. A special emphasis is also put on creating a flexible simulation framework, which allows for the straight- forward incorporation of closely related physical effects such as frictional sliding, abrasive wear and adhesion. Furthermore, all devised algorithms are designed to meet the requirements of high performance computing systems with regard to parallel efficiency and scalability. Finally, the present work demonstrates that dual mortar methods also provide a convenient framework for many other single-field and multi-field problems of computational engineering i beyond solid and contact mechanics. Exemplarily, mortar finite element discretization in com- putational fluid dynamics, mortar-based interface treatment in fluid-structure interaction and a coupled fluid-structure-contact interaction approach are outlined. Altogether, the results obtained give rise to the hope that the numerical methods and algorithms developed in this thesis will not only prove helpful for the investigation of highly nonlinear contact phenomena, but also that they may lay the foundation for a general purpose analysis tool providing all capabilities of non-conforming discretization and interface coupling for complex multiphysics simulations. ii Zusammenfassung Es existiert eine große Bandbreite an Anwendungen, angefangen bei Maschinenelementen und tribologischen Systemen bis hin zur Beurteilung von Crashsicherheit oder biomedizinischen Fragestellungen, die allesamt sehr allgemeingultige¨ und leistungsfahige¨ Simulationswerkzeuge aus dem Bereich der numerischen Kontaktmechanik benotigen.¨ Kontaktvorgange¨ deformierbarer Korper¨ und weitere eng damit verknupfte¨ Effekte, wie beispielsweise das reibungsbehaftete Gleiten, bedeuten ein erhebliches Maß an zusatzlicher¨ Komplexitat¨ fur¨ die typischerweise zum Einsatz kommenden numerischen Modellierungsverfahren der Strukturmechanik. Der Haupt- grund hierfur¨ ist die inharente¨ Nichtlinearitat¨ und mangelnde Glattheit der zugrundeliegenden physikalischen Wirkprinzipien. Zur Entwicklung eines wirklich allgemeingultigen¨ Simulations- werkzeugs fur¨ die numerische Kontaktmechanik mussen¨ daher eine ganze Reihe von Heraus- forderungen bewaltigt¨ werden, wobei die beiden wichtigsten Aspekte in der robusten raumlichen¨ und zeitlichen Diskretisierung sowie in der prazisen¨ und effizienten Einbringung von Nebenbe- dingungen zu sehen sind. Im Rahmen der vorliegenden Arbeit wird die Anwendbarkeit von Mortar-basierten Finite- Elemente-Methoden fur¨ Problemstellungen aus der numerischen Kontaktmechanik erforscht. Insbesondere wird ein neuartiger Mortar-Ansatz fur¨ vollstandig¨ nichtlineare Kontaktvorgange¨ (d.h. unter Berucksichtigung¨ großer Deformationen und nichtlinearen Materialverhaltens) vor- gestellt, der seine Leistungsfahigkeit¨ aus einem tragfahigen¨ mathematischen Fundament mit Erkenntnissen aus den Bereichen der Gebietszerlegungsmethoden sowie der nicht-konformen Diskretisierungsverfahren bezieht. Die vorteilhaften Eigenschaften von Mortar-basierten Finite- Elemente-Methoden umfassen deren numerische Stabilitat¨ fur¨ einfach zu definierende diskrete Lagrange-Multiplikator-Raume¨ und die Erhaltung der optimalen raumlichen¨ Konvergenzraten von konformen Diskretisierungsverfahren. Zusatzlich¨ wird jedoch auch der Umgang mit be- liebigen nicht-passenden Interface-Netzen ermoglicht.¨ Mortar-Methoden werden hier zunachst¨ zur Netzkopplung (Mesh Tying) in der nichtlinearen Strukturmechanik untersucht, bevor dann der eigentliche Kontaktfall betrachtet wird. Dabei liegt ein Hauptaugenmerk auf dem sogenannten dualen Lagrange-Multiplikator-Ansatz, bei dem die diskreten Kopplungsvariablen basierend auf einer Biorthogonalitatsbedingung¨ mit den primaren¨ Unbekannten (d.h. Verschiebungen) definiert sind. Im Vergleich zu Standard-Lagrange-Multi- plikator-Ansatzen¨ erreicht man eine lokale Beschrankung¨ der Kopplungsbedingungen am Inter- face. Somit kann der duale Lagrange-Multiplikator-Ansatz eine erhebliche Vereinfachung der resultierenden Algorithmen bewirken ohne sich dabei negativ auf die Optimalitat¨ von Mortar- Methoden auszuwirken. Die hier prasentierte¨ Methodik und Implementierung umfasst zum ers- ten Mal geeignete duale Lagrange-Multiplikator-Ansatze¨ fur¨ samtliche¨ linearen und quadra- tischen Mortar-basierten Elementtypen. Der wichtigste Beitrag der vorliegenden Dissertation liegt jedoch in der konsistenten Er- weiterung einer Mortar-Methode mit dualen Lagrange-Multiplikatoren auf nichtlineare Kon- taktvorgange.¨ Dabei wird zum ersten Mal eine vollstandige¨ Linearisierung des dualen Mortar- Ansatzes im Rahmen von impliziten Zeitintegrationsverfahren vorgestellt, mit deren Hilfe sehr effiziente nichtlineare Losungsmethoden¨ abgeleitet werden konnen.¨ Des Weiteren wird dem Ungleichungs-Charakter der Kontaktbedingungen durch eine entsprechende Erganzung¨ von so- genannten primal-dualen Aktive-Mengen-Strategien oder nichtglatten Newton-Methoden Rech- nung getragen, wodurch eine Kombination der Suche nach den aktiven Kontaktbedingungen iii sowie aller anderer Quellen von Nichtlinearitat¨ innerhalb eines einzigen nichtlinearen Iterations- schemas ermoglicht¨ wird. Infolgedessen zeichnen sich die vorgestellten Ansatze¨ neben der Lo-¨ sungsgenauigkeit von Lagrange-Multiplikator-Methoden und der Robustheit Mortar-basierter Diskretisierung auch durch eine wohl einzigartige numerische Effizienz aus, die sich
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