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Dual Domain Decomposition Methods in Structural Dynamics, Efficient Fakultät für Maschinenwesen Lehrstuhl für Angewandte Mechanik Dual Domain Decomposition Methods in Structural Dynamics Efficient Strategies for Finite Element Tearing and Interconnecting Michael Christopher Dominik Leistner Vollständiger Abdruck der von der Fakultät für Maschinenwesen der Technischen Universität München zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) genehmigten Dissertation. Vorsitzender: Prof. Dr.-Ing. Michael W. Gee Prüfer der Dissertation: 1. Prof. dr. ir. Daniel Rixen 2. Dr. habil. à diriger des recherches Pierre Gosselet Die Dissertation wurde am 15.11.2018 bei der Technischen Universität München eingereicht und durch die Fakultät für Maschinenwesen am 15.02.2019 angenommen. Abstract Parallel iterative domain decomposition methods are an essential tool to simulate structural mechanics because modern hardware takes its computing power almost exclusively from parallelization. These methods have undergone enormous develop- ment in the last three decades but still some important issues remain to be solved. This thesis focuses on the application of dual domain decomposition to problems of linear structural dynamics and the two main issues that appear in this context. First, the properties and the performance of known a priori coarse spaces are assessed for the first time as a function of material heterogeneity and the time step size. Second, strategies for the efficient repeated solution of the same operator with multiple right- hand sides, as it appears in linear dynamics, are investigated and developed. While direct solution methods can easily reuse their factorization of the operator, it is more difficult in iterative methods to efficiently reuse the gathered information of for- mer solution steps. The ability of known recycling methods to construct tailor-made coarse spaces is assessed and improved. In particular, this work and the according preceding papers present the first highly efficient recycling strategies for the recently published adaptive multipreconditioning framework. Zusammenfassung Parallele iterative Gebietszerlegungsverfahren sind ein essenzielles Werkzeug zur Si- mulation von Strukturmechanik, denn moderne Hardware schöpft ihre Rechenleis- tung fast ausschließlich aus Parallelisierung. Diese Methoden haben in den letzten drei Jahrzehnten eine enorme Entwicklung durchlaufen, doch es gibt noch einige wichtige Probleme, die gelöst werden müssen. Diese Arbeit konzentriert sich auf die Anwendung von Gebietszerlegungsverfahren auf lineare Probleme der Strukturdy- namik und die beiden Hauptprobleme die sich in diesem Zusammenhang ergeben. Dies sind erstens die Eigenschaften und die Leistungsfähigkeit von bekannten a priori Coarse Spaces, welche erstmalig in Abhängigkeit der Material-Heterogenität und der Zeitschrittgröße untersucht werden. Der zweite Punkt ist die Untersuchung und Ent- wicklung von Strategien für die effiziente mehrfache Lösung von konstanten Opera- toren mit wechselnder rechter Seite. Während direkte Lösungsverfahren einfach die Faktorisierung des Operators wiederverwenden können, ist es in iterativen Verfahren schwieriger die gesammelten Informationen aus vorhergehenden Lösungsschritten effizient wiederzuverwenden. Die Fähigkeit bekannter Recyclingstrategien, maßge- schneiderte Coarse Spaces aufzubauen, wird untersucht und verbessert. Insbeson- dere werden in dieser Arbeit und in den entsprechenden vorangegangenen Publika- tionen die ersten hocheffizienten Recyclingstrategien für das kürzlich veröffentlichte multipreconditioning Verfahren präsentiert. Vorwort Die vorliegende Arbeit entstand während meiner Tätigkeit als wissenschaftlicher Mit- arbeiter am Lehrstuhl für Angewandte Mechanik der Technischen Universität Mün- chen unter der Leitung meines Doktorvaters Prof. dr. ir. Daniel Rixen. An erster Stelle gilt ihm mein besonderer Dank für seine immer wohlwollende Unterstützung und für die stets exzellenten Rahmenbedingungen in fachlichen wie menschlichen Belangen. Sein mir entgegengebrachtes Vertrauen, die von ihm geförderten persönlichen Ent- wicklungsmöglichkeiten und Gestaltungsfreiräume sowie natürlich die steten fachli- chen Diskussionen und Anregungen haben maßgeblich zum Gelingen meiner Arbeit beigetragen. Ebenso danke ich Dr. habil. Pierre Gosselet für sein großes Interesse an meiner Arbeit und für seine Tätigkeit als Zweitprüfer. Prof. Dr.-Ing. Michael W. Gee danke ich für die Übernahme des Vorsitzes der Prüfungskommission. Das Gelingen meiner Arbeit und dass ich die Zeit am Lehrstuhl in besonders positiver Erinnerung behalten werde, ist auch maßgeblich den vielen aktuellen und ehema- ligen Mitarbeitern des Lehrstuhls, mit denen ich dort zusammenarbeiten durfte, zu verdanken. Viele wertvolle fachliche Diskussionen, die gemeinsame Teilnahme an Tagungen, die vielen gemeinsamen sportlichen Aktivitäten, sowie das stets freund- schaftliche und von Hilfsbereitschaft geprägte Klima haben sehr dazu beigetragen. Ganz besonderer Dank gilt meinem langjährigen Kollegen Oliver Hofmann, dessen einzigartig motivierende, positive Persönlichkeit ich sehr schätzen gelernt habe und der mich in zahlreichen Situationen in besonderer Weise unterstützt hat. Schließlich möchte ich meiner gesamten Familie und ganz besonders meinen Eltern für die stets umfassende und bedingungslose Unterstützung während meines bisheri- gen Werdegangs danken, der ohne diese Unterstützung nicht möglich gewesen wäre. München, im März 2019 Michael Leistner Contents 1 Introduction 1 1.1 Problem Statement . .2 1.2 Objective and Outline . .5 1.3 Scientific Contributions . .7 1.4 Linear Static Elasticity . .8 2 Finite Element Tearing and Interconnecting 11 2.1 Non-overlapping Domain Decomposition . 13 2.1.1 Local Problems . 14 2.1.2 Condensation on the Interface . 19 2.1.3 Primal Interface Problem . 22 2.1.4 Dual Interface Problem . 23 2.1.5 Redundant Lagrange Multipliers . 28 2.2 Iterative Solution . 30 2.2.1 Conjugate Directions . 31 2.2.2 Conjugate Gradients . 33 2.2.3 Projected Preconditioned Conjugate Gradient . 34 2.3 Preconditioning . 40 2.3.1 Basic Preconditioners . 40 2.3.2 Scaling . 41 2.4 Deflation by Auxiliary Coarse Spaces . 43 2.5 Convergence and spectral properties. 46 2.5.1 General Spectral Properties of the Conjugate Gradient Algorithm 46 2.5.2 Spectral properties of the FETI operator. 48 2.5.3 Desireable Coarse Spaces . 51 2.6 Generalized Eigenvalues in the Overlaps . 51 3 Application to Dynamic Elasticity 53 3.1 Linear Dynamic Elasticity . 54 3.2 Time Integration of Constrained Dynamical Problems . 54 3.2.1 Overview and Classification . 55 3.2.2 Requirements . 56 3.2.3 Generalized-α ............................... 58 3.2.4 Constrained dynamics . 62 3.3 Construction of the Interface Problem and Iterative Solution . 63 3.3.1 Preconditioning and Scaling . 63 3.3.2 Initial Acceleration . 64 vii viii Contents 3.3.3 Spectral Properties . 65 3.3.4 Iterative Solution . 66 3.4 Fundamental Concepts for Improving Convergence . 67 3.4.1 Preconditioner Based Methods . 69 3.4.2 Deflation by a Priori Spaces . 69 3.4.3 Deflation by Recycled Spaces . 71 4 Deflation Strategies on Different Scales of Time 73 4.1 Spectral Properties of the Interface Problem . 75 4.1.1 Relation to the Equivalent Static Problem . 75 4.1.2 Conditioning and Convergence on Relevant Time Scales . 78 4.2 A Priori Coarse Spaces . 80 4.2.1 Rigid Body Modes . 81 4.2.2 Generalized Eigenvalues in the Overlaps . 81 4.2.3 Local Mode Shapes and Subspace Angles . 84 4.2.4 Spectral Efficiency . 92 4.2.5 Computational Efficiency . 95 4.2.6 Conclusions . 97 4.3 Recycling Strategies . 98 4.3.1 Plain Reuse . 99 4.3.2 Global Ritz Approximation . 100 4.3.3 Accuracy of Ritz Approximation . 103 4.3.4 Appropriate Coarse Space Sizes . 107 4.4 Conclusion . 110 5 Multipreconditioning 113 5.1 Adaptive Multipreconditioning . 115 5.2 Degeneration of the Search Space . 117 5.2.1 Minimization by a Rank Revealing Matrix Factorization . 117 5.2.2 Zero Contributions due to Inactive Parts . 118 5.2.3 Activation Strategies . 121 5.3 Recycling Strategies . 125 5.3.1 Plain Reuse . 126 5.3.2 Local Ritz Approximation . 127 5.3.3 Computational and Spectral Efficiency . 130 5.3.4 Effect of Increasing the Ritz Space Size . 136 5.3.5 Direct Employment of Ritz Spaces as Coarse Space . 137 5.4 Conclusion . 139 6 Closure 143 6.1 Conclusion . 143 6.2 Outlook . 145 A Example Structures and Further Results 149 Bibliography 157 Nomenclature Acronyms FEM Finite element method dof Degree of freedom BDD Balancing domain decomposition FETI Finite element tearing and interconnecting AMP-FETI Adaptive multipreconditioned FETI FETI-DP FETI dual-primal GenEO Generalized eigenvalues in the overlaps TRKS Total reuse of Krylov subspaces SRKS Selective reuse of Krylov subspaces CG Conjugate gradient PPCG Projected preconditioned conjugate gradient GMRES Generalized minimal residual LMS Linear multistep GSSSS Generalized single-step single-solve RK Runge-Kutta Gmsh A finite element mesh generator software METIS A mesh partitioning algorithm ix x Nomenclature Continuum Mechanics F Deformation gradient S Second Piola-Kirchhoff stresses E Green-Lagrange strain W Strain energy N Normal vector of the surface u Exact continuous displacements u¨ Exact continuous accelerations uh Approximated continuous displacements ¨ uh Approximated continuous accelerations v Continuous virtual displacements ρ0 Density in initial configuration ρ Density ν Poisson ratio E Young’s modulus b Body forces t Traction forces Ω Computational domain in inital configuration Γ Boundary of the computational domain ΓN Neumann
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