Scalable Algorithms for Contact Problems
This book presents a comprehensive and self-contained treatment of the authors' newly developed scalable algorithms for the solutions of multibody contact problems of linear elasticity. The brand new feature of these algorithms is theoretically supported numerical scalability and parallel scala...
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| Hlavní autoři: | , , , |
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| Médium: | E-kniha |
| Jazyk: | angličtina |
| Vydáno: |
New York, NY
Springer Nature
2017
Springer New York Springer |
| Vydání: | 1 |
| Edice: | Advances in Mechanics and Mathematics |
| Témata: | |
| ISBN: | 9781493968343, 1493968343, 9781493968329, 1493968327 |
| ISSN: | 1571-8689, 1876-9896 |
| On-line přístup: | Získat plný text |
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- 5.6 Comments and Conclusions -- References -- 6 Gradient Projection for Separable Convex Sets -- 6.1 Separable Convex Constraints and Projections -- 6.2 Conjugate Gradient Step Versus Gradient Projections -- 6.3 Quadratic Functions with Identity Hessian -- 6.4 Subsymmetric Sets -- 6.5 Dominating Function and Decrease of the Cost Function -- 6.6 Comments and References -- References -- 7 MPGP for Separable QCQP -- 7.1 Projected Gradient, Reduced Gradient, and KKT Conditions -- 7.2 Reduced Projected Gradient -- 7.3 MPGP Scheme -- 7.4 Rate of Convergence -- 7.5 Bound on Norm of Projected Gradient -- 7.6 Implementation -- 7.6.1 Projection Step with Feasible Half-Step -- 7.6.2 MPGP Algorithm in More Detail -- 7.7 Comments and References -- References -- 8 MPRGP for Bound-Constrained QP -- 8.1 Specific Form of KKT Conditions -- 8.2 MPRGP Algorithm -- 8.3 Rate of Convergence -- 8.4 Identification Lemma and Finite Termination -- 8.5 Implementation of MPRGP -- 8.6 Preconditioning -- 8.7 Comments and References -- References -- 9 Solvers for Separable and Equality QP/QCQP Problems -- 9.1 KKT Conditions -- 9.2 Penalty and Method of Multipliers -- 9.3 SMALSE-M -- 9.4 Inequalities Involving the Augmented Lagrangian -- 9.5 Monotonicity and Feasibility -- 9.6 Boundedness -- 9.7 Convergence -- 9.8 Optimality of the Outer Loop -- 9.9 Optimality of the Inner Loop -- 9.10 SMALBE for Bound and Equality Constrained QP Problems -- 9.11 R-Linear Convergence of SMALBE-M -- 9.12 SMALSE-Mw -- 9.13 Solution of More General Problems -- 9.14 Implementation -- 9.15 Comments and References -- References -- Part III Scalable Algorithms for Contact Problems -- 10 TFETI for Scalar Problems -- 10.1 Two Membranes in Unilateral Contact -- 10.2 Variational Formulation -- 10.3 Tearing and Interconnecting -- 10.4 Discretization -- 10.5 Dual Formulation -- 10.6 Natural Coarse Grid
- 14.5 Operators of Elasticity -- 14.6 Decomposed Contact Problem on Skeleton -- 14.7 TBETI Discretization of Contact Problem -- 14.8 Dual Formulation -- 14.9 Bounds on the Spectrum -- 14.10 Optimality -- 14.11 Numerical Experiments -- 14.11.1 Academic Benchmark -- 14.11.2 Comparison TFETI and TBETI -- 14.11.3 Ball Bearing -- 14.12 Comments -- References -- 15 Mortars -- 15.1 Variational Non-penetration Conditions -- 15.2 Variationally Consistent Discretization -- 15.3 Conditioning of Mortar Non-penetration Matrix -- 15.4 Combining Mortar Non-penetration with FETI Interconnecting -- 15.5 Numerical Experiments -- 15.5.1 3D Hertz Problem with Decomposition -- 15.6 Comments and References -- References -- 16 Preconditioning and Scaling -- 16.1 Reorthogonalization-Based Preconditioning -- 16.2 Renormalization-Based Stiffness Scaling -- 16.3 Lumped and Dirichlet Preconditioners in Face -- 16.4 Numerical Experiments -- 16.4.1 3D Heterogeneous Beam -- 16.4.2 Contact Problem with Coulomb Friction -- 16.5 Comments and References -- References -- Part IV Other Applications and Parallel Implementation -- 17 Contact with Plasticity -- 17.1 Algebraic Formulation of Contact Problem for Elasto-Plastic Bodies -- 17.2 Semismooth Newton Method for Optimization Problem -- 17.3 Algorithms for Elasto-Plasticity -- 17.4 TFETI Method for Inner Problem and Benchmark -- 17.5 Numerical Experiments -- 17.6 Comments -- References -- 18 Contact Shape Optimization -- 18.1 Introduction -- 18.2 Discretized Minimum Compliance Problem -- 18.3 Sensitivity Analysis -- 18.4 Numerical Experiments -- 18.5 Comments -- References -- 19 Massively Parallel Implementation -- 19.1 Stiffness Matrix Factorization and Action of K+ -- 19.2 Coarse Problem Implementation -- Action of P -- 19.2.1 Assembling GGT in Parallel -- 19.2.2 Parallel Explicit Inverse -- 19.2.3 Parallel Direct Solution
- 10.7 Bounds on the Spectrum -- 10.8 Optimality -- 10.9 Numerical Experiments -- 10.10 Comments and References -- References -- 11 Frictionless Contact Problems -- 11.1 Linearized Non-penetration Conditions -- 11.2 Equilibrium of a System of Elastic Bodies in Contact -- 11.3 Variational Formulation -- 11.4 Tearing and Interconnecting -- 11.5 Discretization -- 11.6 Dual Formulation -- 11.7 Stable Evaluation of K+x by Using Fixing Nodes -- 11.8 Preconditioning by Projectors to Rigid Body Modes -- 11.9 Bounds on the Spectrum -- 11.10 Optimality -- 11.11 Numerical Experiments -- 11.11.1 Academic Benchmark -- 11.11.2 Roller Bearings of Wind Generator -- 11.12 Comments and References -- References -- 12 Contact Problems with Friction -- 12.1 Equilibrium of Bodies in Contact with Coulomb Friction -- 12.2 Variational Formulation -- 12.3 Tresca (Given) Isotropic Friction -- 12.4 Orthotropic Friction -- 12.5 Domain Decomposition and Discretization -- 12.6 Dual Formulation -- 12.7 Preconditioning by Projectors to Rigid Body Modes -- 12.8 Optimality -- 12.9 Numerical Experiments -- 12.9.1 Academic Benchmark -- 12.9.2 Yielding Clamp Connection -- 12.10 Comments and References -- References -- 13 Transient Contact Problems -- 13.1 Transient Multibody Frictionless Contact Problem -- 13.2 Variational Formulation and Domain Decomposition -- 13.3 Discretization -- 13.4 Dual Formulation of Time Step Problems -- 13.5 Bounds on the Spectrum of Dual Energy Function -- 13.6 Preconditioning by Conjugate Projector -- 13.7 Optimality -- 13.8 Numerical Experiments -- 13.8.1 Academic Benchmark -- 13.8.2 Impact of Three Bodies -- 13.9 Comments -- References -- 14 TBETI -- 14.1 Green's Representation Formula for 2D Laplace Operator -- 14.2 Steklov--Poincaré Operator -- 14.3 Decomposed Boundary Variational Inequality -- 14.4 Boundary Discretization and TBETI
- 19.3 Hybrid TFETI (HTFETI) -- 19.3.1 Description of TFETI Method -- 19.3.2 Parallel Implementation -- 19.3.3 Numerical Experiment -- 19.4 Communication Layer Optimization -- 19.4.1 TFETI Hybrid Parallelization -- 19.5 MatSol, PERMON, and ESPRESO Libraries -- 19.5.1 MatSol -- 19.5.2 PERMON -- 19.5.3 ESPRESO -- 19.6 Numerical Experiments -- References -- Bibliography -- Index
- Intro -- Preface -- Acknowledgments -- Contents -- 1 Contact Problems and Their Solution -- 1.1 Frictionless Contact Problems -- 1.2 Contact Problems with Friction -- 1.3 Transient Contact Problems -- 1.4 Numerical Solution of Contact Problems -- References -- Part I Basic Concepts -- 2 Linear Algebra -- 2.1 Vectors and Matrices -- 2.2 Matrices and Mappings -- 2.3 Inverse and Generalized Inverse -- 2.4 Direct Methods for Solving Linear Equations -- 2.5 Norms -- 2.6 Scalar Products -- 2.7 Eigenvalues and Eigenvectors -- 2.8 Matrix Decompositions -- 2.9 Graphs, Walks, and Adjacency Matrices -- References -- 3 Optimization -- 3.1 Optimization Problems and Solutions -- 3.2 Unconstrained Quadratic Programming -- 3.2.1 Quadratic Cost Functions -- 3.2.2 Unconstrained Minimization of Quadratic Functions -- 3.3 Convexity -- 3.3.1 Convex Quadratic Functions -- 3.3.2 Minimizers of Convex Function -- 3.3.3 Existence of Minimizers -- 3.3.4 Projections to Convex Sets -- 3.4 Equality Constrained Problems -- 3.4.1 Optimality Conditions -- 3.4.2 Existence and Uniqueness -- 3.4.3 Sensitivity -- 3.5 Inequality Constrained Problems -- 3.5.1 Optimality Conditions for Linear Constraints -- 3.5.2 Optimality Conditions for Bound Constrained Problems -- 3.5.3 Optimality Conditions for More General Constraints -- 3.5.4 Existence and Uniqueness -- 3.6 Equality and Inequality Constrained Problems -- 3.6.1 Optimality Conditions -- 3.7 Duality for Quadratic Programming Problems -- 3.7.1 Uniqueness of a KKT Pair -- References -- 4 Analysis -- 4.1 Sobolev Spaces -- 4.2 Trace Spaces -- 4.3 Variational Inequalities -- References -- Part II Optimal QP and QCQP Algorithms -- 5 Conjugate Gradients -- 5.1 First Observations -- 5.2 Conjugate Gradient Method -- 5.3 Rate of Convergence -- 5.4 Preconditioned Conjugate Gradients -- 5.5 Convergence in Presence of Rounding Errors