Scalable Algorithms for Contact Problems (Advances in Mechanics and Mathematics, 36)

Preface
New Features of the Second Edition
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
		1.4.1 Continuous Formulation
		1.4.2 Discretization
		1.4.3 Solution and Scalable Algorithms
	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 Schur Complement
		2.6 Norms
		2.7 Scalar Products
		2.8 Orthogonalization
		2.9 Eigenvalues and Eigenvectors
		2.10 SVD and CS Decompositions
		2.11 Angles of Subspaces
		2.12 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 QuadraticFunctions
		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 GeneralConstraints
			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 the Presence of Rounding Errors
		5.6 Comments
		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
		References
	7 MPGP for Separable QCQP
		7.1 Projected Gradient and KKT Conditions
		7.2 Reduced Gradients
		7.3 Reduced Projected Gradient
		7.4 MPGP Scheme
		7.5 Rate of Convergence
		7.6 Bound on Norm of Projected Gradient
		7.7 Implementation
			7.7.1 Projection Step with Feasible Half-Step
			7.7.2 MPGP Algorithm in More Detail
		7.8 Comments
		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
		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 QPProblems
		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
		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
		10.7 Reducing the Analysis to Subdomains
		10.8 H-h Bounds on the Schur Complements' Spectrum
			10.8.1 An Upper Bound
			10.8.2 A Lower Bound
		10.9 Optimality
		10.10 Numerical Experiments
		10.11 Comments
		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 Preconditioning by Projectors to Rigid Body Modes
		11.8 Stable Evaluation of K+ x by Using Fixing Nodes
		11.9 Fast Reconstruction of Displacements
		11.10 Reducing the Analysis to Subdomains
		11.11 H-h Bounds on the Schur Complement's Spectrum
			11.11.1 An Upper Bound
			11.11.2 A Lower Bound
		11.12 Optimality
		11.13 Numerical Experiments
			11.13.1 Cantilever Beams
			11.13.2 Roller Bearings of Wind Generator
		11.14 Comments
		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 Cantilever Beams with Isotropic Friction
			12.9.2 Cantilever Beams with Coulomb OrthotropicFriction
			12.9.3 Yielding Clamp Connection
		12.10 Comments
		References
	13 Transient Contact Problems
		13.1 Transient Multibody Frictionless Contact Problem
		13.2 Variational Formulation and Domain Decomposition
		13.3 Space Discretization
		13.4 Time Discretization
		13.5 Dual Formulation of Time-Step Problems
		13.6 Bounds on the Spectrum of Relevant Matrices
			13.6.1 Bounds on the Mass Matrix Spectrum
			13.6.2 Bounds on the Dual Stiffness Matrix Spectrum
		13.7 Preconditioning by Conjugate Projector
		13.8 Optimality
		13.9 Numerical Experiments
			13.9.1 Academic Benchmark
			13.9.2 Impact of Three Bodies
		13.10 Comments
		References
	14 TBETI
		14.1 Green's Representation Formula for 2D Laplacian
		14.2 Steklov-Poincaré Operator
		14.3 Decomposed Boundary Variational Inequality
		14.4 Boundary Discretization and TBETI
		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 Dual Stiffness Matrix Spectrum
		14.10 Optimality
		14.11 Numerical Experiments
			14.11.1 Academic Benchmark
			14.11.2 Comparing TFETI and TBETI
			14.11.3 Ball Bearing
		14.12 Comments
		References
	15 Hybrid TFETI and TBETI
		15.1 Hybrid TFETI for 2D Scalar Problems
		15.2 Hybrid TFETI Problems
		15.3 Orthogonal Rigid Body Modes
			15.3.1 Rigid Body Modes of the Interiors of Faces
			15.3.2 Rigid Body Modes of Subdomains
		15.4 Joining Subdomains by the Face Rigid Body Modes
			15.4.1 Coupling Two Adjacent Faces
			15.4.2 Interconnecting General Clusters
		15.5 General Bounds on the Spectrum of Clusters
		15.6 Bounds on the Spectrum of Chained Clusters
			15.6.1 Angle Between the Kernel and Complement of Feasible Vectors
			15.6.2 The Norm of Face's Rigid Body Modes
			15.6.3 The Norm of Subdomain's Rigid Body Modes
			15.6.4 An Estimate
		15.7 Bounds on the Spectrum of Cube Clusters
		15.8 Hybrid TFETI and TBETI for Contact Problems
		15.9 Optimality
		15.10 Numerical Experiments
			15.10.1 Scalar Variational Inequality and Clustering Effect
			15.10.2 Clumped Elastic Cube
			15.10.3 Clumped Cube Over Obstacle
		15.11 Comments
		References
	16 Mortars
		16.1 Variational Non-penetration Conditions
		16.2 Variationally Consistent Discretization
		16.3 Conditioning of Mortar Non-penetration Matrix
		16.4 Combining Mortar Non-penetration with TFETI
		16.5 Numerical Experiments
		16.6 Comments
		References
	17 Preconditioning and Scaling
		17.1 Reorthogonalization-Based Preconditioning
		17.2 Renormalization-Based Stiffness Scaling
		17.3 Lumped and Dirichlet Preconditioners in Face
		17.4 Adaptive Augmentation
			17.4.1 Algorithm
			17.4.2 Convergence of Adaptive SMALBE-M
		17.5 Numerical Experiments
			17.5.1 Reorthogonalization—3D Heterogeneous Beam
			17.5.2 Reorthogonalization—Contact Problem with Coulomb Friction
			17.5.3 Adaptive Augmentation—3D Hertz Problem
		17.6 Comments
		References
Part IV Other Applications and Parallel Implementation
	18 Contact with Plasticity
		18.1 Algebraic Formulation of Contact Problem for Elasto-Plastic Bodies
		18.2 Semi-smooth Newton Method for Optimization Problem
		18.3 Algorithms for Elasto-Plasticity
		18.4 TFETI Method for Inner Problem and Benchmark
		18.5 Numerical Experiments
		18.6 Comments
		References
	19 Contact Shape Optimization
		19.1 Introduction
		19.2 Discretized Minimum Compliance Problem
		19.3 Sensitivity Analysis
		19.4 Numerical Experiments
		19.5 Comments
		References
	20 Massively Parallel Implementation
		20.1 Parallel Loading and Decomposition of Meshes
		20.2 Interconnecting Clusters by Local Saddle Point Problems
		20.3 When to Use TFETI and H-TFETI
		20.4 Parallel Preprocessing of FETI Objects
			20.4.1 Factorization of K
			20.4.2 Assembling of GGT
		20.5 Factorization of GGT
		20.6 Parallel H-TFETI Solver
		20.7 MatSol, PERMON, and ESPRESO Libraries
			20.7.1 MatSol
			20.7.2 PERMON
			20.7.3 ESPRESO
		20.8 Numerical Experiments
			20.8.1 H-TFETI on Large Linear Problem
		20.9 Two-Body Contact Problem
		References
Notation and List of Symbols
Index