Finite Element Approximation of Contact and Friction in Elasticity

Preface
Acknowledgments
Contents
Acronyms
Notations
Part I Basic Concepts
1 Introduction
	1.1 An Overview of the Content
	1.2 Prerequisites
	1.3 An Informal Presentation of the Methods
		1.3.1 The Signorini Problem
		1.3.2 Some Numerical Approximations for Signorini Contact
	1.4 How to Use This Book
2 Sobolev Spaces
	2.1 Distributions
		2.1.1 Continuous and Differentiable Functions
		2.1.2 Test Functions
		2.1.3 Distributions and Distributional Derivatives
		2.1.4 Regular Distributions and Dirac Distribution
		2.1.5 Distributional Gradient and Divergence
		2.1.6 Level Sets of Locally Integrable Functions
	2.2 Fractional Order Sobolev Spaces
		2.2.1 Square-Integrable Functions
		2.2.2 Sobolev Spaces of Fractional Order
		2.2.3 The Slobodeckij Semi-norm
		2.2.4 Some Useful Properties
	2.3 Lipschitz Domains
		2.3.1 A First Definition Using Lipschitz Hypographs
		2.3.2 Examples and Counterexamples of Lipschitz Domains
		2.3.3 Bi-Lipschitz Homeomorphisms
		2.3.4 Lipschitz Domains as a Collection of Lipschitz Mappings
		2.3.5 Partition of Unity for a Lipschitz Boundary
		2.3.6 Sobolev Spaces on the Boundary
		2.3.7 Sobolev Spaces on a Part of the Boundary
		2.3.8 Other Density Theorems
		2.3.9 The Slobodeckij Semi-norm, Once Again
		2.3.10 Domains with Smoother Boundary
	2.4 Trace and Lifting Operators
		2.4.1 The Trace on a Hyperplane
		2.4.2 The Trace on a Lipschitz Boundary
		2.4.3 The Lifting Operator
		2.4.4 First Consequences of Trace and Lifting Theorems
	2.5 Green Formulas
		2.5.1 Preliminaries
		2.5.2 Green Formulas
		2.5.3 Green Formulas on a Part of the Boundary
	2.6 Polynomial Approximation in Fractional Sobolev Spaces
		2.6.1 A Fractional Poincaré-Friedrichs Inequality
		2.6.2 Fractional Deny–Lions Lemma
	2.7 Further Comments
		2.7.1 Distributions and Sobolev Spaces
		2.7.2 Lipschitz Boundaries, Traces and Green Formulas
		2.7.3 Deny–Lions Lemma
		2.7.4 Differential Operators
3 Signorini\'s Problem
	3.1 Presentation
		3.1.1 The Domain and Its Boundaries
		3.1.2 Small Strain Elasticity
		3.1.3 The Nonpenetration Condition
		3.1.4 A First Formulation of Signorini\'s Problem
	3.2 Weak Form and Contact Conditions
		3.2.1 The Weak Formulation for Signorini\'s Problem
		3.2.2 Displacement and Stress on the Contact Boundary
		3.2.3 Green Formula in Elasticity
		3.2.4 Contact Conditions and Strong Form of Signorini
	3.3 Well-posedness
		3.3.1 Ellipticity of the Bilinear Form
		3.3.2 The Well-posedness Result
	3.4 Regularity
		3.4.1 Global Regularity of the Solution
		3.4.2 Binding/Nonbinding Transitions on the Contact Set
	3.5 Further Comments
		3.5.1 About Signorini Contact
		3.5.2 About the Second Korn Inequality
4 Lagrange Finite Elements and Interpolation
	4.1 Lagrange Finite Elements on Simplices
		4.1.1 Simplicial Meshes
		4.1.2 Lagrange Finite Elements
		4.1.3 Conformity
		4.1.4 Basis of Shape Functions
		4.1.5 The Reference Element
	4.2 Some Basic Interpolation Estimates
		4.2.1 The Lagrange Interpolation Operator
		4.2.2 Lagrange Interpolation on the Boundary
		4.2.3 Estimates for Lagrange Interpolation
	4.3 Other Useful Results
		4.3.1 Interpolation Estimate for the Gradient on the Boundary
		4.3.2 Some Discrete Inverse Inequalities
		4.3.3 Discrete Liftings
		4.3.4 Properties of Projection Operators
	4.4 Further Comments
		4.4.1 FEM and Lagrange FEM
		4.4.2 Other Approximation Methods
Part II Numerical Approximation for Signorini
5 Finite Elements for Signorini
	5.1 Preliminaries
		5.1.1 Finite Element Spaces
		5.1.2 A Discrete Variational Inequality
		5.1.3 A Preliminary Error Estimate
	5.2 Finite Element Approximation with Various Cones
		5.2.1 The Convex Cones for Linear Lagrange Finite Elements
		5.2.2 The Convex Cones for Quadratic Lagrange Finite Elements
		5.2.3 The Discrete Problems
		5.2.4 An Abstract Lemma
	5.3 Error Analysis in the Two-Dimensional Case
		5.3.1 Some Local Estimates for the Contact Stress and the Normal Displacement
		5.3.2 Error Analysis for Linear Finite Elements
			5.3.2.1 Conforming Approximation
			5.3.2.2 Nonconforming Approximation
		5.3.3 Error Analysis for Quadratic Finite Elements
			5.3.3.1 Sobolev Regularity 3/2 < s < 5/2
			5.3.3.2 Sobolev Regularity s=5/2
	5.4 Error Analysis in the Three-Dimensional Case
		5.4.1 Extreme Points and New Discrete Convex Cones
		5.4.2 Main Results
		5.4.3 A Quasi-Interpolation Operator
		5.4.4 Error Analysis for Linear Finite Elements
		5.4.5 Error Analysis for Quadratic Finite Elements
			5.4.5.1 Sobolev Regularity 3/2 < s < 5/2
			5.4.5.2 Sobolev Regularity s = 5/2
	5.5 Further Comments
		5.5.1 First Results of Numerical Approximation
		5.5.2 Towards Optimal Rates (Higher Sobolev Regularities)
		5.5.3 The Case of Lower Sobolev Regularities
		5.5.4 Errors in the L2-Norm and Aubin–Nitsche
		5.5.5 Related Results and Other Approximation Methods
6 Nitsche\'s Method
	6.1 A First Derivation of Nitsche\'s Method for Signorini Problem
		6.1.1 The Positive Part Operator
		6.1.2 A Reformulation of the Signorini Conditions
		6.1.3 An Incomplete Nitsche Formulation
	6.2 Nitsche Discrete Formulations and Variants
		6.2.1 A Family of Methods
		6.2.2 The Symmetric Nitsche\'s Method
		6.2.3 Link with Barbosa and Hughes Stabilization
	6.3 Consistency, Well-posedness and Optimal Error Estimates
		6.3.1 Consistency
		6.3.2 Well-posedness
		6.3.3 An Abstract a priori Error Estimate
		6.3.4 Optimal a priori Error Estimate
	6.4 Implementation
	6.5 Further Comments
		6.5.1 About Nitsche\'s Method
		6.5.2 The First Application to Bilateral Contact
		6.5.3 Nitsche for Unilateral Contact
		6.5.4 Symmetry, Skew-symmetry, Etc.
		6.5.5 Lower Sobolev Regularity
		6.5.6 Link with Stabilized Methods and the Augmented Lagrangian
		6.5.7 Other Discretization Methods
7 Mixed Methods
	7.1 Duality Principle and Mixed Weak Form of Signorini
		7.1.1 Obtention of a Lagrangian
		7.1.2 Mixed Problem as a Saddle-Point
		7.1.3 An inf-sup Condition
	7.2 A Mixed Finite Element Method
		7.2.1 The Mixed Method
		7.2.2 Well-Posedness
		7.2.3 An Equivalent Discrete Variational Inequality
		7.2.4 A Discrete inf-sup Condition
	7.3 An a priori Error Estimate for the Mixed Formulation
		7.3.1 An Abstract Lemma
		7.3.2 An Optimal Error Estimate
	7.4 Other Mixed Methods
		7.4.1 A Stabilized Mixed Method
		7.4.2 Mortar and LAC Methods
			7.4.2.1 Formulations
			7.4.2.2 The inf-sup Condition
			7.4.2.3 Optimal Error Estimates
	7.5 Proximal Augmented Lagrangian
		7.5.1 Obtention of an Augmented Lagrangian
		7.5.2 An Augmented Mixed Method
		7.5.3 From Nitsche to Augmented Lagrangian Formulations
	7.6 Implementation
		7.6.1 Semi-Smooth Newton for the Augmented Lagrangian
		7.6.2 Uzawa\'s Algorithm
		7.6.3 A Penalty Formulation from Uzawa
	7.7 Further Comments
		7.7.1 About Mixed Methods
		7.7.2 Discrete inf-sup Conditions
		7.7.3 About the Mixed Methods in This Chapter
		7.7.4 Augmented Lagrangian
Part III Extension to Frictional Contact and Large Strain
8 Tresca Friction
	8.1 Setting
	8.2 Discrete Variational Inequality
		8.2.1 A Discrete Variational Inequality
		8.2.2 A Preliminary Convergence Result
	8.3 Nitsche for Tresca Friction
		8.3.1 Setting
		8.3.2 Well-Posedness and Error Estimates
			8.3.2.1 Well-Posedness
			8.3.2.2 Error Analysis
	8.4 Mixed and Augmented Lagrangian Methods
		8.4.1 Global Setting
		8.4.2 A Mixed Method
		8.4.3 An Augmented Lagrangian Formulation
	8.5 Penalized Frictional Contact
		8.5.1 The Penalty Method
		8.5.2 Convergence Analysis
	8.6 Further Comments
		8.6.1 Tresca Friction
		8.6.2 First Error Estimates for Tresca
		8.6.3 Further Error Estimates for Tresca
		8.6.4 Implementation
9 Coulomb Friction
	9.1 The Frictional Contact Problem in Elasticity
	9.2 Weak Formulation and Existence of Solutions
		9.2.1 Weak Formulations
		9.2.2 Existence, Uniqueness, and Non-uniqueness of Solutions
	9.3 Mixed Finite Element Approximation
		9.3.1 Setting
		9.3.2 Existence and Uniqueness
		9.3.3 An a priori Error Estimate
	9.4 Finite Element Approximation with Nitsche
		9.4.1 Preliminaries
		9.4.2 Nitsche Formulation
		9.4.3 Existence and Uniqueness Results
	9.5 Further Comments
		9.5.1 Mixed Methods and Error Estimates
		9.5.2 Nitsche Method
		9.5.3 Bifurcation Tracking
10 Contact Between Two Elastic Bodies
	10.1 Setting
		10.1.1 The General Configuration
		10.1.2 Biased Contact
		10.1.3 Contact Pairing
		10.1.4 The Contact Conditions
		10.1.5 Friction
		10.1.6 Equilibrium
		10.1.7 Weak Form
	10.2 Finite Element Approximations
		10.2.1 Discrete Variational Inequality, Mortar and LAC
		10.2.2 Nitsche\'s Method
		10.2.3 Mixed Methods
		10.2.4 Augmented Lagrangian
		10.2.5 Penalty
	10.3 Unbiased Formulation for Self- and Multi-body Contact
		10.3.1 Derivation
		10.3.2 The Unbiased Method
	10.4 Further Comments
		10.4.1 Semi-smooth Newton Once Again
		10.4.2 Numerical Integration
11 Contact and Self-contact in Large Strain
	11.1 Setting
		11.1.1 Hyperelasticity
		11.1.2 The Contact Mapping and the Gap Function
		11.1.3 Contact and Friction Conditions
	11.2 Augmented Lagrangian Formulations
		11.2.1 A First Augmented Lagrangian Formulation
		11.2.2 Another Augmented Lagrangian Formulation
	11.3 Unbiased and Biased Nitsche Formulations
		11.3.1 Frictionless Contact
		11.3.2 Frictional Contact
		11.3.3 A Biased Nitsche Method
	11.4 Directional Derivatives of the Gap
		11.4.1 A Preliminary Result
		11.4.2 Derivatives of the Gap
		11.4.3 More and More Derivatives
		11.4.4 Implications in Terms of Numerical Robustness
	11.5 Numerical Results
		11.5.1 Elastic Half-Ring
		11.5.2 Crossed Tubes with Self-contact
	11.6 Further Comments
		11.6.1 Existence Results
		11.6.2 Numerical Methods
A Test-Cases for Verification
	A.1 Scalar Signorini
	A.2 A Manufactured Solution for Tresca Friction
References
Index