/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Robust stability and control for systems that combine continuous-time and discrete-time dynamics. This article is a tutorial on modeling the dynamics of hybrid systems, on the elements of stability theory for hybrid systems, and on the basics of hybrid control. The presentation and selection of material is oriented toward the analysis of asymptotic stability in hybrid systems and the design of stabilizing hybrid controllers. Our emphasis on the robustness of asymptotic stability to data perturbation, external disturbances, and measurement error distinguishes the approach taken here from other approaches to hybrid systems. While we make some connections to alternative approaches, this article does not aspire to be a survey of the hybrid system literature, which is vast and multifaceted.

Hybrid dynamical systems

This paper describes modern techniques used to solve contact problems within Computational Mechanics. On the basis of a continuum description of contact, the mathematical structure of the contact formulation for finite element methods is derived. Emphasis is also placed on the constitutive behavior at the contact interface for normal and tangential (frictional) contact. Furthermore, different discretization schemes currently applied to solve engineering problems are formulated for small and finite strain problems. These include isoparametric interpolations, node-to-segment discretizations and also mortar and Nitsche techniques. Furthermore, several algorithms related to spatial contact search and fulfillment of the inequality constraints at the contact interface are discussed. Here, especially the penalty and Lagrange multiplier schemes are considered and also SQP- and linear-programming methods are reviewed.


Keywords:

contact mechanics;
friction;
penalty method;
Lagrange multiplier method;
contact algorithms;
finite element method;
finite deformations;
discretization methods

Computational Contact Mechanics

Basic error estimates for elliptic problems

This paper gives an extensive documentation of applications of finite-dimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.

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Engineering and Economic Applications of Complementarity Problems

Inequality problems in mechanics and applications

Introductory material pseudo-monotonicity and generalized pseudo-monotonicity hemivariational inequalities for static one-dimensional nonconvex superpotential laws hemivariational inequalities for locally Lipschitz functionals hemivariational inequalities for multidimensional superpotential law noncoercive hemivariational inequalities related to free boundary problems constrained problems for nonconvex star-shaped admissible sets.

Mathematical Theory of Hemivariational Inequalities and Applications

1. Introductory Topics.- 1. Essential Notions and Propositions of Functional Analysis.- 1.1 Topological Vector Spaces and Related Subjects.- 1.1.1 Topological Spaces and Continuous Mappings.- 1.1.2 Locally Convex Topological Vector Spaces, Normed Spaces and Linear Mappings.- 1.2 Duality in Topological Vector Spaces.- 1.2.1 Duality. Weak and Strong Topologies.- 1.2.2 Topologically Dual Pairs of Vector Spaces.- 1.2.3 Duality in Normed and Hilbert Spaces.- 1.2.4 Transpose of a Continuous Linear Mapping Scales of Hilbert Spaces. The Lax-Milgram Theorem.- 1.3 Certain Function Spaces and Their Properties.- 1.3.1 The Spaces $$ {C^m}\left( \Omega \right),{C^m}\left( {\overline \Omega } \right),D\left( \Omega \right),D\left( {\overline \Omega } \right)and{L^p}\left( \Omega \right) $$.- 1.3.2 Spaces of Distributions.- 1.3.3 Sobolev Spaces.- 1.3.4 Trace Theorem. Imbedding Properties of Sobolev Spaces.- 1.3.5 The Space of Functions of Bounded Deformation.- 1.4 Additional Topics.- 1.4.1 Elements of the Theory of Vector-valued Functions and Distributions.- 1.4.2 Elements of Differential Calculus.- 1.4.3 Supplementary Notions and Propositions.- 2. Elements of Convex Analysis.- 2.1 Convex Sets and Functionals.- 2.1.1 Definitions.- 2.1.2 Lower Semicontinuous Convex Functionals.- 2.2 Minimization of Convex Functionals.- 2.2.1 Existence of a Minimum.- 2.2.2 Variational Inequalities.- 2.3 Subdifferentiability.- 2.3.1 Definitions and Related Propositions.- 2.3.2 One-Sided Directional Gateaux-Differential.- 2.4 Subdifferential Calculus.- 2.4.1 The Subdifferential of a Sum of Functionals and of a Composite Functional.- 2.4.2 The Relative Interior of R(?f).- 2.5 Conjugates of Convex Functionals.- 2.5.1 The Classes ?(X) and ?0(X).- 2.5.2 The Conjugacy Operation.- 2.6 Maximal Monotone Operators.- 2.6.1 Definitions and Fundamental Results.- 2.6.2 Maximal Monotone Graphs in ?2.- 2. Inequality Problems.- 3. Variational Inequalities and Superpotentials.- 3.1 Mechanical Laws and Constraints.- 3.1.1 Generalized Forces and the Principle of Virtual Power.- 3.1.2 Multivalued Laws and Constraints in Mechanics.- 3.1.3 Minimization Problems and Variational Inequalities Characterizing the Equilibrium Configurations.- 3.1.4 Dissipative Laws. A Note on the Eigenvalue Problem for Superpotential Laws.- 3.2 Superpotentials and Duality.- 3.2.1 The Hypothesis of Normal Dissipation.- 3.2.2 Duality of Variational Principles.- 3.3 Subdifferential Boundary Conditions and Constitutive Laws.- 3.3.1 Subdifferential Boundary Conditions.- 3.3.2 Subdifferential Constitutive Laws I.- 3.3.3 Subdifferential Constitutive Laws II.- 3.3.4 Extension of Subdifferential Relations to Function Spaces.- 4. Variational Inequalities and Multivalued Convex and Nonconvex Problems in Mechanics.- 4.1 Two General Variational Inequalities and the Derivation of Variational Inequality "Principles" in Mechanics.- 4.1.1 Variational Inequalities of the Fichera Type.- 4.1.2 Variational Inequalities of Other Types.- 4.1.3 The Derivation of Variational Inequality "Principles" in Mechanics.- 4.2 Coexistent Phases. The Morphology of Material Phases.- 4.2.1 Neoclassical Processes and Gibbsian States. Rules for Coexistent Phases.- 4.2.2 Minimum Problems for Gibbsian States.- 4.2.3 Comparison of Gibbsian States. Some Results of the Dynamic Problem.- 4.3 Nonconvex Superpotentials.- 4.3.1 Introduction and Brief Survey of the Basic Mathematical Properties.- 4.3.2 Nonconvex Superpotentials. Hemivariational Inequalities and Substationarity Principles.- 4.3.3 Generalizations of the Hypothesis of Normal Dissipation.- 4.4 The Integral Inclusion Approach to Inequality Problems.- 5. Friction Problems in the Theory of Elasticity.- 5.1 The Static B.V.P.- 5.1.1 The Classical Formulation.- 5.1.2 The Variational Formulation.- 5.2 Existence and Uniqueness Propositions.- 5.2.1 Equivalent Minimum Problem. The case mes ?U>0.- 5.2.2 Study of the Case ?U=O.- 5.2.3 Further Properties of the Solution.- 5.3 Dual Formulation. Complementary Energy.- 5.3.1 Minimization of the Complementary Energy.- 5.3.2 Duality.- 5.4 The Dynamic B.V.P.- 5.4.1 Classical and Variational Formulations of the Problem.- 5.4.2 Existence of Solution.- 5.4.3 The Regularized Problem.- 5.4.4 The Uniqueness of the Solution.- 5.5 A Note on Other Types of Friction Problems.- 6. Subdifferential Constitutive Laws and Boundary Conditions.- 6.1 Subdifferential Material Laws and Classical Boundary Conditions.- 6.1.1 Formulation of the Problem.- 6.1.2 The Existence and Uniqueness of the Solution.- 6.1.3 Duality.- 6.2 Linear Elastic Material Law and Subdifferential Boundary Conditions.- 6.2.1 Formulation of the Boundary Conditions.- 6.2.2 Existence and Uniqueness Propositions.- 6.2.3 Duality.- 6.3 Subdifferential Material Laws and Subdifferential Boundary Conditions. Minimum Propositions for Nonmonotone Laws.- 6.3.1 Formulation and Study of the Problem.- 6.3.2 Nonmonotone Laws.- 6.4 The Corresponding Dynamic and Incremental Problems.- 7. Inequality Problems in the Theory of Thin Elastic Plates.- 7.1 Static Unilateral Problems of von Karman Plates.- 7.1.1 Generalities.- 7.1.2 Boundary Conditions and Corresponding Variational Formulations.- 7.1.3 The Existence of the Solution.- 7.1.4 In-Plane Unilateral Boundary Conditions.- 7.2 The Unilateral Buckling Problem. Eigenvalue Problems for Variational Inequalities.- 7.2.1 Formulation of the Problem.- 7.2.2 A General Proposition on the Existence of the Solution.- 7.2.3 Application to the Buckling Problem.- 7.2.4 Extension of the Rayleigh-Quotient Rule to Unilateral Problems.- 7.3 Dynamic Unilateral Problems of von Karman Plates.- 7.3.1 Boundary Conditions and Variational Inequalities.- 7.3.2 Existence Proposition.- 7.3.3 Uniqueness Proposition.- 8. Variational and Hemivariational Inequalities in Linear Thermoelasticity.- 8.1 B.V.P.s and their Variational Formulations.- 8.1.1 Classical Formulations.- 8.1.2 Variational Formulations.- 8.2 Existence and Uniqueness Propositions.- 8.2.1 Study of Problem 1.- 8.2.2 Study of Problem 2. Some Remarks on Related Problems.- 8.3 Generalizations and Related Variational Inequalities.- 8.4 Hemivariational Inequalities in Linear Thermoelasticity.- 8.4.1 Formulation of Certain General Problems.- 8.4.2 An Existence Result for a Hemivariational Inequality.-A Model Problem.- 9. Variational Inequalities in the Theory of Plasticity and Viscoplasticity.- 9.1 Elastic Viscoplastic Materials.- 9.1.1 Formulation of the Dynamic Problem, Existence and Uniqueness of the Solution.- 9.1.2 The Quasi-static Problem.- 9.2 Elastic Perfectly Plastic Materials.- 9.2.1 Formulation of the Quasi-static Problem.- 9.2.2 Existence and Uniqueness Propositions.- 9.3 Rigid Viscoplastic Flow Problems.- 9.3.1 Classical Formulation of the General Dynamic Problem.- 9.3.2 The Functional Framework and Existence Propositions.- 9.3.3 The Relation Between Velocity and Stress Fields.- 9.4 Other Problems on Bingham Fluids.- 9.4.1 Laminar Flow in a Cylindrical Pipe.- 9.4.2 Heat Transfer in Rigid Viscoplastic Flows.- 9.4.3 Variational Inequalities in the Case of the General Law ? ? ?w(D).- 3. Numerical Applications.- 10. The Numerical Treatment of Static Inequality Problems.- 10.1 Unilateral Contact and Friction Problems.- 10.1.1 Discrete Forms of the Problems of Minimum Potential and Complementary Energy.- 10.1.2 Applications.- 10.2 Torsion of Cylindrical or Prismatic Bars With Convex Strain-Energy Density.- 10.2.1 Formulation of the Problem.- 10.2.2 Discretization and Numerical Application.- 10.3 A Linear Analysis Approach to Certain Classes of Inequality Problems.- 10.3.1 Description of the Method.- 10.3.2 Applications.- 11. Incremental and Dynamic Inequality Problems.- 11.1 The Elastoplastic Calculation of Cable Structures.- 11.1.1 Formulation of the Problem as a Linear Complementarity Problem and Related Expressions.- 11.1.2 Multilevel Decomposition Techniques.- 11.1.3 Application.- 11.2 Incremental Elastoplastic Analysis.L.C.P.s, Variational Inequalities and Minimum Propositions.- 11.3 Dynamic Unilateral Contact Problems.- Epilogue.- Appendices.- Appendix I. Some Basic Notions [20] [112] [321] [322].- Appendix II. Rigidifying Velocity Fields. Objectivity [112] [197] [322].- Appendix III. Dissipation [112]..- Appendix IV. Plasticity and Thermodynamics [75] [196].- List of Notations.- References.

Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions

The aim of this text is the formulation, mathematical study and numerical treatment of static and dynamic problems in mechanical and engineering sciences involving nonconvex and nonsmooth energy functions, or nonmonotone and multivalued stress-strain laws. Such problems lead to a new type of variational forms, the hemivariational inequalities, which also lead to multivalued differential or integral equations. Innovative numerical methods are presented for the treatment of realistic engineering problems. The text deals with the variational theory of engineering problems involving nonmonotone multivalue relations, their mechanical foundation, their mathematical study (existence and certain approximation results) and the corresponding eigenvalue and optimal control problems. All the numerical applications give innovative answers to as yet unsolved or partially solved engineering problems, such as the adhesive contact in cracks, the delamination problem, the sawtooth stress-strain laws in composites, the shear connectors in composite beams, the semirigid connections in steel structures, the adhesive grasping in robotics, etc. The book closes with a consideration of hemivariational inequalities for fractal type geometries and with the neural network approach to the numerical treatment of hemivariational inequalities.

Hemivariational Inequalities: Applications in Mechanics and Engineering

Preface. 1. Elements of Nonsmooth Analysis. Hemivariational Inequalities. 2. Nonsmooth Critical Point Theory. 3. Minimax Methods for Variational-Hemivariational Inequalities. 4. Eigenvalue Problems for Hemivariational Inequalities. 5. Multiple Solutions of Eigenvalue Problems for Hemivariational Inequalities. 6. Eigenvalue Problems for Hemivariational Inequalities on the Sphere. 7. Resonant Eigenvalue Problems for Hemivariational Inequalities. 8. Double Eigenvalue Problems for Hemivariational Inequalities. 9. Periodic and Dynamic Problems.

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Panagiotis D. Panagiotopoulos

Papers

Inequality problems in mechanics and applications

Mathematical Theory of Hemivariational Inequalities and Applications

Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions

Hemivariational Inequalities: Applications in Mechanics and Engineering

Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities