Weimin Han

Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity

Nonlinear variational problems and numerical approximation: Preliminaries of functional analysis Function spaces and their properties Introduction to finite difference and finite element approximations Variational inequalities Mathematical modelling in contact mechanics: Preliminaries of contact mechanics of continua Constitutive relations in solid mechanics Background on variational and numerical analysis in contact mechanics Contact problems in elasticity Contact problems in viscoelasticity: A frictionless contact problem Bilateral contact with slip dependent friction Frictional contact with normal compliance Frictional contact with normal damped response Other viscoelastic contact problems Contact problems in visocplasticity: A Signorini contact problem Frictionless contact with dissipative potential Frictionless contact between two viscoplastic bodies Bilateral contact with Trescas friction law Other viscoelastic contact problems Bibliography Index.

Spherical harmonics and approximations on the unit sphere : an introduction

1 Preliminaries.- 2 Spherical Harmonics.- 3 Differentiation and Integration over the Sphere.- 4 Approximation Theory.- 5 Numerical Quadrature.- 6 Applications: Spectral Methods.

Error analysis of the reproducing kernel particle method

Interest in meshfree (or meshless) methods has grown rapidly in recent years in solving boundary value problems (BVPs) arising in mechanics, especially in dealing with difficult problems involving large deformation, moving discontinuities, etc. In this paper, we provide a theoretical analysis of the reproducing kernel particle method (RKPM), which belongs to the family of meshfree methods. One goal of the paper is to set up a framework for error estimates of RKPM. We introduce the concept of a regular family of particle distributions and derive optimal order error estimates for RKP interpolants on a regular family of particle distributions. The interpolation error estimates can be used to yield error estimates for RKP solutions of BVPs.

Analysis and approximation of contact problems with adhesion or damage

Preface List of Symbols Modeling and Mathematical Background Basic Equations and Boundary Conditions Physical Setting and Evolution Equations Boundary Conditions Contact Processes with Adhesion Constitutive Equations with Damage Preliminaries on Functional Analysis Function Spaces and Their Properties Elements of Nonlinear Analysis Standard Results on Variational Inequalities and Evolution Equations Elementary Inequalities Preliminaries on Numerical Analysis Finite Difference and Finite Element Discretizations Approximation of Displacements and Velocities Estimates on the Discretization of Adhesion Evolution Estimates on the Discretization of Damage Evolution Estimates on the Discretization of Viscoelastic Constitutive Law Estimates on the Discretization of Viscoplastic Constitutive Law Frictionless Contact Problems with Adhesion Quasistatic Viscoelastic Contact with Adhesion Problem Statement Existence and uniqueness Continuous Dependence on the Data Spatially Semidiscrete Numerical Approximation Fully Discrete Numerical Approximation Dynamic Viscoelastic Contact with Adhesion Problem Statement Existence and Uniqueness Fully Discrete Numerical Approximation Quasistatic Viscoplastic Contact with Adhesion Problem Statement Existence and Uniqueness for the Signorini Problem Numerical Approximation for the Signorini Problem Existence and Uniqueness for the Problem with Normal Compliance Numerical Approximation of the Problem with Normal Compliance Relation between the Signorini and Normal Compliance Problems Contact Problems with Damage Quasistatic Viscoelastic Contact with Damage Problem Statement Existence and Uniqueness Fully Discrete Numerical Approximation Dynamic Viscoelastic Contact with Damage Problem Statement Existence and Uniqueness Fully Discrete Numerical Approximation Quasistatic Viscoplastic Contact with Damage Problem Statement Existence and Uniqueness for the Signorini Problem Numerical Approximation for the Signorini Problem Existence and Uniqueness for the Problem with Normal Compliance Numerical Approximation of the Problem with Normal Compliance Relation between the Signorini and Normal Compliance Problems Notes, Comments, and Conclusions Bibliographical Notes, Problems for Future Research, and Conclusions Bibliographical Notes Problems for Future Research Conclusions References Index

Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage

We consider a model for quasistatic frictional contact between a viscoelastic body and a foundation. The material constitutive relation is assumed to be nonlinear. The mechanical damage of the material, caused by excessive stress or strain, is described by the damage function, the evolution of which is determined by a parabolic inclusion. The contact is modeled with the normal compliance condition and the associated version of Coulombs law of dry friction. We derive a variational formulation for the problem and prove the existence of its unique weak solution. We then study a fully discrete scheme for the numerical solutions of the problem and obtain error estimates on the approximate solutions.

Mathematical theory and numerical analysis of bioluminescence tomography

Molecular imaging is widely recognized as the main stream in the next generation of biomedical imaging. Bioluminescence tomography (BLT) is a rapidly developing new area of molecular imaging. The goal of BLT is to provide quantitative three-dimensional reconstruction of a bioluminescent source distribution within a small animal from optical signals on the surface of the animal body. In this paper, a mathematical framework is established for BLT. Solution existence and uniqueness are established. Continuous dependence of the solution is demonstrated with respect to data. Stable BLT schemes are studied, leading to error estimates and convergence of the methods. A numerical example is presented to illustrate the algorithmic performance.

Evolutionary Variational Inequalities Arising in Viscoelastic Contact Problems

We consider a class of evolutionary variational inequalities arising in various frictional contact problems for viscoelastic materials. Under the smallness assumption of a certain coefficient, we prove an existence and uniqueness result using Banachs fixed point theorem. We then study two numerical approximation schemes of the problem: a semidiscrete scheme and a fully discrete scheme. For both schemes, we show the existence of a unique solution and derive error estimates. Finally, all these results are applied to the analysis and numerical approximations of a viscoelastic frictional contact problem, with the finite element method used to discretize the spatial domain.

On the finite element method for mixed variational inequalities arising in elastoplasticity

We analyze the finite-element method for a class of mixed variational inequalities of the second kind, which arises in elastoplastic problems. An abstract variational inequality, of which the elast...

A frictionless contact problem for elastic-viscoplastic materials with normal compliance and damage

Abstract We study a quasistatic frictionless contact problem with normal compliance and damage for elastic–viscoplastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modelled by an inclusion of parabolic type. We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution to the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. An optimal order error estimate is derived for the approximate solutions under suitable solution regularity. Numerical examples are presented to show the performance of the method.

Discontinuous Galerkin Methods for Solving Elliptic Variational Inequalities

We study discontinuous Galerkin methods for solving elliptic variational inequalities of both the first and second kinds. Analysis of numerous discontinuous Galerkin schemes for elliptic boundary value problems is extended to the variational inequalities. We establish a priori error estimates for the discontinuous Galerkin methods, which reach optimal order for linear elements. Results from some numerical examples are reported.