This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

A Posteriori Error Estimation in Finite Element Analysis

In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuscka, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

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A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data

This paper discusses the homogenization method to determine the effective average elastic constants of linear elasticity of general composite materials by considering their microstructure. After giving a brief theory of the homogenization method, a finite element approximation is introduced with convergence study and corresponding error estimate. Applying these, computer programs PREMAT and POSTMAT are developed for preprocessing and postprocessing of material characterization of composite materials. Using these programs, the homogenized elastic constants for macroscopic stress analysis are obtained for typical composite materials to show their capability. Finally, the adaptive finite element method is introduced to improve the accuracy of the finite element approximation.

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Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods

Basic error estimates for elliptic problems

Linear elasticity is one of the more successful theories of mathematical physics Its pragmatic success in describing the small deformations of many materials is uncontested The origins of the three-dimensional theory go back to the beginning of the 19th century and the derivation of the basic equations by Cauchy, Navier, and Poisson The theoretical development of the subject continued at a brisk pace until the early 20th century with the work of Beltrami, Betti, Boussinesq, Kelvin, Kirchhoff, Lame, Saint-Venant, Somigliana, Stokes, and others These authors established the basic theorems of the theory, namely compatibility, reciprocity, and uniqueness, and deduced important general solutions of the underlying field equations In the 20th century the emphasis shifted to the solution of boundary-value problems, and the theory itself remained relatively dormant until the middle of the century when new results appeared concerning, among other things, Saint-Venant’s principle, stress functions, variational principles, and uniqueness

The Linear Theory of Elasticity

Mathematical Theory of Elastic and Elasto Plastic Bodies: An Introduction

Etude des problemes unilateraux pour les fonctions scalaires. Contact unilateral de corps elastiques. Problemes de la theorie de la plasticite

Ivan Hlaváček

Papers

Mathematical Theory of Elastic and Elasto Plastic Bodies: An Introduction

Solution of Variational Inequalities in Mechanics

Numerical methods for unilateral problems in solid mechanics

On inequalities of Korn's type

On inequalities of Korn's type: I. Boundary-value problems for elliptic systems of partial differential equations