Overview

In the previous two chapters we have formulated and analyzed the primal and dual variational formulations of the elastoplasticity problem. Later on, we will study various numerical methods to solve the variational problems. In all the numerical methods to be considered, we will use finite differences to approximate the time derivative and use the finite element method to discretize the spatial variables. The finite element method is widely used for solving boundary value problems of partial differential equations arising in physics and engineering, especially solid mechanics. The method is derived from discretizing the weak formulation of a boundary value problem. The analysis of the finite element method is closely related to that of the boundary value problem.

The development of a finite element algorithm for solving a boundary value problem includes four main steps. First, the boundary value problem is reformulated into an equivalent variational problem. Second, the domain of the independent variables (or usually the domain of the spatial variables, for a time-dependent problem) is partitioned into subdomains called finite elements, and then a finite-dimensional space, called the finite element space, is constructed as a collection of piecewise smooth functions with a certain degree of global smoothness. Third, the variational problem is projected to the finite element space, and in this way, a finite element system is formed. Finally, the finite element system is solved, say by some iterative method, and various conclusions are drawn from the solution of the finite element system. The mathematical theory of the...