The aim of this chapter is to provide the necessary numerical concepts for the practical implementation of the finite-element method. First, we review the notion of isoparametric elements; such elements are important when the elements are distorted and the shape functions cannot be easily expressed in term of the physical coordinates. This is always the case in the numerical simulation of metal-forming processes, where significant deformation of the initial mesh cannot be avoided. Second, we consider in some detail the procedure for numerical integration, which is essential for many applications, particularly when the constitutive equation is nonlinear. We resolve the resulting finite-element equation into two types: linear and nonlinear. Finally, we introduce a simple, one-dimensional mechanical example with linear material behavior. This example is treated completely and a nonlinear material is also used to illustrate the differences in the methods.

Isoparametric Elements

The physical problem under consideration is posed in a domain and is discretized into small subdomains, called elements, as introduced in Chapter 2. Elements are physical in the sense that they are defined themselves in real space. So far we have used the physical domains of those elements directly, by defining shape functions inside an element in terms of nodal values and positions. It is often more convenient to map the physical element domains to even simpler shapes and thus to formulate the discretized problem on these shapes. The expression of the shape functions is much simpler (for nonlinear ones).