A “Mechanics” Approximation and Numerical Implementation

Abstract

This chapter is concerned, above all, with approximations on Ω × [0,T] having a strong mechanics content that are introduced to solve the global step in the method of large time increments (Principle P₃). The linear global step is, even for sophisticated models of behavior, by far the costliest. For small-disturbance static problems, the approximation used at each iteration is an extension of the “radial loading” approximation, which works remarkably well in a number of plastic and viscoplastic problems. This consists of approximating a function defined on Ω × [0,T] by a sum of products, each product being formed by a scalar function of the time variable multiplied by a function of the space variable. A sum of n products defines an approximation of order n. This nonclassical mode of approximation is studied as such in this chapter, where the best approximation to order n is characterized and some general convergence properties are given. The numerical treatment of different steps is also detailed. The discretizations used here in space as well as in time are classical.