Numerical Analysis of General Return Mapping Algorithms

Abstract

In this chapter, we present a rigorous, nonlinear, stability analysis of a class of time discretizations of the weak form of the initial boundary-value problem for both rate-independent and rate-dependent infinitesimal elastoplasticity. To motivate the notion of nonlinear stability appropriate for elastoplasticity, first we consider the simpler model problem of nonlinear heat conduction and give a rigorous proof of nonlinear stability for the generalized midpoint rule. The analysis that follows differs in several aspects from previous treatments of algorithmic stability; in particular: i. The stability analysis is performed directly on the system of variational equations discretized in time, not on the algebraic system arising from both temporal and spatial discretizations. The results carry over immediately to the finite-dimensional problem obtained via a Galerkin (spatial) discretization. ii. Previous stability analyses employ either the notion of A-stability, introduced by Dahlquist [1963] in the context of linear, multistep methods for systems of ODEs or the concept of linearized stability; see, e.g., Hughes [1983] and references therein. The results given below prove nonlinear stability in the sense that arbitrary perturbations in the initial data are attenuated by the algorithm relative to a certain algorithmic-independent norm associated with the continuum problem called the natural norm. See also Dahlquist [1975]. For nonlinear systems of ODEs, this notion of nonlinear stability reduces to the concept of A-contractivity or B-stability introduced by Butcher [1975] in the context of implicit Runge-Kutta methods. A-contractivity is widely accepted now as the proper definition of nonlinear stability; see, e.g., Burrage and Butcher [1979,1980]; and Dahlquist and Jeltsch [1979]. For linear semigroups, the definition of stability employed in this chapter coincides with the notion of Lax stability; see Richtmyer and Morton [1967]. A key step in the stability analysis given below is identifying the natural norm for the continuum problem relative to which the crucial contractivity property holds. For nonlinear heat conduction, this norm is a weighted L 2-norm whose weighting factor is the specific heat capacity times the density. For the semidiscrete version of this problem obtained via a Galerkin spatial discretization, the natural norm 219