STABILITY AND ERROR ANALYSIS FOR TIME INTEGRATORS APPLIED TO STRAIN-SOFTENING MATERIALS

The behavior of explicit time integrators is examined for the field eq uations of dynamic viscoplasticity in one dimension. The analysis proc eeds by linearizing the field equations about the current state and fr eezing coefficients so that the solution to the system of ordinary dif ferential equations can be computed explicitly. Amplification matrices for the central difference method (with rate tangent constitutive upd ate) and fourth-order Runge-Kutta schemes are obtained by applying the se schemes to the resulting linear, constant coefficient equations. Th ese amplification matrices lead to local truncation error estimates fo r the temporal integrators as well as to estimates of the resulting gr owth rate. Analysis of the cental difference method with a forward Eul er constitutive update indicates that the local truncation error invol ves as the product of the third power of the strain-rate and the squar e of the time step. In viscoplastic models, the viscoplastic strain-ra te is an increasing (decreasing) function of the viscoplastic strain i n softening (hardening). Also, the eigenvalues of the rate tangent met hod increase with strain-rate in softening. Thus, it is recommended th at the time step be reduced in localization problems to keep the strai n-increment within a prescribed tolerance. As a byproduct of the analy sis, conditions for numerical stability for the central difference met hod with a rate tangent constitutive update are obtained for strain-ha rdening dynamic viscoplasticity. In the softening regime, the resultin g linearized differential equations admit exponentially growing soluti on. Thus, conventional definitions of stability such as Neumann stabil ity and absolute stability are inappropriate. In order to characterize stability of such systems, we introduce the concept of g-rel stabilit y which combines the concept of relative and absolute stability. We sh ow conditions under which the resulting system is g-rel stable and thu s characterize the nature of the numerical stability that can be expec ted in such problems.