Staggering error control of a class of inelastic processes in random microheterogeneous solids

In this work a staggering solution strategy for the simulation of the time-dependent inelastic mechanical deformation of a class of solids, possessing irregular heterogeneous microstructure, is developed. The system of coupled equations involved consists of (1) a dynamic equation of momentum balance, where the primary field variable is the displacement, (2) an evolution equation for material degradation, where the primary field variable is a state damage function, and (3) an evolution equation for the inelastic strains in the solid where the primary field variable is a plastic strain field. Clearly, the damage and plasticity variables are implicit functions of the displacement, however, for the staggering scheme strategy, it is convenient to formulate them as individual fields during the solution process. The key concept for the strategy to operate efficiently is to estimate and control the so-called staggering error, i.e. the error due to incompletely resolving the coupling between the field equations in a staggering process. This error is a function of the time step size. However, because the coupling is temporally variable, possibly becoming stronger, weaker, or oscillatory, it is extremely difficult to ascertain a priori the time step size needed for prespecified error control. In the present work, to induce desired staggering rates of convergence within each time step, thus controlling the staggering error, an adaptive strategy is developed whereby the time step size is manipulated, enlarged or reduced, to control the intrinsic contraction mapping constant of the staggering system operator. The overall goal is to deliver accurate solutions where temporal discretization error control dictates the upper limits on the time step size, while the iterative staggering strategy refines the step size further to control the staggering error. Three-dimensional numerical experiments are performed to illustrate the solution strategy.