Towards an unconditionally stable numerical scheme for continuum dislocation transport

Abstract

Recent developments in plasticity modeling for crystalline materials are based on dislocations transport models, formulated for computational efficiency in terms of their densities. This leads to sets of coupled partial differential equations in a continuum description involving diffusion and convection-like processes combined with non-linearity. The properties of these equations cause the most traditional numerical methods to fail when applied to solve them. Therefore, dedicated stabilization techniques must be developed in order to obtain physically meaningful and numerically stable approximations. The objective of this paper is to present a dedicated stabilization technique and to apply it to a system of dislocation transport equations in one dimension. This stabilization technique, based on coefficient perturbations, successfully provides unconditional stability with respect to the spatial discretization. Several of its favorable characteristics are discussed, providing evidence of its versatility and effectiveness through a thorough numerical assessment.

A stabilization technique for coupled convection‐diffusion‐reaction equations

“…In this paper, a stabilization technique for general systems of one-dimensional coupled convection-diffusion-reaction equations with constant coefficients was developed. It is a heuristic extension, from the single equation case toward the multiple equations case, of the methodology presented in the work of Hernández et al 50 Thus, for uncoupled systems of equations, the proposed stabilization technique recovers the single equation case approach. Stabilization is achieved by effectively perturbing the transport coefficients of the system of differential equations to be discretized.…”

“…This has the goal of keeping the perturbed, ie, modified, problem as similar as possible to the original physical problem. This is the working mechanism of the stabilization technique based on coefficient perturbations for the single convection-diffusion-reaction equation presented in the work of Hernández et al 50 It is therefore possible to classify the proposed perturbation-based stabilization technique as one using the modified equation approach. 13,16 This technique will be heuristically extended to systems containing several coupled differential equations in Section 3.1.…”

“…In this section, a stabilization technique for a single convection-diffusion-reaction equation, proposed in the work of Hernández et al, 50 is heuristically extended to the case of multiple coupled equations. This extension is built, as for the single equation case, on the injection of the analytical solution of the linearized, homogeneous, and steady-state version of the problem at hand into the numerical stencil obtained by the finite element discretization.…”

Stabilization of coupled convection–diffusion-reaction equations for continuum dislocation transport

“…However, the stabilization technique proposed in [13] cannot be straightforwardly applied to dislocation transport equations involving total and geometrically necessary dislocation densities as the field variables, rather than positive and negative dislocation densities. Such formulations however are more common in the field [2,3,5].…”

“…Section 3 is devoted first to the numerical treatment of a general system of convection-diffusion-reaction equations with constant coefficients in a one-dimensional setting using finite elements. Subsequently, a stabilization technique for a single steady state and linear convection-diffusion-reaction equation, originally proposed in [13], is reviewed. The extension of the stabilization technique extension to systems of coupled convection-diffusion-reaction equations, as proposed in [34], is discussed at the end of this section.…”