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.

No abstract

The plasticity of crystalline materials can be described at the meso-scale by dislocations transport models, typically formulated in terms of dislocation densities. This leads to sets of coupled non-linear partial differential equations involving diffusive and convective transport mechanisms. Since exact solutions for these systems are not available, numerical approximations are needed to efficiently solve them. The properties of these systems of equations cause most traditional numerical methods to fail, even for the case of a single equation. For systems of equations the problem is even more challenging due to the lack of fundamental principles guiding numerical discretization strategies. Special strategies must be developed and carefully applied to obtain physically meaningful and numerically stable approximations. The objective of this paper is to construct a coefficient perturbation-based stabilization technique for general systems of equations and to apply it to the modelling of one-dimensional dislocation transport. A detailed numerical study is carried out in order to demonstrate its ability to render well-behaved and physically admissible numerical approximations.