A Line-free Method of Monopoles for 3D Dislocation Dynamics

TL;DR

This paper introduces a novel line-free approximation method for 3D dislocation dynamics using monopoles, simplifying computations by avoiding explicit line tracking while maintaining accuracy through a weak divergence constraint.

Contribution

The method replaces traditional line segment models with monopoles that move unstructured, improving computational robustness and efficiency in simulating dislocation dynamics.

Findings

Demonstrates computational advantages over traditional methods

Maintains accuracy without explicit line connectivity

Offers a simpler, more robust simulation framework

Abstract

We develop an approximation scheme for three-dimensional dislocation dynamics in which the dislocation line density is concentrated at points, or monopoles. Every monopole carries a Burgers vector and an element of line. The monopoles move according to mobility kinetics driven by elastic and applied forces. The divergence constraint, expressing the requirement that the monopoles approximate a boundary, is enforced weakly. The fundamental difference with traditional approximation schemes based on segments is that in the present approach an explicit linear connectivity, or 'sequence', between the monopoles need not be defined. Instead, the monopoles move as an unstructured point set subject to the weak divergence constraint. In this sense, the new paradigm is 'line-free', i. e., it sidesteps the need to track dislocation lines. This attribute offers significant computational advantages in terms of simplicity, robustness and efficiency, as demonstrated by means of selected numerical examples.