A Theoretical and Computational Framework for Modeling Diffusion-Driven Boundary Motion Without Remeshing

Abstract

Diffusion-driven phase boundary motion influences both shape and topologies in many microstructural systems. In this paper, we discuss two aspects. First we present a theoretical framework, in which we permit in the models (i) bulk fields such as displacements and bulk mass density (ii) surface fields such as surface energy and surface mass density. The resulting equation for the free energy rate contains appropriate coupling between physical fields. We postulate a minimization principle and consequently obtain updated boundary locations. Second, we use the hierarchical partition of unity meshless compositional procedure as the computational framework for modeling evolving phases. Non-Uniform Rational B-Splines or NURBS are used for discretizing the geometrical model, distribution of material properties, and the accompanying behavioral field. This is a geometry-based alternative to the finite element method for modeling moving boundary problems. Within this framework, computational geometry inspired ideas such as parametric embedding and boolean compositions enable modeling of moving boundaries and changing topologies without remeshing. We demonstrate evolution towards optimal slit and circular shapes of single void systems. We also demonstrate dissolution, growth, coalescence, break-up and disappearance in multi-void systems. In all cases energetic trade-offs between surface and strain energies and their interactions with surface and bulk diffusions dominate the final shapes.