Extended smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries

By Hui-Chia Yu, Hsun-Yi Chen, K. Thornton. 2009

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In this paper, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A continuous function, the domain parameter, is used to modify the original differential equations such that the equations are solved in the region where a domain parameter takes a specified value while boundary conditions are imposed on the region where the value of the domain parameter varies smoothly across a short distance. The mathematical derivations are straightforward and applicable to a wide variety of partial differential equations. To demonstrate the general applicability of the approach, we provide four examples herein: (1) the diffusion equation with both Neumann and Dirichlet boundary conditions; (2) the diffusion equation with both surface diffusion and reaction; (3) the mechanical equilibrium equation; and (4) the equation for phase transformation with the presence of additional boundaries. The solutions for several of these cases are validated against numerical solutions of the corresponding sharp-interface equations. The potential of the approach is demonstrated with five applications: surface-reaction–diffusion kinetics with a complex geometry, Kirkendall-effect-induced deformation, thermal stress in a complex geometry, phase transformations affected by substrate surfaces and relaxation of a droplet on irregular surfaces.

Subfields: Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/03c112d46af5c018e9c16e07cbcdd90b4cc6f193