Extended Smoothed Boundary Method for Solving Partial Differential Equations with General Boundary Conditions on Complex Boundaries

Abstract

In this article, 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 generically 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 corresponding analytical and semi-analytical solutions. 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 a self-propelled droplet.

This document is the revised version of arXiv:0912.1288v1

References in corpus (3)

• A variational approach to the moving contact line hydrodynamics
• Phase-Field Model of Dendritic Sidebranching with Thermal Noise
• Phase Field Theory of Heterogeneous Crystal Nucleation

Cited by in corpus (9)

• Paradigm shift in diffusion-mediated surface phenomena
• Efficient solutions for nonlocal diffusion problems via boundary-adapted spectral methods
• Multiphase mean curvature flows with high mobility contrasts: a phase-field approach, with applications to nanowires
• A Hybrid Semi-Lagrangian Cut Cell Method for Advection-Diffusion Problems with Robin Boundary Conditions in Moving Domains
• Robust, strong form mechanics on an adaptive structured grid: efficiently solving variable-geometry near-singular problems with diffuse interfaces
• Self-similar diffuse boundary method for phase boundary driven flow
• Surface-directed and bulk spinodal decomposition compete to decide the morphology of bimetallic nanoparticles
• Phase-field Modelling of Anisotropic Solid-State Dewetting on Patterned Substrates
• evoxels: A differentiable physics framework for voxel-based microstructure simulations