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

TL;DR

This paper introduces a versatile smoothed boundary method for solving various partial differential equations with complex boundary conditions, demonstrated through multiple examples and applications involving irregular geometries.

Contribution

The paper presents a novel smoothed boundary approach that simplifies applying boundary conditions on complex geometries for a wide range of PDEs.

Findings

Method accurately solves PDEs with complex boundaries

Validated against analytical solutions in several cases

Demonstrated applicability in diverse physical problems

Abstract

In this article, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A function that has a prescribed value on the domain in which a differential equation is valid and smoothly but rapidly varying values on the boundary where boundary conditions are imposed is used to modify the original differential equations. The mathematical derivations are straight forward, and generically applicable to a wide variety of partial differential equations. To demonstrate the general applicability of the approach, we provide four examples: (1) the diffusion equation with both Neumann and Dirichlet boundary conditions, (2) the diffusion equation with surface diffusion, (3) the mechanical equilibrium equation, and (4) the equation for phase transformation with additional boundaries. The solutions for a few 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-propelling droplet.