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

TL;DR

This paper introduces an extended smoothed boundary method that effectively solves various partial differential equations with complex boundary conditions on arbitrarily shaped domains, demonstrated through multiple practical examples.

Contribution

The paper presents a novel, generalizable smoothed boundary approach for PDEs with complex boundaries, applicable to diverse equations and boundary conditions.

Findings

Validated solutions against analytical benchmarks.

Successfully applied to complex geometries and physical phenomena.

Demonstrated versatility across multiple PDE types.

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.