PDEs on deformable domains: Boundary Arbitrary Lagrangian-Eulerian (BALE) and Deformable Boundary Perturbation (DBP) methods

TL;DR

This paper introduces two novel methods, BALE and DBP, for solving PDEs on deformable domains, enabling boundary tracking and perturbation analysis, with applications demonstrated through finite element implementation.

Contribution

The paper presents the Boundary Arbitrary Lagrangian-Eulerian (BALE) and Deformable Boundary Perturbation (DBP) methods, extending PDE solutions to deformable domains with boundary tracking and perturbation techniques.

Findings

BALE effectively tracks boundary deformations in PDE problems.

DBP simplifies boundary perturbation analysis for small deformations.

Finite element method is suitable for implementing these boundary-focused approaches.

Abstract

Many physical problems can be modelled by partial differential equations on unknown domains. Several examples can easily be found in the dynamics of free interfaces in fluid dynamics, solid mechanics or in fluid-solid interactions. To solve these equations in an arbitrary domain with nonlinear deformations, we propose a mathematical approach allowing to track the boundary of the domain, analogue of, and complementary to, the Arbitrary Lagrangian-Eulerian (ALE) method for the interior of the domain. We name this method as the Boundary Arbitrary Lagrangian-Eulerian (BALE) method. Additionally, in many situations nonlinear deformations can be avoided with the help of some analyses which rely on small deformations of the boundary, such as stability analysis, asymptotic expansion and gradient-based shape optimisation. For these cases, we propose an approach to perturb the domain and its boundaries and write the partial differential equations at the unperturbed domain together with the boundary conditions at the unperturbed boundary, instead of at the perturbed ones, which are a priori unknown. We name this method as the Deformable Boundary Perturbation (DBP) method. These two proposed methods rely on the boundary exterior differential operator, whose relevant properties for the present work are evidenced. We show an example for which the BALE and DBP methods are applied, and for which we include the weak formulation revealing the appropriateness of the finite element method in this context.