An alternative choice of the boundary condition for the arbitrary Lagrangian-Eulerian method

Many physical problems can be modelled by partial differential equations (PDEs) on unknown domains. Several examples can be easily 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, the Arbitrary Lagrangian-Eulerian (ALE) method is in widespread use. For this method, a change of spatial variables is used, which should fulfil a given partial differential equation in the domain together with boundary conditions at the boundary. In the normal direction, the boundary should follow the material point, thus using a Lagrangian approach, whereas in the tangential direction, there are infinite possibilities. In this direction, whereas in solid mechanics a Lagrangian approach is common, in fluid mechanics an Eulerian approach is usually more convenient. Alternative approaches are also usually taken which are problem (physics and geometry) dependent, not always valid and may eventually lead to large mesh deformations. We propose a method to determine the tangential displacement of the boundary based on the boundary displacement fulfilling a boundary partial differential equation, which does not depend on the problem and leads to an uniform deformation of the mesh. Due to the analogy with the ALE method, we refer to this as Differential Boundary Arbitrary Lagrangian-Eulerian (DBALE) method. We illustrate the advantages of the method concerning problem independency and capabilities to control the mesh distortion by means of reference and real problems. (C) 2021 Elsevier Inc. All rights reserved.