Arbitrary Lagrangian–Eulerian methods for analysis of regressing solid domains and interface tracking

We present the details of the formulation and implementation of the arbitrary Lagrangian–Eulerian (ALE) finite element method for three-dimensional problems involving regressing solid domains and moving boundaries. An example of such problems is the simulation of solid-propellant rockets in which the evolution of a fluid–solid interface is governed by a combustion law and the transfer of mass and momentum across it. The ALE method, while providing a means to track the location of the interface, allows the adaptation of the finite element mesh to the constantly changing solid domain. In this study, the mesh adaptation is achieved via a novel smoothing technique in which the shape of finite elements with smaller volumes, which are more susceptible to mesh-entanglement, are better preserved compared to those with larger volumes. An analysis of the stability of the ALE computations, under certain simplifying assumptions, is also performed. The stability limits determined from this analysis can be utilized as constraints for adjusting mesh velocities or time increments in the convective mesh-motion phase of the ALE computations. In addition, a method is provided for generating verification problems with moving interfaces from those with known solutions on stationary material domains. A problem in which the prescribed growth of a cavity in an infinite medium under a time-varying pressure loading is used to verify the implementation and to demonstrate the verification technique.