On the numerical implementation of variational arbitrary Lagrangian-Eulerian (VALE) formulations

This paper is concerned with the implementation of variational arbitrary Lagrangian–Eulerian formulations, also known as variational r-adaption methods. These methods seek to minimize the energy function with respect to the finite-element mesh over the reference configuration of the body. We propose a solution strategy based on a viscous regularization of the configurational forces. This procedure eliminates the ill-posedness of the problem without changing its solutions, i.e. the minimizers of the regularized problems are also minimizers of the original functional. We also develop strategies for optimizing the triangulation, or mesh connectivity, and for allowing nodes to migrate in and out of the boundary of the domain. Selected numerical examples demonstrate the robustness of the solution procedures and their ability to produce highly anisotropic mesh refinement in regions of high energy density.