A novel mesh regularization approach based on finite element distortion potentials: Application to material expansion processes with extreme volume change

The accuracy of finite element solutions is closely tied to the mesh quality. In particular, geometrically nonlinear problems involving large and strongly localized deformations often result in prohibitively large element distortions. In this work, we propose a novel mesh regularization approach allowing to restore a non-distorted high-quality mesh in an adaptive manner without the need for expensive re-meshing procedures. The core idea of this approach lies in the definition of a finite element distortion potential considering contributions from different distortion modes such as skewness and aspect ratio of the elements. The regularized mesh is found by minimization of this potential. Moreover, based on the concept of spatial localization functions, the method allows to specify tailored requirements on mesh resolution and quality for regions with strongly localized mechanical deformation and mesh distortion. In addition, while existing mesh regularization schemes often keep the boundary nodes of the discretization fixed, we propose a mesh-sliding algorithm based on variationally consistent mortar methods allowing for an unrestricted tangential motion of nodes along the problem boundary. Especially for problems involving significant surface deformation (e.g., frictional contact), this approach allows for an improved mesh relaxation as compared to schemes with fixed boundary nodes. To transfer data such as tensor-valued history variables of the material model from the old (distorted) to the new (regularized) mesh, a structure-preserving invariant interpolation scheme for second-order tensors is employed, which has been proposed in our previous work and is designed to preserve important properties of tensor-valued data such as objectivity and positive definiteness. As a practically relevant application scenario, we consider the thermo-mechanical expansion of materials such as foams involving extreme volume changes by up to two orders of magnitude along with large and strongly localized strains as well as thermo-mechanical contact interaction. For this scenario, it is demonstrated that the proposed regularization approach preserves a high mesh quality at small computational costs. In contrast, simulations without mesh adaption are shown to lead to significant mesh distortion, deteriorating result quality, and, eventually, to non-convergence of the numerical solution scheme.