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 mechanical properties of tensor-valued data such as objectivity and positive definiteness... {continued see pdf}

翻译：有限元解的精度与网格质量密切相关。特别是在涉及大变形和强局部化变形的几何非线性问题中，通常会导致元素畸变过大而无法计算。本文提出了一种新型网格正则化方法，能够以自适应方式恢复无畸变的高质量网格，而无需昂贵的重新网格划分过程。该方法的核心思想在于定义一个有限元畸变势，该势能考虑了元素偏斜度和纵横比等不同畸变模式的贡献。正则化网格通过最小化该势能获得。此外，基于空间局部化函数的概念，该方法允许为具有强局部化机械变形和网格畸变的区域指定关于网格分辨率和质量的定制化要求。另外，虽然现有的网格正则化方案通常固定离散化边界节点，但我们提出了一种基于变分一致 mortar 方法的网格滑动算法，允许节点沿问题边界进行无约束的切向运动。特别是对于涉及显著表面变形（例如摩擦接触）的问题，与固定边界节点的方案相比，该方法可以实现更好的网格松弛。为了将材料模型的张量值历史变量等数据从旧（畸变）网格转移到新（正则化）网格，我们采用了一种针对二阶张量的结构保持不变插值方案。该方案在我们先前的工作中提出，旨在保持张量值数据的重要力学性质，如客观性和正定性... {详见pdf}