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。特别是涉及大变形及强局部化变形的几何非线性问题，常常导致难以接受的单元畸变。本文提出一种新型网格正则化方法，能够在无需昂贵网格重划分流程的情况下，以自适应方式恢复非畸变的高质量网格。该方法的核心思想在于定义一个考虑单元倾斜度和纵横比等不同畸变模式贡献的有限元畸变势，通过最小化该势能获得正则化网格。此外，基于空间定位函数的概念，该方法能够针对机械变形和网格畸变高度局部化的区域，指定网格分辨率与质量的定制化要求。同时，现有网格正则化方案常固定离散化边界节点，而本文基于变分一致性的无覆层（mortar）方法提出了一种网格滑动算法，允许节点沿问题边界进行无约束切向运动。对于涉及显著表面变形（如摩擦接触）的问题，该方法相较于固定边界节点的方案能够实现更优的网格松弛。为将材料模型的张量值历史变量等数据从旧（畸变）网格传输至新（正则化）网格，我们采用了一种先前工作中提出的基于结构保持不变性的二阶张量插值方案，该方案旨在保持张量值数据的重要力学特性（如客观性和正定性）…（详见PDF）