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

IF 6.9 1区 工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY Computer Methods in Applied Mechanics and Engineering Pub Date : 2024-10-15 DOI:10.1016/j.cma.2024.117444
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引用次数: 0

Abstract

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.
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基于有限元畸变势的新型网格正则化方法:应用于具有极端体积变化的材料膨胀过程
有限元求解的精度与网格质量密切相关。特别是涉及大变形和强局部变形的几何非线性问题,往往会导致令人望而却步的大面积元素失真。在这项工作中,我们提出了一种新颖的网格正则化方法,它能以自适应的方式恢复无扭曲的高质量网格,而无需昂贵的重新网格化程序。这种方法的核心思想在于定义有限元畸变势能,同时考虑不同畸变模式的贡献,如元素的倾斜度和长宽比。正则化网格是通过最小化该变形势来实现的。此外,基于空间局部化函数的概念,该方法可针对具有强烈局部机械变形和网格畸变的区域,对网格分辨率和质量提出量身定制的要求。此外,现有的网格正则化方案通常会固定离散化的边界节点,而我们提出的网格滑动算法基于变化一致的迫击炮方法,允许节点沿问题边界无限制地切向运动。特别是对于涉及重大表面变形(如摩擦接触)的问题,与采用固定边界节点的方案相比,这种方法可以改善网格松弛。为了将材料模型的张量值历史变量等数据从旧的(扭曲的)网格转移到新的(正则化的)网格,我们采用了二阶张量的结构保持不变插值方案,该方案已在我们之前的工作中提出,旨在保持张量值数据的重要特性,如客观性和正确定性。作为一种实际应用场景,我们考虑了泡沫等材料的热机械膨胀,其中涉及高达两个数量级的极端体积变化、大而强的局部应变以及热机械接触相互作用。在这种情况下,所提出的正则化方法以较小的计算成本保持了较高的网格质量。相比之下,不进行网格自适应的模拟则会导致严重的网格畸变、结果质量下降,最终导致数值求解方案不收敛。
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来源期刊
CiteScore
12.70
自引率
15.30%
发文量
719
审稿时长
44 days
期刊介绍: Computer Methods in Applied Mechanics and Engineering stands as a cornerstone in the realm of computational science and engineering. With a history spanning over five decades, the journal has been a key platform for disseminating papers on advanced mathematical modeling and numerical solutions. Interdisciplinary in nature, these contributions encompass mechanics, mathematics, computer science, and various scientific disciplines. The journal welcomes a broad range of computational methods addressing the simulation, analysis, and design of complex physical problems, making it a vital resource for researchers in the field.
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