基于有限元算法的三角形和四面体网格变形中单元反转的分析与解决方法
Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes
AI总结：
本文分析FEMWARP算法在三角形和四面体网格变形中产生单元反转的原因，并提出结合优化修复器等方法增强稳健性。
摘要：
我们考虑一种名为FEMWARP的算法，用于对三角形和四面体有限元网格进行变形，该算法利用有限元方法本身来计算变形。该算法输入一个由边界网格（一维中的线段或二维中的三角形）定义的二维或三维域，其内部具有体积网格（二维中的三角形或三维中的四面体）。它还输入边界网格的规定运动。它输出体积网格顶点更新后的位置。算法的第一步是根据初始网格为每个内部顶点确定一组局部权重，这些权重根据其邻居的位置来描述每个内部顶点。这些权重使用有限元刚度矩阵计算。在应用边界变换后，求解基于权重的线性方程组以确定内部顶点的最终位置。FEMWARP算法已在先前的文献中考虑过（例如，Baker在2001年的一篇论文中）。FEMWARP在特定应用中成功计算了变形网格。然而，有时FEMWARP会反转单元；这是本文的主要关注点。我们分析了这种不良行为的原因，并提出了几种技术使该方法对反转更加稳健。所提出方法中最成功的一种包括将FEMWARP与基于优化的网格修复器相结合。
英文摘要：
We consider an algorithm called FEMWARP for warping triangular and tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a two- or three-dimensional domain defined by a boundary mesh (segments in one dimension or triangles in two dimensions) that has a volume mesh (triangles in two dimensions or tetrahedra in three dimensions) in its interior. It also takes as input a prescribed movement of the boundary mesh. It computes as output updated positions of the vertices of the volume mesh. The first step of the algorithm is to determine from the initial mesh a set of local weights for each interior vertex that describes each interior vertex in terms of the positions of its neighbors. These weights are computed using a finite element stiffness matrix. After a boundary transformation is applied, a linear system of equations based upon the weights is solved to determine the final positions of the interior vertices. The FEMWARP algorithm has been considered in the previous literature (e.g., in a 2001 paper by Baker). FEMWARP has been succesful in computing deformed meshes for certain applications. However, sometimes FEMWARP reverses elements; this is our main concern in this paper. We analyze the causes for this undesirable behavior and propose several techniques to make the method more robust against reversals. The most successful of the proposed methods includes combining FEMWARP with an optimization-based untangler.
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