Triangulations are an ubiquitous input for the finite element community. However, most raw triangulations obtained by imaging techniques are unsuitable as-is for finite element analysis. In this paper, we give a robust pipeline for handling those triangulations, based on the computation of a one-to-one parametrization for automatically selected patches of input triangles, which makes each patch amenable to remeshing by standard finite element meshing algorithms. Using only geometrical arguments, we prove that a discrete parametrization of a patch is one-to-one if (and only if) its image in the parameter space is such that all parametric triangles have a positive area. We then derive a non-standard linear discretization scheme based on mean value coordinates to compute such one-to-one parametrizations, and show that the scheme does not discretize a Laplacian on a structured mesh. The proposed pipeline is implemented in the open source mesh generator Gmsh, where the creation of suitable patches is based on triangulation topology and parametrization quality, combined with feature edge detection. Several examples illustrate the robustness of the resulting implementation.

中文翻译：

---

离散模型的自动表面网格生成：基于重新参数化的完整自动管道

三角剖分是有限元社区普遍存在的输入。然而，大多数通过成像技术获得的原始三角剖分不适用于有限元分析。在本文中，我们基于对自动选择的输入三角形补丁进行一对一参数化的计算，给出了处理这些三角剖分的强大管道，这使得每个补丁都可以通过标准有限元网格划分算法重新划分网格。仅使用几何参数，我们证明当（且仅当）其在参数空间中的图像使得所有参数三角形具有正面积时，补丁的离散参数化是一对一的。然后我们基于平均值坐标推导出非标准线性离散化方案来计算这种一对一的参数化，并表明该方案不会在结构化网格上离散拉普拉斯算子。提议的管道在开源网格生成器 Gmsh 中实现，其中合适的补丁的创建基于三角剖分拓扑和参数化质量，并结合特征边缘检测。几个例子说明了最终实现的稳健性。