I present a 3D advancing-front mesh refinement algorithm that generates a constrained Delaunay mesh for any piecewise linear complex (PLC) and extend this algorithm to produce truly Delaunay meshes for any PLC. First, as in my recently published 2D algorithm, I split the input line segments such that the length of the subsegments is asymptotically proportional to the local feature size (LFS). For each facet, I refine the mesh such that the edge lengths and the radius of the circumcircle of every triangular element are asymptotically proportional to the LFS. Finally, I refine the volume mesh to produce a constrained Delaunay mesh whose tetrahedral elements are well graded and have a radius-edge ratio less than some $\omega^* > 2/\sqrt{3}$ (except ``near'' small input angles). I extend this algorithm to generate truly Delaunay meshes by ensuring that every triangular element on a facet satisfies Gabriel's condition, i.e., its diametral sphere is empty. On an ``apex'' vertex where multiple facets intersect, Gabriel's condition is satisfied by a modified split-on-a-sphere (SOS) technique. On a line where multiple facets intersect, Gabriel's condition is satisfied by mirroring meshes near the line of intersection. The SOS technique ensures that the triangles on a facet near the apex vertex have angles that are proportional to the angular feature size (AFS), a term I define in the paper. All tetrahedra (except ``near'' small input angles) are well graded and have a radius-edge ratio less than $\omega^* > \sqrt{2}$ for a truly Delaunay mesh. The upper bounds for the radius-edge ratio are an improvement by a factor of $\sqrt{2}$ over current state-of-the-art algorithms.

中文翻译：

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3D前进-前Delaunay网格细化算法

我提出了一种3D超前网格细化算法，该算法可为任何分段线性复合体（PLC）生成约束Delaunay网格，并将此算法扩展为可为任何PLC生成真正的Delaunay网格。首先，就像我最近发布的2D算法一样，我将输入线段分开，以使子段的长度渐近地与局部特征尺寸（LFS）成比例。对于每个小平面，我都会细化网格，以使每个三角形元素的边长和外接圆的半径与LFS渐近成比例。最后，我优化体积网格以生成约束的Delaunay网格，该网格的四面体元素渐变良好，并且半径-边缘比率小于$ \ omega ^ *> 2 / \ sqrt {3} $（``near''除外）小输入角度）。通过确保小平面上的每个三角形元素都满足Gabriel的条件（即，其径向球面是空的），我将此算法扩展为生成真正的Delaunay网格。在多个小面相交的``顶点''顶点上，Gabriel的条件通过修改后的球面分割（SOS）技术得以满足。在多个面相交的直线上，通过在相交线附近镜像网格来满足Gabriel的条件。SOS技术确保顶点顶点附近的小平面上的三角形具有与角度特征尺寸（AFS）成比例的角度，这是我在本文中定义的术语。所有的四面体（``近''小输入角除外）的坡度都很好，并且对于真正的Delaunay网格而言，其半径-边缘比率小于$ \ omega ^ *> \ sqrt {2} $。