通过径向划分逼近的非单纯Delaunay网格生成
Non-simplicial Delaunay meshing via approximation by radical partitions
AI总结：
本文提出通过径向划分逼近构造非单纯Delaunay网格，将问题转化为对偶凸多面体序列极限，并引入离散Dirichlet泛函度量偏差，实验验证了方法的可行性。
摘要：
我们考虑将多面体Delaunay剖分构造为幂图（径向划分）序列的极限。对偶Voronoi图作为加权Delaunay剖分序列的极限获得。该问题被归结为构造一对对偶凸多面体，分别内接和外接于一个圆抛物面，作为一般对偶凸多面体对序列的极限。原始多面体序列应收敛于外接多面体，而对偶多面体序列收敛于内接多面体。我们感兴趣的情况是原始多面体的顶点可以移动或合并，即对偶多面体不允许出现新的面。这些规则定义了通过半径变化、球体移动和消除将初始球体集合转换为Delaunay球体集合的变换。存在性定理尚不可用，但我们提出一个泛函来度量凸多面体与内接于抛物面的多面体的偏差。它是幂函数的离散Dirichlet泛函，该幂函数是对偶顶点到抛物面垂直距离的线性插值。该泛函的绝对极小值在恒定幂场上取得，这意味着内接多面体可以通过简单平移获得。由于未知量是原始多面体的顶点，该泛函在对偶曲面上的表述不是二次的，因此将球体集合转换为Delaunay球体的变换不是唯一的。我们专注于实验验证该方法的可行性，并将网格质量问题搁置一旁。所提出泛函的梯度为零定义了一个描述Delaunay球体演化的流形。因此，Delaunay-Voronoi网格可以使用该流形作为约束进行优化。
英文摘要：
We consider the construction of a polyhedral Delaunay partition as a limit of the sequence of power diagrams (radical partitions). The dual Voronoi diagram is obtained as a limit of the sequence of weighted Delaunay partitions. The problem is reduced to the construction of two dual convex polyhedra, inscribed and superscribed around a circular paraboloid, as a limit of the sequence of pairs of general dual convex polyhedra. The sequence of primal polyhedra should converge to the superscribed polyhedron and the sequence of the dual polyhedra converges to the inscribed polyhedron. We are interested in the case when the vertices of primal polyhedra can move or merge together, i.e., no new faces are allowed for dual polyhedra. These rules define the transformation of the set of initial spheres into the set of Delaunay spheres using radius variation and sphere movement and elimination. Existence theorems are still unavailable but we suggest a functional measuring the deviation of the convex polyhedron from the one inscribed into the paraboloid. It is the discrete Dirichlet functional for the power function which is a linear interpolant of the vertical distance of the dual vertices from the paraboloid. The functional's absolute minimizer is attained on the constant power field, meaning that the inscribed polyhedron can be obtained by a simple translation. This formulation of the functional for the dual surface is not quadratic since the unknowns are the vertices of the primal polyhedron, hence, the transformation of the set of spheres into Delaunay spheres is not unique. We concentrate on the experimental confirmation of the approach viability and put aside mesh quality problems. The zero value of the gradient of the proposed functional defines a manifold describing the evolution of Delaunay spheres. Hence, Delaunay-Voronoi meshes can be optimized using the manifold as a constraint.
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