We present Laguerre Voronoi based subdivision algorithms for the quadrilateral and hexahedral meshing of particle systems within a bounded region in two and three dimensions, respectively. Particles are smooth functions over circular or spherical domains. The algorithm first breaks the bounded region containing the particles into Voronoi cells that are then subsequently decomposed into an initial quadrilateral or an initial hexahedral scaffold conforming to individual particles. The scaffolds are subsequently refined via applications of recursive subdivision (splitting and averaging rules). Our choice of averaging rules yield a particle conforming quadrilateral/ hexahedral mesh, of good quality, along with being smooth and differentiable in the limit. Extensions of the basic scheme to dynamic re-meshing in the case of addition, deletion, and moving particles are also discussed. Motivating applications of the use of these static and dynamic meshes for particle systems include the mechanics of epoxy/glass composite materials, bio-molecular force field calculations, and gas hydrodynamics simulations in cosmology.

中文翻译：

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基于 Laguerre Voronoi 的粒子系统网格划分方案

我们提出了基于 Laguerre Voronoi 的细分算法，分别用于二维和三维有界区域内粒子系统的四边形和六面体网格划分。粒子是圆形或球形域上的平滑函数。该算法首先将包含粒子的有界区域分解为 Voronoi 单元，然后将其分解为符合单个粒子的初始四边形或初始六面体支架。随后通过应用递归细分（拆分和平均规则）对支架进行细化。我们选择的平均规则产生了一个符合粒子的四边形/六面体网格，质量好，并且在极限上是平滑和可微的。在添加、删除、还讨论了运动粒子。将这些静态和动态网格用于粒子系统的激励应用包括环氧树脂/玻璃复合材料的力学、生物分子力场计算和宇宙学中的气体流体动力学模拟。