The Visual Computer ( IF 3.0 ) Pub Date : 2021-07-05 , DOI: 10.1007/s00371-021-02189-0 Dennis R. Bukenberger 1 , Hendrik P. A. Lensch 1
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Abstract
We propose concepts to utilize basic mathematical principles for computing the exact mass properties of objects with varying densities. For objects given as 3D triangle meshes, the method is analytically accurate and at the same time faster than any established approximation method. Our concept is based on tetrahedra as underlying primitives, which allows for the object’s actual mesh surface to be incorporated in the computation. The density within a tetrahedron is allowed to vary linearly, i.e., arbitrary density fields can be approximated by specifying the density at all vertices of a tetrahedral mesh. Involved integrals are formulated in closed form and can be evaluated by simple, easily parallelized, vector-matrix multiplications. The ability to compute exact masses and centroids for objects of varying density enables novel or more exact solutions to several interesting problems: besides the accurate analysis of objects under given density fields, this includes the synthesis of parameterized density functions for the make-it-stand challenge or manufacturing of objects with controlled rotational inertia. In addition, based on the tetrahedralization of Voronoi cells we introduce a precise method to solve \(L_{2|\infty }\) Lloyd relaxations by exact integration of the Chebyshev norm. In the context of additive manufacturing research, objects of varying density are a prominent topic. However, current state-of-the-art algorithms are still based on voxelizations, which produce rather crude approximations of masses and mass centers of 3D objects. Many existing frameworks will benefit by replacing approximations with fast and exact calculations.
Graphic abstract
中文翻译：
变密度四面体及其应用
摘要
我们提出了利用基本数学原理来计算具有不同密度的物体的精确质量属性的概念。对于以 3D 三角形网格形式给出的对象，该方法在分析上是准确的，同时比任何已建立的近似方法都快。我们的概念基于四面体作为底层基元，它允许将对象的实际网格表面纳入计算。四面体内的密度允许线性变化，即可以通过指定四面体网格所有顶点的密度来近似任意密度场。涉及的积分以封闭形式表示，可以通过简单、易于并行化的向量矩阵乘法来计算。计算不同密度物体的精确质量和质心的能力为几个有趣的问题提供了新颖或更精确的解决方案：除了在给定密度场下准确分析物体外，这包括合成参数化密度函数以实现挑战或制造具有受控转动惯量的物体。此外，基于 Voronoi 单元的四面体化，我们引入了一种精确的方法来解决\(L_{2|\infty }\)劳埃德松弛通过切比雪夫范数的精确积分。在增材制造研究的背景下，不同密度的物体是一个突出的话题。然而，当前最先进的算法仍然基于体素化，它产生了相当粗略的 3D 对象质量和质心的近似值。许多现有框架将通过用快速准确的计算替换近似值而受益。
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