We propose finite-time measures to compute the divergence, the curl and the velocity gradient tensor of the point particle velocity for two- and three-dimensional moving particle clouds. For this purpose, a tessellation of the particle positions is performed to assign a volume to each particle. We introduce a modified Voronoi tessellation which overcomes some drawbacks of the classical construction. Instead of the circumcenter we use the center of gravity of the Delaunay cell for defining the vertices. Considering then two subsequent time instants, the dynamics of the volume can be assessed. Determining the volume change of tessellation cells yields the divergence of the particle velocity. Reorganizing the various velocity coefficients allows computing the curl and even the velocity gradient tensor. The helicity of particle velocity can be likewise computed and swirling motion of particle clouds can be quantified. First we assess the numerical accuracy for randomly distributed particles. We find a strong Pearson correlation between the divergence computed with the the modified tessellation, and the exact value. Moreover, we show that the proposed method converges with first order in space and time in two and three dimensions. Then we consider particles advected with random velocity fields with imposed power-law energy spectra. We study the number of particles necessary to guarantee a given precision. Finally, applications to fluid particles advected in three-dimensional fully developed isotropic turbulence show the utility of the approach for real world applications to quantify self-organization in particle clouds and their vortical or even swirling motion.

翻译：我们提出有限时间度量方法，用于计算二维和三维运动粒子云中点粒子速度的散度、旋度和速度梯度张量。为此，对粒子位置进行镶嵌剖分，为每个粒子分配一个体积。我们引入了一种改进的沃罗诺伊镶嵌，克服了经典构造的一些缺陷：采用德劳内胞的重心而非外心定义顶点。通过考虑两个连续时刻，可评估体积的动力学变化。确定镶嵌胞体积变化可得到粒子速度的散度；重组各类速度系数可计算旋度乃至速度梯度张量。同样可计算粒子速度的螺旋度，并量化粒子云的旋涡运动。我们首先评估随机分布粒子的数值精度，发现改进镶嵌计算的散度与精确值之间存在强皮尔逊相关性。此外，该方法在二维和三维空间中呈现空间与时间的一阶收敛性。随后，我们考虑在具有幂律能谱的随机速度场中输运的粒子，研究保证给定精度所需的最小粒子数量。最后，将方法应用于三维充分发展各向同性湍流中输运的流体粒子，展示了该方法在真实场景中量化粒子云自组织及其涡旋或旋转运动的实用性。