The cost of tracking Lagrangian particles in a domain discretized on an unstructured grid can become prohibitively expensive as the number of particles or elements grows. A major part of the cost in these calculations is spent on locating the element that hosts a particle and detecting binary collisions, with the latter traditionally requiring $\mathcal{O}(N^2)$ operations, N being the number of particles. This paper introduces an optimal search box strategy to significantly reduce the cost of these two operations, ensuring a nearly $\mathcal{O}(N)$ scaling of the cost of collision detection for large-scale simulations. The particle localization strategy is constructed by obtaining an a priori estimate for the optimal number of search boxes as a function of the number of elements, particles, and time steps. The introduced method is generic, as it must be tuned only once for a given implementation and element type. The optimal number of search boxes for collision detection, although complex in form, can be reasonably approximated as the number of particles. The optimality of our method is shown using three drastically varying geometries.

随着粒子或元素数量的增加，跟踪在非结构化网格上离散的域中的拉格朗日粒子的成本可能变得过高。在这些计算中，大部分成本用于定位包含粒子的元素并检测二进制碰撞，而传统上后者需要$\mathcal{Ø}（{ñ}^2）$运算，N是粒子数。本文介绍了一种最佳搜索框策略，可以显着降低这两种操作的成本，从而确保$\mathcal{Ø}（ñ）$大规模仿真中碰撞检测成本的缩放比例。通过获取最佳搜索框数量的先验估计值（作为元素，粒子和时间步长的函数）来构造粒子定位策略。引入的方法是通用的，因为对于给定的实现和元素类型，它只能被调整一次。用于碰撞检测的搜索框的最佳数量，尽管形式复杂，但可以合理地近似为粒子数量。使用三种急剧变化的几何形状显示了我们方法的最优性。