# A Fast Particle-Locating Method for the Arbitrary Polyhedral Mesh
## 1 Introduction  
The hybrid Euler-Lagrangian models are widely used in simulating the flows in the dispersed systems. In these models, the continuous phase is described through spatial field distribution in Eulerian framework, while the dispersed phase is considered through tracking the movements of the dispersed phase entities (particles, bubbles, or droplets, which we shall call particles collectively). During simulations, the interactions between continuous and dispersed phases are realized by computing the movements of the particles using the own parameters at the particle centers and evaluating the effect of the particles (in terms of their volumes, velocities, and so on) on the dynamic behaviors of the continuous phase. As a result, locating the host cell of each particle at every time step is required. Although locating the host cell on a uniform Cartesian mesh is easy, it is not the case for the unstructured meshes, especially for those with the arbitrary polyhedral cells, where the particle locating is usually a time-consuming task and brings a heavy burden on the overall computational efficiency. In previous studies [1; 2], it was found a faster particle-locating method is necessary for simulations within the complex geometries using the arbitrary polyhedral meshes. Several methods have been proposed for the particle locating on irregular meshes. Existing methods can be categorized into two types, i.e., the known vicinity face-to-face searching methods, such as [PERSON] and [PERSON] [3], [PERSON] et al. [4], [PERSON] et al. [5], [PERSON] et al. [6], [PERSON] et al. [7], [PERSON] [8], and [PERSON] et al. [9], among others, and the auxiliary grid methods, such as [PERSON] and [PERSON] [10], [PERSON] and [PERSON] [11], [PERSON] et al. [12], and [PERSON] et al. [13], among others. Using the known vicinity face-to-face searching methods, a starting point along with its host cell as the starting host cell is given for each tracked particle at first. The point-in-cell test is conducted in the starting host cell at first and then carried out in one of its neighbor cells, determined by some prescribed rules if the starting host cell does not contain the particle. This procedure continues until the host cell is found [14]. The computational efficiency of the known vicinity face-to-face searching method highly depends on the choice of the starting point. Unfortunately, it is difficult to find the optimal starting point to lower the computational expense. The auxiliary grid methods construct a regular background grid, e.g., a tree-type grid or a uniform Cartesian grid with a larger grid spacing than that of the computational mesh, to efficiently locate the host cells for tracked particles. The mapping relation between the background grid and computational mesh is specified before the simulation. It stores, for each background cell, the indices of the several computational cells which are fully or partially contained by the background cell are found. The particle is located within a background cell at first, and then the point-in-cell tests are conducted within only several computational cells related to that background cell to determine the host cell. Since locating the particle on a regular background grid is considerably simple and the point-in-cell tests are only conducted in several most possible cells, the computational efficiency of particle locating is significantly improved by using an auxiliary regular grid. So far, the auxiliary grid method with a uniform background grid has been used for particle locating on two-dimensional curvilinear mesh [11; 12]. In this study, we extend the auxiliary grid method to the three-dimensional arbitrary polyhedral mesh, which is a practical choice for numerical simulations in complex geometries and more challenging for particle-locating methods. Furthermore, we study the influence of the resolution of the background grid on the performance of the particle-locating method.