# A Local Adaptive Mesh Refinement for JFO Cavitation Model on Cartesian Meshes
## 1 Introduction
The algorithm accounted for cavitation by introducing additional nonlinearity to the discretized Reynolds equation in an unconstrained equation system. The FBNS algorithm was significantly faster (two orders of magnitude) than the traditional algorithms, as compared in their work. [PERSON] and [PERSON] [22] employed the multigrid method in the Elrod algorithm. They found that the multigrid method was almost 10 times faster than alternating direction implicit (ADI) and 20-30 times faster than the Gauss-Seidel method. [PERSON] et al. [23] proposed an alternative solution scheme based on the finite volume method (FVM). The specially designed algorithm accelerated convergence speed and eliminated the scheme dependency on the choice of lubricant bulk modulus. To further improve the computational efficiency, the paper employed the adaptive mesh refinement (AMR) method in the bearing cavitation simulation. Although the AMR method has been developed for nearly 40 years (the block-structured AMR was first reported by [PERSON] and [PERSON] [24] in 1984), few articles focused on this topic. AMR is the method of numerical discretization that allows different numerical resolutions to exist in the computational domain, i.e., nonuniform mesh. AMR allows local mesh refinements and coarsening during the solution process, which heavily speeds up calculation and saves computational cost. AMR is often essential for multiscale problems such as global mantle convection simulation [25; 26], time-dependent shock hydrodynamics [27; 28], and sharp interface capture between two phases [29; 30]. The AMR method has different approaches for managing nonuniform meshes. There are two main kinds of AMRs based on the Cartesian grid, which are suitable for a regular shape-bearing simulation, as shown in Figure 1. (1) Block-based AMR refines a predefined block when the cell within the block is mostly tagged to be refined. The advantages of the block-based AMR include the nature of structured meshes in each block and relatively simple data structures (permit reuse of uniform mesh code). However, it becomes chal lenging to encode as grid hierarchy increases. (2) Cell-based AMR refines only those cells that are supposed to be refined. The nature of structured meshes is lost due to independent cell management. However, a tree-based data structure is often used to manage the meshes, making the neighbor relation efficiently obtained [31]. The unstructured AMR also provides geometric flexibility at the cost of explicitly storing all neighborhood relations [32]. In the paper, a cell-based AMR algorithm was developed for bearing cavitation simulation. The aim is twofold: (1) The AMR method is introduced into cavitation problem. It provides another way to significantly improve the computational efficiency. This method exhibits potential advantages in analysis of bearings with complex geometric shapes. (2) Due to the complexity of the AMR method, the basic AMR algorithm was explained in the paper. The algorithm may provide guidance for people interested in AMR. The AMR algorithm was constructed based on a quadtree data structure with a \(Z\)-order filling curve managing cells. The hybrids of interpolation schemes on hanging nodes were applied. A cell matching method was used to handle periodic boundary condition. The difference schemes at nonuniform mesh for the universal Reynolds equation were derived. [PERSON] cavitation algorithm was integrated into the AMR algorithm for the calculation of the bearing cavitation solution. The Richardson extrapolation method was employed as an a posteriori error estimation to tag the areas where they need to be refined. The AMR procedure was programmed in MATLAB. The details of the mesh management method, main algorithmic logics, nonuniform difference schemes, and error estimation were presented below.