# A Local Adaptive Mesh Refinement for JFO Cavitation Model on Cartesian Meshes
## 1 Introduction  
Hydrodynamic bearings are widely used in machines, particularly in engines and power plants. The bearing consists of a rotating journal inserting a bore or a sleeve [1]. To ensure adequate lubrication, the journal wall driving velocity should be high enough to force oil into a converging clearance so that the load can be supported by the increasing hydrodynamic pressure [2]. The load-carrying capacity is one of the critical performance parameters and is treated as the prime objective in journal bearing design [1]. Cavitation can occur at particular kinds of bearings where the variable-shaped divergence clearance exists in the oil passage. Typically, cavitation occurs at the divergence clearance region in full circle bearings. Finger-shaped voids that cover most of the divergence clearance region can be observed [3]. Another specific case is textured thrust bearings (such as dimple-enhanced seal-like thrust bearings), in which cavitation occurs at the tail end of each dimple [4]. Although cavitation does not necessarily have a deleterious effect upon the load-carrying capacity, the cavitation algorithm's predicted load-carrying capacity is significantly affected [5]. Cavitation is essentially characterized as the two-phase flow of liquid and gas [6]. The liquid phase is oil. The gas phase is the air which escapes from oil when the pressure is below saturation pressure, or the oil vapor where the oil boils to form bubbles if the pressure is lower than the vapor pressure [7]. No matter what kind of cavitation, themultiscale density exists throughout the bearing clearance region. The oil density ratio, for example, changed from 1.0000 to 1.0002 (under the bulk of 1.72 \(\times\) 10\({}^{9}\) N/m\({}^{2}\)), while the gas density ratio changed from 1.0000 to 0.1754 [8; 9; 10]. Moreover, the stepwise nonlinearity of the density appearing at the oil film reformation boundary essentially increases the difficulty in the numerical solution. As a pioneer, [PERSON] [11] proposed a novel universal partial differential equation (PDE) which overcame the numerical difficulty in the implementation of the Jakobsson-Floberg-Olsson (JFO) cavitation theory [12; 13]. The algorithm was recognized as the Elrod algorithm. The main issue that existed in the Elrod algorithm was the occasional instability that occurs during the abrupt change of switch functions. It sometimes limited the application in practical cavitation problems. Many pieces of research were devoted to improving the stability of the algorithm [14; 15]. An effective way to relieve the instability of the Elrod algorithm was to employ a modified switch function algorithm proposed by [PERSON] and [PERSON] [16]. The algorithm allowed the presence of an intermediate switch function between 0 and 1, and the stepwise nonlinearity was primarily reduced. Similarly, a regularized cavitation algorithm developed by [PERSON] et al. [17] was also applicative due to the same idea of employing smooth switch functions. Another way to overcome the instability was to use the non-switch function algorithm proposed by [PERSON] et al. [18; 19]. In this algorithm, the pressure and density ratio were solved simultaneously by an explicit iterative scheme without employing switching functions. It was found that the algorithm was very stable for various microtextured bearings. Based on the concept of complementarity, a novel comprehensive finite element method (FEM) derivation of the Reynolds equation was developed by [PERSON] et al. [20]. The algorithm can naturally detect the rupture and reformation boundaries without the need for additional boundary conditions. The algorithm was beneficial for studying two-dimensional textured bearings and complex three-dimensional problems without nonconvergence or instability issues. Another research topic on the cavitation algorithm is to improve its computational efficiency. [PERSON] et al. [21] developed an efficient algorithm, called Fischer-Burmeister-Newton-Schur (FBNS), by reformulating the discretized cavitating flow.