# Numerical Analysis via Mixed Inverse Hydrodynamic
## 1 Introduction  
The reciprocating rod seal comprises a rod made of a hard material and a seal made of a relatively soft material, which is embedded via plastic deformation between the rod and a housing. The seal prevents the working fluid from leaking and maintains the pressure inside the housing; it is an important component that is widely used in various industries, such as automobiles and aviation. However, even if a seal is installed, leakage of the hydraulic cylinder is mechanically unavoidable and has a significant effect on the efficiency of the system, as well as causing environmental pollution. Therefore, it is essential to accurately analyze and characterize seal behavior in hydraulic systems. The seal mainly operates in the area of mixed hydrodynamic lubrication, in which the asperities that are simultaneously present between the surfaces and fluid support the external force. However, depending on its operating conditions, the reciprocating motion seal can be operated in severe lubrication regimes as its range of use expands, such as fully hydrodynamic lubrication, where there is almost no contact between the asperities and boundary lubrication. The behavior of rubber seals operating in such a wide range of lubrication conditions has been analyzed through elastohydrodynamic lubrication (EHL) theory or inverse hydrodynamic lubrication (IHL) theory. [PERSON] et al. [1] predicted the dynamic sealing performance of O-ring and rectangular seals through EHL analysis, including friction due to fluid shear stress. [PERSON] al. presented an EHL model that considered many temperature- and pressure-dependent parameters across a broad range of temperatures and sealed pressures [2] and verified it through experimental comparison [3]. Researchers have continued to analyze the behavior of rubber seals operating under mixed lubrication conditions. Using a mixed EHL model, [PERSON] et al. analyzed the behavior of single-lip [4] and double-lip U-cup seals [5]. Subsequently, they expanded the existing model to a thermal mixed EHL model that considered the surface temperature rise due to frictional heat generated in the sealing area [6] and applied it to the O-ring and U-cup seal [7]. To analyze the effect of stretching on the O-ring seal, [PERSON] et al. [8]. simulated the EHL model, which was solved by simplifying it into an axisymmetric (2D) model by extending it in three dimensions. General hydrodynamic lubrication (HL) analysis is mainly applied to sliding bearings, and the oil film gradient or squeeze between two hard surfaces is the main pressure-increasing mechanism, i.e., HL analysis involves calculating the oil film pressure according to the operating conditions, such as viscosity, speed, and load. However, it is impossible to analyze the behavior of rubber seals through HL analysis, such as sliding bearings, because large deformations occur even with small pressures. If EHL analysis is performed to solve this problem, the analysis becomes increasingly difficult and time-consuming. To address difficulties in analyzing the behavior of rubber seals, [PERSON] et al. [9]. first presented IHL theory. Whereas HL analysis calculates the pressure distribution from a specific oil film distribution, IHL analysis calculates the oil film distribution from the pressure distribution. In the IHL, it is assumed that the contact pressure obtained through the finite element (FE) analysis model is the same as the lubrication analysis pressure calculated by the oil film. Because the elastic deformation of the rubber surface is much larger than the fluid film thickness, it is hardly affected by the presence of a thin oil film [10].