Research On Adaptive Variable Step Difference Algorithm For Reynolds Equation Based On Hierarchical Model

As a basic differential equation in hydromechanics,the Reynolds equation can be seen in many engineering problems.The most common numerical methods for solving are the finite difference methods(FDM for short),in which the whole models discrete by grid.The difference quotient is used instead of the derivative to transform the partial differential equation into a linear equation.And in order to obtain high-precision calculation results,a large-scale grid is often needed to discretize the entire model.But this treatment will lead to the poor flexibility and computing resources wastes.The adaptive variable step difference algorithm based on the hierarchical model was developed to dealing with this problem mentioned above.In this method,a new refinement region algorithm based on the pressure gradient change and the pressure integral value of the cell grid region was introduced to refinement.Instead of the global refinement,the refinement was taken place in local region resolved on a preprocessing stage using the result from the basic grid.The main work of this paper is as follows:(1)A variable step difference algorithm based computing framework for solving the Reynolds equation was constructed.By deriving the general form of Reynolds equation and performing dimensionless processing,the difference scheme between the uniform mesh and the non-uniform mesh discrete was established,and the coefficient iterative matrix and the right term are integrated according to the determined boundary conditions,and the variable step size was used in the differential algorithm.(2)A region extraction algorithm based on pressure gradient was proposed.Taking the radial sliding bearing as an example,the calculation of multiple sets of parameters was carried out.According to the uniform oil grid,the oil film pressure of the lower node was weakened to the cell grid to obtain the gradient density information of each cell,and finally the effective subdivision area can be extracted.(3)Based on the cell grid pressure gradient information as the criterion to determine the subdivision region,the unit grid pressure value of each grid was studied to help the determination of subdivision area.Using the interpolation shape function of the four-node isoparametric element,the Gaussian integral method was used to calculate the oil film pressure integral of the unit mesh in the solution domain,which was used as the basic science to effectively extract the subdivision.Grid area,the hierarchical model construction process and the corresponding segmentation strategy were also discussed.(4)The radial sliding bearing calculation model and the piston ring-cylinder liner model of the internal combustion engine were introduced,and the solution was solved by the hierarchical adaptive variable step size difference algorithm.The results of the adaptive variable step size difference method were compared with the results of the traditional finite difference method.The results show that the adaptive variable step size difference algorithm based on the hierarchical model can express the oil film pressure surface with less degree of freedom,and in the iterative process the number of iteration steps required is also small,and the convergence speed and solution efficiency of the solution are significantly improved compared with the traditional halved grid.The flexibility and adaptability of the algorithm in different models are also verified along with the accuracy and efficiency of the algorithm.