Research On Solving Reynolds Equation By Method Of Combining IGA And Multi-grid

As the basic equation of fluid mechanics,the solution of Reynolds equation has very important reference value for some engineering designs.At present,there are many numerical methods for solving Reynolds equation,and some of them are proved to be feasible.The Finite Difference Method is a fast numerical method,but it has some shortcomings in accuracy.The Finite Element Method has high accuracy,but its calculation efficiency is low.Although many scholars have improved it,it is still difficult to solve the problem of calculation accuracy and efficiency.In order to satisfy the computational accuracy and improve the efficiency of solving linear equations,this paper researched the Igeometric(IGA)and multi-grid method,and combines the two methods to solve the Reynolds equation.Compared with many traditional numerical methods,Igeometric analysis can achieve higher calculation accuracy with fewer degrees of freedom,and this numerical algorithm avoids the process of regional discretization from the Finite Element Analysis,and achieves seamless docking between CAD and CAE.Therefore,while ensuring the accuracy of calculation,the efficiency of geometric analysis is greatly improved compared with other numerical algorithms,and the multi-grid method can be used to accelerate the solution,which can further improve the efficiency of its calculation.In order to match Reynolds equation with Igeometric,the basic form of Reynolds equation is deduced and a solution model suitable for geometric analysis was established.According to the characteristics of Igeometric,due to the non-interpolation of NURBS basis function,the boundary conditions can not be loaded by the conventional method of the Finite Element Analysis.In order to effectively load the boundary conditions in geometric analysis,a collocation method is proposed to load the boundary conditions.On this basis,for the calculation of linear equations,this paper first researchs the Gauss-Seidel iteration method and SOR iteration method,and then focuses on the multi-grid method.In this paper,a multi-grid calculation model based on h-refinement is established,and a solution method of mapping matrix between layers of grid based on h-refinement is proposed,which is used to accelerate the solution of linearequations.A numerical example is introduced to verify the results.It is found that the efficiency of the Multi-grid method is much faster than that of the Gauss-Seidel method.However,in the iteration process of the Multi-grid method,the speed of reduction for error suddenly slows down at a certain error value.In this paper,the phenomenon is analyzed and studied,and an automatic adjustment Multi-grid method is proposed.This method obviously improves the phenomenon of slow convergence and makes the convergence speed of the Multi-grid method faster.Based on this situation,compared with SOR iteration method,it is found that the efficiency of Multi-grid method is better than SOR iteration method in most cases.Only when the value of relaxation factor is close to the optimal relaxation factor,the convergence speed of SOR iteration method is slightly faster than that of Multi-grid method.However,in view of the lack of an effective method to solve the optimal relaxation factor,the efficiency of the Multi-grid method is generally better than that of the SOR iteration method.