Application Of RBF-PUM In Two-Dimensional Profile Steady Seepage Problem

In recent year,the seepage problem with a free surface is one of the key problems in the research of Groundwater Engineering.The method used in the previous research is mainly the finite element method,the one is the Moving Mesh method,and the other is the Fixed Grid method.Although two methods have achieved effective results,there are many shortcomings because of their dependence on meshes:each iteration of the Moving Grid method will generate new grids,which can lead to a large calculation.If the initial position of the free surface is larger than the final position,it is easy to cause the grid deformity,and the accuracy will be reduced;In spite of the Fixed Grid methods need not generate new mesh,it can not accurately calculate the permeability matrix of the units which are split by the free surface.At present,the developing meshless methods can overcome these shortcomings.Radial basis function collocation method is a typical representative of the meshless method.It has a lot of advantages,for example,simple algorithm,no integral,allowing local refinement,high calculation accuracy and so on.So radial basis function collocation method is the first choice for researchers to study numerical solution of partial differential equations.However,it is easy to lead to dense linear equations in the global approximation of large-scale problems,which not only increases the amount of calculation,but also reduces the computational efficiency.And partition of unity is the method which can transform global approximation into local approximation.Therefore,this paper combine the radial basis function collocation method and the partition of unity method to solve the problem of the location of the seepage free surface.Radial basis function partition of unity collocation method(RBF-PUM)is a kind of meshless method.Its essence is estimating the global approximation with the local approximation by using the principle of finite covering.This method can not only guarantee the precision of calculation,but also generate sparse matrix for coefficient in the process of discrete differential equations.The prominent advantages of this kind of matrix are: saving the storage space and fast calculation.Therefore,RBF-PUM has the characteristics of time and space efficiency,better accuracy in solving large-scale problems.In this paper,firstly we summarize not only the background and the research progress of the meshless method,but also current situation of research on seepage free surface.Secondly,starting from the related basic theory of seepage,this paper describes the basic principle ofRBF-PUM,establishes a numerical model of profile seepage problem.Then,this paper uses RBF-PUM to solve the plane seepage problem,and satisfactory result illustrates the feasibility and accuracy of the method in solving seepage problem.Finally,the RBF-PUM is applied to solve the problem about position of the free surface in the two-dimensional profile steady seepage flow,and the ideal result is obtained by the MATLAB program and the actual calculation.So RBF-PUM provides a new method for hydrogeological calculation.Combined with the case we analyze the advantages and disadvantages of this method,and hope for the future development.