Multiscale Radial Basis Functions Collocation Method For The Fourth Order Thin Plate Problem

The problem of thin plate bending is of great significance in theory and practice.Meshfree methods are widely used in this problem,most of which include basic solution method,boundary node method,etc.In this paper,multiscale collocation method is used to study the problem of thin plate bending.This method can not only maintain good approximation accuracy of radial basis function,but also have sparse linear algebraic equations.The basis function used in the multiscale collocation method is compactly supported radial basis function,which functions are used to solve partial differential equations,it is deduced that the matrix of discrete system is sparse and has good condition numbers.In this paper,the fourth order thin plate problem is briefly introduced.Secondly,the basic principles of non-symmetric radial basis function collocation method and symmetric radial basis function collocation method are introduced.Because the symmetric radial basis function collocation method requires a higher derivative,the non-symmetric radial basis function collocation method is used to solve the thin plate problem.This method requires no the solution domain discretization,no constant approximation,and no numerical integration.Therefore,there is no need to construct mesh during the calculation,which greatly reduces the amount of calculation and the cost of calculation,and the accuracy is very high.However,the non-symmetric radial basis function collocation method often leads to highly ill-conditioned discrete algebraic systems in practical numerical calculations.Finally,in order to deal with the contradiction between the approximation accuracy and numerical stability in the discrete method.A multiscale radial basis function collocation method is introduced and used in this paper to solve the fourth order thin plate problem.The first method is the multilevel correction method,proposed by Floater and Iske[16].The second method is the local multilevel correction method,which is proposed in reference[24].Multilevel correction method and local multilevel correction method are collectively called a kind of multiscale collocation method.Wendland's C6 and C8 compactly supported radial basis functions are used in the program implementation to derive the derivative of Wendland's C6 function.According to the integral operator definition,Wendland's C6 function is used to derive Wendland's C8 function,and the derivative of Wendland's C8 function is derived.Numerical experiments are carried out to verify the effectiveness of the proposed method.