The Application Of Meshless Generalized Finite Difference Method In Helmholtz's Positive And Negative Problems

Meshless methods are the new numerical methods for solving initial boundary value problems of partial differential equations.Different from mesh-based numerical methods such as finite element,finite difference and boundary element method and so on,the meshless methods construct interpolated basis functions from nodes information,which gets rid of the dependence of traditional methods on grids in great degree.Therefore,the meshless methods have obvious advantages in solving the problems involving large deformations,high-dimensional complex domains,crack dynamic expansion,and moving boundaries and so on.The generalized finite difference method has just emerged for a few years as a new type of meshless methods.The method,based on Taylor series expansion and weighted least squares fitting of multivariate function,expresses the partial derivatives of the unknown parameters in the control equation as the linear combination of the function values of adjacent nodes,which gets rid of the dependence of traditional methods such as finite element method on grids.The coefficient matrix generated by this method is a sparse matrix and can be easily solved by various sparse matrix solvers.At present,this method has developed rapidly at home and abroad and is widely used to solve various scientific and engineering problems.This paper discusses the application of the generalized finite difference method in acoustic problems and applies it to solve the inverse Helmholtz problems for the first time.Firstly,assuming that all boundary conditions in the solution domain are measurable,the author solves the two-dimensional and three-dimensional Helmholtz direct problems on the basis and analyzes the stability and accuracy of the method.In the inverse Helmholtz problems this paper assumes that some boundary conditions in the computational domain are unmeasured(inverse Cauchy problem).By introducing additional boundary conditions on the remaining boundaries,the physical parameters on the unmeasured boundaries are predicted and simulated by the generalized finite difference method.The numerical examples show that the generalized finite difference method can avoid the ill-conditioned characteristics of the coefficient matrix to the utmost when it simulates the inverse problems.Meanwhile,we will still obtain the accurate and stable numerical solutions even if there are large noises in the boundary conditions.