The Numerical Method Of Radial Basis Function For Laplace Eigenvalue Problem

The Laplace eigenvalue problem is a classical research topic with rich contents,wide range of application and multidisciplinary intersection.It is widely used in electromagnetic propagation theory,quantum mechanics,riemann manifold topology,optical film vibration,thermal radiation and other engineering fields.It is the basis of many eigenvalue problem research.The common numerical methods for solving Laplace eigenvalues,such as FEM,FDM,BEM,etc.But the approximate effect is not ideal,because of those methods are rely on some mesh,so it will take a large costs when discrete the complex regions,then this conditions need to be improved urgently.It is critical need to explore a new theories of high-precision approximation.At present,the RBF meshless method for solving ellipse,parabolic,hyperbolic PDE equations,as well as high-speed collision,large deformation,fracture mechanics,fatigue damage and other problems have a good numerical effects,but the solution to eigenvalue problems is still in the stage of scarcity and development.In this paper,we use the RBF meshless collocation method with strong form,this method include asymmetric Kansa method and symmetric Hermite method.Because of considering the Drichlet boundary condition u=0,which will cause the discrete system singularity.We proposed a new way to combination RBF meshless method and numerical iteration theory to solve Laplace eigenvalue,which expanded the range of RBF Meshfree's applications,and finally this problem is converted to solve the weak singularity generalized eigenvalue problem Ax=?Bx.It not only effectively to suppresse the singularity but also improve the accuracy by the improved Arnoldi and Jacobi-Davidson iterative algorithm.The Laplace's analytical solutions can be solved by variables separation method in this papers.We respectively make the numerical experiments of RBF meshless in square domain,circular domain and L-shaped region,the numerical results verify our method is effective.The thesis use the RBF meshless to approximate Laplace eigenvalues.We analyzed and compared the function's basis and the numerical error graph and convergence graph of Kansa and Hermite.It is obviously to get the conclusions that IMQ is the best numerical result among those three function basis,the Kansa method has a higher approximation accuracy,and have the both characteristics is that the convergence is not stability.The results are also compared with the traditional FDM and FEM,and it is found that our my method is more effective.In addition,about the truncation problem of the eigenvalue approximation interval,our numerical conclusion is that the first three eigenvalues can be approximated very well and with a high accuracy by RBF Meshless.This result is similar to the five-point center difference method,but RBF meshless's accuracy more higher.Finally,we use the MATLAB GUI developed the Laplace eigenvalue numerical software,which can output the Laplace's exact solution when users have a demand,The system can also output some information:the error graph,convergence graph and the two-dimensional projection of eigenfunction.The numerical method includes:RBF meshless method,finite difference,finite element,spectrum,practical strong,more convenient,beautiful interface,etc.