Research On Radial Basis Function Method And Its Key Issues In Solving Boundary Value Problems

In electromagnetic calculations, grid methods which are based on meshing andlocal approximation techniques usually face difficulties in field split and contradictionsbetween accuracy and amount of calculation, and radial basis function method has beenwidely used in data interpolation and solving differential equations, and graduallyformed a kind of collocation meshless method. Due to the complexity and diversity ofelectromagnetic problems in engineering, research and application of RBF meshlessmethod in the electromagnetic field has not yet been systematic and perfect. Therefore,the paper will study research on RBF method and its key issues in solving BVP, in orderto solve part of the problem of RBF method applied to electromagnetic calculations,which has great significance for extensions and applications of meshless method as wellas computational electromagnetics method.According to these ideas, research results and major work done in the paper are asfollows:①The basic principle and implementation process research of RBF interpolationand solving BVP, numerical examples illustrate the effectiveness of the RBF collocationmethod in data interpolation and solving BVP. With fewer nodes in the configuration, itcan not only obtain high accuracy, but also the implementation process is simple, whichfully embodies the advantages of it in numerical computation.②There is no consensus in the shape parameter selection for MQ interpolation upto now. Starting from the stiffness matrix which is the common problem in interpolation,this thesis pointed out that the interpolation accuracy and stability is closely related withcondition number of the stiffness matrix, and then study the relationship of conditionnumber versus node numbers and shape parameters from the numerical aspect. Finally,the node number and shape parameters can be selected according to the scope ofcondition number. An new MQ parameter setting method was provided and itsapplicability was verified by interpolation experiments.③Combination of the RBF method and virtual boundary element method formsRBF-virtual boundary method in electromagnetic calculation. This method combinesthe characteristics of RBF method and virtual boundary element method, breaking somelimitations of the traditional boundary element method. It not only avoids singularintegral and boundary effect existing in the traditional boundary element method, but also gets rid of limitations by the boundary element mesh. So it only focus on thediscrete node information of the boundary and simplifies the calculation process.Meanwhile, examples show that the structural information of the virtual boundary hasno significant effect on the results, reflecting the flexibility of the method.④In multi-medium BVP, for the presence of derivative discontinuity of function atthe interface, the thesis proposes subzone scheme and verifies its effectiveness throughtypical examples. To effectively solve the unbounded problem in electromagnetic fields,the paper attempts to combine RBF and asymptotic boundary conditions to give fullplay to both their own characteristics, and finally reflects its applicability in unboundedproblems through some typical examples.