电磁场边值问题求解的径向基函数方法

Radial Basis Function Method for Boundary Value Problems in Electromagnetic Field

为了解决网格法在电磁场计算中面临的网格剖分和场函数不连续问题,引径向基函数插值方法入电磁场边值问题计算,形成基于径向基函数的配点型无网格解法,它在求解中仅需节点信息,彻底摆脱了网格对求解的制约;在径向基函数插值原理基础上给出电磁场边值问题的径向基函数求解方法,建立了相应的离散模型。在1维和2维边值问题求解中,径向基函数法和网格法(有限差分法)对比表明:径向基函数方法不仅在配置点处有更高的计算精度,而且插值函数光滑性更好,对准确解的逼近能力更强。该方法结合了无网格和径向函数的特点,具有实施方便灵活、精度高和易于处理高维问题的优势。

Traditional numerical methods in computational electromagnetic like FEM,FDM,BEM and MOM are based on meshes,which means the meshes generating procedure is necessary in their implementation,as a result,region subdivision and discontinuity of field function become extreme challenges in mesh methods.Therefore,a radial basis function based interpolation method is introduced into electromagnetic field computation.A new meshless method with collocation type is founded,which only needs node information and gets rid of restricts of meshes thoroughly in numerical computation.After a detailed discussion of interpolation principle based on radial basis function,radial basis function（RBF） method for boundary value problems in electromagnetic is formed and corresponding discrete model is given.Then numerical experiments on one-and two-dimensional boundary problems are carried out.The RBF method and mesh method especially finite difference method are compared both at the collocation nodes＇ errors and the smoothness of numerical surface.Numerical results show that the RBF method not only gains higher accuracy at collocation nodes but also has smoother solutions.The RBF method combines the virtues of meshless idea and radial function,so the solutions are of high accuracy and the implementation is convenient and flexible,especially it takes great advantage in high dimension problems.