Application Of MLPG Method In Three Dimension Helmholtz Equation

In the fields of computational electromagnetics and computational acoustics,when using traditional finite elements methods to solve Helmholtz equations,most of these meth-ods are based on meshing.These methods divide the whole calculation area into discrete elements.Then the information of nodes and connection information is recorded.The ap-proximate solution at each element node can be given.However,the traditional methods are highly mesh dependent,which causes difficulties when dealing with complex three-dimensional problems,such as large deformation problems and etc.The reconstruction of the mesh not only adds up the computational complexity but also reduces the computational accuracy and efficiency.Meshless methods only require discrete nodes in the domain to construct basis func-tions,which reduces the reliance on meshes.Meshless local Petorv-Galerkin method is considered as one of the truly meshless method,since it ignores meshes and is based on local weak form over local sub-domains,such as spheres for three dimensional case.In this method,the compact function is used to collect and integrate the node information.Both the approximation function and integration are performed only in the local sub-domains near the nodes,which greatly reduces the workload of constructing and reconstructing mesh in the traditional finite element method.At the same time,the method is more conductive to dealing with complex situations where the mesh may need to be reconstructed.In this paper,we study the application of the MLPG in solving three dimensional Helmholtz equation with the following workflow.First,we use moving lest square(MLS)method to construct the shape functions of MLPG.Then the local weak form corresponding to the three dimensional Helmholtz equation is derived,in which the essential boundary con-dition is be imposed by the penalty method.Furthermore,Gussian points are set in the local sphere sub-domain,and the integral can be solved by means of the coordinate transforma-tion.Finally,we investigate the numerical experiments and consider the influences of the radiuses of the influence and support domains on the numerical solution of 3D Helmholtz.The reliability and validity are verified by these numerical examples.