Theapplication Research On Two Categories Of Meshless Method

In the past twenty years, the meshless methods have been developing very rapidly,and they have been effectively used in solving many partial differential equations whichproblems in the fields of science and engineering. As meshless method is independentof the grid, it can avoid mesh distortion and twisting arising from the the traditionalnumerical methods based on meshes such as finite element and boundary elementmethod. The unique advantages in the meshless methods make their implement areeasier than the finite element method, the boundary element methods.At present, the development of meshless methods includes Smoothed ParticleHydrodynamics, reproducing kernel particle method, multiple scale reproducing kernelparticle method, diffuse element method, Element-free Galerkin Method, Hp-cloudsMethod, meshless local Petrov-Galerkin method, the finite point method， waveletparticle method, radial basis function method, the meshless method with complexvariables and the meshless method of boundary integral equation.In this paper, we introduce two important meshless methods, i.e. Smoothed ParticleHydrodynamics (SPH) and Kansa ’s method, and the application in functioninterpolation and obstacles.Smoothed Particle Hydrodynamics(SPH) is a meshless method developedgradually in recent30years. In this paper, we simply introduces the basic idea behindSPH method and some common weight functions. We also propose an infinitydifferentiable weight function, which satisfies the demanded properties. Furthermore,the function posses consistency and the order of interpolation error is.. Weemploy the property of Dirac function and the given weight function to study a integraltype interpolation.As a meshless method, Kansa ’s method is independent of meshes and units. Radialbasis functions such as MQ function is used to meet the relevant conditions at the node.Kansa ’s method is an important method used to solve the PDEs, and it is widely used incomputing science. Kansa’s method has been widely applied in solving ordinary orpartial differential equations, including two-phase and three-phase hybrid model tissueengineering problems, the shallow water equations, tides and currents in the simulation, the heat conduction equation, the free boundary problem, the the fractional diffusionequation. In recent years, many experts and scholars have made great efforts in thisrespect.In this paper, we use Kansa’s method to solve the obstacle problem whichcontroled by a Possion equation, and we combine the Kansa’s method with give theUzawa algorithm.Numerical examples are given to verify the efficiency of thismethod.