The Study of the Radial Basis Function and the Radial Point Interpolation Meshless Method

Abstract: Meshless method is a new numerical analysis method which has rapidly developed in recent years. This method based on the information of the nodes, thoroughly or partly cancelled the meshes, avoiding the difficulities which are broughted by creating the meshes. Because of the advantages of no meshes ,the meshless method is superior than the FEM in large deformation problems and so on . There are several developing meshless methods nowadays, such as:Element-free Galerkin method(EFG),Reproducing kernel particle meshless method(RKP),Partition of unity meshless method and so on.Compared with the FEM, the apprximate functions have almost no interpolation property (the Kronecker╬┤property)in many meshless method. So it is difficulty to solve the essential boundaries. Point interpolation method(PIM) is a new meshless method which is advanced to solve this problem. Its shape function has the╬┤property, so it could solve the essentian boundaries easily as the FEM. But it also has a disadvantage: the singularity of the matrix in computing the interpolation function sometimes.Actually , Radial Point Interpolation Meshless Method(RPIM) can avoid this problem. It has used the radial basis function, represented by properly scattered points to constract the function, not only has the advantage of the PIM, but also solved the singualrity of the matrix efficiently.According to this ,this paper can be devided into three parts:The first part is the fundmental theory of the meshless method. This part introduced the basic theory of the mechanics for solids, the concept of the radial basis function, and the theory of the strong-forms and weak-forms.The second part is the theory of the meshless method.mainly introduced the theory of the radial point interpolation meshless method based on the global weak-form(RPIM).In the third part of the paper, several examples are calculated and analysised and also compared the result with the theoretical solutions and the ANSYS solutions. They show good agreement and verifies the reliability of the theory in the present paper…
Key words: meshless method; radial basis function; shape function; Radial Point Interpolation Method(RPIM)

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