Smoothed Point Interpolation Meshfree Method With Applications To The Engineering Mechanics

Meshfree method is a new kind of numerical methods. It works without the background grids. Hence, it can overcome the shortcoming of FEM. Therefore, it has a unique advantage for the discontinuous and large deformation problems. Point interpolation method is one of the meshfree methods, it is convenient to impose essential boundary conditions, and it has a good fitness with the finite element method. Many techniques based on the finite element method can be easily applied to the point interpolation method, which result in the good potential application to some engineering and scientific problems.In this thesis, we first give an introduction to the smooth point interpolation method. For the node-based smooth point interpolation method (NS-PIM), the problem is first separated into several small smooth domains. For each smooth domain, the smoothed strain is obtained by the following operation: (?)(1) in which,Ψis the smooth operation.Then, we calculate the displacement and strain energy using the smoothed strain. This method can overcome "too hard" problem of the FEM, and the "too soft" phenomena in NS-PIM. Note that the strain energy we obtained was always larger than the exact solutions.Another smoothed PIM is Edge-based smoothed PIM (ES-PIM). In the ES-PIM, The construction of smooth area was not based on nodes, but edges. We connect the two endpoints and the edge and the two regional centers of the two neighbor units to construct the smooth area. ES-PIM eased the "too soft" problem of the NS-PIM. The numerical solution obtained from the ES-PIM was very closer to exact solutions.In this paper we propose hybrid scheme based on the smoothed PIM and the finite element method, which is called HPF. In HPF, an average strain is obtained using the conforming strain from FEM and the smoothed strain in NS-PIM: (?)(2) in which,ε% is the strain in FEM,εis the strain in NS-PIM. Then we calculate the the displacement and strain energy of the system. The format of Galerkin energy function is as follows, (?)(3) where, Dis the elastic constant matrix, b is the vector of the body force in the problem domain, T is the vector of the area force in the problem domain.Theorem 1: When the same meshes are used, the HPF has the same dimension with FEM and NS-PIM. Furthermore, the strain energy obtained in HPF is between the strain energy obtained in FEM and smooth point interpolation. (?)(4) ( V )is the strain energy obtained in NS-PIM: (:)(5)Π% ( V)is the strain energy obtained in FEM: (?)(6)Theorem 2: When the same meshed are used, the strain energy in HPF is not less than the strain energy in FEM, and not greater than the strain energy in NS-PIM, (?)(7) According to the above analysis, the strain energy in HPF is between the strain energy in FEM and the strain energy in NS-PIM. It can overcome the "too hard" problem of the FEM and the "too soft" problem of the NS-PIM. Therefore, HPF method has a higher accuracy.Using the new numerical method proposed in this paper, we have done some numerical simulation. Compared with the traditional PIM and the NS-PIM, the new proposed method HPF has good accuracy and high convergence rate.