The Topology Optimization Design Of Two Dimensional Structures Based On The Meshless LRPIM

The meshless method is a new numerical method with a great prospect developed after traditional numerical methods such as Finite Element Method, Boundary Element Method et al. The meshless method is independent of meshes, and completely or partly eliminates meshing. By using this method, it becomes easy to solve large deformation problems, crack propagation problems and high velocity impact problems et al. The meshless local radial point interpolation method (LRPIM) doesn't need any element or mesh for the purpose of the energy integration or interpolation. Therefore it is a truly meshless method. The shape functions have the Kronecker delta function property, and the essential boundary conditions can be easily imposed.The purpose of topology optimization is to find a distribution of the structure, so that it can meet the volume, displacement constraint conditions et al, and attain some optimal performances. Topology optimization of a continuum structure is essentially a 0-1 discrete variable optimization problem. At present, the topology optimization problems of continuum structures are mostly based on the finite element method. This article explores that the topology optimization method of the two-dimensional linear elastic structure based on the meshless LRPIM.The shape function of the meshless LRPIM is all constructed by using the radial basis functions with polynomial basis functions, the singularity of the system matrix is overcome. The shape functions and their derivatives are simple, consequently, lower computational cost. The efficient and accurate results can be obtained. The discrete system equations of the meshless local radial point interpolation method are derived and a two-dimensional cantilever is analysed. The deflection and stress results obtained are correspond closely with the theoretical solution. Effects of the shape parameters of the radial basis function on the numerical results are discussed. Effects of sizes of the quadrature sub-domain and the influence domain on the numerical results are investigated. In studying the topology optimization based on the finite element method, the implementation of topology optimization methods is further understood. In the topology optimization based on meshless LRPIM, considering the relative density of nodes as design variables, and the minimization of compliance as an objective function, the mathematical formulation of the topology optimization is developed using the SIMP(solid isotropic microstructures with penalization) interpolation scheme. The optimization formulation is solved by the optimality criteria method. Numerical examples show that the proposed approach is feasible and efficient for the topology optimization of the continuum structure, and can effectively overcome the checkerboard phenomenon.