The Research On Meshless Natural Neighbour Petrov-Galerkin Method And Its Applications In Structural Responses And Topology Optimization Of Structures

Topology optimization is a relatively new but extremely rapidly expanding research field. Up to now, the prevailing analysis method in the topology optimization is the finite element method (FEM), and the bulk part of the work focus on static compliance optimization, while limited efforts were put into the dynamic topological optimization. However, for the FEM, there are some shortcomings such as mesh distortion, frequent remeshing when dealing with large deformation or moving boundary problems, etc. The meshless method avoids the dependence of meshes, and thoroughly or partly eliminates meshing. It makes this kind of methods possess great advantages when dealing with crack propagation problems, super-large deformation problems and high velocity impact problems, etc. In this dissertation, the natural neighbor Petrov-Galerkin method (NNPG) is used to study the elasto-static, dynamic problems and the topology optimization of the continuum structures.At the beginning of the dissertation, the history and recent developments of the topology optimization and meshless method are overviewed, and several typical topology optimization methods are introduced particularly. The fundamental theory of topology optimization of continuum structure is discussed, including the formulation and solution of the optimization model, implementation and computational procedure, which based on the SIMP (Solid Isotropic Material with Penalization) density-stiffness interpolation model and the optimality criteria method. Numerical instabilities such as checkerboard and mesh-dependence as well as the available ways to circumvent them are also discussed. A continuous bulk density fields is proposed to eliminate the checkerboard patterns of the material distribution. In order to achieve mesh-independent designs by the topology optimization, a non-local relationship between the density and stiffness is introduced, based on the proposed modified continuous bulk density fields, both of the checkerboard and mesh-dependence problems which pertaining to the conventional numerical FEM-based topology optimization methods are circumvented simultaneously.When dealing with topology optimization problems, it is vital to obtain accurate structural responses. It’s very important to testify the accuracy and convergence of the NNPG method before applying the method to study the topology optimization of the continuum structure. To this end, both of the elasto-static and dynamic problems in2D plane problems as well as the Mindlin plate problems are studied by the natural neighbor Petrov-Galerkin method. In the implementation, based on the well established Delaunay triangulation, the FEM shape functions of three nodes triangular are taken as test functions, which reduce the order of integrands involved in domain integrals and improves the computational efficiency of the method. The trial functions which are constructed based on the natural neighbor interpolations have the Kronecker Delta function property, which facilitate imposition of essential boundary conditions and dealing with discontinuity problems. Numerical examples show that the present method possesses high accuracy and good performance of stability, and is easy to implement.The topology optimization for plane and Mindlin plate problems based on the meshless NNPG method and the proposed continuous bulk density fields, where the objective is to minimize compliance, is investigated. The numerical approach presented here is based on the SIMP approach and the optimality criteria method. A continuous bulk density fields constructed by the natural neighbor interpolation shape functions is proposed to eliminate the checkerboard pattern. This approach is used to get higher resolution solutions with mesh refinement and the checkerboard control. Based on the continuous bulk density fields, the non-local effect is considered. To this end, the original bulk density fields is reconstructed by introducing a influence domain whose radius is Rmin, in this way, the design space is restricted, which is very necessary to ensure the existence of solution. By adjusting Rmin, the complication of the optimized structures is controlled by the proposed method. In this way, the checkerboard and mesh-dependence problems which pertaining to the conventional numerical FEM-based topology optimization methods are all circumvented.The dynamic topological optimization for plane and Mindlin plate problems based on the meshless NNPG method is also studied, which include the eigenvalue optimization problems for free vibration of structures and the frequency response optimization problems for forced vibration of structures. The formulation of the dynamic topological optimization and the possibility of localized modes in low density areas are discussed. While the oscillation of the objective function during the eigenvalue optimization process and the effect of the non-structural mass to optimized structures by the topology optimization are investigated particularly. A heuristic approach, which modifies the sensitivity information to control the optimization path, is presented to avoid the oscillation of the objective function in eigenvalue optimization. By this way, some "optimized" design which closing to the global optimum of the original problem can be obtained.Numerical results show that the present method obtains reasonable, checkerboard controlled, node independent and black-white optimization results for the topology optimization of the continuum structures including the minimum compliance topology optimization, the eigenvalue optimization for the free vibration of structures and the frequency response optimization for the forced vibration of structures, which demonstrate the feasibility and validity of the present method for these problems.