The Research Of Topology Optimization Of Continuum Structures Based On Meshless Numerical Technique

The topology optimization design of the continuum structures is one of the most challenging research topics in the field of the structural optimization. To date, the prevailing numerical method in the topology optimization is the finite element method (FEM). However, for FEM, there are some shortcomings such as mesh distortion, frequent remeshing when dealing with large deformation or moving boundary problems. To ameliorate these difficulties, a group of meshless methods have been developed and achieved remarkable progress. In meshless methods, the approximate solution is constructed entirely in terms of a set of nodes, and no element or characterization of the interrelationship of the nodes is needed, so these methods can reduce the calculated amount while ensure a good accuracy. In this dissertation, the meshless methods are introduced into the topology optimization of the continuum structures, and the relevant technologies are studied systemically.At the beginning of the dissertation, recent developments of structural topology optimization are briefly summarized, and several typical topology optimization methods are introduced particularly. Recent developments of meshless methods and application of meshless methods in structural optimization are also overviewed. After that, the elasto-static, dynamic and geometrically nonlinear topology optimization problems of the continuum structures are studied based on the meshless methods.The global weak-form meshless method and local weak-form meshless method are introduced into the topology optimization of the continuum structures, and the topology optimization choosing the minimization of compliance as objective function is studied. The optimization model, sensitivity analysis and optimality criteria method based on SIMP is also studied. The influence of the parameters in meshless methods and optimization algorithms on the topology results is discussed. Numerical examples show that the proposed approach is feasible and efficient for topology optimization problems of the continuum structures, and both of the checkerboard and mesh-dependence problems which pertaining to the conventional numerical FEM-based topology optimization methods are circumvented simultaneously.The meshless method is combined with evolutionary structural optimization method (ESO) to carry out the topology optimization of the continuum structures. The mathematical formulation of the topology optimization is developed based on the EFG method and the FVMLPG method using the stresses criterion and the displacement criterion, respectively. The influence pf the parameters in the algorithms on the topology results is discussed.The dynamic topology optimization of the continuum structures based on the meshless method is studied, which include the eigenvalue optimization problems for free vibrating structures and the harmonic response optimization problems for forced vibrating structures. For free vibrating structures, maximizing the fundamental eigenvalue is taken as objective function, and the results are discussed for different distributed nodes and different volume constraints. For harmonic response optimization problems, minimizing the dynamical compliance is taken as objective function, and the results are discussed for different harmonic frequency. Numerical examples show that the proposed approach is feasible and efficient for dynamic topology optimization problems. When choosing the relative density of nodes as design variable, a continuous relative density field is constructed using the meshless shape functions which can effectively eliminate the checkerboard and mesh-dependence patterns.The geometrically non-linear topology optimization of the continuum structures based on the meshless method is also studied, which include the linear elastic material geometrically non-linear topology optimization and the hyper-elastic material geometrically non-linear topology optimization. Geometrically non-linear structural response based on meshless method is formulated using a Total-Lagrange technique, and the equilibrium is found by an incremental scheme combined with Newton-Raphson iteration. Considering the relative density of nodes as design variables, the minimization of compliance as an objective function, the mathematical formulation of the topology optimization is developed using the SIMP interpolation scheme. Sensitivity of the objective function is derived based on the adjoint method. Numerical examples show that the proposed approach is feasible and efficient for the topology optimization of the geometrically non-linear continuum structures, and the reasonable, checkerboard controlled, node independent and black-white optimization results are obtained.