Research Of The Continuous Structural Dynamic Topology Optimization Based On The VDR-BESO Method

Structural optimization has made great progress after decades of development,and the topology optimization method has become a very valuable research direction in structural optimization design due to its unique advantages.It is very important to study the theory and method of the topology optimization and further extend the application field of the topology optimization.As one of the most significant methods in topology optimization method,the bi-directional evolutionary structural optimization(BESO)method is one of the hot and difficult problems in current research.At present,there are still some problems in dynamic topology optimization method,such as low computational efficiency,long optimization process and so on.In order to solve these problems,this paper studies the basic principle of the method intensively based on the BESO method and expands its application fields to multi-objective and multi-constraint optimization problems.Firstly,the basic principle of the BESO method is studied,and the basic principle of the static stiffness optimization and the frequency optimization is analysed in emphatically.In order to solve the problem of high computational cost and long process of iteration in current topology optimization problems,an equivalent static load method is proposed to transform the structural dynamic optimization problem under the action of the dynamic load into a static optimization problem with multiple working conditions.Numerical examples show that the equivalent static load can solve the dynamic stiffness optimization problem.Secondly,a method of variable deletion ratio(VDR)is proposed to solve the problem of low computational efficiency of the BESO method.After elaborating the basic idea of the VDR,several VDR functions are constructed.The correctness and validity of the VDR method are verified by numerical examples such as the frequency optimization of the simply supported beam and the dynamic stiffness optimization of the long cantilever beam.Thirdly,the multi-objective optimization problem under the action of the dynamic load is studied on the basis of the VDR method.And the basic theory and solving method of the multi-objective optimization problems are introduced in detail.The mathematical model of the multi-objective dynamic optimization which minimizes the average dynamic compliance and maximizes the natural frequency of the structure is established using the weighted coefficient method.Numerical examples are given to verify the correctness of the multi-objective dynamic optimization,proving that the VDR method is also applicable to the multi-objective optimization problems.At the same time,the multiplication and division method is applied to the multi-objective optimization problem,and the result is compared with the result of the weighted coefficient method.Finally,considering the complexity of the current engineering problems,the multi-constraint optimization problems of volume and displacement constraints are studied.The basic theory of multi-constraint optimization is introduced in detail.The Lagrange multiplier is introduced to deal with the displacement constraint.Combined with the VDR method,the static stiffness multi-constraint optimization problem and the dynamic stiffness multi-constraint optimization problem based on the equivalent static load are studied.The validity of the displacement constraint is verified by numerical examples.