The Parameterized Level Set Topology Optimization Method Based On Reaction Diffusion Equation

Structural topology optimization is to determine the optimal distribution of materials in the design domain under certain boundary conditions,loads and constraints.The structural topology optimization based level set method uses the zero value curve of the level set function to implicitly define the structural boundary,which can clearly describe the structural boundary.However,the traditional level set method uses the upwind method to solve the Hamilton-Jacobi partial differential equation(PDE),the time step is limited by the CourantFriedrichs-Lewy(CFL)condition,the computational efficiency is low,and holes cannot be generated in the structure.Therefore,this paper improves the traditional level set method,and proposes a more stable and efficient parameterized level set topology optimization method,and uses some examples to verify the effectiveness of this method.Firstly,a level set method based on reaction diffusion equation(RDE)is proposed,which can control the geometric configuration complexity of the structure,generate holes in the structure,and do not need the tedious re-initialization operation.RDE is the evolution equation of level set function,the diffusion term can ensure the smoothness of level set function,and the reaction term is the topological derivative of the objective function,which can drive the level set function to evolve.At the same time,by using the variation of objective function when the structure is disturbed,the calculation formula of topological derivative is derived,which simplifies the solution process.Secondly,using the advantages of sparse coefficient matrix and local correlation of parameters of compactly supported radial basis function(CS-RBF),in the process of solving RDE,CS-RBF is used to interpolate the level set function to get the improved RDE,and the diffusion term in RDE is modified to make the structure boundary smoother.The examples show that the combination of CS-RBF and RDE can keep the advantages of RDE and improve the optimization efficiency.Thirdly,a new method to deal with volume constraints is proposed by combining fuzzy adaptive PID control algorithm with structural topology optimization.According to the given initial PID parameters,the updated formula of Lagrange multiplier is obtained,so that the value of Lagrange multiplier can be adjusted,which obtains better convergence effect of volume constraint.An example is used to verify the influence of different initial PID parameters on the process of structural topology optimization.After that,the proposed level set method is applied to the topology optimization of two-dimensional structure,which proves that the proposed algorithm has greatly improved the computational efficiency and the numerical stability.Finally,the parameterized level set method(PLSM)proposed in this paper is extends to the three-dimensional field,and the problem of minimum structural compliance under volume constraints is studied.The mathematical model of three-dimensional structure topology optimization is established,and the updating formula of topological derivative and level set function are derived.The feasibility and effectiveness of the proposed method in threedimensional structural topology optimization are verified by several examples.