A Hierarchical Double Penalty Method of Gray-Scale Elements for SIMP in Topology Optimization

In the topology optimization for continuum structures,the sensitivity filtering methods can effectively eliminate the problem of numerical instability,but it is easy to cause the gray-scale diffusion phenomenon at the boundary of the final optimized structure.In order to obtain a crisp topological configuration,a hierarchical double penalty method of gray-scale elements for SIMP(solid isotropic microstructures with penalization)is proposed.The method applies a penalty factor in the SIMP method without sensitivity filter to modify sensitivity of the element from the standard SIMP method,which accelerates the approximation of the intermediate density units to the discrete state of 0 or 1.To further speed up the process,the method is implemented in a hierarchical mesh subdivision strategy.Starting from a coarse finite element mesh and using the element density equivalent mapping method,the solution with the coarse mesh as a starting input for the same problem but with a refined mesh.By reducing the computational cost of the units in the optimization process,the convergence rate of the optimization process is improved while achieving the distinct topology structure.Different methods are used to solve the MBB beam problem.The number of intermediate density elements contained in the optimized structures obtained by different methods and their time consumption in the entire optimization process were compared,respectively.The mesh dependency of this method is verified by using cantilever beam examples under different mesh divisions.The results show that the SIMP algorithm combined with hierarchical double penalty obtains a topological configuration with clear boundaries and improves the convergence rate while retaining the stability of the original solution.