Structural Topology Optimization Based On Proportional Topology Optimization Method

Topology optimization design is a conceptual method for structural pre-design in which the optimal layout of structure is obtained by topology optimization algorithm within the given parameters and constraints.In this paper,the continuum structure is considered as the investigating object,with the aim of finding the optimal distribution of single-material and multi-material with the minimum compliance in the design domain.Considering the deficiency of power law penalty method,a density interpolation approach based on the logistic function is introduced in the present study.This approach can not only establish the relationship between the material density and Young's modulus more reasonably,but also effectively realize the polarization of the intermediate density elements.Furthermore,a large number of high-relative-density elements are participating in the optimization process which can help algorithm to find the optimal topology design.Moreover,an optimization algorithm which we called proportional topology optimization(PTO)has been studied in detail in this paper.When the PTO algorithm is used to solve minimum compliance problem,the design variables are assigned to elements proportionally by the value of compliance during optimization process.Based on this algorithm,a dynamic historical balance coefficient is introduced to reduce the objective function value and be away from unstable values.It is worth to mentioning that PTO algorithm does not incorporate sensitivities,the complication associated with sensitivities can be avoided by the modified interpolation scheme in conjunction with PTO algorithm,which can improve the efficiency and accuracy of the algorithm.For multi-material topology optimization,based on the recursive multiphase material interpolation(RMMI)model,a modified multiphase material interpolation scheme is proposed.We aim at the problem of material boundary overlaps caused by the intermediate density in the original algorithm,a density-filter based simple Heaviside threshold function combined with the modified interpolation is introduced in this work to obtain clear 0/1 optimal topology design.Moreover,in order to solve the problem of long solution time in topology optimization due to the increase of material phase,the proportional topology optimization is extended to multi-material topology optimization,which can improve the efficiency of multi-phase material topology optimization by distributing the materials proportionally of the compliance to each phase material.Then,a multi-material topology optimization method based on ordered SIMP interpolation is proposed,in which can solve multi-phase material topology optimization problem without increasing any design variables.In order to make up for the insufficiency of punishment for the intermediate density element in each material by the power function,a continuation method of penalty factor is proposed in this paper.It can concentrate the elemental density value toward the solid material density value by increasing the penalty factor gradually in the topology optimization process.Moreover,in order to obtain more accurate topology design,the PTO algorithm is used to replace the original optimization criterion(OC)method to solve each material phase.Finally,in order to extend the optimization algorithm to manufacturing problems in practical engineering application,a modification of optimization criterion method(OC)is proposed.The optimization result is approximated to solid or void element in the process of updating design variables by a gray scale suppression operator which is based on exponential function.Moreover,to verify the effectiveness of the algorithm in the multi-material topology optimization,the alternating active-phase algorithm is combined to solve multi-material problem.The effectiveness and feasibility of the proposed method are demonstrated by several typical numerical examples of multi-material topology optimization,in which the optimal design with distinct boundaries can be obtained.