# Proportional Topology Optimization algorithm for two-scale concurrent design of lattice structures

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https://doi.org/10.15625/0866-7136/18368## Keywords:

proportional topology optimization, PTO, two-scale, lattice structures## Abstract

In this paper, the Proportional Topology Optimization (PTO) algorithm is extended for the two-scale concurrent topology optimization, in which both the structure and material cellular micro-structure are subject to design. PTO was originally developed on the concept that the amount of material being distributed to an element would be proportional to the contribution of that element in the objective function. Sensitivity analysis is not required. In a two-scale concurrent topology optimization problem, two sets of design variables are defined, one for macro-structure and one for micro-structure. Here, the objective function is reformulated such that the contribution of each micro-scale design variable can be determined, facilitating the employment of PTO. The macroscopic effective elastic tensor is evaluated by the energy-based homogenization method (EBHM), providing a link between micro-structure and macro-structure. Feasibility and efficiency of the proposed PTO approach are demonstrated via several benchmark examples of both two and three dimensional structures.

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