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A method for setting the objective function in the problem of the topological optimization of three-dimensional structures using a density gradient

Vladimir A. Levin, Anatoly V. Vershinin, Konstantin M. Zingerman and Petr A. Vasilyev

Vol. 11 (2023), No. 4, 451–480
Abstract

We give a method and algorithm that optimizes the shape of structural elements. The goal is to minimize the mass of the structural element and to prevent stresses from exceeding the specified limit.

The proposed approach is based on an iterative finite-element analysis of structural elements. At each iteration, the material is redistributed within the design space using the gradient method, which solves the problem of minimizing a specially designed target function. This function includes three terms, one of which depends on the density gradient. The aim of introducing this target function is to improve the robustness of optimization procedure and to avoid the occurrence of checkerboard structures.

The developed algorithm is implemented as a software module that performs the topological optimization of structures. Several model problems are described.

A comparative analysis of the results obtained using this module and the results obtained earlier shows that the proposed approach revealed their common features, and in some cases eliminated their shortcomings. The sharpness of boundaries of optimized structures is improved.

Keywords
topological optimization, finite-element analysis, stress–strain state, setting the objective function, density gradient, CAE system
Mathematical Subject Classification
Primary: 74P15, 74S05
Milestones
Received: 26 June 2022
Revised: 4 December 2022
Accepted: 1 February 2023
Published: 1 December 2023

Communicated by Martin Ostoja-Starzewski
Authors
Vladimir A. Levin
Faculty of Mechanics and Mathematics
Lomonosov Moscow State University
Moscow
Russia
Anatoly V. Vershinin
Faculty of Mechanics and Mathematics
Lomonosov Moscow State University
Moscow
Russia
Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences
Moscow
Russia
Konstantin M. Zingerman
Department of Applied Mathematics and Cybernetics
Tver State University
Tver
Russia
Institute of Nuclear Physics and Engineering
National Research Nuclear University MEPhI
Moscow
Russia
Petr A. Vasilyev
Faculty of Mechanics and Mathematics
Lomonosov Moscow State University
Moscow
Russia