chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2022-23-4-308-326cheb-1396Research ArticleИстория математики и приложенийСomputer scienceТопологическая оптимизация элементов конструкций с учетом структурной неоднородности материала с использованием градиентного методаTopology optimization of structural elements using gradient method with account for the material’s structural inhomogeneityЛевинВладимир АнатольевичLevinVladimir Anatol’evich

доктор физико-математических наук, профессор

doctor of physical and mathematical sciences, professor

v.a.levin@mail.ruЗингерманКонстантин МоисеевичZingermanKonstantin Moiseevich

доктор физико-математических наук, профессор

 doctor of physical and mathematical sciences, professor

zingerman@rambler.ruВершининАнатолий ВикторовичVershininAnatoliy Victorovich

доктор физико-математических наук, профессор

doctor of physical and mathematical sciences, professor

versh1984@mail.ruВасильевПетр АндреевичVasilyevPetr Andreevich

аспирант

postgraduate student

petr.vasilyev.1996@yandex.ruМосковский государственный университет им. М. В. ЛомоносоваРоссияLomonosov Moscow State UniversityRussian FederationТверской государственный университетРоссияTver State UniversityRussian Federation202216012023234308326Copyright © Левин В.А., Зингерман К.М., Вершинин А.В., Васильев П.А., 20222022Левин В.А., Зингерман К.М., Вершинин А.В., Васильев П.А.Levin V.A., Zingerman K.M., Vershinin A.V., Vasilyev P.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/1396

В данной работе представлен и реализован алгоритм, выполняющий топологическую оптимизацию распределения массы двумерного тела под нагрузкой. Конечной целью алгоритма является минимизация веса тела при ограничении на максимальные напряжения в его точках. За основу взята идея переменной плотности, а также алгоритм BESO, добавляющий и удаляющий элементы в зависимости от узловых напряжений.Алгоритм использует метод конечных элементов и представляет из себя итеративный процесс, на каждом шаге которого сначала происходит вычисление напряжений в теле при помощи CAE Fidesys, а затем результаты расчёта анализируются. По результатам анализа модули Юнга в узлах конечноэлементной сетки изменяются, чтобы отражать новое распределение массы, скорректированное для лучшего соответствия нагрузкам. Особенностью используемого подхода является использование целевой функции со слагаемым, которое представляет из себя сумму квадратов разностных производных плотности по четырём направлениям. Это позволяет избежать резких изменений плотности и возникновения решётчатых структур на ранних итерациях. Для определения плотностей на каждой итерации используется модификация метода градиентного спуска Adam.Реализованный алгоритм верифицирован на ряде тестовых примеров для плоских статических задач теории упругости. Приведены результаты расчетов, выполнено сравнение срезультатами, полученными другими авторами. Для одной из задач представлены результаты расчетов на разных сетках, которые позволяют сделать вывод о сеточной сходимостиалгоритма.

This paper presents and implements an algorithm that performs topological optimization of the mass distribution of a two-dimensional body under load. The ultimate goal of the algorithm is to minimize body weight under stress constraints at the points of the body. The approach is based on the idea of variable density and the BESO algorithm that adds and deletes elements depending on stresses.The algorithm uses the finite element method and is an iterative process. At each iteration the stresses in the body are calculated using CAE Fidesys, and then the calculation results are analyzed. According to the analysis, Young’s moduli at the nodes of the finite element mesh are changed to reflect new mass distribution adjusted for better compliance with loads.The specific feature of the used approach is utilization of objective function with the special term. This term is the sum of the squares of the differential derivatives of density in four directions. This feature permits one to avoid sharp changes in density and the appearanceof lattice structures in the early iterations. The Adam gradient method is used to determine densities at each iteration.The implemented algorithm is verified on a number of test cases for plane static problems of the theory of elasticity. The results of computations are presented. A comparison is made with the results obtained by other authors. For one of the problems, the results of calculations on different grids are given. These results allows one to conclude about the grid convergence of the algorithm.

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The authors declare that there are no conflicts of interest present.