Geodesic
Topology optimization of structural elements using gradient method with account for the material's structural inhomogeneity
Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 308-326

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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 appearance of 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.
Keywords: topological optimization, structurally inhomogeneous material, engineering strength analysis, Fidesys, gradient descent. 
@article{CHEB_2022_23_4_a25,
     author = {V. A. Levin and K. M. Zingerman and A. V. Vershinin and P. A. Vasilyev},
     title = {Topology optimization of structural elements using gradient method with account for the material's structural inhomogeneity},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {308--326},
     publisher = {mathdoc},
     volume = {23},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a25/}
}
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V. A. Levin; K. M. Zingerman; A. V. Vershinin; P. A. Vasilyev. Topology optimization of structural elements using gradient method with account for the material's structural inhomogeneity. Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 308-326. http://geodesic.mathdoc.fr/item/CHEB_2022_23_4_a25/