Topology optimization of structural elements using gradient method with account for the material's structural inhomogeneity

V. A. Levin ; K. M. Zingerman ; A. V. Vershinin ; P. A. Vasilyev

Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 308-326.

Résumé

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.

Détail

Keywords: topological optimization, structurally inhomogeneous material, engineering strength analysis, Fidesys, gradient descent.

Bibliographie

[1] Bendsøe M.P., “Optimal shape design as a material distribution problem”, Structural Optimization, 1:4 (1989), 193–202 | DOI

[2] Bendsøe M.P., Sigmund O., Topology Optimization. Theory, Methods and Applications, Springer, Berlin, 2003 | MR

[3] Sysoeva V.V., Chedrik V.V., “Algoritmy optimizatsii topologii silovykh konstruktsii”, Uchenye zapisi TsAGI, XLII:2 (2011), 91–102

[4] Oganesyan P.A., Shevtsov S.N., “Optimizatsiya topologii konstruktsii v pakete Abaqus”, Izvestiya Samarskogo nauchnogo tsentra Rossiiskoi akademii nauk, 16:6(2) (2014), 543–549

[5] Borovikov A.A., Tushev O.N., “Razrabotka silovoi konstruktsii kosmicheskogo apparata s ispolzovaniem topologicheskoi optimizatsii dlya dvukh variantov tekhnologii izgotovleniya”, Inzhenernyi zhurnal: nauka i innovatsii, 2018, no. 9, 1–13

[6] Research Report: Generative Design and Topology Optimization: In-Depth Look at the Two Latest Design Technologies (data obrascheniya: 22.10. 2019) https://www.engineering.com/ResourceMain.aspx?resid=826

[7] Abbey T., Topology Optimization, Digital Engineering, , June 2017 https://www.digitalengineering247.com/article/topology-optimization/

[8] Abbey T., Topology Optimization Methods, Digital Engineering, , September 2017 https://www.digitalengineering247.com/article/topology-optimization-methods

[9] Abbey T., Topology Optimization. Part 3, Digital Engineering, , October 2017 https://www.digitalengineering247.com/article/topology-optimization-2

[10] Levin V.A., Vershinin A.V., Nelineinaya vychislitelnaya mekhanika prochnosti, Tsikl monografii v 5 tomakh, v. 2, Chislennye metody. Parallelnye vychisleniya na EVM, ed. V.A. Levin, Fizmatlit, M., 2015

[11] Vasilevskii Yu.V., Danilov A.A., Lipnikov K.N., Chugunov V.N., Nelineinaya vychislitelnaya mekhanika prochnosti, Tsikl monografii v 5 tomakh, v. 4, Avtomatizirovannye tekhnologii postroeniya nestrukturirovannykh raschetnykh setok, ed. V.A. Levin, Fizmatlit, M., 2015

[12] Charara M., Vershinin A., Sabitov D., Pekar G., “SEM wave propagation in complex media with tetrahedral to hexahedral mesh”, 73-rd European Association of Geoscientists and Engineers Conference and Exhibition (Vienna, Austria, 2011), 41–45

[13] Kukushkin A.V., Konovalov D.A., Vershinin A.V., Levin V.A., “Numerical simulation in CAE Fidesys of bonded contact problems on non-conformal meshes”, Journal of Physics: Conference Series, 1158:2 (2019), 032022 | DOI

[14] Koga J.-I., Koga J., Homma S., “Checkerboard problem to topology optimization of continuum structures”, Computational Engineering, Finance, and Science, 2013, arXiv: 1309.5677 [cs.CE]

[15] Boldyrev A.V., “Topologicheskaya optimizatsiya silovykh konstruktsii na osnove modeli peremennoi plotnosti”, Izvestiya Samarskogo nauchnogo tsentra Rossiiskoi akademii nauk, 13:1–3 (2011), 670–673

[16] Brackett D., Ashcroft I., Hague R., “Topology optimization for additive manufacturing”, Proc. SFF Symp. Austin Texas, 2011, 348–362

[17] Cheng L., Liang X., Bai J., Chen Q., Lemon J., To A., “On utilizing topology optimization to design support structure to prevent residual stress induced build failure in laser powder bed metal additive manufacturing”, Additive Manufacturing, 27 (2019), 290–304 | DOI

[18] Diaz A.R., Bendsøe M.P., “Shape optimization of structures for multiple loading conditions using a homogenization method”, Structural Optimization, 4 (1992), 17–22 | DOI

[19] Lizin V.T., Pyatkin V.A., Proektirovanie tonkostennykh konstruktsii, 4-e izd., Mashinostroenie, M., 2003, 448 pp.

[20] Badriev I.B., Banderov V.V., Zadvornov O.A., “On the solving of equilibrium problem for the soft network shell with a load concentrated at the point”, PNIPU Mechanics Bulletin, 2013, no. 3, 16–34 | MR

[21] Badriev I.B., Korablev A.I., Makarov M.V., Smirnova E.V., “Mathematical simulation of the bending problem of the hinged sandwich plate in physically nonlinear statement”, Journal of Physics: Conference Series, 1158:2 (2019), 022013 | DOI

[22] Badriev I.B., Banderov V.V., Paimushin V.N., Gazizullin R.K., “Contact interaction of the plate with supporting deformable bases under the conditions of bending”, Journal of Physics: Conference Series, 1158:2 (2019), 022014 | DOI

[23] Nikishkov G.P., Vershinin A.V., Nikishkov Y.G., “Mesh-independent equivalent domain integral method for j-integral evaluation”, Advances in Engineering Software, 100 (2016), 308–318 | DOI

[24] Levin V.A., Zingerman K.M., Vershinin A.V., Yakovlev M.Ya., “Numerical analysis of effective mechanical properties of rubber-cord composites under finite strains”, Compos. Struct., 131 (2015), 25–36 | DOI

[25] Vershinin A.V., Levin V.A., Zingerman K.M., Sboychakov A.M., Yakovlev M.Ya., “Software for estimation of second order effective material properties of porous samples with geometrical and physical nonlinearity accounted for”, Adv. Eng. Softw., 86 (2015), 80–84 | DOI

[26] Levin V.A., Zingermann K.M., “Effective Constitutive Equations for Porous Elastic Materials at Finite Strains and Superimposed Finite Strains”, Journal of Applied Mechanics, Transactions ASME, 70:6 (2003), 809–816 | DOI | Zbl

[27] Konovalov D.A., Yakovlev M.Ya., “O chislennoi otsenke effektivnykh uprugikh kharakteristik elastomernykh kompozitov pri konechnykh deformatsiyakh s ispolzovaniem metoda spektralnykh elementov s pomoschyu CAE Fidesys”, Chebyshevskii sbornik, 18:13 (2017), 316–329

[28] Yakovlev M.Ya, Lukyanchikov I.S., Levin V.A., Vershinin A.V., Zingerman K.M., “Calculation of the effective properties of the prestressed nonlinear elastic heterogeneous materials under finite strains based on the solutions of the boundary value problems using finite element method”, Journal of Physics: Conference Series, 1158:4 (2019), 042037 | DOI

[29] Levin V., Vdovichenko I., Vershinin A., Yakovlev M., Zingerman K., “Numerical Estimation of Effective Mechanical Properties for Reinforced Plexiglas in the Two-Dimensional Case”, Modelling and Simulation in Engineering, 2016, 9010576, 10 pp.

[30] Levin V.A., Vdovichenko I.I., Vershinin A.V., Yakovlev M.Y., Zingerman K.M., “An approach to the computation of effective strength characteristics of porous materials”, Letters on materials, 7:4 (2017), 452–454 | DOI

[31] Eglit M. E., Lektsii po osnovam mekhaniki sploshnykh sred, Izd. stereotip, LENAND, M., 2016, 208 pp.

[32] Morozov E.M., Levin V.A., Vershinin A.V., Prochnostnoi analiz. Fidesis v rukakh inzhenera, URSS, M., 2015, 408 pp.

[33] Konovalov D., Vershinin A., Zingerman K., Levin V., “The implementation of spectral element method in a CAE system for the solution of elasticity problems on hybrid curvilinear meshes”, Modelling and Simulation in Engineering, 2017 (2017), 1797561 | DOI | MR

[34] Karpenko V.S., Vershinin A.V., Levin V.A., Zingerman K.M., “Some results of mesh convergence estimation for the spectral element method of different orders in fidesys industrial package”, IOP Conference Series: Materials Science and Engineering, 158 (2016), 012049 | DOI

[35] Kukushkin A.V., Konovalov D.A., Vershinin A.V., Levin V.A., “Numerical simulation in CAE Fidesys of bonded contact problems on non-conformal meshes”, Journal of Physics: Conference Series, 1158:2 (2019), 032022 | DOI

[36] Kingma D.P., Ba J.L., “Adam: a Method for Stochastic Optimization”, Proc. 3-rd International Conference on Learning Representations, 2015, 1–13, arXiv: 1412.6980 [cs.LG]

[37] Ogden R., Non-linear elastic deformations, Ellis Horwood, Chichester, 1984 | MR

[38] Lurie A.I., Nonlinear Theory of Elasticity, North-Holland, Amsterdam, 1990 | MR | Zbl

[39] Kislitsyn V.D., Mokhireva K.A., Shadrin V.V., Svistkov A.L., “Research and modeling of viscoelastic behavior of elastomeric nanocomposites”, PNRPU Mechanics Bulletin, 2021, no. 2, 76–87 | DOI

[40] Levin V.A., Zubov L.M., Zingerman K.M., “An exact solution for the problem of flexure of a composite beam with preliminarily strained layers under large strains”, International Journal of Solids and Structures, 67–68 (2015), 244–249 | DOI | MR

[41] Zingerman K.M., Levin V.A., “Redistribution of finite elastic strains after the formation of inclusions. Approximate analytical solution”, Journal of Applied Mathematics and Mechanics, 73:6 (2009), 710–721 | DOI | MR | Zbl

[42] Levin V.A., Zingerman K.M., “A class of methods and algorithms for the analysis of successive origination of holes in a pre-stressed viscoelastic body. Finite strains”, Communications in Numerical Methods in Engineering, 24:12 (2008), 2240–2251 | DOI | MR | Zbl