Topology optimization of structural elements using gradient method with account for the material’s structural inhomogeneity
https://doi.org/10.22405/2226-8383-2022-23-4-308-326
Abstract
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.
About the Authors
Vladimir Anatol’evich Levin
Lomonosov Moscow State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Konstantin Moiseevich Zingerman
Tver State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Anatoliy Victorovich Vershinin
Lomonosov Moscow State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Petr Andreevich Vasilyev
Lomonosov Moscow State University
Russian Federation
Russian Federation
postgraduate student
References
1. Bendsøe M.P. Optimal shape design as a material distribution problem // Structural
2. Optimization. 1989. 1 (4), pp. 193–202.
3. Bendsøe M.P., Sigmund O. Topology Optimization. Theory, Methods and Applications. Berlin:
4. Springer, 2003.
5. Sysoeva V.V., Chedrik V.V. Algorithms for structural topology opttimization // TsAGI Science
6. Journal. 2011. V. XLII, No. 2. P. 91–102. (in Russian).
7. Oganesyan P.A., Shevtsov S.N. Topology designs optimization in Abaqus package // Izvestia of
8. Samara Scientific Center of the Russian Academy of Sciences 2014. V. 16, No. 6(2). P. 543–549.
9. Borovikov A.A., Tushev O.N. Development of a spacecraft load bearing structures using
10. topology optimization for two versions of manufacturing technologies // Engineering Journal:
11. Science and Innovation. 2018. No. 9. P. 1–13. (in Russian).
12. Research Report: Generative Design and Topology Optimization: In-Depth Look
13. at the Two Latest Design Technologies [Электронный ресурс] URL: https://www.
14. engineering.com/ResourceMain.aspx?resid=826. (дата обращения: 22.10.2019).
15. Abbey T. Topology Optimization // Digital Engineering. June 2017. [Электронный ресурс]
16. URL: https://www.digitalengineering247.com/article/topology-optimization/
17. Abbey T. Topology Optimization Methods. Digital Engineering. September 2017. [Элек-
18. тронный ресурс] URL: https://www.digitalengineering247.com/article/topology-optimizationmethods
19. Abbey T. Topology Optimization. Part 3. Digital Engineering. October 2017. [Электронный
20. ресурс] URL: https://www.digitalengineering247.com/article/topology-optimization-2
21. Levin V.A., Vershinin A.V. Numerical methods. Parallel computing. (Nonlinear computational
22. strength mechanics. Series of monographs in 5 volumes. Ed. by V.A. Levin. V. 2). — Moscow:
23. Fizmatlit, 2015. (in Russian).
24. Vassilevski Yu.V., Danilov A.A., Lipnikov K.N., Chugunov V.N. Automatized technologies
25. of generation of unstructured computational meshes . (Nonlinear computational strength
26. mechanics. Series of monographs in 5 volumes. Ed. by V.A. Levin. V. 4). — Moscow: Fizmatlit,
27. (in Russian).
28. Charara M., Vershinin A., Sabitov D., Pekar G. SEM wave propagation in complex media with
29. tetrahedral to hexahedral mesh. [73-rd European Association of Geoscientists and Engineers
30. Conference and Exhibition] Vienna, Austria, 2011. P. 41—45.
31. Kukushkin A.V., Konovalov D.A., Vershinin A.V., Levin V.A. Numerical simulation in CAE
32. Fidesys of bonded contact problems on non-conformal meshes. Journal of Physics: Conference
33. Series. — 2019. — V. 1158, No. 2. Art. 032022.
34. Koga J.-I., Koga J., Homma S. Checkerboard problem to topology optimization of continuum
35. structures // Computational Engineering, Finance, and Science. 2013. arXiv:1309.5677 [cs.CE].
36. Boldyrev A.V. Topological optimization of load-bearing constructions on the basis of variable
37. density model // Izvestia of Samara Scientific Center of the Russian Academy of Sciences 2011.
38. V. 13. No. 1–3. P. 670–673. (in Russian).
39. Brackett D., Ashcroft I., Hague R. Topology optimization for additive manufacturing // Proc.
40. SFF Symp. Austin Texas. — 2011. — P. 348—362.
41. Cheng L., Liang X., Bai J., Chen Q., Lemon J., To A. On utilizing topology optimization
42. to design support structure to prevent residual stress induced build failure in laser powder
43. bed metal additive manufacturing // Additive Manufacturing. 2019. V. 27. P. 290–304. Doi:
44. 1016/j.addma.2019.03.001.
45. Diaz A.R., Bendsøe M.P. Shape optimization of structures for multiple loading conditions
46. using a homogenization method // Structural Optimization. — 1992. — V. 4. — P. 17–22.
47. Lizin V.T., Pyatkin V.A. Design of thin-walled structures. 4-th ed. Moscow: Mashinostroenie,
48. 448 p. (in Russian).
49. Badriev I.B., Banderov V.V., Zadvornov O.A. On the solving of equilibrium problem for the
50. soft network shell with a load concentrated at the point // PNIPU Mechanics Bulletin. – 2013.
51. – No. 3. – P. 16–34.
52. Badriev I.B., Korablev A.I., Makarov M.V., Smirnova E.V. Mathematical simulation of the
53. bending problem of the hinged sandwich plate in physically nonlinear statement // Journal of
54. Physics: Conference Series. — 2019. — V. 1158, No. 2. — Art. 022013.
55. Badriev I.B., Banderov V.V., Paimushin V.N., Gazizullin R.K. Contact interaction of the
56. plate with supporting deformable bases under the conditions of bending // Journal of Physics:
57. Conference Series. — 2019. — V. 1158, No. 2. — Art. 022014.
58. Nikishkov G.P., Vershinin A.V., Nikishkov Y.G. Mesh-independent equivalent domain integral
59. method for j-integral evaluation // Advances in Engineering Software. –— 2016. — V. 100. —–
60. P. 308-–318.
61. Levin V.A., Zingerman K.M., Vershinin A.V., Yakovlev M.Ya. Numerical analysis of effective
62. mechanical properties of rubber-cord composites under finite strains // Compos. Struct. —
63. — V. 131. –– P. 25—36.
64. Vershinin A.V., Levin V.A., Zingerman K.M., Sboychakov A.M., Yakovlev M.Ya. Software for
65. estimation of second order effective material properties of porous samples with geometrical and
66. physical nonlinearity accounted for // Adv. Eng. Softw. 2015. V. 86. –– P. 80–84.
67. Levin V.A., Zingermann K.M. Effective Constitutive Equations for Porous Elastic Materials at
68. Finite Strains and Superimposed Finite Strains // Journal of Applied Mechanics, Transactions
69. ASME. – 2003. – V. 70 (6). – P. 809-816.
70. Konovalov D. A., Yakovlev M. Ya. Numerical estimation of effective elastic properties of
71. elastomer composites under finite strains using spectral element method with CAE Fidesys //
72. Chebyshevskii sbornik. 2017. V. 18, No. 3. P. 316-–329. (in Russian).
73. Yakovlev M.Ya, Lukyanchikov I.S., Levin V.A., Vershinin A.V., Zingerman K.M. Calculation
74. of the effective properties of the prestressed nonlinear elastic heterogeneous materials under
75. finite strains based on the solutions of the boundary value problems using finite element method
76. // Journal of Physics: Conference Series. 2019. V. 1158, No. 4. Art. 042037.
77. Levin V., Vdovichenko I., Vershinin A., Yakovlev M., Zingerman K. Numerical Estimation
78. of Effective Mechanical Properties for Reinforced Plexiglas in the Two-Dimensional Case //
79. Modelling and Simulation in Engineering, Art. 9010576, 2016. – 10 p.
80. Levin V.A., Vdovichenko I.I., Vershinin A.V., Yakovlev M.Y., Zingerman K.M. An approach
81. to the computation of effective strength characteristics of porous materials // Letters on
82. materials. 2017. V. 7, No. 4. –– P. 452–454.
83. Eglit M.E. Lectures on foundations of continuum mechanics. Moscow: LENAND, 2016. — 208
84. p. (in Russian).
85. Morozov E.M., Levin V.A., Vershinin A.V. Strength analysis: Fidesys in the hands of an
86. engineer. Moscow: LENAND, 2015. – 408 p. (in Russian).
87. Konovalov D., Vershinin A., Zingerman K., Levin V. The implementation of spectral element
88. method in a CAE system for the solution of elasticity problems on hybrid curvilinear meshes
89. // Modelling and Simulation in Engineering. –— 2017. — V. 2017. Art. 1797561. DOI:
90. 1155/2017/1797561.
91. Karpenko V.S., Vershinin A.V., Levin V.A., Zingerman K.M. Some results of mesh
92. convergence estimation for the spectral element method of different orders in fidesys industrial
93. package // IOP Conference Series: Materials Science and Engineering. –— 2016. — V. 158. Art.
94.
95. Kukushkin A.V., Konovalov D.A., Vershinin A.V., Levin V.A. Numerical simulation in CAE
96. Fidesys of bonded contact problems on non-conformal meshes // Journal of Physics: Conference
97. Series. –— 2019. –— V. 1158, No. 2. —– Art. 032022.
98. Kingma D.P., Ba J.L. Adam: a Method for Stochastic Optimization // Proc. 3-rd International
99. Conference on Learning Representations. 2015. arXiv:1412.6980 [cs.LG]. P. 1—13
100. Ogden R. Non-linear elastic deformations. Ellis Horwood: Chichester, 1984.
101. Lurie A.I. Nonlinear Theory of Elasticity. North-Holland, Amsterdam, 1990.
102. Kislitsyn V.D., Mokhireva K.A., Shadrin V.V., Svistkov A.L. Research and modeling of
103. viscoelastic behavior of elastomeric nanocomposites// PNRPU Mechanics Bulletin, 2021,
104. (2), p. 76–87.
105. Levin V.A., Zubov L.M., Zingerman K.M. An exact solution for the problem of flexure of a
106. composite beam with preliminarily strained layers under large strains// International Journal
107. of Solids and Structures. 2015. V. 67–68. P. 244–249.
108. Zingerman K.M., Levin V.A. Redistribution of finite elastic strains after the formation of
109. inclusions. Approximate analytical solution// Journal of Applied Mathematics and Mechanics.
110. V. 73 (6). P. 710–721.
111. Levin V.A., Zingerman K.M. A class of methods and algorithms for the analysis of successive
112. origination of holes in a pre-stressed viscoelastic body. Finite strains// Communications in
113. Numerical Methods in Engineering. 2008. V. 24 (12). P. 2240–2251.
Review
For citations:
Levin V.A., Zingerman K.M., Vershinin A.V., Vasilyev P.A. Topology optimization of structural elements using gradient method with account for the material’s structural inhomogeneity. Chebyshevskii Sbornik. 2022;23(4):308-326. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-4-308-326
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