№2 2022


Designing various-purpose subassemblies based on topological optimization


D.V. Sorokin, L.A. Babkina, O.V. Brazgovka


Reshetnev Siberian State University of Science and Technology
Krasnoyarsk, Russian Federation


In recent decades, theory, algorithms and topological optimization technologies have been actively developing to solve problems in various fields. This technique is widely used and used in the design of subassemblies in engineering tasks. Of particular interest is the implementation of topological optimization methods in the aerospace industry. The main result of topological optimization is the determination of the effective distribution of material(s) in a part. The article provides an overview of recent developments related to the design of broad-purpose products, including for space and rocket technology based on topological optimization. The main algorithms for solving topological optimization problems are briefly listed. A separate emphasis in the review is placed on the opening qualitative new opportunities in modeling cellular and mesh structures, products from composite materials and multimaterials. Examples of topological optimization solutions that have already become classic are given. Many design results are closely related to the introduction of various modern additive manufacturing technologies. The review concludes with topical issues that will ensure future growth potential and prospects for the application of the entire range of design and additive production technologies in the aerospace industry.


topological optimization, geometric modeling, cellular structure, engineering analysis, composite, additive technologies


[1] Bendsoe M. P., Kikuchi N. Generating optimal topologies in structural design using a homogenization method // Computer Methods in Applies Mechanics and Engineering, 1988, vol. 71, issue 2, рр. 197–224. doi: 10.1016/0045-7825(88)90086-2.

[2] Bendsoe M. P. Optimal shape design as a material distribution problem // Structural optimization, 1989, vol. 1, рр. 193–202. doi: 10.1007/BF01650949.

[3] Bendsoe M. P. Optimization of Structural Topology, Shape and Material. Berlin, Springer, 1995. doi: 10.1007/978-3-662-03115-5.

[4] Bendsoe M. P., Sigmund O. Topology optimization: theory, methods, and applications. Berlin, Heidelberg, New York, Barcelona, Hong Kong, London, Milan, Paris, Tokyo, Springer Science & Business Media, 2003, 370 p.

[5] Topology Optimization Guide. Available at: (accessed 13.04.2022).

[6] Structures Optimization for Additive Manufacturing. Available at: (accessed 13.04.2022).

[7] Giga-voxel computational morphogenesis for structural design. Available at: (accessed 13.04.2022).

[8] Bendsoe M. P., Sigmund O. Topology optimization: theory, methods, and applications. Berlin, Heidelberg, New York, Barcelona, Hong Kong, London, Milan, Paris, Tokyo, Springer Science & Business Media, 2003, 370 p.

[9] Huang X., Xie M. Evolutionary Topology Optimization of Continuum Structures: Methods and Applications. 2010, 240 p. doi: 10.1002/9780470689486.

[10] Xiaoying Yang B. E. Bi-directional Evolutionary Method for Stiffness and Displacement Optimisation. Thesis of the degree of Master of Engineering. Australia, Melbourne. 1999.

[11] Xia L., Xia Q., Huang X., Xie Y. M. Bi-directional Evolutionary Structural Optimization on Advanced Structures and Materials // Archives of Computational Methods in Engineering, 2016, vol. 25, pp. 437–478. doi: 10.1007/s11831-016-9203-2.

[12] Rozvany G. The SIMP method in topology optimization – Theoretical background, advantages and new applications. 2012. doi: 10.2514/6.2000-4738.

[13] Zhang W., Zhong W., Guo X. An explicit length scale control approach in SIMP-based topology optimization // Computer Methods in Applied Mechanics and Engineering, 2014, vol. 282, pp. 71–86. doi: 10.1016/j.cma.2014.08.027.

[14] Cheng W., Wang H., Zhang M., Du R. Improved proportional topology optimization algorithm for minimum volume problem with stress constraints // Engineering Computations, 2020. doi: 10.1108/EC-12-2019-0560.

[15] Prager W. Optimality criteria in structural design // Proc. Natl. Acad. Sci., 1968, vol. 3, issue 3, pp. 794–794. doi: 10.1073/pnas.61.3.794.

[16] Song L., Gao T., Tang L., Du X., Zhu J., Lin Y., Shi G., Liu H., Zhou G., Zhang W. An all-movable rudder designed by thermo-elastic topology optimization and manufactured by additive manufacturing // Computers and Structures, 2021, vol. 243. doi: 10.1016/j.compstruc.2020.106405.

[17] Li L., Liu Ch., Zhang W., Du Z., Guo X. Combined model-based topology optimization of stiffened plate structures via MMC approach // International Journal of Mechanical Sciences, 2021, vol. 208. doi: 10.1016/j.ijmecsci.2021.106682.

[18] Li B., Liu H., Yang Z., Zhang J. Stiffness design of plate/shell structures by evolutionary topology optimization // Thin-Walled Structures, 2019, vol. 141, pp. 232–250. doi: 10.1016/j.tws.2019.04.012.

[19] Sohouli A., Kefal A., Abdelhamid A., Yildiz M., Suleman A. Continuous density-based topology optimization of cracked structures using peridynamics // Structural and Multidisciplinary Optimization, 2020, vol. 62, pp. 2375–2389. doi: 10.1007/s00158-020-02608-1.

[20] Habibian A., Sohoulia A., Kefal A., Nadler B., Yildiz M., Suleman A. Multi-material topology optimization of structures with discontinuities using Peridynamics // Composite Structures, 2021, vol. 258, 113345. doi: 10.1016/j.compstruct.2020.113345.

[21] Gao J., Xiao M., Zhou M., Gao L. Isogeometric topology and shape optimization for composite structuresusing level-sets and adaptive Gauss quadrature // Composite Structures, 2022, vol. 285, 115263.

[22] Lieu Q. X., Lee J. A multi-resolution approach for multi-material topology optimization based on isogeometric analysis // Computer Methods in Applied Mechanics and Engineering, 2017, vol. 323, pp. 272–302.

[23] Sandersa E. D., Aguilob M. A., Paulinoa G. H. Multi-material continuum topology optimization with arbitrary volume and mass constraints // Computer Methods in Applied Mechanics and Engineering, 2018, vol. 340, pp. 798–823.

[24] Robbins J., Owen S. J., Clark B. W., Voth T. E. An efficient and scalable approach for generating topologically optimized cellular structures for additive manufacturing // Addit. Manuf., 2016, vol. 12, pp. 296–304.

[25] Azarov A. V., Antonov F. K., Vasiliev V. V., Golubev M. V., Krasovskii D. S., Razin A. F., Salov V. A., Stupnikov V. V., Khaziev A. R. Development of a two-matrix composite material fabricated by 3d printing // Polymer Science, Series D, 2017, vol. 10, no. 1, pp. 87–90.

[26] Xia L., Breitkopf P. Concurrent topology optimization design of material and structure within FE2 nonlinear multiscale analysis framework // Computer Methods in Applied Mechanics and Engineering, 2014, vol. 278, pp. 524–542.

[27] Xia L., Breitkopf P. A reduced multiscale model for nonlinear structural topology optimization // Computer Methods in Applied Mechanics and Engineering, 2014, vol. 280, pp. 117–134.

[28] Xia L., Breitkopf P. Design of of materials using topology optimization and energy-based homogenization approach in Matlab // Structural and Multidisciplinary Optimization, 2015. doi: 10.1007/s00158-015-1294-0.

[29] Xia L., Breitkopf P. Multiscale structural topology optimization with an approximate constitutive model for local material microstructure // Computer Methods in Applied Mechanics and Engineering, 2015, vol. 286, pp. 147–167.

[30] Xia L., Breitkopf P. Recent advances on topology optimization of multiscale nonlinear structures // Archives of Computational Methods in Engineering, 2017. doi: 10.1007/S11831-016-9170-7.

[31] Theocaris P. S., Stavroulaki G. E. Optimal material design in composites: an iterative approach based on homogenized cells // Computer Methods in Applied Mechanics and Engineering, 1999, vol. 169, no. 1–2, pp. 31–42.

[32] Rodrigues H., Guedes J. M., Bendsoe M. P. Hierarchical optimization of material and structure // Structural and Multidisciplinary Optimization, 2002, vol. 24, no. 1, pp. 1–10.

[33] Zhang W., Sun S. Scale-related topology optimization of cellular materials and structures // International Journal for Numerical Methods in Engineering, 2006, vol. 68, no. 9, pp. 993–1011.

[34] da Silva A. L. F., Salas R. A., Nelli Silvaa E. C., Reddy J. N. Topology optimization of fibers orientation in hyperelastic composite material // Composite Structures, 2020, vol. 231, 111488. doi: 10.1016/j.compstruct.2019.111488.

[35] Hozic D., Thore C.-J., Cameron Ch., Loukil M. A new method for simultaneous material and topology optimization of composite laminate structures using Hyperbolic Function Parametrization // Composite Structures, 2021, vol. 276, 114374. doi: 10.1016/j.compstruct.2021.114374.

[36] Huang Y., Tian X., Zheng Z., Li D., Malakhov A. V., Polilov A. N. Multiscale concurrent design and 3D printing of continuous fiber reinforced thermoplastic composites with optimized fiber trajectory and topological structure // Composite Structures, 2022, vol. 285, 115241.

[37] Next-generation engineering design software. Available at: (accessed 13.04.2022).

[38] 3D printing a rocket engine. Available at: (accessed 13.04.2022).

[39] Bohara R. P., Linforth S., Nguyen T., Ghazlan A., Ngo T. Novel lightweight high-energy absorbing auxetic structures guided by topology optimization // International Journal of Mechanical Sciences, 2021, vol. 211, 106793.

[40] Choi J. B., Lakes R. S. Non-linear properties of metallic cellular materials with a negative Poisson's ratio // Journal of Materials Science, 1992, vol. 27, pp. 5375–5381.

[41] Davood M., Babak H., Ranajay G., Magid H. A., Hamid N.-H., Ashkan V. Elastic properties of chiral, anti-chiral, and hierarchical honeycombs: A simple energy-based approach // Theoretical and Applied Mechanics Letters, 2016, vol. 6, issue 2, pp. 81–96.

[42] Li T., Liu F., Wang L. Enhancing indentation and impact resistance in auxetic composite materials // Composites, Part B: Engineering, 2020, vol. 198, 108229.

[43] Xu M., Xu Z., Zhang Z., Lei H., Bai Y., Fang D. Mechanical properties and energy absorption capability of AuxHex structure under in-plane compression: theoretical and experimental studies // International Journal of Mechanical Sciences, 2019, vol. 159, pp. 43–57. doi: 10.1016/j.ijmecsci.2019.05.044.

[44] Guo M.-F., Yang H., Ma L. Design and characterization of 3D AuxHex lattice structures // International Journal of Mechanical Sciences, 2020, vol. 181, 105700. doi: 10.1016/j.ijmecsci.2020.105700.

[45] Wang H., Lu Z., Yang Z., Li X. A novel re-entrant auxetic honeycomb with enhanced in-plane impact resistance // Composite Structures, 2019, vol. 208, pp. 758–770. doi: 10.1016/j.compstruct.2018.10.024.

[46] Khakhlenkova A. A., Lopatin A. V. Obzor konstrukcij adapterov sovremennyh kosmicheskih apparatov [Overview of adapter designs for modern spacecrafts] // Spacecrafts & Technologies, 2018, vol. 2, no. 3, pp. 134–146. doi: 10.26732/2618-7957-2018-3-134-146. (In Russian)

[47] Razin A. F., Vasiliev V. V. Development of composite anisogrid spacecraft attach fitting // Proc. of 11th European Conf. on Composite Materials, 2004, 9 p.

[48] Vasiliev V. V., Barynin V. A., Razin A. F. Anisogrid composite lattice structures Development and aerospace applications // Composite structures, 2012, vol. 94, issue 3, pp. 1117–1127. doi. 10.1016/j.compstruct.2011.10.023.

[49] Frketic J., Dickens T., Ramakrishnan S. Automated manufacturing and processing of fiber- reinforced polymer (FRP) composites: An additive review of contemporary and modern techniques for advanced materials manufacturing // Additive Manufacturing, 2017, vol. 14, pp. 69–86. doi: 10.1016/j.addma.2017.01.003.

[50] Fedulov B., Fedorenko A., Khaziev A., Antonov F. Optimization of parts manufactured using continuous fiber threedimensional printing technology // Composites, Part B: Engineering, 2021, vol. 227, 109406.

[51] Baldzhiev R. S., Alekseyev A. A., Azarov A. V. Topology optimization of the lattice payload adapter for carrier rocket // Advances in Composite Science and Technology, 2018, vol. 683, 012061. doi: 10.1088/1757-899X/683/1/012061.

For citing this article

Sorokin D.V., Babkina L.A., Brazgovka O.V. Designing various-purpose subassemblies based on topological optimization // Spacecrafts & Technologies, 2022, vol. 6, no. 2, pp. 61-82. doi: 10.26732/

Creative Commons License
This Article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0).