Theory and analysis of the topology optimization methods
Authors: Kosykh P.A., Azarov A.V.
Published in issue: #4(136)/2023
DOI: 10.18698/2308-6033-2023-4-2264
Category: Mechanics | Chapter: Mechanics of Deformable Solid Body
As of today, additive technologies are making it possible to create products being close to the optimal shape. Topological optimization is widely used to design such products. The paper considers two common approaches to solving this problem: SIMP and BESO methods. Essence of the topological optimization problem, its formulation in a general form and typical examples to demonstrate this problem are described. Theoretical foundations and implementation features are presenting each method; algorithms sensitivity to the initial settings is analyzed. The problems that arise within the solution, such as the chessboard problem and dependence on the finite element mesh, are analyzed; options for solving these problems are provided. Two approaches were compared. It is concluded that the BESO method is offering more efficient and more design-friendly solutions.
References
[1] Azarov A.V., Antonov F.K., Golubev M.V., Khaziev A.R., Ushanov S.A. Composite 3D printing for the small size unmanned aerial vehicle structure. Composites Part B: Engineering, 2019, vol. 169, pp. 157–163. https://doi.org/10.1016/j.compositesb.2019.03.073
[2] Blakey-Milner B., Gradl P., Snedden G. Metal additive manufacturing in aerospace: A review. Materials & Design, 2021, vol. 209. https://doi.org/10.1016/j.matdes.2021.110008
[3] Bendsoe M.P., Sigmund O. Topology Optimization: Theory, Methods and Applications. New York, Springer Verlag, 2003, 271 p.
[4] Huang X., Xie Y.M. Evolutionary topology optimization of continuum structures: methods and applications. Chichester, John Wiley & Sons, 2010, 228 p.
[5] Allaire G., Gournay F., Jouve F., Toader A.-M. Structural optimization using topological and shape sensitivity via a level set method. Control and Cybernetics, 2005, vol. 34 (1), pp. 59–80.
[6] Challis V.J. A discrete level-set topology optimization code written in MATLAB. Struct Multidisc Optim, 2010, vol. 41, pp. 453–464. https://doi.org/10.1007/s00158-009-0430-0
[7] Biyikli E., To A.C. Proportional Topology Optimization: A New Non-Sensitivity Method for Solving Stress Constrained and Minimum Compliance Problems and Its Implementation in MATLAB. PLoS ONE, 2015, vol. 10 (12). https://doi.org/10.1371/journal.pone.0145041
[8] Komarov V.A. Proektirovanie silovykh additivnykh konstruktsiy: teoreticheskie osnovy [Theoretical basis for design of load-bearing structures produced using additive technologies]. Ontologiya proektirovaniya — Ontology of designing, 2017, no. 2 (24), pp. 191–206. https://doi.org/10.18287/2223-9537-2017-7-2-191-206
[9] Kishov E.A., Komarov V.A. Topologiya optimizatsii silovykh konstruktsiy metodom vypukloy linearizatsii [Topology optimization of a load-bearing structure via the method of convex linearization]. Vestnik Samarskogo universiteta. Aerokosmicheskaya tekhnika, tekhnologii i mashinostroenie — Vestnik of Samara University. Aerospace and Mechanical Engineering, 2018, no. 1, pp. 137–149. https://doi.org/10.18287/2541-7533-2018-17-1-137-149
[10] Papapetrou V.S., Patel C., Tamijani A.Y. Stiffness-based optimization framework for the topology and fiber paths of continuous fiber composites. Composites Part B, 2020. https://doi.org/10.1016/j.compositesb.2019.107681
[11] Gandhi Y., Minak G. A Review on topology optimization strategies for additively manufactured continuous fiber-reinforced composite structures. Appl. Sci., 2022, vol. 12. https://doi.org/10.3390/app122111211
[12] Fedulov B.N., Fedorenko A.N., Antonov F.K., Lomakin E.V. Algoritm topologicheskoy optimizatsii konstruktsii, vypolnennoy iz anizitropnogo materiala s uchetom parametrov orientatsii armirovaniya [Algorithm for Topology Optimization of a Structure Made of Anisotropic Material with Consideration of the Reinforcement Orientation Parameters]. Vestnik Permskogo natsionalnogo issledovatelskogo politekhnicheskogo universiteta. Mekhanika — PNRPU Mechanics Bulletin, 2021, no. 3, pp. 182–189. https://doi.org/10.15593/perm.mech/2021.3.17
[13] Azarov A.V., Latysheva T.A., Khaziev A.R. Optimal design of advanced 3D printed composite parts of rocket and space structures. IOP Conference Series: Materials Science and Engineering, 2020. https://doi.org/0.1088/1757-899X/934/1/012062
[14] Sigmund O. A 99 line topology optimization code written in matlab. Structural and Multidisciplinary Optimization, 2001, vol. 21, no. 2, pp. 120–127.