Abstract
This paper presents a density-based shape optimization using an interface motion scheme as in level set methods. The aim is to generate optimal material distributions with high-quality interfaces within a uniform geometric representation for topology and shape optimization. This reduces the effort for post-processing and facilitates an automated conversion to CAD models. By using a density function, the proposed method can seamlessly adopt density-based topology optimization results as the initial design. Finite element analyses are performed using the same mesh and density field as for optimization. The interface motion is based on an advection equation and shape derivatives without a penalization of intermediate densities. This prevents the formation of large grey transition regions while avoiding mesh-dependent spatial oscillations of the interface. Thus, boundary-fitted meshes with smooth surfaces and sufficiently retained stiffness and volume can be extracted. In addition, an optional constraint for the mean curvature of the surface as well as the necessary shape derivative is introduced. Thus, the surface curvature can be limited to a technically justifiable value, which improves, for instance, the manufacturability and fatigue strength. The evaluation is carried out using a two-dimensional example starting from a topology-optimized design as well as four three-dimensional examples starting from a trivial design. The considered optimization objective is to minimize the volume with respect to compliance constraints and, partially, additional mean curvature constraints.
Similar content being viewed by others
References
Abe K, Koro K (2006) A topology optimization approach using VOF method. Struct Multidisc Optim 31:470–479. https://doi.org/10.1007/s00158-005-0582-5
Allaire G, Jouve F, Toader A-M (2002) A level-set method for shape optimization. CR Math 334:1125–1130. https://doi.org/10.1016/S1631-073X(02)02412-3
Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194:363–393. https://doi.org/10.1016/j.jcp.2003.09.032
Altair Engineering Inc. (2020a) HyperWorks. https://www.altair.de/hyperworks/
Altair Engineering Inc. (2020b) OptiStruct. https://www.altair.de/optistruct/
Amir O, Lazarov BS (2018) Achieving stress-constrained topological design via length scale control. Struct Multidisc Optim 58:2053–2071. https://doi.org/10.1007/s00158-018-2019-y
Andreasen CS, Elingaard MO, Aage N (2020) Level set topology and shape optimization by density methods using cut elements with length scale control. Struct Multidisc Optim 62:685–707. https://doi.org/10.1007/s00158-020-02527-1
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202. https://doi.org/10.1007/BF01650949
Bendsøe MP, Sigmund O (2004) Topology optimization. Springer, Berlin
Bertsekas DP (1976) On the Goldstein-Levitin-Polyak gradient projection method. IEEE Trans Automat Contr 21:174–184. https://doi.org/10.1109/TAC.1976.1101194
Bletzinger K-U (2014) A consistent frame for sensitivity filtering and the vertex assigned morphing of optimal shape. Struct Multidisc Optim 49:873–895. https://doi.org/10.1007/s00158-013-1031-5
Bruns TE, Tortorelli DA (2003) An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms. Int J Numer Meth Eng 57:1413–1430. https://doi.org/10.1002/nme.783
Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194:344–362. https://doi.org/10.1016/j.jcp.2003.09.033
Cai S, Zhang W, Zhu J, Gao T (2014) Stress constrained shape and topology optimization with fixed mesh: A B-spline finite cell method combined with level set function. Comput Methods Appl Mech Eng 278:361–387. https://doi.org/10.1016/j.cma.2014.06.007
Cea J (1986) Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût. ESAIM: M2AN 20:371–402. https://doi.org/10.1051/m2an/1986200303711
Chen S, Wang MY, Liu AQ (2008) Shape feature control in structural topology optimization. Comput Aided Des 40:951–962. https://doi.org/10.1016/j.cad.2008.07.004
Chen L, Bletzinger K-U, Geiser A, Wüchner R (2019) A modified search direction method for inequality constrained optimization problems using the singular-value decomposition of normalized response gradients. Struct Multidisc Optim 60:2305–2323. https://doi.org/10.1007/s00158-019-02320-9
Cho S, Ha S-H (2009) Isogeometric shape design optimization: Exact geometry and enhanced sensitivity. Struct Multidisc Optim 38:53–70. https://doi.org/10.1007/s00158-008-0266-z
Cottet G-H, Maitre E (2011) A level set method for fluid-structure interactions with immersed surfaces. Math Models Methods Appl Sci 16:415–438. https://doi.org/10.1142/S0218202506001212
Courant R, Friedrichs K, Lewy H (1928) Über die partiellen Differenzengleichungen der mathematischen Physik. Math Ann 100:32–74. https://doi.org/10.1007/BF01448839
da Silva GA, Beck AT, Sigmund O (2019) Stress-constrained topology optimization considering uniform manufacturing uncertainties. Comput Methods Appl Mech Eng 344:512–537. https://doi.org/10.1016/j.cma.2018.10.020
Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidisc Optim 49:1–38. https://doi.org/10.1007/s00158-013-0956-z
Debruge LL (1980) The aerodynamic significance of fillet geometry in turbocompressor blade rows. J Eng Power 102:984–993. https://doi.org/10.1115/1.3230371
Dunning PD (2018) Minimum length-scale constraints for parameterized implicit function based topology optimization. Struct Multidisc Optim 58:155–169. https://doi.org/10.1007/s00158-017-1883-1
Duysinx P, van Miegroet L, Jacobs T, Fleury C (2006) Generalized shape optimization using X-FEM and level set methods. In: IUTAM symposium on topological design optimization of structures, machines and materials, pp 23–32. https://doi.org/10.1007/1-4020-4752-5_3
Eschenauer HA, Kobelev VV, Schumacher A (1994) Bubble method for topology and shape optimization of structures. Struct Optim 8:42–51. https://doi.org/10.1007/BF01742933
Fiebig S, Axmann J (2013) Using a binary material Using a binary material model for stress constraints and nonlinearities up to crash in topology optimization. In: Proceedings of the 10th world congress on structural and multidisciplinary optimization
Franke T, Fiebig S, Bartz R, Vietor T, Hage J, Vom Hofe A (2018a) Adaptive topology and shape optimization with integrated casting simulation. In: Proceedings of the 6th international conference on engineering optimization. https://doi.org/10.1007/978-3-319-97773-7_109
Franke T, Fiebig S, Paul K, Vietor T, Sellschopp J (2018b) Topology optimization with integrated casting simulation and parallel manufacturing process improvement. In: Schumacher A, Vietor T, Fiebig S, Bletzinger K-U, Maute K (eds) Advances in structural and multidisciplinary optimization: Proceedings of the 12th world congress of structural and multidisciplinary optimization (WCSMO12). Springer, Cham
Gladwell GML, Haftka RT, Gürdal Z (eds) (1992) Elements of structural optimization, vol 11. Springer, Netherlands
Goldman R (2005) Curvature formulas for implicit curves and surfaces. Comput Aided Geom Des 22:632–658. https://doi.org/10.1016/j.cagd.2005.06.005
Grote K-H, Antonsson EK (eds) (2009) Springer handbook of mechanical engineering. Springer, Berlin
Guo X, Zhang W, Zhong W (2014a) Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. J Appl Mech 81:197. https://doi.org/10.1115/1.4027609
Guo X, Zhang W, Zhong W (2014b) Explicit feature control in structural topology optimization via level set method. Comput Methods Appl Mech Eng 272:354–378. https://doi.org/10.1016/j.cma.2014.01.010
Ha S-H, Cho S (2008) Level set based topological shape optimization of geometrically nonlinear structures using unstructured mesh. Comput Struct 86:1447–1455. https://doi.org/10.1016/j.compstruc.2007.05.025
Haber RB, Jog CS, Bendsøe MP (1996) A new approach to variable-topology shape design using a constraint on perimeter. Struct Optim 11:1–12. https://doi.org/10.1007/BF01279647
Haftka RT, Grandhi RV (1986) Structural shape optimization—a survey. Comput Methods Appl Mech Eng 57:91–106. https://doi.org/10.1016/0045-7825(86)90072-1
Haftka RT, Gürdal Z (1992) Constrained optimization. In: Gladwell GML, Haftka RT, Gürdal Z (eds) Elements of structural optimization. Springer, Netherlands, pp 159–207
Han Y, Xu B, Wang Q, Liu Y, Duan Z (2021) Topology optimization of material nonlinear continuum structures under stress constraints. Comput Methods Appl Mech Eng 378:113731. https://doi.org/10.1016/j.cma.2021.113731
Hirt C, Nichols B (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39:201–225. https://doi.org/10.1016/0021-9991(81)90145-5
Hsu Y-L (1994) A review of structural shape optimization. Comput Ind 25:3–13. https://doi.org/10.1016/0166-3615(94)90028-0
Jansen M (2019) Explicit level set and density methods for topology optimization with equivalent minimum length scale constraints. Struct Multidisc Optim 59:1775–1788. https://doi.org/10.1007/s00158-018-2162-5
Kohn R, Allaire G (1993) Topology optimization and optimal shape design using homogenization. In: Topology design of structures, pp 207–218
Lange R-J (2015) Distribution theory for Schrödinger’s integral equation. J Math Phys 56:122105. https://doi.org/10.1063/1.4936302
Lazarov BS, Wang F, Sigmund O (2016) Length scale and manufacturability in density-based topology optimization. Arch Appl Mech 86:189–218. https://doi.org/10.1007/s00419-015-1106-4
LeVeque RJ (2012) Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge
Liu J, Yu H, Ma Y (2016) Minimum void length scale control in level set topology optimization subject to machining radii. Comput Aided Des 81:70–80. https://doi.org/10.1016/j.cad.2016.09.007
Lorensen WE, Cline HE (1987) Marching cubes: A high resolution 3D surface construction algorithm. In: ACM SIGGRAPH computer graphics, pp 163–169. https://doi.org/10.1145/37401.37422
Maitre E, Milcent T, Cottet G-H, Raoult A, Usson Y (2009) Applications of level set methods in computational biophysics. Math Comput Model 49:2161–2169. https://doi.org/10.1016/j.mcm.2008.07.026
Maitre E, Misbah C, Peyla P, Raoult A (2012) Comparison between advected-field and level-set methods in the study of vesicle dynamics. Physica D 241:1146–1157. https://doi.org/10.1016/j.physd.2012.03.005
Manson J, Smith J, Schaefer S (2011) Contouring discrete indicator functions. Comput Graph Forum 30:385–393. https://doi.org/10.1111/j.1467-8659.2011.01869.x
Mattheck C, Burghardt S (1990) A new method of structural shape optimization based on biological growth. Int J Fatigue 12:185–190. https://doi.org/10.1016/0142-1123(90)90094-U
Milcent T (2011) Shape derivative of the Willmore functional and applications to equilibrium shapes of vesicles. INRA Research Report
Mlejnek HP (1992) Some aspects of the genesis of structures. Struct Optim 5:64–69. https://doi.org/10.1007/BF01744697
Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Meth Eng 46:131–150. https://doi.org/10.1002/(SICI)1097-0207(19990910)46:1%3c131:AID-NME726%3e3.0.CO;2-J
Najian Asl R, Shayegan S, Geiser A, Hojjat M, Bletzinger K-U (2017) A consistent formulation for imposing packaging constraints in shape optimization using Vertex Morphing parametrization. Struct Multidisc Optim 56:1507–1519. https://doi.org/10.1007/s00158-017-1819-9
París J, Navarrina F, Colominas I, Casteleiro M (2009) Topology optimization of continuum structures with local and global stress constraints. Struct Multidisc Optim 39:419–437. https://doi.org/10.1007/s00158-008-0336-2
Parvizian J, Düster A, Rank E (2007) Finite cell method. Comput Mech 41:121–133. https://doi.org/10.1007/s00466-007-0173-y
Parvizian J, Düster A, Rank E (2012) Topology optimization using the finite cell method. Optim Eng 13:57–78. https://doi.org/10.1007/s11081-011-9159-x
Petersson J (1999) Some convergence results in perimeter-controlled topology optimization. Comput Methods Appl Mech Eng 171:123–140. https://doi.org/10.1016/S0045-7825(98)00248-5
Pironneau O (1984) Optimal shape design for elliptic systems. Springer, Berlin
Rosen JB (1960) The gradient projection method for nonlinear programming. Part I. Linear constraints. J Soc Ind Appl Math 8:181–217. https://doi.org/10.1137/0108011
Schevenels M, Lazarov BS, Sigmund O (2011) Robust topology optimization accounting for spatially varying manufacturing errors. Comput Methods Appl Mech Eng 200:3613–3627. https://doi.org/10.1016/j.cma.2011.08.006
Sethian JA (1996) A fast marching level set method for monotonically advancing fronts. Proc Natl Acad Sci USA 93:1591–1595. https://doi.org/10.1073/pnas.93.4.1591
Sharma A, Maute K (2018) Stress-based topology optimization using spatial gradient stabilized XFEM. Struct Multidisc Optim 57:17–38. https://doi.org/10.1007/s00158-017-1833-y
Sharp N, Soliman Y, Crane K (2019) The vector heat method. ACM Trans Graph 38:1–19. https://doi.org/10.1145/3243651
Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidisc Optim 21:120–127. https://doi.org/10.1007/s001580050176
Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33:401–424. https://doi.org/10.1007/s00158-006-0087-x
Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidisc Optim 48:1031–1055. https://doi.org/10.1007/s00158-013-0978-6
Simon J (2010) Differentiation with respect to the domain in boundary value problems. Numer Funct Anal Optim 2:649–687. https://doi.org/10.1080/01630563.1980.10120631
Sokolowski J, Zolesio J-P (eds) (1992I) Introduction to shape optimization. Springer Series in Computational Mathematics, Berlin
Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim 12:555–573. https://doi.org/10.1137/S1052623499362822
van Dijk NP, Langelaar M, Keulen F (2012) Explicit level-set-based topology optimization using an exact Heaviside function and consistent sensitivity analysis. Int J Numer Meth Eng 91:67–97. https://doi.org/10.1002/nme.4258
van Dijk NP, Maute K, Langelaar M, van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidisc Optim 48:437–472. https://doi.org/10.1007/s00158-013-0912-y
Villanueva CH, Maute K (2014) Density and level set-XFEM schemes for topology optimization of 3-D structures. Comput Mech 54:133–150. https://doi.org/10.1007/s00466-014-1027-z
Wallin M, Ristinmaa M, Askfelt H (2012) Optimal topologies derived from a phase-field method. Struct Multidisc Optim 45:171–183. https://doi.org/10.1007/s00158-011-0688-x
Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246. https://doi.org/10.1016/S0045-7825(02)00559-5
Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidisc Optim 43:767–784. https://doi.org/10.1007/s00158-010-0602-y
Xia L, Fritzen F, Breitkopf P (2017) Evolutionary topology optimization of elastoplastic structures. Struct Multidisc Optim 55:569–581. https://doi.org/10.1007/s00158-016-1523-1
Xia L, Zhang L, Xia Q, Shi T (2018) Stress-based topology optimization using bi-directional evolutionary structural optimization method. Comput Methods Appl Mech Eng 333:356–370. https://doi.org/10.1016/j.cma.2018.01.035
Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896. https://doi.org/10.1016/0045-7949(93)90035-C
Yamasaki S, Nishiwaki S, Yamada T, Izui K, Yoshimura M (2010) A structural optimization method based on the level set method using a new geometry-based re-initialization scheme. Int J Numer Meth Eng 83:1580–1624. https://doi.org/10.1002/nme.2874
Yoely YM, Amir O, Hanniel I (2018) Topology and shape optimization with explicit geometric constraints using a spline-based representation and a fixed grid. Procedia Manuf 21:189–196. https://doi.org/10.1016/j.promfg.2018.02.110
Yoely YM, Hanniel I, Amir O (2020) Structural optimization with explicit geometric constraints using a B-spline representation. Mech Based Des Struct Mach 7:1–32. https://doi.org/10.1080/15397734.2020.1824793
Young V, Querin OM, Steven GP, Xie YM (1999) 3D and multiple load case bi-directional evolutionary structural optimization (BESO). Struct Optim 18:183–192. https://doi.org/10.1007/BF01195993
Zhang W, Chen J, Zhu X, Zhou J, Xue D, Lei X, Guo X (2017a) Explicit three dimensional topology optimization via Moving Morphable Void (MMV) approach. Comput Methods Appl Mech Eng 322:590–614. https://doi.org/10.1016/j.cma.2017.05.002
Zhang W, Zhou J, Zhu Y, Guo X (2017b) Structural complexity control in topology optimization via moving morphable component (MMC) approach. Struct Multidisc Optim 56:535–552. https://doi.org/10.1007/s00158-017-1736-y
Zhang W, Li D, Zhou J, Du Z, Li B, Guo X (2018) A Moving Morphable Void (MMV)-based explicit approach for topology optimization considering stress constraints. Comput Methods Appl Mech Eng 334:381–413. https://doi.org/10.1016/j.cma.2018.01.050
Zhou M, Rozvany G (1991) The COC algorithm, Part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89:309–336. https://doi.org/10.1016/0045-7825(91)90046-9
Zhou M, Shyy Y, Thomas HL (2001) Checkerboard and minimum member size control in topology optimization. Struct Multidisc Optim 21:152–158. https://doi.org/10.1007/s001580050179
Zhou M, Lazarov BS, Wang F, Sigmund O (2015) Minimum length scale in topology optimization by geometric constraints. Comput Methods Appl Mech Eng 293:266–282. https://doi.org/10.1016/j.cma.2015.05.003
Funding
This work was financed by the Volkswagen AG. Results, opinions and conclusions are not necessarily those of the Volkswagen AG.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Replication of results
The method presented in this paper is part of the commercial optimization environment LEOPARD of Volkswagen AG. Nevertheless, using the proposed numerical schemes, integration into other optimization environments and thus the reproduction of the presented results are possible without any restrictions. The example of the wheel carrier is published in the supplementary material of this paper, including the design and non-design space as well as the used load cases and material definitions.
Additional information
Responsible Editor: W. H. Zhang
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Bartz, R., Franke, T., Fiebig, S. et al. Density-based shape optimization of 3D structures with mean curvature constraints. Struct Multidisc Optim 65, 5 (2022). https://doi.org/10.1007/s00158-021-03089-6
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00158-021-03089-6