Abstract
Topology Optimization (TO) represents a relevant tool in the design of mechanical structures and, as such, it is currently used in many industrial applications. However, many TO optimization techniques are still questionable when applied to crashworthiness optimization problems due to their complexity and lack of gradient information. The aim of this work is to describe the Hybrid Kriging-assisted Level Set Method (HKG-LSM) and test its performance in the optimization of mechanical structures consisting of ensembles of beams subjected to both static and dynamic loads. The algorithm adopts a low-dimensional parametrization introduced by the Evolutionary Level Set Method (EA-LSM) for structural Topology Optimization and couples the Efficient Global Optimization (EGO) and the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) to converge towards the optimum within a fixed budget of evaluations. It takes advantage of the explorative capabilities of EGO ensuring a fast convergence at the beginning of the optimization procedure, as well as the flexibility and robustness of CMA-ES to exploit promising regions of the search space Precisely, HKG-LSM first uses the Kriging-based method for Level Set Topology Optimization (KG-LSM) and afterwards switches to the EA-LSM using CMA-ES, whose parameters are initialized based on the previous model. Within the research, a minimum compliance cantilever beam test case is used to validate the presented strategy at different dimensionalities, up to 15 variables. The method is then applied to a 15-variables 2D crash test case, consisting of a cylindrical pole impact on a rectangular beam fixed at both ends. Results show that HKG-LSM performs well in terms of convergence speed and hence represents a valuable option in real-world applications with limited computational resources.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
CalculiX is an open-source, 3D structural FEM software developed at MTU Aero Engines in Munich. CalculiX, Version 2.13, was used in this work: http://www.calculix.de/.
References
Allaire, G., Jouve, F., Toader, A.M.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194(1), 363–393 (2004). https://doi.org/10.1016/j.jcp.2003.09.032
Arsenyev, I.: Efficient Surrogate-based Robust Design Optimization Method. Ph.D. thesis, Technische Universität München (2017)
Aulig, N.: Generic topology optimization based on local state features. Ph.D. thesis, Technische Universität Darmstadt, VDI Verlag, Germany (2017)
Aulig, N., Olhofer, M.: State-based representation for structural topology optimization and application to crashworthiness. In: 2016 IEEE Congress on Evolutionary Computation (CEC), Vancouver, Canada, pp. 1642–1649 (2016). https://doi.org/10.1109/CEC.2016.7743985
Bendsøe, M.P., Sigmund, O.: Topology Optimization - Theory, Methods, and Applications, 2nd edn. Springer, Berlin (2004). http://www.springer.com/cn/book/9783540429920
Bujny, M., Aulig, N., Olhofer, M., Duddeck, F.: Evolutionary level set method for crashworthiness topology optimization. In: VII European Congress on Computational Methods in Applied Sciences and Engineering, Crete Island, Greece (2016)
Bujny, M., Aulig, N., Olhofer, M., Duddeck, F.: Identification of optimal topologies for crashworthiness with the evolutionary level set method. Int. J. Crashworthiness 23(4), 395–416 (2018). https://doi.org/10.1080/13588265.2017.1331493
Cressie, N.: The origins of kriging. Math. Geol. 22(3), 239–252 (1990). https://doi.org/10.1007/BF00889887
Duddeck, F., Volz, K.: A new topology optimization approach for crashworthiness of passenger vehicles based on physically defined equivalent static loads. In: ICrash International Crashworthiness Conference, Milano, Italy (2012)
Duddeck, F., Hunkeler, S., Lozano, P., Wehrle, E., Zeng, D.: Topology optimization for crashworthiness of thin-walled structures under axial impact using hybrid cellular automata. Struct. Multidiscip. Optim. 54(3), 415–428 (2016). https://doi.org/10.1007/s00158-016-1445-y
Eschenauer, H.A., Kobelev, V.V., Schumacher, A.: Bubble method for topology and shape optimization of structures. Struct. Optim. 8(1), 42–51 (1994). https://doi.org/10.1007/BF01742933
Fang, K.T., Li, R., Sudjianto, A.: Design and Modeling for Computer Experiments. CRC Press (2005)
Forrester, A.I.J., Sóbester, A., Keane, A.J.: Engineering Design via Surrogate Modelling - A Practical Guide. John Wiley & Sons Ltd. (2008)
Guo, X., Zhang, W., Zhong, W.: Doing topology optimization explicitly and geometrically - a new moving morphable components based framework. J. Appl. Mech. 81(8), 081009 (2014). https://doi.org/10.1115/1.4027609
Haber, R., Bendsøe, M.P.: Problem formulation, solution procedures and geometric modeling: key issues in variable-topology optimization. In: 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, Missouri, USA (1998)
Hansen, N.: The CMA evolution strategy: a tutorial (2005). https://hal.inria.fr/hal-01297037, hal-01297037f
Hansen, N.: The CMA evolution strategy: a comparing review. In: Towards a New Evolutionary Computation. Studies in Fuzziness and Soft Computing, pp. 75–102. Springer, Berlin (2006). https://doi.org/10.1007/3-540-32494-1_4
Hansen, N., Ostermeier, A.: Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation. In: Proceedings of IEEE International Conference on Evolutionary Computation, pp. 312–317 (1996). https://doi.org/10.1109/ICEC.1996.542381
Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolution strategies. Evol. Comput. 9(2), 159–195 (2001). https://doi.org/10.1162/106365601750190398
Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13(4), 455–492 (1998). https://doi.org/10.1023/A:1008306431147
Kleijnen, J.P.C.: Kriging metamodeling in simulation: a review. Eur. J. Oper. Res. 192(3), 707–716 (2009). https://doi.org/10.1016/j.ejor.2007.10.013
Lee, H.A., Park, G.J.: Nonlinear dynamic response topology optimization using the equivalent static loads method. Comput. Methods Appl. Mech. Eng. 283, 956–970 (2015). https://doi.org/10.1016/j.cma.2014.10.015
Livermore Software Technology Corporation (LSTC), P. O. Box 712 Livermore, California 94551-0712: LS-DYNA KEYWORD USER’S MANUAL, Volume II - Material Models (2014). lS-DYNA R7.1
Livermore Software Technology Corporation (LSTC), P. O. Box 712 Livermore, California 94551-0712: LS-DYNA Theory Manual (2019)
Michell, A.G.M.: LVIII. The limits of economy of material in frame-structures. Philos. Mag. 8(47), 589–597 (1904). https://doi.org/10.1080/14786440409463229
Mohammadi, H., Riche, R.L., Touboul, E.: Making EGO and CMA-ES complementary for global optimization. In: Learning and Intelligent Optimization. Lecture Notes in Computer Science, pp. 287–292. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-19084-6-29
Mozumder, C., Renaud, J.E., Tovar, A.: Topometry optimisation for crashworthiness design using hybrid cellular automata. Int. J. Veh. Des. 60(1–2) (2012). https://trid.trb.org/view.aspx?id=1222579
Ortmann, C., Schumacher, A.: Graph and heuristic based topology optimization of crash loaded structures. Struct. Multidiscip. Optim. 47(6), 839–854 (2013). https://doi.org/10.1007/s00158-012-0872-7
Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988). https://doi.org/10.1016/0021-9991(88)90002-2
Pedersen, C.B.W.: Topology optimization design of crushed 2d-frames for desired energy absorption history. Struct. Multidiscip. Optim. 25(5–6), 368–382 (2003). https://doi.org/10.1007/s00158-003-0282-y
Rao, S.S.: Engineering Optimization: Theory and Practice. Wiley (1996)
Raponi, E., Bujny, M., Olhofer, M., Aulig, N., Boria, S., Duddeck, F.: Kriging-guided level set method for crash topology optimization. In: 7th GACM Colloquium on Computational Mechanics for Young Scientists from Academia and Industry. Stuttgart, Germany (2017)
Raponi, E., Bujny, M., Olhofer, M., Aulig, N., Boria, S., Duddeck, F.: Kriging-assisted topology optimization of crash structures. Comput. Methods Appl. Mech. Eng. 348, 730–752 (2019). https://doi.org/10.1016/j.cma.2019.02.002
Raponi, E., Bujny, M., Olhofer, M., Boria, S., Duddeck, F.: Hybrid kriging-assisted level set method for structural topology optimization. In: Proceedings of the 11th International Joint Conference on Computational Intelligence (IJCCI 2019), Vienna, Austria, pp. 70–81 (2019). https://doi.org/10.5220/0008067800700081
Raponi, E., Wang, H., Bujny, M., Boria, S., Doerr, C.: High dimensional Bayesian optimization assisted by principal component analysis. In: Parallel Problem Solving from Nature, PPSN XVI, pp. 169–183. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58112-1_12
Storn, R., Price, K.: Differential evolution a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11(4), 341–359 (1997). https://doi.org/10.1023/A:1008202821328
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Raponi, E., Bujny, M., Olhofer, M., Boria, S., Duddeck, F. (2021). Hybrid Strategy Coupling EGO and CMA-ES for Structural Topology Optimization in Statics and Crashworthiness. In: Merelo, J.J., Garibaldi, J., Linares-Barranco, A., Warwick, K., Madani, K. (eds) Computational Intelligence. IJCCI 2019. Studies in Computational Intelligence, vol 922. Springer, Cham. https://doi.org/10.1007/978-3-030-70594-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-70594-7_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-70593-0
Online ISBN: 978-3-030-70594-7
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)