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Representative surrogate problems as test functions for expensive simulators in multidisciplinary design optimization of vehicle structures

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Abstract

A large variety of algorithms for multidisciplinary optimization is available, but for various industrial problem types that involve expensive function evaluations, there is still few guidance available to select efficient optimization algorithms. This is also the case for multidisciplinary vehicle design optimization problems involving, e.g., weight, crashworthiness, and vibrational comfort responses. In this paper, an approach for the development of Representative Surrogate Problems (RSPs) as synthetic test functions for a relatively complex industrial problem is presented. The work builds on existing sensitivity analysis and surrogate data generation methods to establish a novel approach to generate surrogate function sets, which are accessible (i.e. not resource demanding) and aim to generate statistically representative instances of specific classes of industrial problems. The approach is demonstrated through the construction of RSPs for multidisciplinary optimization problems that occur in the context of structural car body design. As a “proof of concept” the RSP approach is applied for the selection of suitable optimization algorithms, for several problem formulations and for a meta-optimization (i.e. an optimization of the optimization algorithm parameters) to increase optimization efficiency. The potential of the approach is demonstrated by comparing the efficiency of several optimization algorithms on an RSP and an independent simulation-based vehicle model. The results corroborate the potential of the proposed approach and significant performance gains in optimization efficiency are achieved. Although the approach is developed for the particular application presented, the approach is described in a general way, to encourage readers to use the gist of the concept.

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Notes

  1. In the automotive industry the word “benchmarking” often refers to comparative test for quality assurance of components, systems or vehicles. In the scope of this paper the word benchmarking is however used in a computing context, and refers to the act of assessing the relative performance of computational algorithms w.r.t. each other under comparable conditions.

  2. Approximate CPU time per simulation using a single logical core of a HP Z600 with 2 Intel Xeon E5520 processors, and 24GB DDR3 Memory.

  3. The peak acceleration results are based on SAE 60 Hz low pass filtered acceleration values of an accelerometer element located at the center of the vehicle on the tunnel.

  4. The distribution of the first order sensitivity indices S i are expressed in terms of \( \sqrt{S_i} \) since this is in the opinion of the authors more intuitive for visualization (in a similar manner as standard deviation can be preferred over variance in particular diagrams).

    1. 1)

      The Interior Point (IP) algorithm is commonly used to solve convex problems. The implementation used is included in MATLAB 2013a in the “fmincon” function option 1. For a description of the algorithm see Boyd and Vandenberghe (2009).

    2. 2)

      Sequential Quadratic Programming (SQP) approaches are generally used to solve smooth nonlinear problems, by sequential steps of the Newton method. In this work the implementation included in MATLAB 2013a in the “fmincon” function (option 3) is used. A description of the algorithm is given in Fletcher (2010).

    3. 3)

      Genetic Algorithms (GA) are a class of evolutionary algorithms, which are inspired by the genetic process of reproduction in biological life. The application of such algorithms is proposed in Rechenberg (1973), and a detailed description can be found in the work of Goldberg and Holland (1988). In this work the “ALGA” implementation included in MATLAB 2013a is used.

    4. 4)

      The Non-dominated Sorting Genetic Algorithm (NSGA-2) is a multi-objective evolutionary algorithm developed by Deb et al. (2000). The variant of the algorithm used in this work is Reference-point based NSGA-II implemented by Lin (2011).

    5. 5)

      Differential Evolution (DE) is another evolutionary algorithm used for optimization (Storm and Price (1997)). The implementation used in this work is an adaptation of the code by Buehren (2008), combined with a penalty approach to enforce nonlinear constraint handling.

    6. 6)

      Particle Swarm Optimization (PSO) algorithms are nature inspired meta-heuristics that mimic the movement of groups of organisms such as bird flocks or fish schools. In this work the implementation by Birge (2006) is applied combined with a penalty factor approach to handle nonlinear constraints. A description for the algorithm principles can be found in chapter 8 of Yang (2010a).

    7. 7)

      Simulated Annealing (SA) is an optimization approach inspired by the thermodynamic process used in metallurgic annealing heat treatment (Kirkpatrick et al. (1983)). In Yang (2010b) a description of the algorithm is provided together with an implementation of the algorithm that is used in this work.

    8. 8)

      Fire Fly inspired optimization algorithms are population based algorithms inspired by the behavior of fire flies. A description of the algorithm and implementation used in this work is provided in Yang (2010b).

  5. CPU time for a RSP function evaluation is about 2.5E-2 [s] for the four responses in the example, using a MATLAB 2013a implementation on an Dell T3500 workstation with an Intel Xeon X5650 processor and 12 GB of RAM. The runtime of the optimizations using the RSP is dominated by the overhead of the optimization algorithm and optimization history saving.

References

  • Alimoradi A, Foley CM, Pezeshk S (2010) Benchmark problems in structural design and performance optimization: past, present, and future—part I. Struct Congr 2010:455–466. doi:10.1061/41131(370)40

    Google Scholar 

  • Baldanzini N, Scippa A (2004) Shape and size optimization of an engine suspension system. Proceedings of the 2004 International Conference on Noise and Vibration Engineering, ISMA

  • Baldanzini N, Caprioli D, Pierini M (2001) Designing the dynamic behavior of an engine suspension system through genetic algorithms. J Vib Acoust 123(4):480–486

    Article  Google Scholar 

  • Birge B (2006) Particle swarm optimization toolbox, retrieved from MATLAB file exchange in January 2014 http://www.mathworks.com/matlabcentral/fileexchange/7506-particle-swarm-optimization-toolbox

  • Blumhardt R (2001) FEM-crash simulation and optimisation. Int J Veh Des 26(4):331–347

    Article  Google Scholar 

  • Boyd S, Vandenberghe L (2009) Convex optimization. Cambridge University Press, 2009. https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf

  • Buehren M (2008) Differential evolution, MATLAB file exchange http://www.mathworks.com/matlabcentral/fileexchange/18593-differential-evolution

  • Deb K, Agrawal S, Pratap A, Meyarivan T (2000) A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. Lect Notes Comput Sci 1917:849–858

    Article  Google Scholar 

  • Duddeck F (2008) Multidisciplinary optimization of car bodies. Struct Multidiscip Optim 35(4):375–389. doi:10.1007/s00158-007-0130-6

    Article  Google Scholar 

  • Durgun I, Yildiz AR (2012) Structural design optimization of vehicle components using cuckoo search algorithm. Mater Test 54(3):185–188

    Article  Google Scholar 

  • Fletcher R (2010) The sequential quadratic programming method. Nonlinear Optimization. Springer Berlin Heidelberg 165–214. doi: 10.1007/978-3-642-11339-0_3

  • Goldberg DE, Holland JH (1988) Genetic algorithms and machine learning. Mach Learn 3(2):95–99

    Article  Google Scholar 

  • Hallquist JO (2006) LS-DYNA theory manual. Livermore Software Technology Corporation

  • Haug E, Scharnhorst T, DuBois P (1986) FEM-Crash, Berechnung eines Fahrzeugaufpralls. VDI-Tagung: Berechnung im Automobilbau, Würzburg, Germany, (VDI-Berichte 613), 479–505

  • Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 1948:293–325

    Article  MathSciNet  MATH  Google Scholar 

  • Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680

    Article  MathSciNet  MATH  Google Scholar 

  • Knowles J, Hughes EJ (2005) Multiobjective optimization on a budget of 250 evaluations. In Evolutionary Multi-Criterion Optimization. Springer, Berlin Heidelberg, pp. 176–190

  • Lin S (2011) NGPM A NSGA-II program in Matlab, MATLAB file exchange. http://www.mathworks.com/matlabcentral/fileexchange/31166-ngpm-a-nsga-ii-program-in-matlab-v1-4

  • Mihaylova P, Pratellesi A, Baldanzini N, Pierini M (2012) Optimization of the global static and dynamic performance of a vehicle body by means of response surface models. ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical Engineers

  • NCAC Finite Element Model Archive. retrieved from January 2014, from http://www.ncac.gwu.edu/vml/models.html

  • Plischke E (2010) An effective algorithm for computing global sensitivity indices (EASI). Reliab Eng Syst Saf 95(4):354–360

    Article  Google Scholar 

  • Prichard D, Theiler J (1994) Generating surrogate data for time series with several simultaneously measured variables. Phys Rev Lett 73(7):951

    Article  Google Scholar 

  • Rastrigin LA (1974) Systems of extremal control. Theoretical Foundations of Engineering Cybernetics Series, (In Russian) Nauka, Moscow

  • Ratto M, Pagano A (2010) Using recursive algorithms for the efficient identification of smoothing spline ANOVA models. AStA Adv Stat Anal 94(4):367–388

    Article  MathSciNet  Google Scholar 

  • Rechenberg I (1973) Evolutionsstrategie. Stuttgart: Holzmann-Froboog. ISBN 3-7728-0373-3

  • Rios LM, Sahinidis NV (2012) Derivative-free optimization: a review of algorithms and comparison of software implementations. J Glob Optim 2012:1–47

    MathSciNet  Google Scholar 

  • Rodgers LJ, Nicewander WA (1988) Thirteen ways to look at the correlation coefficient. Am Stat 42(1):59–66

    Article  Google Scholar 

  • Rosenbrock HH (1960) An automatic method for finding the greatest or least value of a function. Comput J 3(3):175–184

    Article  MathSciNet  Google Scholar 

  • Sala R, Pierini M, Baldanzini N (2014) Optimization efficiency in multidisciplinary vehicle design including NVH criteria. Proceedings of the Leuven Conference on Noise and Vibration Engineering (ISMA); 2014 September 15–17; Leuven, Belgium

  • Sala R, Pierini M, Baldanzini N (2014) The development and application of tailored test problems for meta-simulation of multidisciplinary optimization of vehicle structures. Presentation at the (XI) World Congress on Computational Mechanics; 2014 July 20–25; Barcelona, Spain

  • Saltelli A, Annoni P, Azzini I, Campolongo F, Ratto M, Tarantola S (2010) Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Comput Phys Commun 181(2):259–270

    Article  MathSciNet  MATH  Google Scholar 

  • Schramm U, Pilkey WD (1996) Review: optimal design of structures under impact loading. Shock Vib 3:69–81

    Article  Google Scholar 

  • Schreiber T, Schmitz A (1996) Improved surrogate data for nonlinearity tests. Phys Rev Lett 77(4):635

    Article  Google Scholar 

  • Shan S, Wang GG (2010) Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions. Struct Multidiscip Optim 41(2):219–241

    Article  MathSciNet  MATH  Google Scholar 

  • Simpson TW, Toropov V, Balabanov V, Viana FA (2008) Design and analysis of computer experiments in multidisciplinary design optimization: a review of how far we have come or not 12th AIAA/ISSMO multidisciplinary analysis and optimization conference. 5

  • Sobieszczanski-Sobieski J, Haftka RT (1997) Multidisciplinary aerospace design optimization: survey of recent developments. Struct Optim 14(1):1–23

    Article  Google Scholar 

  • Sobieszczanski-Sobieski J, Kodiyalam S, Yang RJ (2001) Optimization of car body under constraints of noise, vibration, and harshness (NVH), and crash. Struct Multidiscip Optim 22(4):295–306

    Article  Google Scholar 

  • Sobol IM (1990) On sensitivity estimation for nonlinear mathematical models. Matem Mod 2(1):112–118

    MathSciNet  MATH  Google Scholar 

  • Sobol IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul 55(1–3):271–280

    Article  MathSciNet  MATH  Google Scholar 

  • Storm R, Price K (1997) Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359

  • Tang WJ, Wu QH (2009) Biologically inspired optimization: a review. Transa Inst Meas Control 31(6):495–515

    Article  Google Scholar 

  • Varis T, Tuovinen T (2012) Open Benchmark database for multidisciplinary optimization problems. Proc Int Conf Model Appl Simul

  • Venema V (2003) IAAFT implementation in MATLAB retrieved January 2014 from http://www.meteo.uni-bonn.de/victor

  • Venkayya VB (1978) Structural optimization: a review and some recommendations. Int J Numer Methods Eng 13(2):203–228

    Article  MATH  Google Scholar 

  • Wolpert DH, Macready WG (1995) No free lunch theorems for search. Technical Report SFI-TR-95-02-010, Santa Fe Institute. 10

  • Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1(1):67–82

    Article  Google Scholar 

  • Wu SR, Gu L (2012) Introduction to the explicit finite element method for nonlinear transient dynamics. Wiley

  • Yang XS (2010a) Nature-inspired metaheuristic algorithms. Luniver press, 2010. http://web.info.uvt.ro/~dzaharie/cne2013/proiecte/tehnici/FireflyAlgorithm/Yang_nature_book_part.pdf

  • Yang XS (2010b) Engineering optimization: an introduction with metaheuristic applications. Wiley

  • Yang RJ, Tseng L, Nagy L, Cheng J (1994) Feasibility study of crash optimization. In: Gilmore BJ, Hoetzel DA, Dutta D, Eschenauer HA (eds.) Advances in design automation, ASME. DE-69–2:549–556

  • Yang RJ, Gu L, Tho CH, Sobieski J (2001) Multi-disciplinary optimization of a full vehicle with high performance computing. In: Conf. of the American Inst. of Aeronautics and Astronautics, pp 688–698, AIAA Paper No. AIAA- 2001–1273

  • Yildiz AR, Solanki KN (2012) Multi-objective optimization of vehicle crashworthiness using a new particle swarm based approach. Int J Adv Manuf Technol 59(1-4):367–376

    Article  Google Scholar 

  • Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical results. Evol Comput 8(2):173–195

    Article  Google Scholar 

Download references

Acknowledgments

This work is performed in the scope of the GRESIMO and ENLIGHT projects, targeting environmentally friendly mobility solutions. The authors have been partially funded by the European Community’s 7th Framework program by means of: an ITN fellowship in the GRESIMO project as part of the People program (Marie Curie Actions) grant agreement no. 290050, and a contribution to the activities in the ENLIGHT project grant agreement no. 314567. Furthermore, the authors are thankful for the publicly available finite element vehicle models used in this work. These models have been developed by the National Crash Analysis Center (NCAC) of The George Washington University under a contract with the FHWA and NHTSA of the US DOT.

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Sala, R., Baldanzini, N. & Pierini, M. Representative surrogate problems as test functions for expensive simulators in multidisciplinary design optimization of vehicle structures. Struct Multidisc Optim 54, 449–468 (2016). https://doi.org/10.1007/s00158-016-1410-9

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