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

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. 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. 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. 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. 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).

  5. 5.
    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).

  6. 6.

    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.

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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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Keywords

  • Multidisciplinary design optimization
  • Test problems
  • Benchmarking
  • Meta-optimization
  • Vehicle design