Surrogate-based optimisation using adaptively scaled radial basis functions

Abstract Aerodynamic shape optimisation is widely used in several applications, such as road vehicles, aircraft and trains. This paper investigates the performance of two surrogate-based optimisation methods; a Proper Orthogonal Decomposition-based method and a force-based surrogate model. The generic passenger vehicle DrivAer is used as a test case where the predictive capability of the surrogate in terms of aerodynamic drag is presented. The Proper Orthogonal Decomposition-based method uses simulation results from topologically different meshes by interpolating all solutions to a common mesh for which the decomposition is calculated. Both the Proper Orthogonal Decomposition- and force-based approaches make use of Radial Basis Function interpolation. The Radial Basis Function hyperparameters are optimised using differential evolution. Additionally, the axis scaling is treated as a hyperparameter, which reduces the interpolation error by more than 50% for the investigated test case. It is shown that the force-based approach performs better than the Proper Orthogonal Decomposition method, especially at low sample counts, both with and without adaptive scaling. The sample points, from which the surrogate model is built, are determined using an optimised Latin Hypercube sampling plan. The Latin Hypercube sampling plan is extended to include both continuous and categorical values, which further improve the surrogate’s predictive capability when categorical design parameters, such as on/off parameters, are included in the design space. The performance of the force-based surrogate model is compared with four other gradient-free optimisation techniques: Random Sample, Differential Evolution, Nelder–Mead and Bayesian Optimisation. The surrogate model performed as good as, or better than these algorithms, for 17 out of the 18 investigated benchmark problems.

[1]  Jianchao Zeng,et al.  Surrogate-Assisted Cooperative Swarm Optimization of High-Dimensional Expensive Problems , 2017, IEEE Transactions on Evolutionary Computation.

[2]  Ernesto Benini,et al.  A surrogate-assisted evolutionary algorithm based on the genetic diversity objective , 2015, Appl. Soft Comput..

[3]  Simone Sebben,et al.  On the Effects of Wind Tunnel Floor Tangential Blowing on the Aerodynamic Forces of Passenger Vehicles , 2017 .

[4]  Vassilios Theofilis,et al.  Modal Analysis of Fluid Flows: An Overview , 2017, 1702.01453.

[5]  Kirthevasan Kandasamy,et al.  Tuning Hyperparameters without Grad Students: Scalable and Robust Bayesian Optimisation with Dragonfly , 2019, J. Mach. Learn. Res..

[6]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[7]  E. Iuliano.  Global optimization of benchmark aerodynamic cases using physics-based surrogate models , 2017 .

[8]  Alan Edelman,et al.  Julia: A Fresh Approach to Numerical Computing , 2014, SIAM Rev..

[9]  Lixing Han,et al.  Implementing the Nelder-Mead simplex algorithm with adaptive parameters , 2010, Computational Optimization and Applications.

[10]  Tom Verstraete,et al.  Differential evolution based soft optimization to attenuate vane-rotor shock interaction in high-pressure turbines , 2013, Appl. Soft Comput..

[11]  Domenico Quagliarella,et al.  Proper Orthogonal Decomposition, surrogate modelling and evolutionary optimization in aerodynamic design , 2013 .

[12]  R. Haftka,et al.  Ensemble of surrogates , 2007 .

[13]  David R. Anderson,et al.  Model selection and multimodel inference : a practical information-theoretic approach , 2003 .

[14]  Patrick Kofod Mogensen,et al.  Optim: A mathematical optimization package for Julia , 2018, J. Open Source Softw..

[15]  Hossein Zare-Behtash,et al.  State-of-the-art in aerodynamic shape optimisation methods , 2018, Appl. Soft Comput..

[16]  Shmuel Rippa,et al.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation , 1999, Adv. Comput. Math..

[17]  Johann Sienz,et al.  Formulation of the Optimal Latin Hypercube Design of Experiments Using a Permutation Genetic Algorithm , 2004 .

[18]  Kyriakos C. Giannakoglou,et al.  Design of optimal aerodynamic shapes using stochastic optimization methods and computational intelligence , 2002 .

[19]  Nitish Srivastava,et al.  Dropout: a simple way to prevent neural networks from overfitting , 2014, J. Mach. Learn. Res..

[20]  S. Jakobsson,et al.  Rational radial basis function interpolation with applications to antenna design , 2009, J. Comput. Appl. Math..

[21]  Nikolaus A. Adams,et al.  Introduction of a New Realistic Generic Car Model for Aerodynamic Investigations , 2012 .

[22]  Nima Amjady,et al.  Short-term load forecast of electrical power system by radial basis function neural network and new stochastic search algorithm , 2016 .

[23]  Valentina Dolci,et al.  Proper Orthogonal Decomposition as Surrogate Model for Aerodynamic Optimization , 2016 .

[24]  Jiaqi Luo,et al.  Design optimization of the last stage of a 4.5-stage compressor using a POD-based hybrid model , 2018 .

[25]  Jonathon Howard,et al.  Drawing an elephant with four complex parameters , 2010 .

[26]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[27]  S. Sarra,et al.  A rational radial basis function method for accurately resolving discontinuities and steep gradients , 2018, Applied Numerical Mathematics.

[28]  Naif Alajlan,et al.  Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems , 2019, Computers, Materials & Continua.

[29]  Peter I. Frazier,et al.  A Tutorial on Bayesian Optimization , 2018, ArXiv.

[30]  Gregory E. Fasshauer,et al.  Meshfree Approximation Methods with Matlab , 2007, Interdisciplinary Mathematical Sciences.