Robust Design of Suspension System with Polynomial Chaos Expansion and Machine Learning

During the early development of a new vehicle project, the uncertainty of parameters should be taken into consideration because the design may be perturbed due to real components’ complexity and manufacturing tolerances. Thus, the numerical validation of critical suspension specifications, such as durability and ride comfort should be carried out with random factors. In this article a multi-objective optimization methodology is proposed which involves the specification’s robustness as one of the optimization objectives. To predict the output variation from a given set of uncertain-but-bounded parameters proposed by optimization iterations, an adaptive chaos polynomial expansion (PCE) is applied to combine a local design of experiments with global response surfaces. Furthermore, in order to reduce the additional tests required for PCE construction, a machine learning algorithm based on inter-design correlation matrix firstly classifies the current design points through data mining and clustering. Then it learns how to predict the robustness of future optimized solutions with no extra simulations. At the end of the optimization, a Pareto front between specifications and their robustness can be obtained which represents the best compromises among objectives. The optimum set on the front is classified and can serve as a reference for future design. An example of a quarter car model has been tested for which the target is to optimize the global durability based on real road excitations. The statistical distribution of the parameters such as the trajectories and speeds is also taken into account. The result shows the natural incompatibility between the durability of the chassis and the robustness of this durability. Here the term robustness does not mean “strength”, but means that the performance is less sensitive to perturbations. In addition, a stochastic sampling verifies the good robustness prediction of PCE method and machine learning, based on a greatly reduced number of tests. This example demonstrates the effectiveness of the approach, in particular its ability to save computational costs for full vehicle simulation.

1. Chatillon M. M. (2005) Méthodologie de conception robuste appliquée aux trains de véhicules de tourisme. Doctoral dissertation, Ecully, Ecole centrale de Lyon.

2. Dessombz O. (2000) Analyse dynamique de structures comportant des paramètres incertains. Doctoral dissertation, Ecully, Ecole centrale de Lyon.

3. Wiener N. (1938) The homogeneous chaos. American Journal of Mathematics, 60 (4), 897-936. https://doi.org/10.2307/2371268

4. Xiu D., Karniadakis G. E. (2002) The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM journal on scientific computing, 24 (2), 619-644. https://doi.org/10.1137/s1064827501387826

5. Wu J., Luo Z., Zhang Y., Zhang N., Chen L. (2013) Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions. International Journal for Numerical Methods in Engineering, 95 (7), 608-630. https://doi.org/10.1002/nme.4525

6. Kim N. H., Wang H., Queipo N. (2004) Adaptive reduction of design variables using global sensitivity in reliability-based optimization. 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conferencep, 4515. https://doi.org/10.2514/6.2004-4515

7. Hu C., Youn B. D. (2011) Adaptive-sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems. Structural and Multidisciplinary Optimization, 43 (3), 419-442. https://doi.org/10.1007/s00158-010-0568-9

8. Knio O. M., Najm H. N., Ghanem R. G. (2001) A stochastic projection method for fluid flow: I. basic formulation. Journal of computational Physics, 173 (2), 481-511. https://doi.org/10.1006/jcph.2001.6889

9. Eddy J., Lewis K. (2001) Effective generation of Pareto sets using genetic programming. Proceedings of DETC’01 ASME 2001 Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Pittsburgh, PA, September 9-12, 2001, 132.

10. Loyer B. (2009) Conception fonctionnelle robuste par optimisation multicritère de systèmes de suspension automobile passifs et semi-actifs. Doctoral dissertation, Ecully, Ecole centrale de Lyon.

11. Allen T. T., Bernshteyn M. A., Kabiri-Bamoradian K. (2003) Constructing meta-models for computer experiments. Journal of Quality Technology, 35 (3), 264-274. https://doi.org/10.1080/00224065.2003.11980220

12. Acar E., Rais-Rohani M. (2009) Ensemble of metamodels with optimized weight factors. Structural and Multidisciplinary Optimization, 37 (3), 279-294. https://doi.org/10.1007/s00158-008-0230-y

13. Dumont E., Khaldi M. (2018) Alternova Layer 1 user guide for Renault. Eurodecision, 2018 May.

14. Di Pierro F, Khu S.T., Djordjevic S., Savic D. (2004) A new genetic algorithm to solve effectively highly multi-objective problems: Poga. Report Nr. 2004/02, Center for WaterSystems, University of Exeter.

15. Wang G. G. (2003) Adaptive response surface method using inherited latin hypercube design points. Journal of Mechanical Design, 125 (2), 210-220. https://doi.org/10.1115/1.1561044