{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:30:07Z","timestamp":1760146207110,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,16]],"date-time":"2024-10-16T00:00:00Z","timestamp":1729036800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"French ANRT","award":["2020\/1218"],"award-info":[{"award-number":["2020\/1218"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this paper, a novel surrogate model for shape-parametrized vehicle drag force prediction is proposed. It is assumed that only a limited dataset of high-fidelity CFD results is available, typically less than ten high-fidelity CFD solutions for different shape samples. The idea is to take advantage not only of the drag coefficients but also physical fields such as velocity, pressure, and kinetic energy evaluated on a cutting plane in the wake of the vehicle and perpendicular to the road. This additional \u201caugmented\u201d information provides a more accurate and robust prediction of the drag force compared to a standard surface response methodology. As a first step, an original reparametrization of the shape based on combination coefficients of shape principal components is proposed, leading to a low-dimensional representation of the shape space. The second step consists in determining principal components of the x-direction momentum flux through a cutting plane behind the car. The final step is to find the mapping between the reduced shape description and the momentum flux formula to achieve an accurate drag estimation. The resulting surrogate model is a space-parameter separated representation with shape principal component coefficients and spatial modes dedicated to drag-force evaluation. The algorithm can deal with shapes of variable mesh by using an optimal transport procedure that interpolates the fields on a shared reference mesh. The Machine Learning algorithm is challenged on a car concept with a three-dimensional shape design space. With only two well-chosen samples, the numerical algorithm is able to return a drag surrogate model with reasonable uniform error over the validation dataset. An incremental learning approach involving additional high-fidelity computations is also proposed. The leading algorithm is shown to improve the model accuracy. The study also shows the sensitivity of the results with respect to the initial experimental design. As feedback, we discuss and suggest what appear to be the correct choices of experimental designs for the best results.<\/jats:p>","DOI":"10.3390\/computation12100207","type":"journal-article","created":{"date-parts":[[2024,10,21]],"date-time":"2024-10-21T04:45:35Z","timestamp":1729485935000},"page":"207","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Enhanced Drag Force Estimation in Automotive Design: A Surrogate Model Leveraging Limited Full-Order Model Drag Data and Comprehensive Physical Field Integration"],"prefix":"10.3390","volume":"12","author":[{"given":"Kalinja","family":"Naffer-Chevassier","sequence":"first","affiliation":[{"name":"Renault Group, Technocentre Renault, 78280 Guyancourt, France"},{"name":"LMAC Lab, Universit\u00e9 de Technologie de Compi\u00e8gne, CS 60319, 60203 Compi\u00e8gne, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0854-4670","authenticated-orcid":false,"given":"Florian","family":"De Vuyst","sequence":"additional","affiliation":[{"name":"BMBI Lab, Universit\u00e9 de Technologie de Compi\u00e8gne, CNRS, CS 60319, 60203 Compi\u00e8gne, France"}]},{"given":"Yohann","family":"Goardou","sequence":"additional","affiliation":[{"name":"Renault Group, Technocentre Renault, 78280 Guyancourt, France"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lions, J.L., and Magenes, E. (1972). Non-Homogeneous Boundary Value Problems and Applications, Springer.","DOI":"10.1007\/978-3-642-65161-8"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1017\/S0022112074002023","article-title":"On optimum design in fluid mechanics","volume":"64","author":"Pironneau","year":"1974","journal-title":"J. Fluid Mech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/BF01061285","article-title":"Aerodynamic design via control theory","volume":"3","author":"Jameson","year":"1988","journal-title":"J. Sci. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Jameson, A. (1995, January 19\u201322). Optimum aerodynamic design using CFD and control theory, AIAA paper 95-1729. Proceedings of the AIAA 12th Computational Fluid Dynamics Conference, San Diego, CA, USA.","DOI":"10.2514\/6.1995-1729"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2514\/1.J057955","article-title":"Shape Optimization Using the Adjoint Lattice Boltzmann Method for Aerodynamic Applications","volume":"57","author":"Cheylan","year":"2019","journal-title":"AIAA J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.1029\/JB076i008p01905","article-title":"Multiquadric equations of topography and other irregular surfaces","volume":"76","author":"Hardy","year":"1971","journal-title":"J. Geophys. Res."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Buhmann, M.D. (2003). Radial Basis Functions: Theory and Implementation, Cambridge University Press.","DOI":"10.1017\/CBO9780511543241"},{"key":"ref_8","first-page":"133","article-title":"Introduction to Gaussian processes","volume":"168","author":"MacKay","year":"1998","journal-title":"NATO ASI Ser. F Comput. Syst. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1146\/annurev-fluid-010719-060214","article-title":"Machine learning for fluid mechanics","volume":"52","author":"Brunton","year":"2020","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1080\/14786440109462720","article-title":"On Lines and Planes of Closest Fit to Systems of Points in Space","volume":"2","author":"Pearson","year":"1901","journal-title":"Philos. Mag."},{"key":"ref_11","unstructured":"Lumley, J. (1970). Stochastic Tools in Turbulence, Academic Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2323","DOI":"10.1126\/science.290.5500.2323","article-title":"Nonlinear Dimensionality Reduction by Locally Linear Embedding","volume":"290","author":"Roweis","year":"2000","journal-title":"Science"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3197517.3201325","article-title":"Learning three-dimensional flow for interactive aerodynamic design","volume":"37","author":"Umetani","year":"2018","journal-title":"ACM Trans. Graph."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Li, X., Xie, C., and Sha, Z. (2021, January 5\u20138). Part-Aware Product Design Agent Using Deep Generative Network and Local Linear Embedding. Proceedings of the Hawaii International Conference on System Sciences, Kauai, HI, USA.","DOI":"10.24251\/HICSS.2021.640"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1002\/nme.6127","article-title":"An augmented reality platform for interactive aerodynamic design and analysis","volume":"120","author":"Curtit","year":"2019","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Song, B., Yuan, C., Permenter, F., Arechiga, N., and Ahmed, F. (2023, January 20\u201323). Surrogate Modeling of Car Drag Coefficient with Depth and Normal Renderings. Proceedings of the ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 3A: 49th Design Automation Conference (DAC), Boston, MA, USA.","DOI":"10.1115\/DETC2023-117400"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Xie, S., Girshick, R., Dollar, P., Tu, Z., and He, K. (2017, January 21\u201326). Aggregated Residual Transformations for Deep Neural Networks. Proceedings of the 30th IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, HI, USA.","DOI":"10.1109\/CVPR.2017.634"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1111\/cgf.14772","article-title":"Interactive design of 2D car profiles with aerodynamic feedback","volume":"42","author":"Rosset","year":"2023","journal-title":"Comput. Graph. Forum"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Guo, X., Li, W., and Iorio, F. (2016, January 13\u201317). Convolutional Neural Networks for Steady Flow Approximation. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining Association for Computing Machinery, New York, NY, USA.","DOI":"10.1145\/2939672.2939738"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"77","DOI":"10.4271\/15-15-02-0006","article-title":"Deep Learning for Real-Time Aerodynamic Evaluations of Arbitrary Vehicle Shapes","volume":"15","author":"Jacob","year":"2022","journal-title":"SAE Int. J. Passeng. Veh. Syst."},{"key":"ref_21","first-page":"234","article-title":"U-Net: Convolutional Networks for Biomedical Image Segmentation","volume":"9351","author":"Ronneberger","year":"2015","journal-title":"Med Image Comput.-Comput.-Assist. Interv."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Heft, A., Indinger, T., and Adams, N. (2012). Introduction of a New Realistic Generic Car Model for Aerodynamic Investigation. SAE Tech. Pape 2012-01-0168.","DOI":"10.4271\/2012-01-0168"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Monti, F., Boscaini, D., Masci, J., Rodola, E., Svoboda, J., and Bronstein, M.M. (2017, January 21\u201326). Geometric Deep Learning on Graphs and Manifolds Using Mixture Model CNNs. Proceedings of the 30th IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, HI, USA.","DOI":"10.1109\/CVPR.2017.576"},{"key":"ref_24","unstructured":"Baqu\u00e9, P., Remelli, E., Fleuret, F., and Fua, P. (2018, January 10\u201315). Geodesic Convolutional Shape Optimization. Proceedings of the 35th ICML, Stockholm, Sweden."},{"key":"ref_25","unstructured":"Durasov, N., Lukoyanov, A., Donier, J., and Fua, P. (2021). DEBOSH: Deep Bayesian Shape Optimization. arXiv."},{"key":"ref_26","first-page":"22468","article-title":"MeshSDF: Differentiable Iso-Surface Extraction","volume":"33","author":"Remelli","year":"2020","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cad.2019.02.003","article-title":"A Generative Design and Drag Coefficient Prediction System for Sedan Car Side Silhouettes based on Computational Fluid Dynamics","volume":"111","author":"Gunpinar","year":"2019","journal-title":"Comput.-Aided Des."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"398","DOI":"10.4271\/2010-01-0513","article-title":"Automotive aerodynamic design exploration employing new optimization methodology based on CFD","volume":"3","author":"Ando","year":"2010","journal-title":"SAE Int. J. Passeng. Cars-Meek Syst."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Bertram, A., Othmer, C., and Zimmermann, R. (2018, January 8\u201312). Towards Real-time Vehicle Aerodynamic Design via Multi-fidelity Data-driven Reduced Order Modeling. Proceedings of the 2018 AIAA\/ASCE\/AHS\/ASC Structures, Structural Dynamics, and Materials Conference, Kissimmee, FL, USA.","DOI":"10.2514\/6.2018-0916"},{"key":"ref_30","unstructured":"Clancy, L.J. (1975). Aerodynamics, Wiley."},{"key":"ref_31","unstructured":"Anderson, J. (2016). Fundamentals of Aerodynamics, McGraw-Hill. [6th ed.]."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Onorato, M., Costelli, A., and Garrone, A. (1984). Drag Measurement Through Wake Analysis. SAE Tech. Pape. 840302, 85\u201393.","DOI":"10.4271\/840302"},{"key":"ref_33","unstructured":"Sagaut, P. (2006). Large Eddy Simulation for Incompressible Flows, Springer."},{"key":"ref_34","unstructured":"Feydy, J., S\u00e9journ\u00e9, T., Vialard, F.-X., Amari, S.-I., Trouve, A., and Peyr\u00e9, G. (2019, January 16\u201318). Interpolating between Optimal Transport and MMD using Sinkhorn Divergences. Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, Naha, Japan."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1093\/biomet\/30.1-2.81","article-title":"A new measure of rank correlation","volume":"30","author":"Kendall","year":"1938","journal-title":"Biometrika"},{"key":"ref_36","first-page":"226","article-title":"Kendall tau metric","volume":"3","author":"Nelsen","year":"2001","journal-title":"Encycl. Math."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/10\/207\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:14:01Z","timestamp":1760112841000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/10\/207"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,16]]},"references-count":36,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["computation12100207"],"URL":"https:\/\/doi.org\/10.3390\/computation12100207","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2024,10,16]]}}}