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Numerical investigation of minimum drag profiles in laminar flow using deep learning surrogates

Published online by Cambridge University Press:  01 June 2021

Li-Wei Chen Open the ORCID record for Li-Wei Chen [Opens in a new window] ,
Berkay A. Cakal ,
Xiangyu Hu  and
Nils Thuerey Open the ORCID record for Nils Thuerey [Opens in a new window]
Li-Wei Chen*
Affiliation:
Department of Informatics, Technical University of Munich, D-85748Garching, Germany
Berkay A. Cakal
Affiliation:
Department of Informatics, Technical University of Munich, D-85748Garching, Germany
Xiangyu Hu
Affiliation:
Department of Mechanical Engineering, Technical University of Munich, D-85748Garching, Germany
Nils Thuerey
Affiliation:
Department of Informatics, Technical University of Munich, D-85748Garching, Germany
*
†Email address for correspondence: liwei.chen@tum.de
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Abstract

Efficiently predicting the flow field and load in aerodynamic shape optimisation remains a highly challenging and relevant task. Deep learning methods have been of particular interest for such problems, due to their success in solving inverse problems in other fields. In the present study, U-net-based deep neural network (DNN) models are trained with high-fidelity datasets to infer flow fields, and then employed as surrogate models to carry out the shape optimisation problem, i.e. to find a minimal drag profile with a fixed cross-sectional area subjected to a two-dimensional steady laminar flow. A level-set method as well as Bézier curve method are used to parameterise the shape, while trained neural networks in conjunction with automatic differentiation are utilised to calculate the gradient flow in the optimisation framework. The optimised shapes and drag force values calculated from the flow fields predicted by the DNN models agree well with reference data obtained via a Navier–Stokes solver and from the literature, which demonstrates that the DNN models are capable not only of predicting flow field but also yielding satisfactory aerodynamic forces. This is particularly promising as the DNNs were not specifically trained to infer aerodynamic forces. In conjunction with a fast runtime, the DNN-based optimisation framework shows promise for general aerodynamic design problems.

JFM classification

Mathematical Foundations: Computational methods Mathematical Foundations: General fluid mechanics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 919 , 25 July 2021 , A34
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

de Avila Belbute-Peres, F., Economon, T. & Kolter, Z. 2020 Combining differentiable PDE solvers and graph neural networks for fluid flow prediction. In Proceedings of the 37th International Conference on Machine Learning (ed. H. Daumé III & A. Singh), pp. 2402–2411, vol. 119. PMLR.Google Scholar
de Avila Belbute-Peres, F., Smith, K., Allen, K., Tenenbaum, J. & Kolter, J.Z. 2018 End-to-end differentiable physics for learning and control. In Advances in Neural Information Processing Systems. (ed. S. Bengio et al. .) vol. 31. Curran Associates, Inc.Google Scholar
Baeza, A., Castro, C., Palacios, F. & Zuazua, E. 2008 2D Navier–Stokes shape design using a level set method. AIAA Paper 2008-172.CrossRefGoogle Scholar
Bhatnagar, S., Afshar, Y., Pan, S., Duraisamy, K. & Kaushik, S. 2019 Prediction of aerodynamic flow fields using convolutional neural networks. Comput. Mech. 64 (2), 525–545.CrossRefGoogle Scholar
Bushnell, D.M. 2003 Aircraft drag reduction–a review. J. Aerosp. Engng 217 (1), 1–18.Google Scholar
Bushnell, D.M. & Moore, K.J. 1991 Drag reduction in nature. Annu. Rev. Fluid Mech. 23 (1), 65–79.CrossRefGoogle Scholar
Chen, J., Viquerat, J. & Hachem, E. 2019 U-net architectures for fast prediction of incompressible laminar flows. arXiv:1910.13532.Google Scholar
Dennis, S.C. & Chang, G. 1970 Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. J. Fluid Mech. 42, 471–489.CrossRefGoogle Scholar
Economon, T.D., Palacios, F. & Alonso, J.J. 2013 A viscous continuous adjoint approach for the design of rotating engineering applications. In 21st AIAA Computational Fluid Dynamics Conference. American Institute of Aeronautics and Astronautics.CrossRefGoogle Scholar
Economon, T.D., Palacios, F., Copeland, S.R., Lukaczyk, T.W. & Alonso, J.J. 2016 SU2: an open-source suite for multiphysics simulation and design. AIAA J. 54 (3), 828–846.CrossRefGoogle Scholar
Eismann, S., Bartzsch, S. & Ermon, S. 2017 Shape optimization in laminar flow with a label-guided variational autoencoder. arXiv:1712.03599.Google Scholar
Gardner, B.A. & Selig, M.S. 2003 Airfoil design using a genetic algorithm and an inverse method. AIAA Paper 2003-0043.CrossRefGoogle Scholar
Giles, M.B. & Pierce, N.A. 2000 An introduction to the adjoint approach to design. Flow Turbul. Combust. 65, 393–415.CrossRefGoogle Scholar
Glowinski, R. & Pironneau, O. 1975 On the numerical computation of the minimum-drag profile in laminar flow. J. Fluid Mech. 72, 385–389.CrossRefGoogle Scholar
Glowinski, R. & Pironneau, O. 1976 Towards the computation of minimum drag profiles in viscous laminar flow. Z. Angew. Math. Model. 1, 58–66.Google Scholar
He, L., Kao, C.-Y. & Osher, S. 2007 Incorporating topological derivatives into shape derivatives based level set methods. Comput. Fluids 225, 891–909.Google Scholar
Holl, P., Thuerey, N. & Koltun, V. 2020 Learning to control PDEs with differentiable physics. In International Conference on Learning Representations, 2020. https://openreview.net/forum?id=HyeSin4FPB.Google Scholar
Jameson, A. 1988 Aerodynamic design via control theory. J. Sci. Comput. 3, 233–260.CrossRefGoogle Scholar
Katamine, E., Azegami, H., Tsubata, T. & Itoh, S. 2005 Solution to shape optimisation problems of viscous fields. Intl J. Comput. Fluid Dyn. 19 (1), 45–51.CrossRefGoogle Scholar
Kim, D.W. & Kim, M.U. 2005 Minimum drag shape in two dimensional viscous flow. Intl J. Numer. Meth. Fluids 21 (2), 93–111.CrossRefGoogle Scholar
Kingma, D.P. & Ba, J. 2014 Adam: a method for stochastic optimization. arXiv:1412.6980.Google Scholar
Kline, H.L., Economon, T.D. & Alonso, J.J. 2016 Multi-objective optimization of a hypersonic inlet using generalized outflow boundary conditions in the continuous adjoint method. In 54th AIAA Aerospace Sciences Meeting. AIAA Paper 2016-0912.CrossRefGoogle Scholar
Kondoh, T., Matsumori, T. & Kawamoto, A. 2012 Drag minimization and lift maximization in laminar flows via topology optimization employing simple objective function expressions based on body force integration. Struct. Multidiscipl. Optim. 45, 693–701.CrossRefGoogle Scholar
Kraft, D. 2017 Self-consistent gradient flow for shape optimization. Optim. Meth. Softw. 32 (4), 790–812.CrossRefGoogle ScholarPubMed
Li, J., Zhang, M., Martins, J.R.R.A. & Shu, C. 2020 Efficient aerodynamic shape optimization with deep-learning-based geometric filtering. AIAA J. 58 (10), 4243–4259.CrossRefGoogle Scholar
Ling, J., Kurzawski, A. & Templeton, J. 2016 Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166.CrossRefGoogle Scholar
Michelassi, V., Chen, L., Pichler, R. & Sandberg, R.D. 2015 Compressible direct numerical simulation of low-pressure turbines-part II: effect of inflow disturbances. Trans. ASME: J. Turbomach. 137, 071005.Google Scholar
Mueller, L. & Verstraete, T. 2019 Adjoint-based multi-point and multi-objective optimization of a turbocharger radial turbine. Intl J. Turbomach. Propul. Power 2, 14–30.CrossRefGoogle Scholar
Osher, S. & Sethian, J.A. 1988 Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49.CrossRefGoogle Scholar
Paszke, A., et al. 2019 PyTorch: an imperative style, high-performance deep learning library. In Advances in Neural Information Processing Systems 32 (ed. H. Wallach et al. ), pp. 8024–8035. Curran Associates, Inc.Google Scholar
Patankar, S.V. & Spalding, D.B. 1983 A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. In Numerical Prediction of Flow, Heat Transfer, Turbulence and Combustion, pp. 54–73. Elsevier.CrossRefGoogle Scholar
Pironneau, O. 1973 On optimum profiles in Stokes flow. J. Fluid Mech. 59, 117–128.CrossRefGoogle Scholar
Pironneau, O. 1974 On optimum design in fluid mechanics. J. Fluid Mech. 64, 97–110.CrossRefGoogle Scholar
Queipo, N.V., Haftka, R.T., Shyy, W., Goel, T., Vaidyanathan, R. & Kevin Tucker, P. 2005 Surrogate-based analysis and optimization. Prog. Aerosp. Sci. 41 (1), 1–28.CrossRefGoogle Scholar
Renganathan, S.A., Maulik, R. & Ahuja, J. 2020 Enhanced data efficiency using deep neural networks and gaussian processes for aerodynamic design optimization. arXiv:2008.06731.CrossRefGoogle Scholar
Ronneberger, O., Fischer, P. & Brox, T. 2015 U-net: convolutional networks for biomedical image segmentation. In Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015 (ed. N. Navab, J. Hornegger, W.M. Wells & A.F. Frangi), pp. 234–241. Springer International Publishing.CrossRefGoogle Scholar
Sen, S., Mittal, S. & Biswas, G. 2009 Steady separated flow past a circular cylinder at low Reynolds numbers. J. Fluid Mech. 620, 89–119.CrossRefGoogle Scholar
Sethian, J.A. 1999 a Computational Methods for Fluid Flow, 2nd edn, chap. 16, 17. Cambridge University Press.Google Scholar
Sethian, J.A. 1999 b Fast marching methods. SIAM Rev. 41, 199–235.CrossRefGoogle Scholar
Sethian, J.A. & Smereka, P. 2003 Level set methods for fluid interface. Annu. Rev. Fluid Mech. 35, 341–372.CrossRefGoogle Scholar
Skinner, S.N. & Zare-Behtash, H. 2018 State-of-the-art in aerodynamic shape optimisation methods. Appl. Softw. Comput. 62, 933–962.CrossRefGoogle Scholar
Sun, G. & Wang, S. 2019 A review of the artificial neural network surrogate modeling in aerodynamic design. J. Aerosp. Engng 233 (16), 5863–5872.Google Scholar
Thévenin, D. & Janiga, G. (Ed.) 2008 Optimization and Computational Fluid Dynamics. Springer-Verlag.CrossRefGoogle Scholar
Thuerey, N., Weissenow, K., Prantl, L. & Hu, X. 2018 Deep learning methods for Reynolds-averaged Navier–Stokes simulations of airfoil flows. arXiv:1810.08217.Google Scholar
Tritton, D.J. 1959 Experiments on the flow past a circular cylinder at low Reynolds numbers. J. Fluid Mech. 6 (4), 547–567.CrossRefGoogle Scholar
Um, K., Brand, R., Holl, P., Fei, R. & Thuerey, N. 2020 Solver-in-the-loop: learning from differentiable physics to interact with iterative PDE-solvers. In Advances in Neural Information Processing Systems (ed. H. Larochelle et al. ), pp. 6111–6122. vol. 33. Curran Associates, Inc.Google Scholar
Versteeg, H.K. & Malalasekera, W. 2007 An Introduction to Computational Fluid Dynamics, 2nd edn, chap. 6. Pearson Education Limited.Google Scholar
Viquerat, J. & Hachem, E. 2019 A supervised neural network for drag prediction of arbitrary 2D shapes in low Reynolds number flows. arXiv:1907.05090.CrossRefGoogle Scholar
Yang, F., Yue, Z., Li, L. & Yang, W. 2018 A new curvature-controlled stacking-line method for optimization design of compressor cascade considering surface smoothness. J. Aerosp. Engng 232, 459–471.Google Scholar
Yondo, R., Andrés, E. & Valero, E. 2018 A review on design of experiments and surrogate models in aircraft real-time and many-query aerodynamic analyses. Prog. Aerosp. Sci. 96, 23–61.CrossRefGoogle Scholar
Zahedi, S. & Tornberg, A.-K. 2010 Delta function approximations in level set methods by distance function extension. J. Comput. Phys. 229, 2199–2219.CrossRefGoogle Scholar
Zhang, X., Qiang, X., Teng, J. & Yu, W. 2020 A new curvature-controlled stacking-line method for optimization design of compressor cascade considering surface smoothness. J. Aerosp. Engng 234, 1061–1074.Google Scholar
Zhou, B.Y., Albring, T., Gauger, N.R., da Silva, C.R.I., Economon, T.D. & Alonso, J.J. 2016 An efficient unsteady aerodynamic and aeroacoustic design framework using discrete adjoint. AIAA Paper 2016-3369.CrossRefGoogle Scholar
Zingg, D.W., Nemec, M. & Pulliam, T.H. 2008 A comparative evaluation of genetic and gradient-based algorithms applied to aerodynamic optimization. Eur. J. Comput. Mech. 17 (1–2), 103–126.CrossRefGoogle Scholar