Adjoint-based aerodynamic shape optimization with hybridized discontinuous Galerkin methods

Aerodynamic shape optimization; Hybridized discontinuous Galerkin; Discrete adjoint; Rational splines parametrization; Inverse design; Compressible flows

Abstract :

[en] We present a discrete adjoint approach to aerodynamic shape optimization (ASO) based on a hybridized discontinuous Galerkin (HDG) discretization. Our implementation is designed to tie in as seamlessly as possible into a solver architecture written for general balance laws, thus adding design capability to a tool with a wide range of applicability. Design variables are introduced on designated surfaces using the knots of a 2D spline-based geometry representation, while gradients are computed from the adjoint solution using a difference approximation of residual perturbations. A suitable optimization algorithm, such as an in-house steepest descent or the Preconditioned Sequential Quadratic Programming (PSQP) approach from the pyOpt framework, is then employed to find an improved geometry. We present verification of the implementation, including drag or heat flux minimization in compressible flows, as well as inverse design.

Disciplines :

Aerospace & aeronautics engineering

Author, co-author :

Balis, Joachim ; Université de Liège - ULiège > Unités de recherche interfacultaires > Space sciences, Technologies and Astrophysics Research (STAR) ; VKI - Von Karman Institute for Fluid Dynamics [BE]

Jacobs, Frederik ; VKI - Von Karman Institute for Fluid Dynamics [BE]

May, Georg ; VKI - Von Karman Institute for Fluid Dynamics [BE]

Language :

English

Title :

Adjoint-based aerodynamic shape optimization with hybridized discontinuous Galerkin methods

Publication date :

2024

Journal title :

Computers and Fluids

ISSN :

0045-7930

Publisher :

Elsevier BV

Volume :

268

Pages :

106116

Peer reviewed :

Peer reviewed

Additional URL :

https://www.sciencedirect.com/science/article/pii/S0045793023003419?via%3Dihub

Available on ORBi :

since 20 November 2023

Statistics

Number of views

43 (18 by ULiège)

Number of downloads

104 (3 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenAlex citations

Bibliography

Skinner, S., Zare-Behtash, H., State-of-the-art in aerodynamic shape optimisation methods. Appl Soft Comput 62 (2018), 933–962, 10.1016/j.asoc.2017.09.030.
Vassberg, J., Jameson, A., Industrial applications of aerodynamic shape optimization. VKI lecture series on optimization and multidisciplinary design, 2018, 10.35294/ls201804.vassberg2.
Jameson, A., Aerodynamic shape optimization using the adjoint method. VKI lecture series on aerodynamic drag prediction and reduction, 2003.
Gunzburger, M.D., Perspectives in flow control and optimization. 2002, Society for Industrial and Applied Mathematics.
Pironneau, O., On optimum design in fluid mechanics. J Fluid Mech 64 (1974), 97–110.
Wang, Z.J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H.T., et al. High-order CFD methods: current status and perspective. Internat J Numer Methods Fluids 72:8 (2013), 811–845, 10.1002/fld.3767.
Huynh, H.T., Wang, Z.J., Vincent, P.E., High-order methods for computational fluid dynamics: A brief review of compact differential formulations on unstructured grids. Comput & Fluids 98 (2014), 209–220, 10.1016/j.compfluid.2013.12.007.
Reed WH, Hill TR. Triangular mesh methods for the neutron transport equation. Los Alamos Report LA-UR-73-479, 1973.
Cockburn, B., Gopalakrishnan, J., Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J Numer Anal 47:2 (2009), 1319–1365, 10.1137/070706616.
Nguyen, N.C., Peraire, J., Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection–diffusion equations. J Comput Phys 228:23 (2009), 8841–8855, 10.1016/j.jcp.2009.08.030.
Nguyen, N.C., Peraire, J., Cockburn, B., A hybridizable discontinuous Galerkin method for Stokes flow. Comput Methods Appl Mech Engrg 199:9 (2010), 582–597, 10.1016/j.cma.2009.10.007.
Nguyen, N.C., Peraire, J., Cockburn, B., A hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations. J Comput Phys 230 (2011), 1147–1170, 10.1016/j.jcp.2010.10.032.
Peraire, J., Nguyen, N.C., Cockburn, B., A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations. 48th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition, 2010, 10.2514/6.2010-363.
Nguyen, N.C., Peraire, J., An adaptive shock-capturing HDG method for compressible flows. 20th AIAA computational fluid dynamics conference, 2011, 10.2514/6.2011-3060.
Lehrenfeld, C., Hybrid discontinuous Galerkin methods for solving incompressible flow problems. (Ph.D. thesis), 2010, Rheinisch-Westfälischen Technischen Hochschule Aachen.
Oikawa, I., A hybridized discontinuous Galerkin method with reduced stabilization. J Sci Comput 65 (2014), 327–340, 10.1007/s10915-014-9962-6.
Oikawa, I., Analysis of a reduced-order HDG method for the Stokes equations. J Sci Comput, 2015, 10.1007/s10915-015-0090-8.
Qiu, W., Shi, K., A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes. IMA J Numer Anal 36:4 (2016), 1943–1967, 10.1093/imanum/drv067.
Jaust, A., Schütz, J., A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows. Comput & Fluids 98 (2014), 177–185, 10.1016/j.compfluid.2014.01.019.
Fidkowski, K.J., A hybridized discontinuous Galerkin method on mapped deforming domains. Comput & Fluids 139 (2016), 80–91, 10.1016/j.compfluid.2016.04.004.
Chaurasia, H.K., Nguyen, N., Peraire, J., A time-spectral hybridizable discontinuous Galerkin method for periodic flow problems. 21st AIAA computational fluid dynamics conference, 2013, 10.2514/6.2013-2861.
Fernandez, P., Nguyen, N.C., Peraire, J., The hybridized discontinuous Galerkin method for implicit large-eddy simulation of transitional turbulent flows. J Comput Phys 336 (2017), 308–329, 10.1016/j.jcp.2017.02.015.
Moro, D., Nguyen, N.C., Peraire, J., Navier-Stokes solution using hybridizable discontinuous Galerkin methods. 20th AIAA computational fluid dynamics conference 2011, 2011, 10.2514/6.2011-3407.
Woopen, M., Ludescher, T., May, G., A hybridized discontinuous Galerkin method for turbulent compressible flow. 44th AIAA fluid dynamics conference, 2014, 10.2514/6.2014-2783.
Terrana, S., Nguyen, N., Peraire, J., GPU-accelerated large eddy simulation of hypersonic flows. AIAA scitech 2020 forum, 2020, 10.2514/6.2020-1062.
Vila-Pérez, J., Giacomini, M., Sevilla, R., Huerta, A., Hybridisable discontinuous Galerkin formulation of compressible flows. Arch Comput Methods Eng, 28, 2020, 10.1007/s11831-020-09508-z.
Cockburn, B., Hybridizable discontinuous Galerkin methods for second-order elliptic problems: overview, a new result and open problems. Jpn J Ind Appl Math, 40, 2023, 10.1007/s13160-023-00603-9.
Wang, L., Mavriplis, D.J., Anderson, W.K., Adjoint sensitivity formulation for discontinuous Galerkin discretizations in unsteady inviscid flow problems. AIAA J 48:12 (2010), 2867–2883, 10.2514/1.J050444.
Giacomini, M., An equilibrated fluxes approach to the certified descent algorithm for shape optimization using conforming finite element and discontinuous Galerkin discretizations. J Sci Comput 75:1 (2018), 560–595, 10.1007/s10915-017-0545-1.
Wang, K., Yu, S., Wang, Z., Feng, R., Liu, T., Adjoint-based airfoil optimization with adaptive isogeometric discontinuous Galerkin method. Comput Methods Appl Mech Engrg 344 (2019), 602–625, 10.1016/j.cma.2018.10.033.
Chen, G., Fidkowski, K.J., Variable-fidelity multipoint aerodynamic shape optimization with output-based adapted meshes. Aerosp Sci Technol, 105, 2020, 106004, 10.1016/j.ast.2020.106004.
Fidkowski, K.J., Gradient-based shape optimization for unsteady turbulent simulations using field inversion and machine learning. Aerosp Sci Technol, 129, 2022, 107843, 10.1016/j.ast.2022.107843.
Sanjaya, D.P., Fidkowski, K., High-order node movement discretization error control in shape optimization. AIAA scitech forum 2023, 2023, 10.2514/6.2023-2367.
Coppeans, A., Fidkowski, K., Martins, J.R., Comparison of finite volume and high order discontinuous Galerkin based aerodynamic shape optimization. AIAA scitech forum 2023, 2023, 10.2514/6.2023-1845.
Woopen, M., Balan, A., May, G., Schütz, J., A comparison of hybridized and standard DG methods for target-based hp-adaptive simulation of compressible flow. Comput & Fluids 98 (2014), 3–16, 10.1016/j.compfluid.2014.03.023.
Woopen, M., May, G., Schütz, J., Adjoint-based error estimation and mesh adaptation for hybridized discontinuous Galerkin methods. Internat J Numer Methods Fluids 76:11 (2014), 811–834, 10.1002/fld.3959.
Balan, A., Woopen, M., May, G., Adjoint-based hp -adaptivity on anisotropic meshes for high-order compressible flow simulations. Comput & Fluids 139 (2016), 47–67, 10.1016/j.compfluid.2016.03.029.
Schütz, J., May, G., An adjoint consistency analysis for a class of hybrid mixed methods. IMA J Numer Anal 34:3 (2014), 1222–1239, 10.1093/imanum/drt036.
May, G., Hybridized discontinuous Galerkin methods : Formulation and discrete adjoint. VKI 38th lecture series on advanced computational fluid dynamics, 2015, 1–21.
Schöberl, J., NETGEN An advancing front 2D/3D-mesh generator based on abstract rules. Comput Vis Sci 1:1 (1997), 41–52, 10.1007/s007910050004.
Scoggins, J.B., Magin, T.E., Development of mutation++ : Multicomponent thermodynamic and transport properties for ionized plasmas written in C++. 11th AIAA/ASME joint thermophysics and heat transfer conference, 2014, 10.2514/6.2014-2966.
Balis, J., Jacobs, F., May, G., Aerodynamic shape optimization with hybridized discontinuous Galerkin schemes. AIAA scitech forum 2023, 2023, 10.2514/6.2023-1422.
Perez, R.E., Jansen, P.W., Martins, J.R.R.A., PyOpt: A Python-based object-oriented framework for nonlinear constrained optimization. Struct Multidiscip Optim 45:1 (2012), 101–118, 10.1007/s00158-011-0666-3.
Woopen, M., Balan, A., May, G., A unifying computational framework for adaptive high-order finite element methods. 22nd AIAA computational fluid dynamics conference, 2015, 10.2514/6.2015-2601.
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal 39:5 (2002), 1749–1779, 10.1137/s0036142901384162.
Hartmann, R., Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier–Stokes equations. Internat J Numer Methods Fluids 51:9–10 (2006), 1131–1156, 10.1002/fld.1134.
May, G., Devesse, K., Rangarajan, A., Magin, T., A hybridized discontinuous Galerkin solver for high-speed compressible flow. Aerospace, 8(11), 2021, 322, 10.3390/aerospace8110322.
Gangl, P., Sturm, K., Neunteufel, M., Schöberl, J., Fully and semi-automated shape differentiation in NGSolve. Struct Multidiscip Optim 63:3 (2020), 1579–1607, 10.1007/s00158-020-02742-w.
Yang, Z., Mavriplis, D.J., Mesh deformation strategy optimized by the adjoint method on unstructured meshes. AIAA J 45:12 (2007), 2885–2896, 10.2514/1.30592.
Luke, E., Collins, E., Blades, E., A fast mesh deformation method using explicit interpolation. J Comput Phys 231:2 (2012), 586–601, 10.1016/j.jcp.2011.09.021.
Secco, N.R., Kenway, G.K.W., He, P., Mader, C.A., Martins, J.R.R.A., Efficient mesh generation and deformation for aerodynamic shape optimization. AIAA J, 2020, 10.2514/1.J059491.
Gill, P.E., Murray, W., Saunders, M.A., SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM J Optim 12:4 (2002), 979–1006, 10.1137/S1052623499350013.
Yu, Y., Lyu, Z., Xu, Z., Martins, J.R., On the influence of optimization algorithm and initial design on wing aerodynamic shape optimization. Aerosp Sci Technol 75:February (2018), 183–199, 10.1016/j.ast.2018.01.016.
Fyrillas, M.M., Shape optimization for 2D diffusive scalar transport. Opt Eng 10:4 (2008), 477–489, 10.1007/s11081-008-9071-1.
Eyi, S., Hanquist, K.M., Boyd, I.D., Shape optimization of reentry vehicles to minimize heat loading. AIAA scitech 2019 forum, 2019, 10.2514/6.2019-0973.