×

zbMATH — the first resource for mathematics

Robust and efficient adjoint solver for complex flow conditions. (English) Zbl 1410.76274
Summary: A key step in gradient-based aerodynamic shape optimisation using the Reynolds-averaged Navier-Stokes equations is to compute the adjoint solution. Adjoint equations inherit the linear stability and the stiffness of the nonlinear flow equations. Therefore for industrial cases with complex geometries at off-design flow conditions, solving the resulting stiff adjoint equation can be challenging. In this paper, Krylov subspace solvers enhanced by subspace recycling and preconditioned with incomplete lower-upper factorisation are used to solve the stiff adjoint equations arising from typical design and off-design conditions. Compared to the baseline matrix-forming adjoint solver based on the generalized minimal residual method, the proposed algorithm achieved memory reduction of up to a factor of two and convergence speedup of up to a factor of three, on industry-relevant cases. These test cases include the DLR-F6 and DLR-F11 configurations, a wing-body configuration in pre-shock buffet and a large civil aircraft with mesh sizes ranging from 3 to 30 million. The proposed method seems to be particularly effective for the more difficult flow conditions.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76G25 General aerodynamics and subsonic flows
Software:
GCROT; LAPACK; TAU
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Othmer, C., Adjoint methods for car aerodynamics, J Math Ind, 4, 1, 1-23, (2014)
[2] Reuther, J.; Jameson, A.; Farmer, J.; Martinelli, L.; Saunders, D., Aerodynamic shape optimization of complex aircraft configurations via an adjoint formulation, AIAA Paper, 94, (1996)
[3] Giles, M.; Pierce, N., An introduction to the adjoint approach to design, Flow Turbul Combust, 65, 3-4, 393-415, (2000) · Zbl 0996.76023
[4] Brezillon, J.; Gauger, N., 2D and 3D aerodynamic shape optimisation using the adjoint approach, Aerosp Sci Technol, 8, 8, 715-727, (2004) · Zbl 1089.76051
[5] Kroll, N.; Gauger, N.; Brezillon, J.; Dwight, R.; Fazzolari, A.; Vollmer, D., Flow simulation and shape optimization for aircraft design, J Comput Appl Math, 203, 2, 397-411, (2007) · Zbl 1110.76048
[6] Campobasso, M.; Duta, M.; Giles, M., Adjoint calculation of sensitivities of turbomachinery objective functions, J Propul Power, 19, 4, 693-703, (2003)
[7] Wang, D.; He, L., Adjoint aerodynamic design optimization for blades in multistage turbomachinesâ;;part i: methodology and verification, J Turbomach, 132, 2, 021011, (2010)
[8] Wang, D.; He, L.; Li, Y.; Wells, R., Adjoint aerodynamic design optimization for blades in multistage turbomachines-part II: validation and application, J Turbomach, 132, 2, 021012, (2010)
[9] Jameson, A., Aerodynamic design via control theory, J Sci Comput, 3, 3, 233-260, (1988) · Zbl 0676.76055
[10] Jameson, A.; Martinelli, L.; Pierce, N., Optimum aerodynamic design using the Navier-Stokes equations, Theor Comput Fluid Dyn, 10, 1-4, 213-237, (1998) · Zbl 0912.76067
[11] Anderson, W.; Venkatakrishnan, V., Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation, Comput Fluids, 28, 4, 443-480, (1999) · Zbl 0968.76074
[12] Nielsen, E.; Lu, J.; Park, M.; Darmofal, D., An implicit, exact dual adjoint solution method for turbulent flows on unstructured grids, Comput Fluids, 33, 9, 1131-1155, (2004) · Zbl 1103.76346
[13] Mavriplis, D., Multigrid solution of the discrete adjoint for optimization problems on unstructured meshes, AIAA J, 44, 1, 42-50, (2006)
[14] Mavriplis, D., Discrete adjoint-based approach for optimization problems on three-dimensional unstructured meshes, AIAA J, 45, 4, 741-750, (2007)
[15] Nambu, T.; Mavriplis, D.; Mani, K., Adjoint-based shape optimization of high-lift airfoils using the NSU2D unstructured mesh solver, 52nd aerospace sciences meeting, 0554, (2014)
[16] Brezillon, J.; Dwight, R.; Widhalm, M., Aerodynamic optimization for cruise and high-lift configurations, MEGADESIGN and MegaOpt-German initiatives for aerodynamic simulation and optimization in aircraft design, 249-262, (2009), Springer
[17] Shroff, G.; Keller, H., Stabilization of unstable procedures: the recursive projection method, SIAM J Numer Anal, 30, 4, 1099-1120, (1993) · Zbl 0789.65037
[18] Saad, Y.; Schultz, M., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J Sci Stat Comput, 7, 3, 856-869, (1986) · Zbl 0599.65018
[19] Dwight, R.; Brezillon, J., Effect of approximations of the discrete adjoint on gradient-based optimization, AIAA J, 44, 12, 3022-3031, (2006)
[20] Knoll, D.; Keyes, D., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J Comput Phys, 193, 2, 357-397, (2004) · Zbl 1036.65045
[21] Chisholm, T.; Zingg, D., A Jacobian-free Newton-Krylov algorithm for compressible turbulent fluid flows, J Comput Phys, 228, 9, 3490-3507, (2009) · Zbl 1319.76033
[22] Osusky, L.; Buckley, H.; Reist, T.; Zingg, D., Drag minimization based on the Navier-Stokes equations using a Newton-Krylov approach, AIAA J, 53, 6, 1555-1577, (2015)
[23] Hicken, J., Output error estimation for summation-by-parts finite-difference schemes, J Comput Phys, 231, 9, 3828-3848, (2012) · Zbl 1242.65223
[24] Lyu, Z.; Kenway, G.; Martins, J., Aerodynamic shape optimization investigations of the common research model wing benchmark, AIAA J, 53, 4, 968-985, (2014)
[25] de Sturler, E., Truncation strategies for optimal Krylov subspace methods, SIAM J Numer Anal, 36, 3, 864-889, (1999) · Zbl 0960.65031
[26] Hicken, J.; Zingg, D., A simplified and flexible variant of GCROT for solving nonsymmetric linear systems, SIAM J Sci Comput, 32, 3, 1672-1694, (2010) · Zbl 1213.65055
[27] Parks, M.; de Sturler, E.; Mackey, G.; Johnson, D.; Maiti, S., Recycling Krylov subspaces for sequences of linear systems, SIAM J Sci Comput, 28, 5, 1651-1674, (2006) · Zbl 1123.65022
[28] Xu, S.; Timme, S.; Badcock, K., Enabling off-design linearised aerodynamics analysis using Krylov subspace recycling technique, Comput Fluids, 140, 385-396, (2016) · Zbl 1390.76267
[29] Schwamborn, D.; Gerhold, T.; Heinrich, R., The DLR TAU-code: recent applications in research and industry, ECCOMAS CFD 2006: Proceedings of the European conference on computational fluid dynamics, Egmond aan Zee, The Netherlands, September 5-8, (2006)
[30] Gerhold, T., Overview of the hybrid RANS code TAU, MEGAFLOW-numerical flow simulation for aircraft design, 81-92, (2005), Springer · Zbl 1273.76313
[31] Jameson, A.; Schmidt, W.; Turkel, E., Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes, (1981)
[32] Swanson, R.; Turkel, E., On central-difference and upwind schemes, J Comput Phys, 101, 2, 292-306, (1992) · Zbl 0757.76044
[33] Allmaras, S.; Johnson, F.; Spalart, P., Modifications and clarifications for the implementation of the Spalart-Allmaras turbulence model, Seventh international conference on computational fluid dynamics (ICCFD7), 1-11, (2012)
[34] Roe, P., Approximate Riemann solvers, parameter vectors, and difference schemes, J Comput Phys, 43, 2, 357-372, (1981) · Zbl 0474.65066
[35] Yoon, S.; Jameson, A., Lower-upper symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations, AIAA J, 26, 9, 1025-1026, (1988)
[36] de Sturler, E., Nested Krylov methods based on GCR, J Comput Appl Math, 67, 1, 15-41, (1996) · Zbl 0854.65026
[37] Fischer, P., An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations, J Comput Phys, 133, 1, 84-101, (1997) · Zbl 0904.76057
[38] Anderson, E.; Bai, Z.; Bischof, C.; Blackford, S.; Dongarra, J.; Du Croz, J., LAPACK users’ guide, 9, (1999), SIAM · Zbl 0934.65030
[39] Osusky, M., A parallel Newton-Krylov-Schur algorithm for the Reynolds-averaged Navier-Stokes equations, (2013), Ph.D. thesis University of Toronto
[40] Pueyo, A.; Zingg, D., Efficient Newton-Krylov solver for aerodynamic computations, AIAA J, 36, 11, 1991-1997, (1998)
[41] Dwight, R.; Brezillon, J., Efficient and robust algorithms for solution of the adjoint compressible Navier-Stokes equations with applications, Int J Numer Methods Fluids, 60, 4, 365-389, (2009) · Zbl 1161.76035
[42] McCracken, A.; Da Ronch, A.; Timme, S.; Badcock, K., Solution of linear systems in Fourier-based methods for aircraft applications, Int J Comput Fluid Dyn, 27, 2, 79-87, (2013)
[43] Ghysels, P.; Ashby, T.; Meerbergen, K.; Vanroose, W., Hiding global communication latency in the GMRES algorithm on massively parallel machines, SIAM J Sci Comput, 35, 1, C48-C71, (2013) · Zbl 1273.65050
[44] Keyes, D., Aerodynamic applications of Newton-Krylov-Schwarz solvers, Fourteenth international conference on numerical methods in fluid dynamics, 1-20, (1995), Springer · Zbl 0854.76073
[45] Saad, Y.; Sosonkina, M., Distributed Schur complement techniques for general sparse linear systems, SIAM J Sci Comput, 21, 4, 1337-1356, (1999) · Zbl 0955.65020
[46] Mohiyuddin, M.; Hoemmen, M.; Demmel, J.; Yelick, K., Minimizing communication in sparse matrix solvers, Proceedings of the conference on high performance computing networking, storage and analysis, 36, (2009), ACM
[47] Reist, T.; Zingg, D., High-fidelity aerodynamic shape optimization of a lifting-fuselage concept for regional aircraft, J Aircr, (2016)
[48] Laflin, K.; Klausmeyer, S.; Zickuhr, T.; Vassberg, J.; Wahls, R.; Morrison, J., Data summary from second AIAA computational fluid dynamics drag prediction workshop, J Aircr, 42, 5, 1165-1178, (2005)
[49] Rossow, C.-C.; Godard, J.-L.; Hoheisel, H.; Schmitt, V., Investigations of propulsion integration interference effects on a transport aircraft configuration, JAircr, 31, 5, 1022-1030, (1994)
[50] Sartor, F.; Timme, S., Reynolds-averaged Navier-Stokes simulations of shock buffet on half wing-body configuration, AIAA Paper 2015-1939, (2015)
[51] Sartor, F.; Timme, S., Delayed detached-eddy simulation of shock buffet on half wing-body configuration, accepted for publication in AIAA J, (2016)
[52] Lawson, S.; Greenwell, D.; Quinn, M., Characterisation of buffet on a civil aircraft wing, AIAA paper 2016-1309, (2016)
[53] Langer, S., Agglomeration multigrid methods with implicit Runge-Kutta smoothers applied to aerodynamic simulations on unstructured grids, J Comput Phys, 277, 72-100, (2014) · Zbl 1349.76353
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.