×

zbMATH — the first resource for mathematics

Stabilisation of discrete steady adjoint solvers. (English) Zbl 1351.76131
Summary: A new implicit time-stepping scheme which uses Runge-Kutta time-stepping and Krylov methods as a smoother inside FAS-cycle multigrid acceleration is proposed to stabilise the flow solver and its discrete adjoint counterpart. The algorithm can fully converge the discrete adjoint solver in a wide range of cases where conventional point-implicit methods fail due to either physical or numerical instability. This enables the discrete adjoint to be applied to a much wider range of flow regimes. In addition, the new algorithm offers improved efficiency when applied to stable cases for which the conventional Block-Jacobi solver can fully converge. Both stable and unstable cases are presented to demonstrate the improved robustness and performance of the new scheme. Eigen-analysis is presented to outline the mechanism of the adjoint stabilisation effect.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
Software:
TAPENADE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Giles, M. B.; Pierce, N. A., An introduction to the adjoint approach to design, Flow Turbul. Combust., 65, 3-4, 393-415, (2000) · Zbl 0996.76023
[2] Anderson, W. K.; Venkatakrishnan, V., Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation, Comput. Fluids, 28, 4, 443-480, (1999) · Zbl 0968.76074
[3] Papadimitriou, D.; Giannakoglou, K., A continuous adjoint method with objective function derivatives based on boundary integrals, for inviscid and viscous flows, Comput. Fluids, 36, 2, 325-341, (2007) · Zbl 1177.76369
[4] Othmer, C., A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows, Int. J. Numer. Methods Fluids, 58, 8, 861-877, (2008) · Zbl 1152.76025
[5] Giles, M. B.; Duta, M. C.; Müller, J.-D.; Pierce, N. A., Algorithm developments for discrete adjoint methods, AIAA J., 41, 2, 198-205, (2003)
[6] Campobasso, M. S.; Giles, M. B., Effects of flow instabilities on the linear analysis of turbomachinery aeroelasticity, J. Propuls. Power, 19, 2, 250-259, (2003)
[7] Campobasso, M. S.; Giles, M. B., Stabilization of a linear flow solver for turbomachinery aeroelasticity using recursive projection method, AIAA J., 42, 9, 1765-1774, (2004)
[8] Dwight, R. P.; 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
[9] Saad, Y., Iterative methods for sparse linear systems, (2003), Society for Industrial and Applied Mathematics · Zbl 1002.65042
[10] Mani, K.; Mavriplis, D. J., Geometry optimization in three-dimensional unsteady flow problems using the discrete adjoint, (2013), AIAA-CP 2013-0662
[11] Krakos, J. A.; Darmofal, D. L., Effect of small-scale unsteadiness on adjoint-based output sensitivity, (2009), AIAA-CP 2009-4274
[12] Othmer, C., Adjoint methods for car aerodynamics, J. Math. Ind., 4, 1, 6, (2014)
[13] Protas, B.; Bewley, T. R.; Hagen, G., A computational framework for the regularization of adjoint analysis in multiscale PDE systems, J. Comput. Phys., 195, 1, 49-89, (2004) · Zbl 1049.65059
[14] Mavriplis, D. J., Solution of the unsteady discrete adjoint for three-dimensional problems on dynamically deforming unstructured meshes, (2008), AIAA-CP 08-727
[15] 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
[16] Moinier, P.; Müller, J.-D.; Giles, M. B., Edge-based multigrid and preconditioning for hybrid grids, AIAA J., 40, 10, 1954-1960, (2002)
[17] Müller, J.-D., Coarsening 3-D hybrid meshes for multigrid methods, (Proceedings of the 9th Copper Mountain Multigrid Conference, (1999), Copper Mountain Colorado, USA)
[18] Dwight, R. P., Efficiency improvements of RANS-based analysis and optimization using implicit and adjoint methods on unstructured grids, (2006), University of Manchester, Ph.D. thesis
[19] Knoll, D. A.; Keyes, D. E., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 2, 357-397, (2004) · Zbl 1036.65045
[20] Van Leer, B., Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys., 32, 101-136, (1979) · Zbl 1364.65223
[21] Smith, T. M.; Hooper, R. W.; Ober, C. C.; Lorber, A. A., Intelligent nonlinear solvers for computational fluid dynamics, (2006), AIAA-CP 2006-1483
[22] McCracken, A. J.; Timme, S.; Badcock, K. J.; Eberthardsteiner, J., Accelerating convergence of the CFD linear frequency domain method by a preconditioned linear solver, (6th European Congress on Computational Methods in Applied Sciences and Engineering, (2012))
[23] Brandt, A., Multigrid techniques: 1984 guide, (14th VKI Lecture Series on Computational Fluid Dynamics 1984-04, (1984), Von Karman Institute for Fluid Dynamics Rhode-St.-Genèse, Belgium) · Zbl 0581.76033
[24] Mavriplis, D. J., An assessment of linear versus nonlinear multigrid methods for unstructured mesh solvers, J. Comput. Phys., 175, 1, 302-325, (2002) · Zbl 0995.65099
[25] Naumovich, A.; Förster, M.; Dwight, R., Algebraic multigrid within defect correction for the linearized Euler equations, Numer. Linear Algebra Appl., 17, 2-3, 307-324, (2010) · Zbl 1240.76016
[26] Swanson, R.; Turkel, E.; Rossow, C.-C., Convergence acceleration of Runge-Kutta schemes for solving the Navier-Stokes equations, J. Comput. Phys., 224, 1, 365-388, (2007) · Zbl 1261.76036
[27] Roberts, T. W.; Swanson, R., A study of multigrid preconditioners using eigensystem analysis, (2005), AIAA-CP 05-5229
[28] Rossow, C.-C., Efficient computation of compressible and incompressible flows, J. Comput. Phys., 220, 2, 879-899, (2007) · Zbl 1235.76096
[29] Lallemand, M.; Steve, H.; Dervieux, A., Unstructured multigridding by volume agglomeration: current status, Comput. Fluids, 21, 3, 397-433, (1992) · Zbl 0753.76136
[30] Langer, S.; Schwöppe, A.; Kroll, N., The DLR flow solver TAU - status and recent algorithmic developments, (2014), AIAA-CP 14-0080
[31] Crumpton, P.; Moinier, P.; Giles, M., An unstructured algorithm for high Reynolds number flows on highly stretched grids, (Numerical Methods in Laminar and Turbulent Flow, (1997)), 561-572
[32] Moinier, P., Algorithm developments for an unstructured viscous flow solver, (1999), Oxford University, Ph.D. thesis
[33] Hascoët, L.; Pascual, V., The tapenade automatic differentiation tool: principles, model, and specification, ACM Transactions on Mathematical Software, 39, 3, (2013) · Zbl 1295.65026
[34] Jameson, A.; Schmidt, W.; Turkel, E., Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes, (1981), AIAA-CP 81-1259
[35] Spalart, P. R.; Allmaras, S. R., A one equation turbulence model for aerodinamic flows, (1992), AIAA-CP 92-439
[36] Müller, J.-D.; Cusdin, P., On the performance of discrete adjoint CFD codes using automatic differentiation, Int. J. Numer. Methods Fluids, 47, 8-9, 939-945, (2005) · Zbl 1134.76431
[37] Christianson, B., Reverse accumulation and implict functions, Optim. Methods Softw., 9, 4, 307-322, (1998) · Zbl 0922.65013
[38] Courty, F.; Dervieux, A.; Koobus, B.; Hascoët, L., Reverse automatic differentiation for optimum design: from adjoint state assembly to gradient computation, Optim. Methods Softw., 18, 5, 615-627, (2003) · Zbl 1142.90524
[39] Christakopoulos, F.; Jones, D.; Müller, J.-D., Pseudo-timestepping and verification for automatic differentiation derived CFD codes, Comput. Fluids, 46, 1, 174-179, (2011) · Zbl 1433.76129
[40] Giles, M. B., On the iterative solution of adjoint equations, (Corliss, G.; Faure, C.; Griewank, A.; Hascoët, L.; Naumann, U., Automatic Differentiation of Algorithms, (2002), Springer New York), 145-151
[41] Swanson, R.; Turkel, E.; Yaniv, S., Analysis of a RK/implicit smoother for multigrid, (Computational Fluid Dynamics 2010, (2011), Springer), 409-417 · Zbl 1346.76096
[42] Martinelli, L., Calculations of viscous flows with a multigrid method, (1987), Dept. of Mech. and Aerospace Eng., Princeton University, Ph.D. thesis
[43] Swanson, R.; Rossow, C.-C., An efficient solver for the RANS equations and a one-equation turbulence model, Comput. Fluids, 42, 1, 13-25, (2011) · Zbl 1271.76196
[44] Mavriplis, D. J., Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes, J. Comput. Phys., 145, 1, 141-165, (1998) · Zbl 0926.76066
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.