×

zbMATH — the first resource for mathematics

An adjoint view on flux consistency and strong wall boundary conditions to the Navier-Stokes equations. (English) Zbl 1349.76390
Summary: Inconsistent discrete expressions in the boundary treatment of Navier-Stokes solvers and in the definition of force objective functionals can lead to discrete-adjoint boundary treatments that are not a valid representation of the boundary conditions to the corresponding adjoint partial differential equations. The underlying problem is studied for an elementary 1D advection-diffusion problem first using a node-centred finite-volume discretisation. The defect of the boundary operators in the inconsistently defined discrete-adjoint problem leads to oscillations and becomes evident with the additional insight of the continuous-adjoint approach. A homogenisation of the discretisations for the primal boundary treatment and the force objective functional yields second-order functional accuracy and eliminates the defect in the discrete-adjoint boundary treatment. Subsequently, the issue is studied for aerodynamic Reynolds-averaged Navier-Stokes problems in conjunction with a standard finite-volume discretisation on median-dual grids and a strong implementation of noslip walls, found in many unstructured general-purpose flow solvers. Going out from a base-line discretisation of force objective functionals which is independent of the boundary treatment in the flow solver, two improved flux-consistent schemes are presented; based on either body wall-defined or farfield-defined control-volumes they resolve the dual inconsistency. The behaviour of the schemes is investigated on a sequence of grids in 2D and 3D.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65N08 Finite volume methods for boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
Software:
TAF; TAU
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 5th drag prediction workshop (DPW-5), (2011), and AIAA-2011-3508
[2] S.R. Allmaras, F.T. Johnson, P.R. Spalart, Modifications and clarifications for the implementation of the Spalart-Allmaras turbulence model, in: ICCFD7-1902, 7th International Conference on Computational Fluid Dynamics, Big Island, Hawaii, 9-13 July 2012.
[3] 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, (2008) · Zbl 1161.76035
[4] Giering, R.; Kaminski, T., Recipes for adjoint code construction, ACM Trans. Math. Softw., 24, 4, 437-474, (1998) · Zbl 0934.65027
[5] Giles, M. B., On the use of Runge-Kutta time-marching and multigrid for the solution of steady adjoint equations, (2000), Oxford University Computing Laboratory Oxford, England, Technical Report no. 00/10
[6] 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)
[7] Giles, M. B.; Müller, J.-D.; Duta, M. C., Adjoint code developments using the exact discrete approach, (2001), AIAA-2001-2596
[8] Hartmann, R., Adjoint consistency analysis of discontinuous Galerkin discretizations, SIAM J. Numer. Anal., 45, 2671-2696, (2007) · Zbl 1189.76341
[9] Hartmann, R.; Held, J.; Leicht, T., Adjoint-based error estimation and adaptive mesh refinement for the RANS and \(k - \omega\) turbulence model equations, J. Comput. Phys., 230, 11, 4268-4284, (2010) · Zbl 1343.76044
[10] Hartmann, R.; Leicht, T., Higher order and adaptive DG methods for compressible flows, (Deconinck, H.; Abgrall, R., 37th Advanced CFD Lecture Series: Recent Developments in Higher Order Methods and Industrial Application in Aeronautics, Dec. 9-12, 2013, VKI LS 2014-03, (2014), Von Karman Institute for Fluid Dynamics Rhode Saint Genèse, Belgium)
[11] Hicken, J.; Zingg, D., The role of dual consistency in functional accuracy: error estimation and superconvergence, (2011), AIAA-2011-3855
[12] Hicken, J. E.; Zingg, D. W., Superconvergent functional estimates from summation-by-parts finite-difference discretizations, SIAM J. Sci. Comput., 33, 893-922, (2011) · Zbl 1227.65102
[13] Hicken, J. E.; Zingg, D. W., Dual consistency and functional accuracy: a finite-difference perspective, J. Comput. Phys., 256, 161-182, (2014) · Zbl 1349.65559
[14] Jameson, A., Aerodynamic design via control theory, J. Sci. Comput., 3, 3, 233-260, (Sep. 1988)
[15] Jameson, A., Optimum aerodynamic design using CFD and control theory, (1995), AIAA-1995-1729-CP
[16] Kroll, N.; Langer, S.; Schwöppe, A., The DLR flow solver TAU - status and recent algorithmic developments, (2014), AIAA-2014-0080
[17] Liu, Z.; Sandu, A., On the properties of discrete adjoints of numerical methods for the advection equation, Int. J. Numer. Methods Fluids, 56, 7, 769-803, (2008) · Zbl 1134.65057
[18] Lu, J. C.-C., An a posteriori error control framework for adaptive precision optimization using discontinuous Galerkin finite element method, (2005), Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Ph.D. thesis
[19] Mavriplis, D. J., Discrete adjoint-based approach for optimization problems on three-dimensional unstructured meshes, AIAA J., 45, 4, 741-750, (2007)
[20] Nielsen, E. J.; James, L. B.; Park, M. A.; Darmofal, D. L., An implicit, exact dual adjoint solution method for turbulent flows on unstructured grids, Comput. Fluids, 33, 1131-1155, (2004) · Zbl 1103.76346
[21] Nordström, J.; Eriksson, S.; Eliasson, P., Weak and strong wall boundary procedures and convergence to the steady-state of the Navier-Stokes equations, J. Comput. Phys., 231, 4867-4884, (2012) · Zbl 1245.76020
[22] Oliver, T. A.; Darmofal, D. L., Analysis of dual consistency for discontinuous Galerkin discretizations of source terms, SIAM J. Numer. Anal., 47, 5, 3507-3525, (2008) · Zbl 1203.65258
[23] Pulliam, T. H., Artificial dissipation models for the Euler equations, AIAA J., 24, 12, 1931-1940, (1986) · Zbl 0611.76075
[24] D. Schwamborn, T. Gerhold, R. Heinrich, The DLR TAU-code: recent applications in research and industry, in: ECCOMAS CFD 2006, Egmond aan Zee, The Netherlands, September 5-8, 2006.
[25] Soto, O.; Löhner, R.; Yang, C., An adjoint-based design methodology for CFD problems, Int. J. Numer. Methods Heat Fluid Flow, 14, 6, 734-759, (2004) · Zbl 1078.76057
[26] Stück, A.; Rung, T., Adjoint complement to viscous finite-volume pressure-correction methods, J. Comput. Phys., 248, 402-419, (2013) · Zbl 1349.76391
[27] Swanson, R. C.; Turkel, E., Multistage schemes with multigrid for Euler and Navier-Stokes equations, (1997), NASA, Tech. Rep. 3631
[28] Zymaris, A. S.; Papadimitriou, D. I.; Giannakoglou, K. C.; Othmer, C., Adjoint wall functions: a new concept for use in aerodynamic shape optimization, J. Comput. Phys., 229, 13, 5228-5245, (2010) · Zbl 1346.76059
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.