zbMATH — the first resource for mathematics

Dual-consistency study for Green-Gauss gradient schemes in an unstructured Navier-Stokes method. (English) Zbl 1380.76068
Summary: A multi-pass reconstruction method for the discrete-adjoint residual is presented that computes the adjoint of the viscous fluxes based on Green-Gauss gradients in an unstructured finite-volume RANS method. The intermediate discrete-adjoint multipliers of the multi-pass reconstruction are the dual viscous stresses containing the dual Green-Gauss gradients. Since the latter are explicitly evaluated on the fly, meaningful discrete-adjoint operators can be identified and compared against their primal counterparts. Numerical experiments are carried out for a 1D diffusion problem, 2D and 3D RANS cases on a sequence of grids to verify the consistency of the dual Green-Gauss gradients. They are compared against a rediscretisation of the adjoint Green-Gauss gradients known from the continuous-adjoint approach. In that sense, the multi-pass residual reconstruction method provides a deeper insight into the effective dual discretisation, an important part of which is the dual Green-Gauss gradient.
76M12 Finite volume methods applied to problems in fluid mechanics
76M25 Other numerical methods (fluid mechanics) (MSC2010)
Full Text: DOI
[1] 5th Drag Prediction Workshop (DPW-5), 2011, https://aiaa-dpw.larc.nasa.gov/Workshop5/workshop5.html and AIAA-2011-3508.
[2] Albring, T.; Sagebaum, M.; Gauger, R. G., Efficient aerodynamic design using the discrete adjoint method in SU2, (2016), AIAA-2016-3518
[3] Allmaras, S. R.; Johnson, F. T.; Spalart, P. R., Modifications and clarifications for the implementation of the Spalart-Allmaras turbulence model, (ICCFD7-1902, 7th International Conference on Computational Fluid Dynamics, Big Island, Hawaii, 9-13 July 2012, (2012))
[4] 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
[5] Giering, R.; Kaminski, T., Recipes for adjoint code construction, ACM Trans. Math. Softw., 24, 4, 437-474, (1998) · Zbl 0934.65027
[6] Giles, M. B.; Pierce, N. A., Analytic adjoint solutions for the quasi-1D Euler equations, (2000), Oxford University Computing Laboratory, Oxford University England, Report No. 00/03
[7] M.B. Giles, On the use of Runge-Kutta time-marching and multigrid for the solution of steady adjoint equations, 2000.
[8] Giles, M. B.; Müller, J.-D.; Duta, M. C., Adjoint code developments using the exact discrete approach, (2001), AIAA-2001-2596
[9] 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)
[10] Hartmann, R.; Leicht, T., Generalized adjoint consistent treatment of wall boundary conditions for compressible flows, J. Comput. Phys., 300, 754-778, (2015) · Zbl 1349.76215
[11] Haselbacher, A.; Blazek, J., On the accurate and efficient discretization of the Navier-Stokes equations on mixed grids, AIAA J., 38, 2094-2102, (2000)
[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., Transonic airfoil calculations using the Euler equations, (Roe, P. L., Numerical Methods in Aeronautical Fluid Dynamics, (1982), Academic Press London) · Zbl 0562.76053
[15] Jameson, A., Aerodynamic design via control theory, J. Sci. Comput., 3, 3, 233-260, (1988) · Zbl 0676.76055
[16] Jameson, A., Optimum aerodynamic design using CFD and control theory, (1995), AIAA-1995-1729-CP
[17] Jameson, A.; Martinelli, L.; Pierce, N. A., Optimum aerodynamic design using the Navier-Stokes equations, Theor. Comput. Fluid Dyn., 10, 213-237, (1998) · Zbl 0912.76067
[18] Kroll, N.; Langer, S.; Schwöppe, A., The DLR flow solver TAU - status and recent algorithmic developments, (2014), AIAA-2014-0080
[19] 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
[20] 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
[21] 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
[22] Luo, H.; Baum, J. D.; Lohner, R.; Cabello, C., Adaptive edge-based finite element schemes for the Euler and Navier-Stokes equations on unstructured grids, (1993), AIAA-93-0336
[23] Mavriplis, D. J., A three dimensional multigrid Reynolds-averaged Navier-Stokes solver for unstructured meshes, (1994), ICASE Report No. 94-29
[24] Mavriplis, D. J., Revisiting the least-squares procedure for gradient reconstruction on unstructured meshes, (2003), AIAA-2003-3986
[25] Mavriplis, D. J., Discrete adjoint-based approach for optimization problems on three-dimensional unstructured meshes, AIAA J., 45, 4, 741-750, (2007)
[26] Nadarajah, S.; Jameson, A., Studies of the continuous and discrete adjoint approaches to viscous automatic aerodynamic shape optimization, (2001), AIAA-2001-2530
[27] Nielsen, E. J.; Lu, J.; 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
[28] Peter, J.; Dwight, R. P., Numerical sensitivity analysis for aerodynamic optimization: a survey of approaches, Comput. Fluids, 39, 3, 373-391, (2010) · Zbl 1242.76301
[29] Schwamborn, D.; Gerhold, T.; Heinrich, R., The DLR TAU-code: recent applications in research and industry, (ECCOMAS CFD 2006, Egmond aan Zee, The Netherlands, September 5-8, (2006))
[30] Schwöppe, A.; Diskin, B., Accuracy of the cell-centered grid metric in the DLR TAU-code, (New Results in Numerical and Experimental Fluid Mechanics VIII, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 121, (2013), Springer-Verlag), 429-437
[31] 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
[32] Stück, A.; Rung, T., Adjoint complement to viscous finite-volume pressure-correction methods, J. Comput. Phys., 248, 402-419, (2013) · Zbl 1349.76391
[33] Stück, A., An adjoint view on flux consistency and strong wall boundary conditions to the Navier-Stokes equations, J. Comput. Phys., 301, 247-264, (2015) · Zbl 1349.76390
[34] Swanson, R. C.; Turkel, E., On central difference and upwind schemes, J. Comput. Phys., 101, 292-306, (1992) · Zbl 0757.76044
[35] Venditti, D. A.; Darmofal, D. L., Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flow, J. Comput. Phys., 187, 22-46, (2003) · Zbl 1047.76541
[36] Zymaris, A. S.; Papadimitriou, D. I.; Giannakoglou, K. C.; Othmer, C., Continuous adjoint approach to the Spalart-Allmaras turbulence model for incompressible flows, Comput. Fluids, 38, 1528-1538, (2009) · Zbl 1242.76064
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.