An implicit, exact dual adjoint solution method for turbulent flows on unstructured grids.

*(English)*Zbl 1103.76346Summary: An implicit algorithm for solving the discrete adjoint system based on an unstructured-grid discretization of the Navier-Stokes equations is presented. The method is constructed such that an adjoint solution exactly dual to a direct differentiation approach is recovered at each time step, yielding a convergence rate which is asymptotically equivalent to that of the primal system. The new approach is implemented within a three-dimensional unstructured-grid framework and results are presented for inviscid, laminar, and turbulent flows. Improvements to the baseline solution algorithm, such as line-implicit relaxation and a tight coupling of the turbulence model, are also presented. By storing nearest-neighbor terms in the residual computation, the dual scheme is computationally efficient, while requiring twice the memory of the flow solution. The current implementation allows for multiple right-hand side vectors, enabling simultaneous adjoint solutions for several cost functions or constraints with minimal additional storage requirements, while reducing the solution time compared to serial applications of the adjoint solver. The scheme is expected to have a broad impact on computational problems related to design optimization as well as error estimation and grid adaptation efforts.

##### MSC:

76M12 | Finite volume methods applied to problems in fluid mechanics |

76F10 | Shear flows and turbulence |

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\textit{E. J. Nielsen} et al., Comput. Fluids 33, No. 9, 1131--1155 (2004; Zbl 1103.76346)

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[1] | Nielsen EJ. Aerodynamic design sensitivities on an unstructured mesh using the Navier-Stokes equations and a discrete adjoint formulation. PhD dissertation, Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, December 1998 |

[2] | Anderson, W.K.; Bonhaus, D.L., Airfoil design on unstructured grids for turbulent flows, Aiaa j., 37, 2, 185-191, (1999) |

[3] | 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 |

[4] | Nielsen, E.J.; Anderson, W.K., Aerodynamic design optimization on unstructured meshes using the navier – stokes equations, Aiaa j., 37, 11, 1411-1419, (1999) |

[5] | Nielsen, E.J.; Anderson, W.K., Recent improvements in aerodynamic design optimization on unstructured meshes, Aiaa j., 40, 6, 1155-1163, (2002) |

[6] | Jameson, A., Aerodynamic design via control theory, J. sci. comput., 3, 233-260, (1988) · Zbl 0676.76055 |

[7] | Jameson A, Pierce NA, Martinelli L. Optimum aerodynamic design using the Navier-Stokes equations. AIAA Paper 97-0101, January 1997 · Zbl 0912.76067 |

[8] | Reuther, J.J.; Jameson, A.; Alonso, J.J.; Rimlinger, M.J.; Saunders, D., Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, J. aircraft, 36, 1, 51-60, (1999) · Zbl 0983.76077 |

[9] | Elliott, J.; Peraire, J., Practical three-dimensional aerodynamic design and optimization using unstructured meshes, Aiaa j., 35, 9, 1479-1485, (1997) · Zbl 0900.76420 |

[10] | Soemarwoto B. Multipoint aerodynamic design by optimization. PhD dissertation, Department of Theoretical Aerodynamics, Delft University of Technology, December 1996 |

[11] | Mohammadi B. Optimal shape design, reverse mode of automatic differentiation and turbulence. AIAA Paper 97-0099, January 1997 |

[12] | Nemec M, Zingg DW. Towards efficient aerodynamic shape optimization based on the Navier-Stokes equations. AIAA Paper 2001-2532, 2001 |

[13] | Kim, H.-J.; Sasaki, D.; Obayashi, S.; Nakahashi, K., Aerodynamic optimization of supersonic transport wing using unstructured adjoint method, Aiaa j., 39, 6, 1011-1020, (2001) |

[14] | Soto O, Lohner R. A mixed adjoint formulation for incompressible turbulent problems. AIAA Paper 2002-0451, 2002 |

[15] | Sung C, Kwon JH. Aerodynamic design optimization using the Navier-Stokes and adjoint equations. AIAA Paper 2001-0266, 2001 |

[16] | Kim, C.S.; Kim, C.; Rho, O.H., Sensitivity analysis for the navier – stokes equations with two-equation turbulence models, Aiaa j., 39, 5, 838-845, (2001) |

[17] | Iollo A, Salas MD, Ta’asan S. Shape optimization governed by the Euler equations using an adjoint method. ICASE Report No. 93-78, November 1993 |

[18] | Newman III JC, Taylor III AC, Burgreen GW. An unstructured grid approach to sensitivity analysis and shape optimization using the Euler equations. AIAA Paper 95-1646, 1995 |

[19] | Giles, M.B.; Pierce, N.A., An introduction to the adjoint approach to design, Flow, turbulence combust., 65, 3 and 4, 393-415, (2000) · Zbl 0996.76023 |

[20] | Giles MB. On the use of Runge-Kutta time-marching and multigrid for the solution of steady adjoint equations. AD2000 Conference in Nice, June 19-23, 2000 |

[21] | Giles MB. Adjoint code developments using the exact discrete approach. AIAA Paper 2001-2596, 2001 |

[22] | Giles, M.B., On the iterative solution of adjoint equations, () · Zbl 1134.76396 |

[23] | Giles, M.B.; Duta, M.C.; Muller, J.-D.; Pierce, N.A., Algorithm developments for discrete adjoint methods, Aiaa j., 41, 2, 198-205, (2003) |

[24] | Monk, P.; Suli, E., The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals, SIAM J. numer. anal., 8, 251-274, (1998) · Zbl 0932.65115 |

[25] | Paraschivoiu, M.; Peraire, J.; Patera, A., A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations, Comput. methods appl. mech. eng., 150, 289-312, (1997) · Zbl 0907.65102 |

[26] | Pierce, N.A.; Giles, M.B., Adjoint recovery of superconvergent functionals from PDE approximations, SIAM rev., 42, 2, 247-264, (2000) · Zbl 0948.65119 |

[27] | Venditti, D.A.; Darmofal, D.L., Adjoint error estimation and grid adaptation for functional outputs: application to quasi-one-dimensional flow, J. comput. phys., 164, 204-227, (2000) · Zbl 0995.76057 |

[28] | Venditti, D.A.; Darmofal, D.L., Grid adaptation for functional outputs: application to two-dimensional inviscid flows, J. comput. phys., 176, 40-69, (2002) · Zbl 1120.76342 |

[29] | Muller JD, Giles MB. Solution adaptive mesh refinement using adjoint error analysis. AIAA Paper 2001-2550, 2001 |

[30] | Venditti DA. Grid adaptation for functional outputs of compressible flow simulations. PhD dissertation, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, June 2002 |

[31] | Park MA. Adjoint-based, three-dimensional error prediction and grid adaptation. AIAA Paper 2002-3286, 2002 |

[32] | Venditti, D.A.; Darmofal, D.L., Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows, J comput phys, 187, 22-46, (2003) · Zbl 1047.76541 |

[33] | Peraire, J.; Vahdati, M.; Morgan, K.; Zienkiewicz, O.C., Adaptive remeshing for compressible flow computations, J. comput. phys., 72, 249-266, (1987) · Zbl 0631.76085 |

[34] | Pirzadeh, S.Z., A solution-adaptive unstructured grid method by grid subdivision and local remeshing, J. aircraft, 37, 5, 818-824, (2000) |

[35] | Park MT, Kwon OJ. Unsteady flow computations using a 3-D parallel unstructured dynamic mesh adaptation algorithm. AIAA Paper 2001-0865, 2001 |

[36] | Warren GP, Anderson WK, Thomas JL, Krist SL. Grid convergence for adaptive methods. AIAA Paper 91-1592, 1991 |

[37] | Elliott J. Discrete adjoint analysis and optimization with overset grid modelling of the compressible high-Re Navier-Stokes equations. In: 6th Overset Grid and Solution Technology Symposium, Fort Walton Beach, FL, October 2002 |

[38] | Anderson, W.K.; Bonhaus, D.L., An implicit upwind algorithm for computing turbulent flows on unstructured grids, Comput. fluids, 23, 1, 1-21, (1994) · Zbl 0806.76053 |

[39] | Anderson, W.K.; Rausch, R.D.; Bonhaus, D.L., Implicit/multigrid algorithms for incompressible turbulent flows on unstructured grids, J. comput. phys., 128, 391-408, (1996) · Zbl 0862.76045 |

[40] | Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. stat. comput., 7, 3, 856-869, (1986) · Zbl 0599.65018 |

[41] | Spalart PR, Allmaras SR. A one-equation turbulence model for aerodynamic flows. AIAA Paper 92-0429, 1992 |

[42] | Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 2, 357-372, (1981) · Zbl 0474.65066 |

[43] | Van Leer, B., Flux vector splitting for the Euler equations, Lect. notes phys., 170, 501-512, (1982) |

[44] | Baldwin BS, Barth TJ. A one-equation turbulence transport model for high Reynolds number wall bounded flows. NASA Technical Memorandum 102847, August 1991 |

[45] | Allmaras SR. Multigrid for the 2-D compressible Navier-Stokes equations. AIAA Paper 99-3336, 1999 |

[46] | Schmitt V, Charpin F. Pressure distributions on the ONERA-M6 wing at transonic Mach numbers, experimental database for computer program assessment. AGARD-AR-138, May 1979. p. B1-1-44 |

[47] | Trottenberg, U.; Oosterlee, C.; Schuller, A., Multigrid, (2001), Academic Press San Diego, p. 134 |

[48] | Venkatakrishnan V. Improved multigrid performance of compressible Navier-Stokes solvers. AIAA Paper 98-2967, 1998 |

[49] | Mavriplis, D., Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes, J. comput. phys., 145, 1, 141-165, (1998) · Zbl 0926.76066 |

[50] | Redeker G. DLR-F4 wing body configuration. AGARD-AR-303, vol. 2, August 1994 |

[51] | Karypis, G.; Kumar, V., A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. sci. comput., 20, 1, 359-392, (1998) · Zbl 0915.68129 |

[52] | Karypis G, Kumar V. Multilevel algorithms for multi-constraint graph partitioning. University of Minnesota Technical Report No. 98-019, 1998 |

[53] | Levy DW, Zickuhr T, Vassberg J, Agrawal S, Wahls RA, Pirzadeh S, et al. Summary of data from the first AIAA CFD drag prediction workshop. AIAA Paper 2002-0841, 2002 |

[54] | Kleb WL, Nielsen EJ, Gnoffo PA, Park MA, Wood WA. Collaborative software development in support of Fast Adaptive AeroSpace Tools (FAAST). AIAA Paper 2003-3978, 2003 |

[55] | Samareh JA. A novel shape parameterization approach. NASA TM-1999-209116, May 1999 |

[56] | Pirzadeh, S., Three-dimensional unstructured viscous grids by the advancing-layers method, Aiaa j., 34, 1, 43-49, (1996) · Zbl 0900.76487 |

[57] | Spaid, F.W., High Reynolds number multielement airfoil flowfield measurements, J. aircraft, 37, 3, 499-507, (2000) |

[58] | Rumsey CL, Lee-Rausch EM, Watson RD. Three-dimensional effects on multi-element high lift computations. AIAA Paper 2002-0845, 2002 |

[59] | Alexandrov N, Alter S, Atkins H, Bey K, Bibb K, Biedron R, et al. Opportunities for breakthroughs in large-scale computational simulation and design. NASA TM-2002-211747, April 2002 |

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.