A coupled fully implicit method for solving drag-adjoint equations of steady incompressible flows. (English) Zbl 07899013
Summary: Numerical instability is one of the main concerns when applying adjoint-based gradient computation in aerodynamic design optimization for a variety of engineering problems. An adjoint pressure-adjoint velocity-coupled fully implicit method for solving the adjoint equations of steady incompressible flows is developed in the present work, taking the aerodynamic drag as the optimization target. The finite volume method for cell-centered, collocated variables on unstructured grids is used for spatial discretization. A stabilization term based on pressure-weighted interpolation (PWI) of the adjoint velocity is added to the adjoint continuity equation to avoid spurious oscillations of adjoint pressure. All the terms of the continuous adjoint equations are discretized implicitly, including the adjoint transpose convection (ATC) term, which is regarded as one of the major sources of numerical instability. The resulting discretized equations are integrated into one linear system, and the adjoint fields can be obtained simultaneously. To prevent the diagonal element of the coefficient matrix of the linear system from getting too small, a singular value decomposition is applied to the diagonal block of the coefficient matrix, and a limiter is applied to the diagonal elements. The condition number of the coefficient matrix may be quite large, so a parallel sparse linear solver that effectively combines direct and iterative methods through a Schur complement approach is applied to solve the linear system. Compared to the staggered iterative solver, no damping or masking is needed for the ATC term in the proposed fully implicit method. There is no need to sacrifice numerical accuracy for numerical stability. This method has been tested in several benchmark cases, and the numerical results show good numerical stability and accuracy. The present method provides a sound basis for industrial applications of adjoint-based aerodynamic optimization methods.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 76N25 | Flow control and optimization for compressible fluids and gas dynamics |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
continuous adjoint; aerodynamic optimization; pressure-weighted interpolation; fully implicit; direct and iterative combined solverReferences:
| [1] | Peter, J.; Dwight, R. P., Numerical sensitivity analysis for aerodynamic optimization: a survey of approaches, Comput. Fluids, 39, 373-391, 2010 · Zbl 1242.76301 |
| [2] | Wang, Q.; Gao, J., The drag-adjoint field of a circular cylinder wake at Reynolds numbers 20, 100 and 500, J. Fluid Mech., 730, 145-161, 2012 · Zbl 1291.76117 |
| [3] | Meliga, P.; Boujo, E.; Pujals, G.; Gallaire, F., Sensitivity of aerodynamic forces in laminar and turbulent flow past a square cylinder, Phys. Fluids, 26, 26-62, 2014 |
| [4] | Jameson, A., Aerodynamic design via control theory, J. Sci. Comput., 3, 233-260, 1988 · Zbl 0676.76055 |
| [5] | Papadimitriou, D. I.; Giannakoglou, K. C., A continuous adjoint method with objective function derivatives based on boundary integrals, for inviscid and viscous flows, Comput. Fluids, 36, 325-341, 2007 · Zbl 1177.76369 |
| [6] | Othmer, C., A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows, Int. J. Numer. Methods Fluids, 58, 861-877, 2008 · Zbl 1152.76025 |
| [7] | Giles, M. B.; Duta, M. C.; Muller, J.; Pierce, N. A., Algorithm developments for discrete adjoint methods, AIAA J., 41, 198-205, 2003 |
| [8] | He, P.; Mader, C. A.; Martins, J. R.; Maki, K. J., An aerodynamic design optimization framework using a discrete adjoint approach with openfoam, Comput. Fluids, 168, 285-303, 2018 · Zbl 1390.76007 |
| [9] | Towara, M.; Naumann, U., A discrete adjoint model for openfoam, Proc. Comput. Sci., 18, 429-438, 2013 |
| [10] | Towara, M., Discrete adjoint optimization with OpenFOAM, 2018, RWTH Aachen University, Ph.D. thesis · Zbl 1453.65397 |
| [11] | Mader, C. A.; Martins, J. R., Adjoint: an approach for the rapid development of discrete adjoint solvers, AIAA J., 46, 863-873, 2008 |
| [12] | Nadarajah, S. K.; Jameson, A., A comparison of the continuous and discrete adjoint approach to automatic aerodynamic optimization, 2014, AIAA-2000-0667 |
| [13] | Othmer, C., Adjoint methods for car aerodynamics, J. Math. Ind., 4, Article 6 pp., 2014 |
| [14] | Popvac, M.; Jasak, H.; Rusche, H.; Reichl, C., Rans turbulence treatment for continuous adjoint optimization, (International Symposium on Turbulence, 2015) |
| [15] | Xu, J., Adjoint-based sensitivity analysis for external aerodynamics, 2017, Chalmers University, Master’s thesis |
| [16] | Karpouzas, G. K.; Papoutsis-Kiachagias, E. M.; Schumacher, T.; de Villiers, E.; Giannakoglou, K. C.; Othmer, C., Adjoint optimization for vehicle external aerodynamics, Int. J. Automot. Eng., 7, 1-7, 2016 |
| [17] | Klaij, C., On the stabilization of finite volume methods with co-located variables for incompressible flow, J. Comput. Phys., 297, 84-89, 2015 · Zbl 1349.76347 |
| [18] | Shidvash Vakilipour, V. B.; Mohammadi, Masoud; Ormiston, S., Developing a physical influence upwind scheme for pressure based cell-centered finite volume methods, Int. J. Numer. Methods Fluids, 89, 1-29.021, 2018 |
| [19] | Jasak, H., Error analysis and estimation for the finite volume method with applications to fluid flows, 1996, University of London, Ph.D. thesis |
| [20] | Soto, O.; Lohner, R., On the boundary computation of flow sensitivities from boundary integrals, (42nd AIAA Aerospace Sciences Meeting and Exhibit. 42nd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 2004) |
| [21] | Matthias, S.; Stoevesandt, B.; Peinke, J., Optimization of airfoils using the adjoint approach and the influence of adjoint turbulent viscosity, Computation, 6, 5-28, 2018 |
| [22] | Bueno-Orovio, A.; Castro, C.; Palacios, F.; Zuazua, E., Continuous adjoint approach for the Spalart-Allmaras model in aerodynamic optimization, AIAA J., 50, 631-646, 2012 |
| [23] | 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 |
| [24] | Balay, S.; Abhyankar, S.; Adams, M. F.; Benson, S.; Brown, J.; Brune, P.; Buschelman, K.; Constantinescu, E. M.; Dalcin, L.; Dener, A.; Eijkhout, V.; Gropp, W. D.; Hapla, V.; Isaac, T.; Jolivet, P.; Karpeev, D.; Kaushik, D.; Knepley, M. G.; Kong, F.; Kruger, S.; May, D. A.; McInnes, L. C.; Mills, R. T.; Mitchell, L.; Munson, T.; Roman, J. E.; Rupp, K.; Sanan, P.; Sarich, J.; Smith, B. F.; Zampini, S.; Zhang, H.; Zhang, H.; Zhang, J., PETSc web page, 2022 |
| [25] | Balay, S.; Gropp, W. D.; McInnes, L. C.; Smith, B. F., Efficient management of parallelism in object oriented numerical software libraries, (Arge, E.; Bruaset, A. M.; Langtangen, H. P., Modern Software Tools in Scientific Computing, 1997, Birkhäuser Press), 163-202 · Zbl 0882.65154 |
| [26] | Li, H.; Xu, X.; Wang, M.; Li, C.; Ren, X.; Yang, X., Insertion of petsc in the openfoam framework, ACM Trans. Model. Perform. Eval. Comput. Syst., 2, 16-35, 2017 |
| [27] | Pimenta, F.; Alves, M. A., A coupled finite volume solver for numerical simulation of electrically-driven flows, Comput. Fluids, 193, Article 104279 pp., 2019 |
| [28] | Yuan, J. Y.; Yalamov, P. Y., A method for constructing diagonally dominant preconditioners based on Jacobi rotations, Appl. Math. Comput., 174, 74-80, 2006 · Zbl 1089.65039 |
| [29] | Gaidamour, J.; Henon, P., A parallel direct/iterative solver based on a Schur complement approach, (11th IEEE International Conference on Computational Science and Engineering, 2008), 98-105 |
| [30] | Henon, P.; Saad, Y., A parallel multistage ilu factorization based on a hierarchical graph decomposition, SIAM J. Sci. Comput., 28, 2266-2293, 2006 · Zbl 1126.65028 |
| [31] | Uroic, T.; Jasak, H., Block-selective algebraic multigrid for implicitly coupled pressure-velocity system, Comput. Fluids, 167, 100-110, 2018 · Zbl 1390.76526 |
| [32] | Hecht, F., New development in freefem++, J. Numer. Math., 20, 251-265, 2012 · Zbl 1266.68090 |
| [33] | Davis, T. A.; Duff, I. S., An unsymmetric-pattern multifrontal method for sparse lu factorization, SIAM J. Matrix Anal. Appl., 18, 140-158, 1997 · Zbl 0884.65021 |
| [34] | Breuer, M.; Bernsdorf, J.; Zeiser, T.; Durst, F., Accurate computations of the laminar flow past a square cylinder based on two different methods: lattice-Boltzmann and finite-volume, Int. J. Heat Fluid Flow, 21, 186-196, 2000 |
| [35] | Meile, W.; Brenn, G.; Reppenhagen, A.; Lechner, B.; Fuchs, A., Experiments and numerical simulations on the aerodynamics of the Ahmed body, CFD Lett., 3, 32-39, 2011 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
