Finite-volume goal-oriented mesh adaptation for aerodynamics using functional derivative with respect to nodal coordinates. (English) Zbl 1349.76396
Summary: A new goal-oriented mesh adaptation method for finite volume/finite difference schemes is extended from the structured mesh framework to a more suitable setting for adaptation of unstructured meshes. The method is based on the total derivative of the goal with respect to volume mesh nodes that is computable after the solution of the goal discrete adjoint equation. The asymptotic behaviour of this derivative is assessed on regularly refined unstructured meshes. A local refinement criterion is derived from the requirement of limiting the first order change in the goal that an admissible node displacement may cause. Mesh adaptations are then carried out for classical test cases of 2D Euler flows. Efficiency and local density of the adapted meshes are presented. They are compared with those obtained with a more classical mesh adaptation method in the framework of finite volume/finite difference schemes [D. A. Venditti and D. L. Darmofal, ibid. 176, No. 1, 40–69 (2002; Zbl 1120.76342)]. Results are very close although the present method only makes usage of the current grid.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 76G25 | General aerodynamics and subsonic flows |
Keywords:
goal oriented mesh adaptation; discrete adjoint method; unstructured meshes; steady compressible equations; finite-volume schemeCitations:
Zbl 1120.76342References:
| [1] | van Albada, G. D.; van Leer, B.; Roberts, W. W., A comparative study of computational methods in cosmic gas dynamics, Astron. Astrophys., 1008, 76-84 (1982) · Zbl 0492.76117 |
| [2] | Anderson, W. K.; Venkatakrishnan, V., Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation, Comput. Fluids, 28, 443-480 (1999) · Zbl 0968.76074 |
| [3] | Baker, T. J., Mesh adaptation strategies for problems in fluid dynamics, Finite Elem. Anal. Des., 25, 243-273 (1997) · Zbl 0914.76047 |
| [4] | Biswas, R.; Strawn, R., Tetrahedral and hexahedral mesh adaptation for CFD computations, Appl. Numer. Math., 26, 131-151 (1998) · Zbl 0905.76044 |
| [5] | Bourasseau, S., Contribution à une méthode de raffinement de maillage basée sur le vecteur adjoint pour le calcul de fonctions aérodynamiques (2015), Thèse de l’Université de Nice-Sophia Antipolis |
| [6] | Boussetta, R.; Coupez, T.; Fourment, L., Adaptive remeshing based on a posteriori error estimation for forging simulation, Comput. Methods Appl. Mech. Eng., 195, 6626-6645 (2006) · Zbl 1120.74807 |
| [7] | Cambier, L.; Heib, S.; Plot, S., The Onera elsA CFD software: input from research and feedback from industry, Mech. Ind., 14, 3, 159-174 (2013) |
| [8] | Coupez, T., Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing, J. Comput. Phys., 230, 2391-2405 (2011) · Zbl 1218.65139 |
| [10] | Diskin, B.; Yamaleev, N. K., Grid adaptation using adjoint-based error minimization (2011), AIAA Paper series: Paper 2011-3986 |
| [11] | Dobrzynski, C., MMG3D. User guide (2012), INRIA Technical Report 422 |
| [12] | Dwight, R. P., Efficient a posteriori error estimation for finite volume methods, (NATO Research and Technology Organisation, Proceedings of AVT-147 on Computational Uncertainty. NATO Research and Technology Organisation, Proceedings of AVT-147 on Computational Uncertainty, Athens (2007)) |
| [13] | Dwight, R. P., Goal-oriented mesh adaptation using a dissipation based error indicator, Int. J. Numer. Methods Fluids, 56, 8, 1193-2000 (2007) · Zbl 1136.76033 |
| [14] | Dwight, R. P., Heuristic a posteriori estimation of error due to dissipation in finite volume schemes and application to mesh adaptation, J. Comput. Phys., 227, 2845-2863 (2008) · Zbl 1169.76039 |
| [15] | Frey, P. J.; Alauzet, F., Anisotropic mesh adaptation for CFD computations, Comput. Methods Appl. Mech. Eng., 194, 5068-5082 (2005) · Zbl 1092.76054 |
| [16] | Gueguan, D.; Allain, O.; Dervieux, A.; Alauzet, F., An \(L^\infty - L^p\) mesh adaptive method for computing unsteady bi-fluid flows, Int. J. Numer. Methods Fluids, 84, 1376-1406 (2010) · Zbl 1202.76094 |
| [17] | Jameson, A.; Schmidt, W.; Turkel, E., Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes (1981), AIAA Paper series: Paper 81-1259 |
| [18] | Jameson, A., Aerodynamic design by control theory, J. Sci. Comput., 3, 3, 233-260 (1988) · Zbl 0676.76055 |
| [19] | Fidkowski, K. J.; Roe, P. L., An entropy adjoint approach to mesh refinement, SIAM J. Sci. Comput., 32, 3, 1261-1287 (2010) · Zbl 1213.65142 |
| [20] | Fidkowski, K. J.; Darmofal, D. L., Output-based error estimation and mesh adaptation in computational fluid dynamics: overview and recent results, AIAA J., 49, 4, 673-694 (2011) |
| [21] | Fidkowski, K. J.; Ceze, M. A.; Roe, P. L., Entropy-based drag error estimation and mesh adaptation in two dimensions, J. Aircr., 49, 5, 1485-1496 (2012) |
| [22] | Fraysse, F.; de Vincente, J.; Valero, E., The estimation of truncation error by \(τ\)-estimation revisited, J. Comput. Phys., 231, 2457-3482 (2012) · Zbl 1242.65222 |
| [23] | Fraysse, F.; Valero, E.; Ponsin, J., Comparison of mesh adaptation using the adjoint methodology and truncation error estimates, AIAA J., 50, 9, 1920-1931 (2005) |
| [24] | Giles, M. B.; Pierce, N. A., Adjoint equations in CFD: duality, boundary conditions, and solution behaviour (1997), AIAA Paper series: Paper 1997-1850 |
| [25] | Habashi, W. G.; Dompierre, J.; Bourgault, Y.; Fortin, M.; Vallet, M.-G., Certifiable computational fluid dynamics through mesh optimization, AIAA J., 36, 5, 703-711 (1998) |
| [26] | Hartmann, R.; Houston, P., Adaptive discontinuous finite element methods for the compressible Euler equations, J. Comput. Phys., 183, 508-532 (2002) · Zbl 1057.76033 |
| [27] | Hartmann, R., Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 51, 1131-1156 (2006) · Zbl 1106.76041 |
| [28] | Hartmann, R.; Held, J.; Leicht, T., Adjoint-based error estimation and adaptive mesh refinement for the RANS and k-\(ω\) turbulence model equations, J. Comput. Phys., 230, 4268-4284 (2011) · Zbl 1343.76044 |
| [29] | Jannoun, G.; Hachem, E.; Veysset, J.; Coupez, T., Anisotropic meshes with time-stepping control for unsteady convection-dominated problems, Appl. Math. Model., 39, 7, 1899-1916 (2015) · Zbl 1443.65205 |
| [30] | Peter, J.; Nguyen-Dinh, M.; Trontin, P., Goal oriented mesh adaptation using total derivative of aerodynamic functions with respect to mesh coordinates. With application to Euler flows, Comput. Fluids, 66, 194-214 (2012) |
| [31] | Loseille, A.; Dervieux, A.; Alauzet, F., Fully anisotropic goal-oriented mesh adaptation for 3D steady Euler equations, J. Comput. Phys., 229, 2866-2897 (2010) · Zbl 1307.76060 |
| [32] | Nguyen-Dinh, M.; Peter, J.; Sauvage, R.; Meaux, M.; Désidéri, J. A., Mesh quality assessment based on aerodynamic functional output total derivatives, Eur. J. Mech. B, Fluids, 45, 51-71 (2014) · Zbl 1408.76385 |
| [33] | Nguyen-Dinh, M., Qualification des simulations numériques par adaptation anisotropique de maillages (2014), Thèse de l’Université de Nice-Sophia Antipolis |
| [34] | Nielsen, E.; Park, M., Using an adjoint approach to eliminate mesh sensitivities in aerodynamic design, AIAA J., 44, 5, 948-953 (2005) |
| [35] | Peraire, J.; Vahdati, M.; Morgan, K.; Zienkiewicz, O. C., Adaptive remeshing for compressible flow computations, J. Comput. Phys., 72, 449-466 (1987) · Zbl 0631.76085 |
| [36] | Peraire, J.; Peiro, J.; Morgan, K., Adaptive remeshing for three-dimensional compressible flow computations, J. Comput. Phys., 103, 269-285 (1992) · Zbl 0764.76037 |
| [37] | Pierce, N. A.; Giles, M. B., Adjoint recovery of superconvergent functionals for PDE approximations, SIAM Rev., 42, 247-264 (2000) · Zbl 0948.65119 |
| [38] | Pierce, N. A.; Giles, M. B., Adjoint and defect error bounding and correction for functional estimates (2003), AIAA Paper series: Paper 2003-3846 |
| [39] | Roache, P. J., Perspective: a method for uniform reporting of grid refinement studies, J. Fluids Eng., 116, 3, 405-413 (1994) |
| [40] | Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372 (1981) · Zbl 0474.65066 |
| [41] | Rumsey, C. L.; Thomas, J. L., Application of FUN3D and CFL3D to the third workshop on CFD uncertainty analysis, (Proceedings of the Third Workshop on CFD Uncertainty Analysis. Proceedings of the Third Workshop on CFD Uncertainty Analysis, Lisbon (2008)) |
| [42] | Todarello, G., Goal oriented adaptation of unstructured meshes. Application to finite volume methods (2014), TU-Delft, MSc Thesis |
| [43] | Vassberg, J. C.; Jameson, A., In pursuit of grid convergence, part I: two-dimensional Euler solution (2009), AIAA Paper series: Paper 2009-4114 |
| [44] | Vassberg, J. C.; Jameson, A., In pursuit of grid convergence for two-dimensional Euler solutions, J. Aircr., 47, 1152-1166 (2010) |
| [45] | 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, 40-69 (2000) · Zbl 0995.76057 |
| [46] | 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 |
| [47] | 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 |
| [48] | Warren, G.; Anderson, W.; Thomas, J.; Krist, S., Grid convergence for adaptive methods (1991), AIAA Paper series: Paper 1991-1592 |
| [49] | Widhalm, M.; Brezillon, J.; Leicht, T., Investigation of adjoint based gradient computation for realistic 3d aero-optimization, (Proceedings of 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference (2010)) |
| [50] | Yamalev, N. K.; Diskin, B.; Pathak, K., Error minimization via adjoint-based anisotropic grid adaptation (2010), AIAA Paper series: Paper 2010-4436 |
| [51] | Yano, M.; Darmofal, D. L., An optimization based framework for anisotropic simplex mesh adaptation, J. Comput. Phys., 231, 7626-7649 (2012) · Zbl 1284.65127 |
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