×

An a priori anisotropic goal-oriented error estimate for viscous compressible flow and application to mesh adaptation. (English) Zbl 1416.65413

Summary: We present a goal-oriented error analysis for the calculation of low Reynolds steady compressible flows with anisotropic mesh adaptation. The error analysis is of a priori type. Its central principle is to express the right-hand side of the error equation, often referred as the local error, as a function of the interpolation error of a collection of fields present in the nonlinear partial differential equations. This goal-oriented error analysis is the extension of A. Loseille et al. [ibid. 229, No. 8, 2866–2897 (2010; Zbl 1307.76060)] done for inviscid flows to laminar viscous flows by adding viscous terms. The main benefits of this approach, in comparison to other error estimates in the literature, is that the optimal anisotropy of the mesh directly appears in the error analysis and is not obtained from an ad hoc variable nor a local analysis. As a consequence, an optimum is obtained and the convergence of the mesh adaptive process is very fast, i.e., generally the convergence is obtained after 5 to 10 mesh adaptation cycle. Then, using the continuous mesh framework, an optimal metric is analytically obtained from the error estimation. Applications to mesh adaptive calculations of flows past airfoils are presented.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
65N08 Finite volume methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Citations:

Zbl 1307.76060
PDFBibTeX XMLCite
Full Text: DOI HAL

References:

[1] Alauzet, F., A parallel matrix-free conservative solution interpolation on unstructured tetrahedral meshes, Comput. Methods Appl. Mech. Eng., 299, 116-142 (2016) · Zbl 1425.65142
[2] Alauzet, F.; Loseille, A., High order sonic boom modeling by adaptive methods, J. Comput. Phys., 229, 561-593 (2010) · Zbl 1253.76052
[3] Alauzet, F.; Loseille, A.; Olivier, G., Multi-Scale Anisotropic Mesh Adaptation for Time-Dependent Problems (June 2016), INRIA, RR-8929
[4] Anderson, W. K.; Venkatakrishnan, V., Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation, Comput. Fluids, 28, 4-5, 443-480 (1999) · Zbl 0968.76074
[5] Arian, E.; Salas, M. D., Admitting the inadmissible: adjoint formulation for incomplete cost functionals in aerodynamic optimization, AIAA J., 37, 1, 37-44 (1999)
[6] Arsigny, V.; Fillard, P.; Pennec, X.; Ayache, N., Log-Euclidean metrics for fast and simple calculus on diffusion tensors, Magn. Reson. Med., 56, 2, 411-421 (2006)
[7] Bank, R. E.; Smith, R. K., A posteriori error estimate based on hierarchical bases, SIAM J. Numer. Anal., 30, 921-935 (1993) · Zbl 0787.65078
[8] Becker, R.; Rannacher, R., A feed-back approach to error control in finite element methods: basic analysis and examples, East-West J. Numer. Math., 4, 237-264 (1996) · Zbl 0868.65076
[9] Belhamadia, Y.; Fortin, A.; Chamberland, E., Three-dimensional anisotropic mesh adaptation for phase change problems, J. Comput. Phys., 201, 753-770 (2004) · Zbl 1061.65095
[10] Belme, A., Unsteady Aerodynamic and Adjoint Method (2011), Université de Nice - Sophia Antipolis: Université de Nice - Sophia Antipolis Nice, France, PhD thesis
[11] Belme, A.; Dervieux, A.; Alauzet, F., Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows, J. Comput. Phys., 231, 6323-6348 (2012) · Zbl 1284.65126
[12] Bottasso, C. L., Anisotropic mesh adaption by metric-driven optimization, Int. J. Numer. Methods Eng., 60, 597-639 (2004) · Zbl 1059.65109
[13] Brèthe, G.; Dervieux, A., Anisotropic norm-oriented mesh adaptation for a Poisson problem, J. Comput. Phys., 322, 804-826 (2016) · Zbl 1352.65472
[14] Brèthe, G.; Dervieux, A., A tensorial-based mesh adaptation for a Poisson problem, Eur. J. Comput. Mech., 26, 3, 245-281 (2017)
[15] Brèthes, G.; Allain, O.; Dervieux, A., A mesh-adaptive metric-based full-multigrid for the Poisson problem, Int. J. Numer. Methods Fluids, 79, 1, 30-53 (2015)
[16] Bristeau, M. O.; Glowinski, R.; Periaux, J.; Viviand, H., Presentation of problems and discussion of results, (Numerical Simulation of Compressible Navier-Stokes Flows: A GAMM Workshop. Numerical Simulation of Compressible Navier-Stokes Flows: A GAMM Workshop, Notes on Numerical Fluid Mechanics, vol. 18 (1987))
[17] Bueno-Orovio, A.; Castro, C.; Palacios, F.; Zuazua, E., Continuous adjoint approach for the Spalart-Allmaras model in aerodynamic optimization, AIAA J., 50, 3, 631-646 (2012)
[18] Carabias, A., Analyse et adaptation de maillage pour des schémas non-oscillatoires d’ordre élevé (2013), Université de Nice - Sophia Antipolis: Université de Nice - Sophia Antipolis Nice, France, PhD thesis
[19] Castro, C.; Lozano, C.; Palacios, F.; Zuazua, E., Systematic continuous adjoint approach to viscous aerodynamic design on unstructured grids, AIAA J., 45, 9, 2125-2139 (2007)
[20] Clément, Ph., Approximation by finite element functions using local regularization, Rev. Fr. Autom. Inform. Rech. Opér., R-2, 77-84 (1975) · Zbl 0368.65008
[21] Cournède, P.-H.; Koobus, B.; Dervieux, A., Positivity statements for a mixed-element-volume scheme on fixed and moving grids, Eur. J. Comput. Mech., 15, 7-8, 767-798 (2006) · Zbl 1208.76088
[22] Dompierre, J.; Vallet, M. G.; Fortin, M.; Bourgault, Y.; Habashi, W. G., Anisotropic mesh adaptation: towards a solver and user independent CFD, (35th AIAA Aerospace Sciences Meeting, AIAA Paper 1997-0861. 35th AIAA Aerospace Sciences Meeting, AIAA Paper 1997-0861, Reno, NV, USA (Jan 1997)) · Zbl 0940.76034
[23] Formaggia, L.; Micheletti, S.; Perotto, S., Anisotropic mesh adaptation in computational fluid dynamics: application to the advection-diffusion-reaction and the Stokes problems, Appl. Numer. Math., 51, 4, 511-533 (2004) · Zbl 1107.65098
[24] Formaggia, L.; Perotto, S., New anisotropic a priori error estimates, Numer. Math., 89, 641-667 (2001) · Zbl 0990.65125
[25] Frey, P. J., About surface remeshing, (Proceedings of the 9th International Meshing Roundtable. Proceedings of the 9th International Meshing Roundtable, New Orleans, LO, USA (2000)), 123-136
[26] Frey, P. J.; Alauzet, F., Anisotropic mesh adaptation for CFD computations, Comput. Methods Appl. Mech. Eng., 194, 48-49, 5068-5082 (2005) · Zbl 1092.76054
[27] George, P. L.; Hecht, F.; Vallet, M. G., Creation of internal points in Voronoi’s type method. Control and adaptation, Adv. Eng. Softw., 13, 5-6, 303-312 (1991) · Zbl 0754.65095
[28] Giles, M. B.; Suli, E., Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, ((A)cta (N)umerica (2002), Cambridge University Press), 145-236 · Zbl 1105.65350
[29] Gruau, C.; Coupez, T., 3D tetrahedral, unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric, Comput. Methods Appl. Mech. Eng., 194, 48-49, 4951-4976 (2005) · Zbl 1102.65122
[30] Hecht, F.; Mohammadi, B., Mesh adaptation by metric control for multi-scale phenomena and turbulence, (35th AIAA Aerospace Sciences Meeting, AIAA Paper 1997-0859. 35th AIAA Aerospace Sciences Meeting, AIAA Paper 1997-0859, Reno, NV, USA (Jan 1997))
[31] Jones, W. T.; Nielsen, E. J.; Park, M. A., Validation of 3D adjoint based error estimation and mesh adaptation for sonic boom reduction, (44th AIAA Aerospace Sciences Meeting, AIAA Paper 2006-1150. 44th AIAA Aerospace Sciences Meeting, AIAA Paper 2006-1150, Reno, NV, USA (Jan 2006))
[32] Kroll, N.; Bieler, H.; Deconinck, H.; Couaillier, V.; van der Ven, H.; Sørensen, K., Adigma - a european initiative on the development of adaptive higher-order variational methods for aerospace applications, (Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 113 (2010), Springer: Springer Berlin, Heidelberg)
[33] Leicht, T.; Hartmann, R., Error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations, J. Comput. Phys., 229, 19, 7344-7360 (2010) · Zbl 1425.76115
[34] Li, X.; Shephard, M. S.; Beal, M. W., 3D anisotropic mesh adaptation by mesh modification, Comput. Methods Appl. Mech. Eng., 194, 48-49, 4915-4950 (2005) · Zbl 1090.76060
[35] Loseille, A., Metric-orthogonal anisotropic mesh generation, Proceedings of the 23rd International Meshing Roundtable. Proceedings of the 23rd International Meshing Roundtable, Proc. Eng., 82, 403-415 (2014)
[36] Loseille, A.; Alauzet, F., Optimal 3D highly anisotropic mesh adaptation based on the continuous mesh framework, (Proceedings of the 18th International Meshing Roundtable (2009), Springer), 575-594
[37] Loseille, A.; Alauzet, F., Continuous mesh framework. Part I: well-posed continuous interpolation error, SIAM J. Numer. Anal., 49, 1, 38-60 (2011) · Zbl 1230.65018
[38] Loseille, A.; Alauzet, F., Continuous mesh framework. Part II: validations and applications, SIAM J. Numer. Anal., 49, 1, 61-86 (2011) · Zbl 1230.65019
[39] 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
[40] Loseille, A.; Dervieux, A.; Frey, P. J.; Alauzet, F., Achievement of global second-order mesh convergence for discontinuous flows with adapted unstructured meshes, (37th AIAA Fluid Dynamics Conference, AIAA Paper 2007-4186. 37th AIAA Fluid Dynamics Conference, AIAA Paper 2007-4186, Miami, FL, USA (Jun 2007))
[41] Loseille, A.; Löhner, R., Cavity-based operators for mesh adaptation, (51st AIAA Aerospace Sciences Meeting (Jan 2013))
[42] Loseille, A.; Marcum, D.; Alauzet, F., Alignment and orthogonality in anisotropic metric-based mesh adaptation, (53rd AIAA Aerospace Sciences Meeting, AIAA Paper 2015-0915. 53rd AIAA Aerospace Sciences Meeting, AIAA Paper 2015-0915, Orlando, FL, USA (Jan 2015))
[43] Marcum, D. L.; Alauzet, F., Aligned metric-based anisotropic solution adaptive mesh generation, Proceedings of the 23rd International Meshing Roundtable. Proceedings of the 23rd International Meshing Roundtable, Proc. Eng., 82, 428-444 (2014)
[44] Marcum, D. L., Adaptive unstructured grid generation for viscous flow applications, AIAA J., 34, 8, 2440-2443 (1996) · Zbl 0900.76489
[45] Menier, V.; Loseille, A.; Alauzet, F., CFD validation and adaptivity for viscous flow simulations, (44th AIAA Fluid Dynamics Conference, AIAA Paper 2014-2925. 44th AIAA Fluid Dynamics Conference, AIAA Paper 2014-2925, Atlanta, GA, USA (Jun 2014))
[46] Michal, T.; Babcock, D.; Kamenetskiy, D.; Krakos, J.; Mani, M.; Glasby, R.; Erwin, T.; Stefanski, D., Comparison of fixed and adaptive unstructured grid results for drag prediction workshop 6, (55th AIAA Aerospace Sciences Meeting (Jan 2017))
[47] Pain, C. C.; Humpleby, A. P.; de Oliveira, C. R.E.; Goddard, A. J.H., Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations, Comput. Methods Appl. Mech. Eng., 190, 3771-3796 (2001) · Zbl 1008.76041
[48] Park, M. A., Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation (2002), AIAA Paper 2002-3286
[49] Pirzadeh, S., Viscous unstructured three dimensional grids by the advancing-layers method, (32nd AIAA Aerospace Sciences Meeting (Jan 1994)) · Zbl 0900.76487
[50] Swanson, R.; Langer, S., Steady-state laminar flow solutions for naca 0012 airfoil, Comput. Fluids, 126, 102-128 (2016) · Zbl 1390.76065
[51] Swanson, R.; Turkel, E., A multistage time-stepping scheme for the Navier-Stokes equations, (23rd AIAA Aerospace Sciences Meeting, AIAA Paper 1985-0035. 23rd AIAA Aerospace Sciences Meeting, AIAA Paper 1985-0035, Reno, NV, USA (Jan 1985))
[52] Tam, A.; Ait-Ali-Yahia, D.; Robichaud, M. P.; Moore, M.; Kozel, V.; Habashi, W. G., Anisotropic mesh adaptation for 3D flows on structured and unstructured grids, Comput. Methods Appl. Mech. Eng., 189, 1205-1230 (2000) · Zbl 1005.76061
[53] Venditti, D. A.; Darmofal, D. L., Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows, J. Comput. Phys., 187, 1, 22-46 (2003) · Zbl 1047.76541
[54] Verfürth, R., A Review of A Posteriori Error Estimation and Adaptative Mesh-Refinement Techniques (1996), Wiley Teubner Mathematics: Wiley Teubner Mathematics New York · Zbl 0853.65108
[55] Wintzer, M.; Nemec, M.; Aftosmis, M. J., Adjoint-based adaptive mesh refinement for sonic boom prediction, (AIAA 26th Applied Aerodynamics Conference, AIAA-2008-6593. AIAA 26th Applied Aerodynamics Conference, AIAA-2008-6593, Honolulu, HI, USA (Aug 2008))
[56] Yano, M.; Darmofal, D. L., An optimization-based framework for anisotropic simplex mesh adaptation, J. Comput. Phys., 231, 22, 7626-7649 (2012) · Zbl 1284.65127
[57] Yano, M.; Modisette, J. M.; Darmofal, D. L., The importance of mesh adaptation for higher-order discretizations of aerodynamics flows, (20th AIAA Computational Fluid Dynamics Conference, AIAA-2011-3852. 20th AIAA Computational Fluid Dynamics Conference, AIAA-2011-3852, Honolulu, HI, USA (June 2011))
[58] Zienkiewicz, O. C.; Zhu, J. Z., The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique, Int. J. Numer. Methods Eng., 33, 7, 1331-1364 (1992) · Zbl 0769.73084
[59] Zienkiewicz, O. C.; Zhu, J. Z., The superconvergent patch recovery and a posteriori error estimates. Part 2: error estimates and adaptivity, Int. J. Numer. Methods Eng., 33, 7, 1365-1380 (1992) · Zbl 0769.73085
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.