Augmented Lagrangian preconditioner for large-scale hydrodynamic stability analysis. (English) Zbl 1441.76069
Summary: Hydrodynamic linear stability analysis of large-scale three-dimensional configurations is usually performed with a“time-stepping” approach, based on the adaptation of existing solvers for the unsteady incompressible Navier-Stokes equations. We propose instead to solve the nonlinear steady equations with the Newton method and to determine the largest growth-rate eigenmodes of the linearized equations using a shift-and-invert spectral transformation and a Krylov-Schur algorithm. The solution of the shifted linearized Navier-Stokes problem, which is the bottleneck of this approach, is computed via an iterative Krylov subspace solver preconditioned by the modified augmented Lagrangian (mAL) preconditioner [M. Benzi and M. A. Olshanskii, SIAM J. Numer. Anal. 49, No. 2, 770–788 (2011; Zbl 1245.76044)]. The well-known efficiency of this preconditioned iterative strategy for solving the real linearized steady-state equations is assessed here for the complex shifted linearized equations. The effect of various numerical and physical parameters is investigated numerically on a two-dimensional flow configuration, confirming the reduced number of iterations over state-of-the-art steady-state and time-stepping-based preconditioners. A parallel implementation of the steady Navier-Stokes and eigenvalue solvers, developed in the FreeFEM language, suitably interfaced with the PETSc/SLEPc libraries, is described and made openly available to tackle three-dimensional flow configurations. Its application on a small-scale three-dimensional problem shows the good performance of this iterative approach over a direct \(L U\) factorization strategy, in regards of memory and computational time. On a large-scale three-dimensional problem with 75 million unknowns, a 80% parallel efficiency on 256 up to 2048 processes is obtained.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Keywords:
Navier-Stokes equations; Newton method; Krylov-Schur method; linear stability analysis; recycled Krylov methods; distributed computingCitations:
Zbl 1245.76044References:
| [1] | Drazin, P. G.; Reid, W. H., Hydrodynamic Stability (2004), Cambridge University Press · Zbl 1055.76001 |
| [2] | Huerre, P.; Monkewitz, P. A., Local and global instabilities in spatially developing flows, Annu. Rev. Fluid Mech., 22, 1, 473-537 (1990) · Zbl 0734.76021 |
| [3] | Theofilis, V., Advances in global linear instability analysis of nonparallel and three-dimensional flows, Prog. Aerosp. Sci., 39, 4, 249-315 (2003) |
| [4] | Chomaz, J.-M., Global instabilities in spatially developing flows: Non-normality and nonlinearity, Annu. Rev. Fluid Mech., 37, 1, 357-392 (2005) · Zbl 1117.76027 |
| [5] | Bagheri, S.; Åkervik, E.; Brandt, L.; Henningson, D. S., Matrix-free methods for the stability and control of boundary layers, AIAA J., 47, 5, 1057-1068 (2009) |
| [6] | Sipp, D.; Marquet, O.; Meliga, P.; Barbagallo, A., Dynamics and control of global instabilities in open-flows: a linearized approach, Appl. Mech. Rev., 63, 3 (2010) |
| [7] | Theofilis, V., Global linear instability, Annu. Rev. Fluid Mech., 43, 1, 319-352 (2011) · Zbl 1299.76074 |
| [8] | Dijkstra, H. A.; Wubs, F. W.; Cliffe, A. K.; Doedel, E.; Dragomirescu, I. F.; Eckhardt, B.; Gelfgat, A. Y.; Hazel, A. L.; Lucarini, V.; Salinger, A. G.; Phipps, E. T.; Juan, S. U.; Schuttelaars, H.; Tuckerman, L. S.; Thiele, U., Numerical bifurcation methods and their application to fluid dynamics: Analysis beyond simulation, Commun. Comput. Phys., 15, 1, 1-45 (2014) · Zbl 1373.76026 |
| [9] | Loiseau, J.-C.; Bucci, M. A.; Cherubini, S.; Robinet, J.-C., Time-stepping and Krylov methods for large-scale instability problems, (Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics (2019), Springer International Publishing), 33-73 · Zbl 1398.76005 |
| [10] | Shroff, G. M.; Keller, H. B., Stabilization of unstable procedures: the recursive projection method, SIAM J. Numer. Anal., 30, 4, 1099-1120 (1993) · Zbl 0789.65037 |
| [11] | Åkervik, E.; Brandt, L.; Henningson, D. S.; Hœpffner, J.; Marxen, O.; Schlatter, P., Steady solutions of the Navier-Stokes equations by selective frequency damping, Phys. Fluids, 18, 6, 1-5 (2006) |
| [12] | Citro, V.; Luchini, P.; Giannetti, F.; Auteri, F., Efficient stabilization and acceleration of numerical simulation of fluid flows by residual recombination, J. Comput. Phys., 344, April, 234-246 (2017) |
| [13] | Karniadakis, G. E.; Israeli, M.; Orszag, S. A., High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97, 2, 414-443 (1991) · Zbl 0738.76050 |
| [14] | Kim, J.; Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59, 2, 308-323 (1985) · Zbl 0582.76038 |
| [15] | Tuckerman, L. S.; Barkley, D., Bifurcation analysis for timesteppers, (Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems (2000), Springer, New York, NY), 453-466 · Zbl 0961.35015 |
| [16] | Caliari, M.; Kandolf, P.; Ostermann, A.; Rainer, S., Comparison of software for computing the action of the matrix exponential, BIT Numer. Math., 54, 1, 113-128 (2014) · Zbl 1290.65042 |
| [17] | Rostami, M. W.; Xue, F., Robust linear stability analysis and a new method for computing the action of the matrix exponential, SIAM J. Sci. Comput., 40, 5, A3344-A3370 (2018) · Zbl 1401.65019 |
| [18] | Amestoy, P.; Duff, I.; L’Excellent, J.-Y.; Koster, J., A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23, 1, 15-41 (2001) · Zbl 0992.65018 |
| [19] | Li, X. S., An overview of SuperLU: Algorithms, implementation, and user interface, ACM Trans. Math. Softw., 31, 3, 302-325 (2005) · Zbl 1136.65312 |
| [20] | Christodoulou, K. N.; Scriven, L. E., Finding leading modes of a viscous free surface flow: An asymmetric generalized eigenproblem, J. Sci. Comput., 3, 4, 355-406 (1988) · Zbl 0677.65032 |
| [21] | Ehrenstein, U.; Gallaire, F., On two-dimensional temporal modes in spatially evolving open flows: The flat-plate boundary layer, J. Fluid Mech., 536, 209-218 (2005) · Zbl 1073.76027 |
| [22] | Sartor, F.; Mettot, C.; Sipp, D., Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile, AIAA J., 53, 7, 1980-1993 (2015) |
| [23] | Cliffe, K. A.; Garratt, T. J.; Spence, A., Eigenvalues of the discretized Navier-Stokes equation with application to the detection of hopf bifurcations, Adv. Comput. Math., 1, 3, 337-356 (1993) · Zbl 0830.76048 |
| [24] | Meerbergen, K.; Roose, D., Matrix transformations for computing rightmost eigenvalues of large sparse non-symmetric eigenvalue problems, IMA J. Numer. Anal., 16, 3, 297-346 (1996) · Zbl 0856.65033 |
| [25] | Mack, C. J.; Schmid, P. J., A preconditioned Krylov technique for global hydrodynamic stability analysis of large-scale compressible flows, J. Comput. Phys., 229, 3, 541-560 (2010) · Zbl 1253.76042 |
| [26] | Saad, Y., Variations on Arnoldi’s method for computing eigenelements of large unsymmetric matrices, Linear Algebra Appl., 34, 269-295 (1980) · Zbl 0456.65017 |
| [27] | Marquet, O.; Larsson, M., Global wake instabilities of low aspect-ratio flat-plates, Eur. J. Mech. B/Fluids, 49, 400-412 (2015) · Zbl 1408.76222 |
| [28] | Bagheri, S.; Schlatter, P.; Schmid, P. J.; Henningson, D. S., Global stability of a jet in crossflow, J. Fluid Mech., 624, 33-44 (2009) · Zbl 1171.76372 |
| [29] | Loiseau, J.-C.; Robinet, J.-C.; Cherubini, S.; Leriche, E., Investigation of the roughness-induced transition: Global stability analyses and direct numerical simulations, J. Fluid Mech., 760, 175-211 (2014) |
| [30] | Citro, V.; Giannetti, F.; Luchini, P.; Auteri, F., Global stability and sensitivity analysis of boundary-layer flows past a hemispherical roughness element, Phys. Fluids, 27, 8, 084110 (2015) |
| [31] | Tuckerman, L. S., Steady-state solving via Stokes preconditioning; Recursion relations for elliptic operators, (Dwoyer, D. L.; Hussaini, M. Y.; Voigt, R. G., 11th International Conference on Numerical Methods in Fluid Dynamics (1989), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 573-577 |
| [32] | Mamun, C. K.; Tuckerman, L. S., Asymmetry and hopf bifurcation in spherical couette flow, Phys. Fluids, 7, 1, 80-91 (1994) · Zbl 0836.76033 |
| [33] | Barkley, D.; Tuckerman, L. S., Stokes Preconditioning for the inverse power method, (Kutler, P.; Flores, J.; Chattot, J.-J., Lecture Notes in Physics: Proceedings of the Fifteenth International Conference on Numerical Methods in Fluid Dynamics (1997), Springer: Springer New York), 75-76 |
| [34] | Bergeon, A.; Henry, D.; Benhadid, H.; Tuckerman, L. S., Marangoni convection in binary mixtures with soret effect, J. Fluid Mech., 375, 143-177 (1998) · Zbl 0936.76019 |
| [35] | Tuckerman, L. S.; Bertagnolio, F.; Daube, O.; Le Quéré, P.; Barkley, D., Stokes preconditioning for the inverse Arnoldi method, (Henry, D.; Bergeon, A., Continuation Methods for Fluid Dynamics (2000)), 241-255 · Zbl 0994.76026 |
| [36] | Mercader, I.; Batiste, O.; Alonso, A., Continuation of travelling-wave solutions of the Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 52, 7, 707-721 (2006) · Zbl 1106.76051 |
| [37] | Tuckerman, L. S., Laplacian Preconditioning for the inverse Arnoldi method, Commun. Comput. Phys., 18, 05, 1336-1351 (2015) · Zbl 1388.65029 |
| [38] | Brynjell-Rahkola, M.; Tuckerman, L. S.; Schlatter, P.; Henningson, D. S., Computing optimal forcing using Laplace preconditioning, Commun. Comput. Phys., 22, 05, 1508-1532 (2017) · Zbl 1488.76100 |
| [39] | Beaume, C., Adaptive Stokes preconditioning for steady incompressible flows, Commun. Comput. Phys., 22, 02, 494-516 (2017) · Zbl 1488.35415 |
| [40] | Patankar, S. V.; Spalding, D. B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Numer. Predict. Flow Heat Transf. Turbul. Combust., 54-73 (1983) |
| [41] | Kay, D.; Loghin, D.; Wathen, A., A preconditioner for the steady-state Navier-Stokes equations, SIAM J. Sci. Comput., 24, 1, 237-256 (2002) · Zbl 1013.65039 |
| [42] | Benzi, M.; Olshanskii, M. A., An augmented Lagrangian-based approach to the oseen problem, SIAM J. Sci. Comput., 28, 6, 2095-2113 (2006) · Zbl 1126.76028 |
| [43] | Benzi, M.; Olshanskii, M. A., Field-of-values convergence analysis of augmented Lagrangian preconditioners for the linearized Navier-Stokes problem, SIAM J. Numer. Anal., 49, 2, 770-788 (2011) · Zbl 1245.76044 |
| [44] | Heister, T.; Rapin, G., Efficient augmented Lagrangian-type preconditioning for the Oseen problem using grad – div stabilization, Internat. J. Numer. Methods Fluids, 71, 1, 118-134 (2013) · Zbl 1430.76115 |
| [45] | Benzi, M.; Olshanskii, M. A.; Wang, Z., Modified augmented Lagrangian preconditioners for the incompressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 66, 4, 486-508 (2011) · Zbl 1421.76152 |
| [46] | Benzi, M.; Wang, Z., Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier-Stokes equations, SIAM J. Sci. Comput., 33, 5, 2761-2784 (2011) · Zbl 1298.76114 |
| [47] | Benzi, M.; Wang, Z., A parallel implementation of the modified augmented Lagrangian preconditioner for the incompressible Navier-Stokes equations, Numer. Algorithms, 64, 1, 73-84 (2013) · Zbl 1426.76222 |
| [48] | Segal, A.; Ur Rehman, M.; Vuik, C., Preconditioners for incompressible Navier-Stokes solvers, Numer. Math., 3, 3, 245-275 (2010) · Zbl 1240.65098 |
| [49] | He, X.; Vuik, C., Comparison of some preconditioners for the incompressible Navier-Stokes equations, Numer. Math., 9, 2, 239-261 (2016) · Zbl 1389.76016 |
| [50] | Farrell, P. E.; Mitchell, L.; Wechsung, F., An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier-Stokes equations at high Reynolds number (2018) |
| [51] | Olshanskii, M. A.; Benzi, M., An augmented Lagrangian approach to linearized problems in hydrodynamic stability, SIAM J. Sci. Comput., 30, 3, 1459-1473 (2008) · Zbl 1162.76031 |
| [52] | Hecht, F., New development in FreeFem++, J. Numer. Math., 20, 3-4, 251-266 (2012) · Zbl 1266.68090 |
| [53] | Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; Eijkhout, V.; Gropp, W. D.; Kaushik, D.; Knepley, M. G.; McInnes, L. C.; Rupp, K.; Smith, B. F.; Zampini, S.; Zhang, H., PETSc web page (2019), http://www.mcs.anl.gov/petsc |
| [54] | Hernandez, V.; Roman, J. E.; Vidal, V., SLEPC: A scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math. Software, 31, 3, 351-362 (2005) · Zbl 1136.65315 |
| [55] | Olshanskii, M.; Lube, G.; Heister, T.; Löwe, J., Grad – div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 198, 49, 3975-3988 (2009) · Zbl 1231.76161 |
| [56] | Brezzi, F.; Fortin, M., (Mixed and Hybrid Finite Element Methods. Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15 (1991), Springer-Verlag, New York) · Zbl 0788.73002 |
| [57] | Olshanskii, M. A., A low order Galerkin finite element method for the Navier-Stokes equations of steady incompressible flow: a stabilization issue and iterative methods, Comput. Methods Appl. Mech. Engrg., 191, 47, 5515-5536 (2002) · Zbl 1083.76553 |
| [58] | Olshanskii, M. A.; Reusken, A., Grad – div stabilization for Stokes equations, Math. Comp., 73, 248, 1699-1718 (2004) · Zbl 1051.65103 |
| [59] | Linke, A.; Rebholz, L. G.; Wilson, N. E., On the convergence rate of grad – div stabilized Taylor-Hood to Scott-Vogelius solutions for incompressible flow problems, J. Math. Anal. Appl., 381, 2, 612-626 (2011) · Zbl 1331.76038 |
| [60] | Stewart, G. W., A Krylov-Schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl., 23, 3, 601-614 (2002) · Zbl 1003.65045 |
| [61] | Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14, 2, 461-469 (1993) · Zbl 0780.65022 |
| [62] | Jolivet, P.; Dolean, V.; Hecht, F.; Nataf, F.; Prud’homme, C.; Spillane, N., High performance domain decomposition methods on massively parallel architectures with FreeFem++, J. Numer. Math., 20, 3-4, 287-302 (2012) · Zbl 1267.65195 |
| [63] | Chan, T. F.; Van Der Vorst, H. A., Approximate and incomplete factorizations, (Parallel Numerical Algorithms (1997), Springer), 167-202 · Zbl 0865.65015 |
| [64] | Adams, M.; Bayraktar, H.; Keaveny, T.; Papadopoulos, P., Ultrascalable implicit finite element analyses in solid mechanics with over a half a billion degrees of freedom, (Proceedings of the 2004 ACM/IEEE Conference on Supercomputing. Proceedings of the 2004 ACM/IEEE Conference on Supercomputing, SC04 (2004), IEEE Computer Society), 34:1-34:15 |
| [65] | Geuzaine, C.; Remacle, J.-F., Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79, 11, 1309-1331 (2009) · Zbl 1176.74181 |
| [66] | Karypis, G.; Kumar, V., A parallel algorithm for multilevel graph partitioning and sparse matrix ordering, J. Parallel Distrib. Comput., 48, 1, 71-95 (1998) |
| [67] | Benzi, M.; Bertaccini, D., Block preconditioning of real-valued iterative algorithms for complex linear systems, IMA J. Numer. Anal., 28, 3, 598-618 (2008) · Zbl 1145.65022 |
| [68] | Tuckerman, L. S.; Langham, J.; Willis, A., Order-of-magnitude speedup for steady states and traveling waves via Stokes preconditioning in channelflow and openpipeflow, Comput. Methods Appl. Sci., 50, 3-31 (2019) |
| [69] | Iorio, M. C.; González, L. M.; Ferrer, E., Direct and adjoint global stability analysis of turbulent transonic flows over a NACA0012 profile, Internat. J. Numer. Methods Fluids, 76, 3, 147-168 (2014) · Zbl 1455.76072 |
| [70] | Roman, J. E.; Campos, C.; Romero, E.; Tomás, A., SLEPc Users manual scalable library for eigenvalue problem computations (2019) |
| [71] | Dolean, V.; Jolivet, P.; Nataf, F., An Introduction to Domain Decomposition Methods: Algorithms, Theory and Parallel Implementation (2015), SIAM · Zbl 1364.65277 |
| [72] | Jolivet, P.; Tournier, P.-H., Block iterative methods and recycling for improved scalability of linear solvers, (Proceedings of the 2016 International Conference for High Performance Computing, Networking, Storage and Analysis. Proceedings of the 2016 International Conference for High Performance Computing, Networking, Storage and Analysis, SC16 (2016), IEEE) |
| [73] | Elman, H. C.; Howle, V.; Shahid, J.; Shuttleworth, R.; Tuminaro, R. S., A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations, J. Comput. Phys., 227, 3, 1790-1808 (2008) · Zbl 1290.76023 |
| [74] | Cahouet, J.; Chabard, J.-P., Some fast 3D finite element solvers for the generalized Stokes problem, Internat. J. Numer. Methods Fluids, 8, 869-895 (1988) · Zbl 0665.76038 |
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