An implicit block ILU smoother for preconditioning of Newton-Krylov solvers with application in high-order stabilized finite-element methods. (English) Zbl 1441.76052
Summary: This paper presents an efficient and highly-parallelizable preconditioning technique for Newton-Krylov solvers. The proposed method can be viewed as a generalization of the implicit line smoothing technique by extending the groups of implicitly-solved unknowns from lines to blocks. The blocks are formed by partitioning the computational domain such that the strong connections between unknowns are not broken by the partition boundaries. The ILU algorithm is used to obtain an approximate (or exact) factorization for each block. Then, a block-Jacobi iteration is formulated in which the degrees of freedom within the blocks are solved implicitly. To stabilize the iterations for high-CFL systems, a dual-CFL strategy, with a lower CFL in the preconditioner matrix, is developed. The performance of the proposed method as a linear preconditioner is demonstrated for second- and third-order steady-state solutions of Reynolds-Averaged Navier-Stokes (RANS) equations on the NASA Common Research Model (CRM), including the high-lift configuration. For the studied test cases, it is shown that in comparison with the traditional ILU(k) method, the proposed preconditioner requires significantly less memory, and it can result in notably faster solutions.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
Keywords:
computational fluid dynamics; Newton-Krylov methods; steady- state RANS simulations; preconditioning techniques; high-order methods; stabilized finite-element methodsReferences:
| [1] | Kamenetskiy, D. S.; Bussoletti, J. E.; Hilmes, C. L.; Venkatakrishnan, V.; Wigton, L. B.; Johnson, F. T., Numerical evidence of multiple solutions for the Reynolds-averaged Navier-Stokes equations, AIAA J., 52, 1686-1698 (2014) |
| [2] | N.K. Burgess, R.S. Glasby, Advances in numerical methods for CREATE \({}^{TM}\)-AV analysis tools, presented at the 52nd AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 2014-0417, January 2014. |
| [3] | Ceze, M.; Fidkowski, K. J., Constrained pseudo-transient continuation, Internat. J. Numer. Methods Engrg., 102, 1683-1703 (2015) · Zbl 1352.65479 |
| [4] | Anderson, W. K.; Newman, J. C.; Karman, S. L., Stabilized finite elements in FUN3D, J. Aircr., 55, 696-714 (2017) |
| [5] | B.R. Ahrabi, M.J. Brazell, D.J. Mavriplis, An investigation of continuous and discontinuous finite-element discretizations on benchmark 3D tturbulent flows (invited), presented at the 2018 AIAA Aerospace Sciences Meeting, AIAA Paper 2018-1569, January 2018. |
| [6] | R.S. Glasby, J.T. Erwin, Introduction to COFFE: The next-generation HPCMP CREATE \({}^{TM}\)-AV CFD solver, presented at the 54th AIAA Aerospace Sciences Meeting, AIAA Paper 2016-0567, January 2016. |
| [7] | M.C. Galbraith, S.R. Allmaras, D.L. Darmofal, SANS RANS solutions for 3D benchmark configurations, presented at the 2018 AIAA Aerospace Sciences Meeting, AIAA Paper 2018-1570, January 2018. |
| [8] | Diosady, L. T.; Darmofal, D. L., Preconditioning methods for discontinuous Galerkin solutions of the Navier-Stokes equations, J. Comput. Phys., 228, 3917-3935 (2009) · Zbl 1185.76812 |
| [9] | B.R. Ahrabi, D.J. Mavriplis, Scalable solution strategies for stabilized finite-element flow solvers on unstructured meshes, presented at the 55th AIAA Aerospace Sciences Meeting, AIAA Paper 2017-0517. |
| [10] | B.R. Ahrabi, D.J. Mavriplis, Scalable solution strategies for stabilized finite-element flow solvers on unstructured meshes, Part II, presented at the 23rd AIAA Computational Fluid Dynamics Conference, AIAA Paper 2017-4275, June 2017. |
| [11] | Ahrabi, B. R.; Mavriplis, D. J., A scalable solution strategy for high-order stabilized finite-element solvers using an implicit line preconditioner, Comput. Methods Appl. Mech. Engrg., 341, 956-984 (2018) · Zbl 1440.76046 |
| [12] | Persson, P.-O.; Peraire, J., Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier-Stokes equations, SIAM J. Sci. Comput., 30, 2709-2733 (2008) · Zbl 1362.76052 |
| [13] | D.J. Mavriplis, B.R. Ahrabi, M.J. Brazell, Strategies for accelerating newton method continuation in CFD problems, presented at the AIAA Science and Technology Forum and Exposition, AIAA Paper 2019-0100, January 2019. |
| [14] | Okusanya, T.; Darmofal, D.; Peraire, J., Algebraic multigrid for stabilized finite element discretizations of the Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 193, 3667-3686 (2004) · Zbl 1068.76051 |
| [15] | Philip, B.; Chartier, T. P., Adaptive algebraic smoothers, J. Comput. Appl. Math., 236, 2277-2297 (2012) · Zbl 1244.65048 |
| [16] | Brooks, A. N.; Hughes, T. J., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32, 199-259 (1982) · Zbl 0497.76041 |
| [17] | T. Tezduyar, T. Hughes, Finite element formulations for convection dominated flows with particular emphasis on the compressible Euler equations, presented at the 21st Aerospace Sciences meeting, AIAA Paper 1983-0125, 1983. |
| [18] | Hughes, T. J.R.; Scovazzi, G.; Tezduyar, T. E., Stabilized methods for compressible flows, J. Sci. Comput., 43, 343-368 (2010) · Zbl 1203.76130 |
| [19] | Shakib, F.; Hughes, T. J.; Johan, Z., A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 89, 141-219 (1991) |
| [20] | Anderson, W. K.; Wang, L.; Kapadia, S.; Tanis, C.; Hilbert, B., Petrov-Galerkin and discontinuous-Galerkin methods for time-domain and frequency-domain electromagnetic simulations, J. Comput. Phys., 230, 8360-8385 (2011) · Zbl 1241.78041 |
| [21] | J.T. Erwin, W.K. Anderson, L. Wang, S. Kapadia, High-order finite-element method for three-dimensional turbulent Navier-Stokes, in: presented at the 21st AIAA Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences, AIAA Paper 2013-2571, June 2013. |
| [22] | Wang, L.; Anderson, W. K.; Erwin, J. T.; Kapadia, S., Discontinuous Galerkin and Petrov Galerkin methods for compressible viscous flows, Comput. & Fluids, 100, 13-29 (2014) · Zbl 1391.76366 |
| [23] | J.C. Newman, W.K. Anderson, Investigation of unstructured higher-order methods for unsteady flow and moving domains, in: Presented at the 22nd AIAA Computational Fluid Dynamics Conference, AIAA Paper 2015-2917, June 2015. |
| [24] | Ahrabi, B. R.; Anderson, W. K.; Newman, J. C., An adjoint-based hp-adaptive stabilized finite-element method with shock capturing for turbulent flows, Comput. Methods Appl. Mech. Engrg., 318, 1030-1065 (2017) · Zbl 1439.76042 |
| [25] | Anderson, W. K.; Ahrabi, B. R.; Newman, J. C., Finite element solutions for turbulent flow over the NACA 0012 airfoil, AIAA J., 54, 9, 2688-2704 (2016) |
| [26] | R.S. Glasby, N. Burgess, W.K. Anderson, L. Wang, D.J. Mavriplis, S.R. Allmaras, Comparison of SU/PG and DG finite-element techniques for the compressible Navier-Stokes equations on anisotropic unstructured meshes, presented at the 51st AIAA Aerospcae Sciences Meeting including the New Horizons Forum and Aerospace Exposition, AIAA Paper 2013-0691, 2013. |
| [27] | C. Rumsey, Turbulence modeling resource website. Available: http://turbmodels.larc.nasa.gov. |
| [28] | B. Diskin, W.K. Anderson, M.J. Pandya, C.L. Rumsey, J. Thomas, Y. Liu, et. al, Grid convergence for three dimensional benchmark turbulent flows, presented at the 2018 AIAA Aerospace Sciences Meeting, AIAA Paper 2018-1102, January 2018. |
| [29] | T.R. Michal, D.S. Kamenetskiy, J. Krakos, Generation of Anisotropic Adaptive Meshes for the first AIAA geometry and mesh generation workshop, in: Presented at the 2018 AIAA Aerospace Sciences Meeting, AIAA Paper 2018-0658, January 2018. |
| [30] | Alauzet, F.; Loseille, A.; Marcum, D.; Michal, T. R., Assessment of anisotropic mesh adaptation for high-lift prediction of the HL-CRM configuration, (23rd AIAA Computational Fluid Dynamics Conference, AIAA Paper 2017-3300 (2017), American Institute of Aeronautics and Astronautics) |
| [31] | Mavriplis, D. J.; Mani, K., Unstructured mesh solution techniques using the NSU3D solver, (52nd Aerospace Sciences Meeting, AIAA Paper 2014-0081 (2014), American Institute of Aeronautics and Astronautics) |
| [32] | Mavriplis, D. J., An assessment of linear versus nonlinear multigrid methods for unstructured mesh solvers, J. Comput. Phys., 175, 302-325 (2002) · Zbl 0995.65099 |
| [33] | S.R. Allmaras, F.T. Johnson, Modifications and clarifications for the implementation of the Spalart-Allmaras turbulence model, in: ICCFD7-1902, 7th International Conference on Computational Fluid Dynamics, Big Island, Hawaii, 2012. |
| [34] | Barth, T. J., Numerical methods for gasdynamic systems on unstructured meshes, (An Introduction to Recent Developments in Theory and Numerics for Conservation Laws (1999), Springer), 195-285 · Zbl 0969.76040 |
| [35] | Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372 (1981) · Zbl 0474.65066 |
| [36] | Shahbazi, K.; Mavriplis, D. J.; Burgess, N. K., Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations, J. Comput. Phys., 228, 7917-7940 (2009) · Zbl 1391.65181 |
| [37] | Barter, G. E.; Darmofal, D. L., Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation, J. Comput. Phys., 229, 1810-1827 (2010) · Zbl 1329.76153 |
| [38] | P.-O. Persson, J. Peraire, Sub-cell shock capturing for discontinuous Galerkin methods, presented at the 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 2006-0112, January 2006. |
| [39] | K.R. Holst, R.S. Glasby, J.T. Erwin, D.L. Stefanski, R.B. Bond, J.D. Schmisseur, High-order simulations of shock problems using HPCMP CREATE(TM)-AV Kestrel COFFE, presented at the 2018 AIAA Aerospace Sciences Meeting, AIAA Paper 2018-1301, January 2018. |
| [40] | Kelley, C. T.; Keyes, D. E., Convergence analysis of pseudo-transient continuation, SIAM J. Numer. Anal., 35, 508-523 (1998) · Zbl 0911.65080 |
| [41] | Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM · Zbl 1002.65042 |
| [42] | Cuthill, E., Several strategies for reducing the bandwidth of matrices, (Rose, D. J.; Willoughby, R. A., Sparse Matrices and their Applications (1972), Springer US: Springer US Boston, MA), 157-166 |
| [43] | Balay, S.; Buschelman, K.; Gropp, W. D.; Kaushik, D.; Knepley, M. G.; McInnes, L. C., PETSc web page, 2001 (2004) |
| [44] | Balay, S.; Gropp, W. D.; McInnes, L. C.; Smith, B. F., Efficient management of parallelism in object-oriented numerical software libraries, (Modern Software Tools for Scientific Computing (1997), Springer), 163-202 · Zbl 0882.65154 |
| [45] | Smith, B., PETSc, the portable, extensible toolkit for scientific computing, (Encyclopedia of Parallel Computing (2011), Springer) |
| [46] | Thomas, L., Elliptic Problems in Linear Differential Equations over a Network (1949), Watson Scientific Computing Laboratory, Columbia Univ.: Watson Scientific Computing Laboratory, Columbia Univ. NY |
| [47] | Liu, C., A Stabilized Finite Element Dynamic Overset Method for the Navier-Stokes Equations (2016), Department of Computational Engineering, University of Tennessee at Chattanooga, (Ph.D.) |
| [48] | Richardson, L. F., The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci., 210, 307-357 (1911) · JFM 42.0873.02 |
| [49] | Mavriplis, D. J.; Pirzadeh, S., Large-scale parallel unstructured mesh computations for three-dimensional high-lift analysis, J. Aircr., 36, 987-998 (1999) |
| [50] | Karypis, G.; Kumar, V., A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. Sci. Comput., 20, 359-392 (1998) · Zbl 0915.68129 |
| [51] | Chow, E.; Patel, A., Fine-grained parallel incomplete LU factorization, SIAM J. Sci. Comput., 37, C169-C193 (2015) · Zbl 1320.65048 |
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