×

Parallel solution of large-scale free surface viscoelastic flows via sparse approximate inverse preconditioning. (English) Zbl 1274.76095

Summary: Though computational techniques for two-dimensional viscoelastic free surface flows are well developed, three-dimensional flows continue to present significant computational challenges. Fully coupled free surface flow models lead to nonlinear systems whose steady states can be found via Newton’s method. Each Newton iteration requires the solution of a large, sparse linear system, for which memory and computational demands suggest the application of an iterative method, rather than the sparse direct methods widely used for two dimensional simulations. The Jacobian matrix of this system is often ill-conditioned, resulting in unacceptably slow convergence of the linear solver; hence preconditioning is essential. We propose a variant sparse approximate inverse preconditioner for the Jacobian matrix that allows for the solution of problems involving more than a million degrees of freedom in challenging parameter regimes. Construction of this preconditioner requires the solution of small least squares problems that can be simply parallelized on a distributed memory machine. The performance and scalability of this preconditioner with the GMRES solver are investigated for two- and three-dimensional free surface flows on both structured and unstructured meshes in the presence and absence of viscoelasticity. The results suggest that this preconditioner is an extremely promising candidate for solving large-scale steady viscoelastic flows with free surfaces.

MSC:

76A10 Viscoelastic fluids
76M10 Finite element methods applied to problems in fluid mechanics
65Y05 Parallel numerical computation

Software:

SPARSKIT
PDFBibTeX XMLCite
Full Text: DOI Link Link

References:

[1] Christodoulou, K. N.; Scriven, L. E.: Discretization of free-surface flows and other moving boundary-problems, J. comput. Phys. 99, 39-55 (1992) · Zbl 0743.76050
[2] V. F. de Almeida, Gas – liquid counterflow through constricted passages, PhD thesis, University of Minnesota, 1995.
[3] J. M. de Santos, Two-phase cocurrent downflow through constricted passages, PhD thesis, University of Minnesota, 1991.
[4] M. S. Carvalho, Roll coating flows in rigid and deformable gaps, PhD thesis, University of Minnesota, 1996.
[5] Sackinger, P. A.; Schunk, P. R.; Rao, R. R.: A Newton – raphson pseudo-solid domain mapping technique for free and moving boundary problems: a finite element implementation, J. comput. Phys. 125, 83-103 (1996) · Zbl 0853.65138
[6] L. C. Musson, Two-layer slot coating, PhD thesis, University of Minnesota, 2001.
[7] M. Pasquali, Polymer molecules in free surface coating flows, PhD thesis, University of Minnesota, 2000.
[8] Pasquali, M.; Scriven, L. E.: Free surface flows of polymer solutions with models based on the conformation tensor, J. non-Newtonian fluid mech. 108, 363-409 (2002) · Zbl 1143.76369
[9] Lee, A. G.; Shaqfeh, E. S. G.; Khomami, B.: A study of viscoelastic free surface flows by the finite element method: Hele – Shaw and slot coating flows, J. non-Newtonian fluid mech. 108, 327-362 (2002) · Zbl 1143.76479
[10] Dimakopoulos, Y.; Tsamopoulos, J.: On the gas-penetration in straight tubes completely filled with a viscoelastic fluid, J. non-Newtonian fluid mech. 117, 117-139 (2004) · Zbl 1130.76315
[11] Zevallos, G. A.; Carvalho, M. S.; Pasquali, M.: Forward roll coating flows of viscoelastic liquids, J. non-Newtonian fluid mech. 130, 96-109 (2005) · Zbl 1195.76121
[12] Cairncross, R. A.; Schunk, P. R.; Baer, T. A.; Rao, R. R.; Sackinger, P. A.: A finite element method for free surface flows of incompressible fluids in three dimensions, Part I: boundary fitted mesh motion. Int. J. Numer. meth. Fluids 33, 375-403 (2000) · Zbl 0989.76043
[13] R. W. Hooper, Drop dynamics in polymer processing flows, PhD thesis, University of Minnesota, 2001.
[14] Xie, X.; Pasquali, M.: Computing 3-D free surface viscoelastic flows, Moving boundaries VII: Computational modeling of free and moving boundary problems, 225-234 (2003)
[15] Guénette, R.; Fortin, M.: A new finite element method for computing viscoelastic flows, J. non-Newtonian fluid mech. 60, 27-52 (1995)
[16] Szady, M.; Salamon, T.; Liu, A.; Armstrong, R.; Brown, R.: A new mixed finite element method for viscoelastic flows governed by differential constitutive equations, J. non-Newtonian fluid mech. 59, 215-243 (1995)
[17] Duff, I. S.; Erisman, A. M.; Reid, J. K.: Direct methods for sparse matrices, (1986) · Zbl 0604.65011
[18] Saad, Y.; Schultz, M. H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. stat. Comput. 7, 856-869 (1986) · Zbl 0599.65018
[19] Baer, T. A.; Cairncross, R. A.; Schunk, P. R.; Rao, R. R.; Sackinger, P. A.: A finite element method for free surface flows of incompressible fluids in three dimensions. Part II: Dynamic wetting lines, Int. J. Numer. meth. Fluids 33, 405-427 (2000) · Zbl 0989.76044
[20] Benzi, M.: Preconditioning techniques for large linear systems: a survey, J. comput. Phys. 182, 418-477 (2002) · Zbl 1015.65018
[21] Benson, M. W.; Frederickson, P. O.: Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas math 22, 127-140 (1982) · Zbl 0502.65020
[22] Wilson, E. B.: Vector analysis: A text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs, vol. 1, (1901)
[23] Renardy, M.; Hrusa, W. J.; Nohel, J. A.: Mathematical problems in viscoelasticity, (1987) · Zbl 0719.73013
[24] Xie, X.; Pasquali, M.: A new, convenient way of imposing open-flow boundary conditions in two- and three-dimensional viscoelastic flows, J. non-Newtonian fluid mech. 122, 159-176 (2004) · Zbl 1143.76352
[25] Kolotilina, L. Yu.; Yeremin, A. Yu.: Factorized sparse approximate inverse preconditionings. I. theory, SIAM J. Matrix anal. Appl. 14, 45-58 (1993) · Zbl 0767.65037
[26] Benzi, M.; Tuma, M.: A sparse approximate inverse preconditioner for nonsymmetric linear systems, SIAM J. Sci. comput. 19, 968-994 (1998) · Zbl 0930.65027
[27] Cosgrove, J. D. F.; Diaz, J. C.; Griewank, A.: Approximate inverse preconditionings for sparse linear systems, Int. J. Comput. math. 44, 91-110 (1992) · Zbl 0762.65025
[28] Grote, M.; Simon, H. D.: Parallel preconditioning and approximate inverses on the connection machine, Parallel processing for scientific computing, 519-523 (1993)
[29] Chow, E.; Saad, Y.: Approximate inverse preconditioners via sparse – sparse iterations, SIAM J. Sci. comput. 19, 995-1023 (1998) · Zbl 0922.65034
[30] Grote, M. J.; Huckle, T.: Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. comput. 18, 838-853 (1997) · Zbl 0872.65031
[31] Chow, E.: A priori sparsity patterns for parallel sparse approximate inverse preconditioners, SIAM J. Sci. comput. 21, 1804-1822 (2000) · Zbl 0957.65023
[32] Tang, W.: Toward an effective sparse approximate inverse preconditioner, SIAM J. Matrix anal. Appl. 20, 970-986 (1999) · Zbl 0937.65056
[33] Kaporin, I. E.: A preconditioned conjugate-gradient method for solving discrete analogs of differential problems, Differ. equat. 26, 897-906 (1990) · Zbl 0712.65022
[34] Lewis, J. G.: Algorithm 582: the Gibbs – poole – stockmeyer and Gibbs – King algorithms for reordering sparse matrices, ACM trans. Math. software 8, 190-194 (1982)
[35] Benzi, M.; Haws, J. C.; Tuma, M.: Preconditioning highly indefinite and nonsymmetric matrices, SIAM J. Sci. comput. 22, 1333-1353 (2000) · Zbl 0985.65036
[36] Duff, I. S.; Koster, J.: On algorithms for permuting large entries to the diagonal of a sparse matrix, SIAM J. Matrix anal. Appl. 22, 973-996 (2001) · Zbl 0979.05087
[37] Noskov, M.; Benzi, M.; Smooke, M. D.: An implicit compact scheme solver for two-dimensional multicomponent flows, Comput. fluids 36, 376-397 (2007) · Zbl 1177.76263
[38] Y. Saad, SPARSKIT: A basic tool kit for sparse matrix computations. Technical report, Center for Supercomputing Research and Development, University of Illinois, Urbana, IL, 1990.
[39] Kelley, C. T.: Iterative methods for linear and nonlinear equations, (1995) · Zbl 0832.65046
[40] Gropp, W.; Lusk, E.; Skjellum, A.: Using MPI: portable parallel programming with the message – passing interface, (1994) · Zbl 0875.68206
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.