In this paper we present effective preconditioning techniques for solving the nonsymmetric systems that arise from the discretization of the Navier–Stokes equations. These linear systems are solved using either Krylov subspace methods or the Richardson scheme. We demonstrate the effectiveness of our techniques in handling time-accurate as well as steady-state solutions. We also compare our solvers with those published previously.
1.
Tezduyar
, T.
, Aliabadi
, S.
, Behr
, M.
, Johnson
, A.
, and Mittal
, S.
, 1993, “Parallel Finite-Element Computation of 3D Flows
,” Computer
0018-9162, 26
(10
), pp. 27
–36
.2.
Tezduyar
, T.
, Aliabadi
, S.
, Behr
, M.
, and Mittal
, S.
, 1994, “Massively Parallel Finite Element Simulation of Compressible and Incompressible Flows
,” Comput. Methods Appl. Mech. Eng.
, 119
, pp. 157
–177
. 0045-78253.
Mittal
, S.
, and Tezduyar
, T.
, 1994, “Massively Parallel Finite Element Computation of Incompressible Flows Involving Fluid-Body Interactions
,” Comput. Methods Appl. Mech. Eng.
0045-7825, 112
, pp. 253
–282
.4.
Mittal
, S.
, and Tezduyar
, T. E.
, 1995, “Parallel Finite Element Simulation of 3D Incompressible Flows: Fluid-Structure Interactions
,” Int. J. Numer. Methods Fluids
0271-2091, 21
, pp. 933
–953
.5.
Johnson
, A.
, and Tezduyar
, T.
, 1999, “Advanced Mesh Generation and Update Methods for 3D Flow Simulations
,” Comput. Mech.
0178-7675, 23
, pp. 130
–143
.6.
Kalro
, V.
, and Tezduyar
, T. E.
, 2000, “A Parallel 3D Computational Method for Fluid-Structure Interactions in Parachute Systems
,” Comput. Methods Appl. Mech. Eng.
, 190
, pp. 321
–332
. 0045-78257.
Stein
, K.
, Benney
, R.
, Kalro
, V.
, Tezduyar
, T. E.
, Leonard
, J.
, and Accorsi
, M.
, 2000, “Parachute Fluid-Structure Interactions: 3-D Computation
,” Comput. Methods Appl. Mech. Eng.
, 190
, pp. 373
–386
. 0045-78258.
Tezduyar
, T.
, and Osawa
, Y.
, 2001, “Fluid-Structure Interactions of a Parachute Crossing the Far Wake of an Aircraft
,” Comput. Methods Appl. Mech. Eng.
, 191
, pp. 717
–726
. 0045-78259.
Ohayon
, R.
, 2001, “Reduced Symmetric Models for Modal Analysis of Internal Structural-Acoustic and Hydroelastic-Sloshing Systems
,” Comput. Methods Appl. Mech. Eng.
0045-7825, 190
, pp. 3009
–3019
.10.
Tezduyar
, T.
, Sathe
, S.
, Keedy
, R.
, and Stein
, K.
, 2004, “Space–time techniques for finite element computation of flows with moving boundaries and interfaces
,” Proceedings of the Third International Congress on Numerical Methods in Engineering and Applied Science
, S.
Gallegos
, I.
Herrera
, S.
Botello
, F.
Zarate
, and G.
Ayala
, eds., Monterrey, Mexico, CD-ROM.11.
Torii
, R.
, Oshima
, M.
, Kobayashi
, T.
, Takagi
, K.
, and Tezduyar
, T.
, 2004, “Influence of Wall Elasticity on Image-Based Blood Flow Simulation
,” Trans. Jpn. Soc. Mech. Eng., Ser. A
, 70
, pp. 1224
–1231
(in Japanese).12.
van Brummelen
, E.
, and de Borst
, R.
, 2005, “On the Nonnormality of Subiteration for a Fluid-Structure Interaction Problem
,” SIAM J. Sci. Comput. (USA)
1064-8275, 27
, pp. 599
–621
.13.
Michler
, C.
, van Brummelen
, E.
, and de Borst
, R.
, 2005, “An Interface Newton–Krylov Solver for Fluid-Structure Interaction
,” Int. J. Numer. Methods Fluids
0271-2091, 47
, pp. 1189
–1195
.14.
Gerbeau
, J. -F.
, Vidrascu
, M.
, and Frey
, P.
, 2005, “Fluid-Structure Interaction in Blood Flow on Geometries Based on Medical Images
,” Comput. Struct.
0045-7949, 83
, pp. 155
–165
.15.
Tezduyar
, T.
, Sathe
, S.
, Keedy
, R.
, and Stein
, K.
, 2006, “Space-Time Finite Element Techniques for Computation of Fluid-Structure Interactions
,” Comput. Methods Appl. Mech. Eng.
, 195
, pp. 2002
–2027
. 0045-782516.
Tezduyar
, T.
, Sathe
, S.
, and Stein
, K.
, 2006, “Solution Techniques for the Fully-Discretized Equations in Computation of Fluid-Structure Interactions With the Space-Time Formulations
,” Comput. Methods Appl. Mech. Eng.
, 195
, pp. 5743
–5753
. 0045-782517.
Torii
, R.
, Oshima
, M.
, Kobayashi
, T.
, Takagi
, K.
, and Tezduyar
, T.
, 2006, “Computer Modeling of Cardiovascular Fluid-Structure Interactions With the Deforming-Spatial-Domain/Stabilized Space-Time Formulation
,” Comput. Methods Appl. Mech. Eng.
, 195
, pp. 1885
–1895
. 0045-782518.
Tezduyar
, T.
, Sathe
, S.
, Stein
, K.
, and Aureli
, L.
, 2006, “Modeling of Fluid-Structure Interactions With the Space-Time Techniques
,” Fluid-Structure Interaction
(Lecture Notes in Computational Science and Engineering
), H. -J.
Bungartz
and M.
Schafer
, eds., Springer
, New York
, Vol. 53
, pp. 50
–81
.19.
Torii
, R.
, Oshima
, M.
, Kobayashi
, T.
, Takagi
, K.
, and Tezduyar
, T.
, 2006, “Fluid-Structure Interaction Modeling of Aneurysmal Conditions With High and Normal Blood Pressures
,” Comput. Mech.
, 38
, pp. 482
–490
. 0178-767520.
Dettmer
, W.
, and Peric
, D.
, 2006, “A Computational Framework for Fluid-Structure Interaction: Finite Element Formulation and Applications
,” Comput. Methods Appl. Mech. Eng.
, 195
, pp. 5754
–5779
. 0045-782521.
Bazilevs
, Y.
, Calo
, V.
, Huhes
, T.
, and Zhang
, Y.
, 2006, “Isogeometric Fluid-Structure Interaction Analysis With Applications to Arterial Blood Flow
,” Comput. Mech.
0178-7675, 38
, pp. 310
–322
.22.
Khurram
, R.
, and Masud
, A.
, 2006, “A Multiscale/Stabilized Formulation of the Incompressible Navier–Stokes Equations for Moving Boundary Flows and Fluid-Structure Interaction
,” Comput. Mech.
0178-7675, 38
, pp. 403
–416
.23.
Kuttler
, U.
, Forster
, C.
, and Wall
, W.
, 2006, “A Solution for the Incompressibility Dilemma in Partitioned Fluid-Structure Interaction With Pure Dirichlet Fluid Domains
,” Comput. Mech.
, 38
, pp. 417
–429
. 0178-767524.
Lohner
, R.
, Cebral
, J.
, Yang
, C.
, Baum
, J.
, Mestreau
, E. L.
, and Soto
, O.
, 2006, “Extending the Range of Applicability of the Loose Coupling Approach for FSI Simulations
,” Fluid-Structure Interaction
(Lecture Notes in Computational Science and Engineering
), H. -J.
Bungartz
and M.
Schafer
, eds., Springer
, New York
, Vol. 53
, pp. 82
–100
.25.
Bletzinger
, K. -U.
, Wuchner
, R.
, and Kupzok
, A.
, 2006, “Algorithmic Treatment of Shells and Free Form-Membranes in FSI
,” Fluid-Structure Interaction
(Lecture Notes in Computational Science and Engineering
), H. -J.
Bungartz
and M.
Schafer
, eds., Springer
, New York
, Vol. 53
, pp. 336
–355
.26.
Torii
, R.
, Oshima
, M.
, Kobayashi
, T.
, Takagi
, K.
, and Tezduyar
, T.
, 2007, “Influence of Wall Elasticity in Patient-Specific Hemodynamic Simulations
,” Comput. Fluids
0045-7930, 36
, pp. 160
–168
.27.
Masud
, A.
, Bhanabhagvanwala
, M.
, and Khurram
, R.
, 2007, “An Adaptive Mesh Rezoning Scheme for Moving Boundary Flows and Fluid-Structure Interaction
,” Comput. Fluids
, 36
, pp. 77
–91
. 0045-793028.
Sawada
, T.
, and Hisada
, T.
, 2007, “Fuid-Structure Interaction Analysis of the Two Dimensional Flag-in-Wind Problem by an Interface Tracking ALE Finite Element Method
,” Comput. Fluids
, 36
, pp. 136
–146
. 0045-793029.
Wall
, W.
, Genkinger
, S.
, and Ramm
, E.
, 2007, “A Strong Coupling Partitioned Approach for Fluid-Structure Interaction With Free Surfaces
,” Comput. Fluids
, 36
, pp. 169
–183
. 0045-793030.
Tezduyar
, T.
, and Sathe
, S.
, 2007, “Modeling of Fluid-Structure Interactions With the Space-Time Finite Elements: Solution Techniques
,” Int. J. Numer. Methods Fluids
0271-2091, 54
, pp. 855
–900
.31.
Tezduyar
, T.
, Sathe
, S.
, Cragin
, T.
, Nanna
, B.
, Conklin
, B.
, Pausewang
, J.
, and Schwaab
, M.
, 2007, “Modeling of Fluid-Structure Interactions With the Space-Time Finite Elements: Arterial Fluid Mechanics
,” Int. J. Numer. Methods Fluids
0271-2091, 54
, pp. 901
–922
.32.
Torii
, R.
, Oshima
, M.
, Kobayashi
, T.
, Takagi
, K.
, and Tezduyar
, T.
, 2007, “Numerical Investigation of the Effect of Hypertensive Blood Pressure on Cerebral Aneurysm — Dependence of the Effect on the Aneurysm Shape
,” Int. J. Numer. Methods Fluids
0271-2091, 54
, pp. 995
–1009
.33.
Tezduyar
, T.
, Sathe
, S.
, Schwaab
, M.
, and Conklin
, B.
, 2007, “Arterial Fluid Mechanics Modeling With the Stabilized Space-Time Fluid-Structure Interaction Technique
,” Int. J. Numer. Methods Fluids
0271-2091, 57
(5
), pp. 601
–629
.34.
Tezduyar
, T.
, Sathe
, S.
, Pausewang
, J.
, Schwaab
, M.
, Christopher
, J.
, and Crabtree
, J.
, 2008, “Interface Projection Techniques for Fluid–Structure Interaction Modeling With Moving-Mesh Methods
,” Comput. Mech.
0178-7675, 43
(1
), pp. 39
–49
.35.
Tezduyar
, T.
, Sathe
, S.
, Pausewang
, J.
, Schwaab
, M.
, Christopher
, J.
, and Crabtree
, J.
, 2008, “Fluid-Structure Interaction Modeling of Ringsail Parachutes
,” Comput. Mech.
0178-7675, 43
(1
), pp. 133
–142
.36.
Bazilevs
, Y.
, Calo
, V.
, Hughes
, T.
, and Zhang
, Y.
, 2008, “Isogeometric Fluid-Structure Interaction: Theory, Algorithms and Computations
,” unpublished.37.
Baggag
, A.
, and Sameh
, A.
, 2004, “A Nested Iterative Scheme for Indefinite Linear Systems in Particulate Flows
,” Comput. Methods Appl. Mech. Eng.
, 193
, pp. 1923
–1957
. 0045-782538.
Tezduyar
, T.
, and Sathe
, S.
, 2005, “Enhanced-Discretization Successive Update Method (EDSUM)
,” Int. J. Numer. Methods Fluids
0271-2091, 47
, pp. 633
–654
.40.
Golub
, G.
, and Wathen
, A.
, 1998, “An Iteration for Indefinite Systems and Its Application to the Navier–Stokes Equations
,” SIAM J. Sci. Comput. (USA)
1064-8275, 19
, pp. 530
–539
.41.
Elman
, H.
, 1999, “Preconditioning for the Steady-State Navier–Stokes Equations With Low Viscosity
,” SIAM J. Sci. Comput. (USA)
1064-8275, 20
(4
), pp. 1299
–1316
.42.
Silvester
, D.
, Elman
, H.
, Kay
, D.
, and Wathen
, A.
, 2001, “Efficient Preconditioning of the Linearized Navier–Stokes Equations for Incompressible Flow
,” J. Comput. Appl. Math.
0377-0427, 128
, pp. 261
–279
43.
Elman
, H.
, Silvester
, D.
, and Wathen
, A.
, 2005, Finite Elements and Fast Iterative Solvers
, Oxford University Press
, New York
.44.
Elman
, H.
, Howle
, V.
, Shadid
, J.
, Shuttleworth
, R.
, and Tuminaro
, R.
, 2006, “Block Preconditioners Based on Approximate Commutators
,” SIAM J. Sci. Comput. (USA)
1064-8275, 27
(5
), pp. 1651
–1668
.45.
Vainikko
, E.
, and Graham
, I.
, 2004, “A Parallel Solver for PDE Systems and Application to the Incompressible Navier–Stokes Equations
,” Appl. Numer. Math.
0168-9274, 49
(1
), pp. 97
–116
.46.
van der Vorst
, H.
, 1992, “BI-CGSTAB: A Fast and Smoothly Converging Variant of BI-CG for the Solution of Nonsymmetric Linear Systems
,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput.
0196-5204, 13
(2
), pp. 631
–644
.47.
Schenk
, O.
, Gärtner
, K.
, Fichtner
, W.
, and Stricker
, A.
, 2001, “PARDISO: A High-Performance Serial and Parallel Sparse Linear Solver in Semiconductor Device Simulation
,” FGCS, Future Gener. Comput. Syst.
0167-739X, 18
(1
), pp. 69
–78
.48.
Axelsson
, O.
, and Kolotilina
, L.
, 2005, “Diagonally Compensated Reduction and Related Preconditioning Methods
,” Numer. Linear Algebra Appl.
1070-5325, 1
(2
), pp. 155
–177
.49.
Saad
, Y.
, and Schultz
, M.
, 1986, “GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput.
0196-5204, 7
, pp. 856
–869
.50.
Cuthill
, E.
, and McKee
, J.
, 1969, “Reducing the Bandwidth of Sparse Symmetric Matrices
,” Proceedings of the 1969 24th National Conference
, ACM
, New York, NY
, pp. 157
–172
.51.
Saad
, Y.
, 1990, “SPARSKIT: A Basic Tool Kit for Sparse Matrix Computations
,” NASA Ames Research Center
, Technical Report No. 90-20.52.
Manguoglu
, M.
, Sameh
, A. H.
, Tezduyar
, T. E.
, and Sathe
, S.
, 2008, “A Nested Iterative Scheme for Computation of Incompressible Flows in Long Domains
,” Comput. Mech.
0178-7675, 43
(1
), pp. 73
–80
.Copyright © 2009
by American Society of Mechanical Engineers
You do not currently have access to this content.
