Even though computer modeling of spacecraft parachutes involves a number of numerical challenges, advanced techniques developed in recent years for fluid-structure interaction (FSI) modeling in general and for parachute FSI modeling specifically have made simulation-based design studies possible. In this paper we focus on such studies for a single main parachute to be used with the Orion spacecraft. Although these large parachutes are typically used in clusters of two or three parachutes, studies for a single parachute can still provide valuable information for performance analysis and design and can be rather extensive. The major challenges in computer modeling of a single spacecraft parachute are the FSI between the air and the parachute canopy and the geometric complexities created by the construction of the parachute from “rings” and “sails” with hundreds of gaps and slits. The Team for Advanced Flow Simulation and Modeling has successfully addressed the computational challenges related to the FSI and geometric complexities, and has also been devising special procedures as needed for specific design parameter studies. In this paper we present parametric studies based on the suspension line length, canopy loading, and the length of the overinflation control line.

References

1.
Hughes
,
T. J. R.
,
Liu
,
W. K.
, and
Zimmermann
,
T. K.
, 1981, “
Lagrangian-Eulerian Finite Element Formulation for Incompressible Viscous Flows
,”
Comput. Methods Appl. Mech. Eng.
,
29
, pp.
329
349
.
2.
Tezduyar
,
T.
,
Aliabadi
,
S.
,
Behr
,
M.
,
Johnson
,
A.
, and
Mittal
,
S.
, 1993, “
Parallel Finite-Element Computation of 3D Flows
,”
Computer
,
26
, pp.
27
36
.
3.
Tezduyar
,
T. E.
,
Aliabadi
,
S. K.
,
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
.
4.
Mittal
S.
, and
Tezduyar
,
T. E.
, 1994, “
Massively Parallel Finite Element Computation of Incompressible Flows Involving Fluid-Body Interactions
,”
Comput. Methods Appl. Mech. Eng.
,
112
, pp.
253
282
.
5.
Mittal
,
S.
, and
Tezduyar
,
T. E.
, 1995, “
Parallel Finite Element Simulation of 3D Incompressible Flows—Fluid-Structure Interactions
,”
Int. J. Numer. Methods Fluids
,
21
, pp.
933
953
.
6.
Tezduyar
,
T.
,
Aliabadi
,
S.
,
Behr
,
M.
,
Johnson
,
A.
,
Kalro
,
V.
, and
Litke
,
M.
, 1996, “
Flow Simulation and High Performance Computing
,”
Comput. Mech.
,
18
, pp.
397
412
.
7.
Johnson
,
A. A.
, and
Tezduyar
,
T. E.
, 1997, “
Parallel Computation of Incompressible Flows With Complex Geometries
,”
Int. J. Numer. Methods Fluids
,
24
, pp.
1321
1340
.
8.
Johnson
,
A. A.
, and
Tezduyar
,
T. E.
, 1999, “
Advanced Mesh Generation and Update Methods for 3D Flow Simulations
,”
Comput. Mech.
,
23
, pp.
130
143
.
9.
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
.
10.
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
.
11.
Tezduyar
,
T. E.
, 2001, “
Finite Element Methods for Flow Problems With Moving Boundaries and Interfaces
,”
Arch. Comput. Methods Eng.
,
8
, pp.
83
130
.
12.
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
.
13.
Ohayon
,
R.
, 2001, “
Reduced Symmetric Models for Modal Analysis of Internal Structural-Acoustic and Hydroelastic-Sloshing Systems
,”
Comput. Methods Appl. Mech. Eng.
,
190
, pp.
3009
3019
.
14.
Stein
,
K.
,
Tezduyar
,
T.
, and
Benney
,
R.
, 2003, “
Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements
,”
J. Appl. Mech.
,
70
, pp.
58
63
.
15.
Stein
,
K.
,
Tezduyar
,
T. E.
, and
Benney
,
R.
, 2004, “
Automatic Mesh Update With the Solid-Extension Mesh Moving Technique
,”
Comput. Methods Appl. Mech. Eng.
,
193
, pp.
2019
2032
.
16.
Torii
,
R.
,
Oshima
,
M.
,
Kobayashi
,
T.
,
Takagi
,
K.
, and
Tezduyar
,
T. E.
, 2004, “
Influence of Wall Elasticity on Image-Based Blood Flow Simulation
,”
Jpn. Soc. Mech. Eng. J. Ser.
,
70
, pp.
1224
1231
, in Japanese.
17.
Tezduyar
,
T. E.
,
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 III International Congress on Numerical Methods in Engineering and Applied Science,
S.
Gallegos
,
I.
Herrera
,
S.
Botello
,
F.
Zarate
, and
G.
Ayala
, editors, CD-ROM,
Monterrey, Mexico
.
18.
van Brummelen
,
E. H.
, and
de Borst
,
R.
, 2005, “
On the Nonnormality of Subiteration for a Fluid-Structure Interaction Problem
,”
SIAM J. Sci. Comput. (USA)
,
27
, pp.
599
621
.
19.
Tezduyar
,
T. E.
,
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
.
20.
Tezduyar
,
T. E.
,
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
.
21.
Torii
,
R.
,
Oshima
,
M.
,
Kobayashi
,
T.
,
Takagi
,
K.
, and
Tezduyar
,
T. E.
, 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
.
22.
Torii
,
R.
,
Oshima
,
M.
,
Kobayashi
,
T.
,
Takagi
,
K.
, and
Tezduyar
,
T. E.
, 2006, “
Fluid-Structure Interaction Modeling of Aneurysmal Conditions With High and Normal Blood Pressures
,”
Comput. Mech.
,
38
, pp.
482
490
.
23.
Bazilevs
,
Y.
,
Calo
,
V. M.
,
Zhang
,
Y.
, and
Hughes
,
T. J. R.
, 2006, “
Isogeometric Fluid-Structure Interaction Analysis With Applications to Arterial Blood Flow
,”
Comput. Mech.
,
38
, pp.
310
322
.
24.
Khurram
,
R. A.
, and
Masud
,
A.
, 2006, “
A Multiscale/Stabilized Formulation of the Incompressible Navier-Stokes Equations for Moving Boundary Flows and Fluid-Structure Interaction
,”
Comput. Mech.
,
38
, pp.
403
416
.
25.
Tezduyar
,
T. E.
,
Sathe
,
S.
,
Cragin
,
T.
,
Nanna
,
B.
,
Conklin
,
B. S.
,
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
,
54
, pp.
901
922
.
26.
Torii
,
R.
,
Oshima
,
M.
,
Kobayashi
,
T.
,
Takagi
,
K.
, and
Tezduyar
,
T. E.
, 2007, “
Influence of Wall Elasticity in Patient-Specific Hemodynamic Aimulations
,”
Comput. Fluids
,
36
, pp.
160
168
.
27.
Sawada
,
T.
, and
Hisada
,
T.
, 2007, “
Fluid-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
.
28.
Tezduyar
,
T. E.
, and
Sathe
,
S.
, 2007, “
Modeling of Fluid-Structure Interactions With the Space-Time Finite Elements: Solution Techniques
,”
Int. J. Numer. Methods Fluids
,
54
, pp.
855
900
.
29.
Torii
,
R.
,
Oshima
,
M.
,
Kobayashi
,
T.
,
Takagi
,
K.
, and
Tezduyar
,
T. E.
, 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
,
54
, pp.
995
1009
.
30.
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.
,
43
, pp.
73
80
.
31.
Tezduyar
,
T. E.
,
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.
,
43
, pp.
39
49
.
32.
Tezduyar
,
T. E.
,
Sathe
,
S.
,
Pausewang
,
J.
,
Schwaab
,
M.
,
Christopher
,
J.
, and
Crabtree
,
J.
, 2008, “
Fluid-Structure Interaction Modeling of Ringsail Parachutes
,”
Comput. Mech.
,
43
, pp.
133
142
.
33.
Tezduyar
,
T. E.
,
Sathe
,
S.
,
Schwaab
,
M.
, and
Conklin
,
B. S.
, 2008, “
Arterial Fluid Mechanics Modeling With the Stabilized Space-Time Fluid-Structure Interaction Technique
,”
Int. J. Numer. Methods Fluids
,
57
, pp.
601
-
629
.
34.
Sathe
,
S.
, and
Tezduyar
,
T. E.
, 2008, “
Modeling of Fluid-Structure Interactions With the Space-Time Finite Elements: Contact Problems
,”
Comput. Mech.
,
43
, pp.
51
60
.
35.
Torii
,
R.
,
Oshima
,
M.
,
Kobayashi
,
T.
,
Takagi
,
K.
, and
Tezduyar
,
T. E.
, 2008, “
Fluid-Structure Interaction Modeling of a Patient-Specific Cerebral Aneurysm: Influence of Structural Modeling
,”
Comput. Mech.
,
43
, pp.
151
159
.
36.
Bazilevs
,
Y.
,
Calo
,
V. M.
,
Hughes
,
T. J. R.
, and
Zhang
,
Y.
, 2008, “
Isogeometric Fluid-Structure Interaction: Theory, Algorithms, and Computations
,”
Comput. Mech.
,
43
, pp.
3
37
.
37.
Isaksen
,
J. G.
,
Bazilevs
,
Y.
,
Kvamsdal
,
T.
,
Zhang
,
Y.
,
Kaspersen
,
J. H.
,
Waterloo
,
K.
,
Romner
,
B.
, and
Ingebrigtsen
,
T.
, 2008, “
Determination of Wall Tension in Cerebral Artery Aneurysms By Numerical Simulation
,”
Stroke
,
39
, pp.
3172
3178
.
38.
Kuttler
,
U.
, and
Wall
,
W. A.
, 2008, “
Fixed-Point Fluid-Structure Interaction Solvers With Dynamic Relaxation
,”
Comput. Mech.
,
43
, pp.
61
72
.
39.
Dettmer
,
W. G.
, and
Peric
,
D.
, 2008, “
On the Coupling Between Fluid Flow and Mesh Motion in the Modelling of Fluid-Structure Interaction
,”
Comput. Mech.
,
43
, pp.
81
90
.
40.
Tezduyar
,
T. E.
,
Schwaab
,
M.
, and
Sathe
,
S.
, 2009, “
Sequentially-Coupled Arterial Fluid-Structure Interaction (SCAFSI) Technique
,”
Comput. Methods Appl. Mech. Eng.
,
198
, pp.
3524
3533
.
41.
Torii
,
R.
,
Oshima
,
M.
,
Kobayashi
,
T.
,
Takagi
,
K.
, and
Tezduyar
,
T. E.
, 2009, “
Fluid-Structure Interaction Modeling of Blood Flow and Cerebral Aneurysm: Significance of Artery and Aneurysm Shapes
,”
Comput. Methods Appl. Mech. Eng.
,
198
, pp.
3613
3621
.
42.
Manguoglu
,
M.
,
Sameh
,
A. H.
,
Saied
,
F.
,
Tezduyar
,
T. E.
, and
Sathe
,
S.
, 2009, “
Preconditioning Techniques for Nonsymmetric Linear Systems in Computation of Incompressible Flows
,”
J. Appl. Mech.
,
76
, p.
021204
.
43.
Bazilevs
,
Y.
,
Gohean
,
J. R.
,
Hughes
,
T. J. R.
,
Moser
,
R. D.
, and
Zhang
,
Y.
, 2009, “
Patient-Specific Isogeometric Fluid-Structure Interaction Analysis of Thoracic Aortic Blood Flow Due to Implantation of the Jarvik 2000 Left Ventricular Assist Device
,”
Comput. Methods Appl. Mech. Eng.
,
198
, pp.
3534
3550
.
44.
Bazilevs
,
Y.
,
Hsu
,
M.-C.
,
Benson
,
D.
,
Sankaran
,
S.
, and
Marsden
,
A.
, 2009, “
Computational Fluid-Structure Interaction: Methods and Application to a Total Cavopulmonary Connection
,”
Comput. Mech.
,
45
, pp.
77
89
.
45.
Takizawa
,
K.
,
Christopher
,
J.
,
Tezduyar
,
T. E.
, and
Sathe
,
S.
, 2010, “
Space-Time Finite Element Computation of Arterial Fluid-Structure Interactions With Patient-Specific Data
,”
Int. J. Numer. Methods Biomedical Eng.
,
26
, pp.
101
116
.
46.
Tezduyar
,
T. E.
,
Takizawa
,
K.
, and
Christopher
,
J.
, 2009, “
Multiscale Sequentially-Coupled Arterial Fluid-Structure Interaction (SCAFSI) Technique
,”
International Workshop on Fluid-Structure Interaction—Theory, Numerics and Applications
,
S.
Hartmann
,
A.
Meister
,
M.
Schaefer
, and
S.
Turek
, eds.,
Kassel University Press
, pp.
231
253
.
47.
Takizawa
,
K.
,
Moorman
,
C.
,
Wright
,
S.
,
Christopher
,
J.
, and
Tezduyar
,
T. E.
, 2010, “
Wall Shear Stress Calculations in Space-Time Finite Element Computation of Arterial Fluid-Structure Interactions
,”
Comput. Mech.
,
46
, pp.
31
41
.
48.
Tezduyar
,
T. E.
,
Takizawa
,
K.
,
Moorman
,
C.
,
Wright
,
S.
, and
Christopher
,
J.
, 2010, “
Multiscale Sequentially-Coupled Arterial FSI Technique
,”
Comput. Mech.
,
46
, pp.
17
29
.
49.
Torii
,
R.
,
Oshima
,
M.
,
Kobayashi
,
T.
,
Takagi
,
K.
, and
Tezduyar
,
T. E.
, 2010, “
Influence of Wall Thickness on Fluid-Structure Interaction Computations of Cerebral Aneurysms
,”
Int. J. Numer. Methods Biomedical Eng.
,
26
, pp.
336
347
.
50.
Manguoglu
,
M.
,
Takizawa
,
K.
,
Sameh
,
A. H.
, and
Tezduyar
,
T. E.
, 2010, “
Solution of Linear Systems in Arterial Fluid Mechanics Computations With Boundary Layer Mesh Refinement
,”
Comput. Mech.
,
46
, pp.
83
89
.
51.
Torii
,
R.
,
Oshima
,
M.
,
Kobayashi
,
T.
,
Takagi
,
K.
, and
Tezduyar
,
T. E.
, 2010, “
Role of 0D Peripheral Vasculature Model in Fluid-Structure Interaction Modeling of Aneurysms
,”
Comput. Mech.
,
46
, pp.
43
52
.
52.
Bazilevs
,
Y.
,
Hsu
,
M.-C.
,
Zhang
,
Y.
,
Wang
,
W.
,
Liang
,
X.
,
Kvamsdal
,
T.
,
Brekken
,
R.
, and
Isaksen
,
J.
, 2010, “
A Fully-Coupled Fluid-Structure Interaction Simulation of Cerebral Aneurysms
,”
Comput. Mech.
,
46
, pp.
3
16
.
53.
Tezduyar
,
T. E.
,
Takizawa
,
K.
,
Moorman
,
C.
,
Wright
,
S.
, and
Christopher
,
J.
, 2010, “
Space-Time Finite Element Computation of Complex Fluid-Structure Interactions
,”
Int. J. Numer. Methods Fluids
,
64
, pp.
1201
1218
.
54.
Bazilevs
,
Y.
,
Hsu
,
M.-C.
,
Zhang
,
Y.
,
Wang
,
W.
,
Kvamsdal
,
T.
,
Hentschel
,
S.
, and
Isaksen
,
J.
, 2010, “
Computational Fluid-Structure Interaction: Methods and Application to Cerebral Aneurysms
,”
Biomech. Model. Mechanobiol.
,
9
, pp.
481
498
.
55.
Bazilevs
,
Y.
,
Hsu
,
M.-C.
,
Akkerman
,
I.
,
Wright
,
S.
,
Takizawa
,
K.
,
Henicke
,
B.
,
Spielman
,
T.
, and
Tezduyar
,
T. E.
, 2011, “
3D Simulation of Wind Turbine Rotors at Full Scale. Part I: Geometry Modeling and Aerodynamics
,”
Int. J. Numer. Methods Fluids
,
65
, pp.
207
235
.
56.
Bazilevs
,
Y.
,
Hsu
,
M.-C.
,
Kiendl
,
J.
,
Wüchner
,
R.
, and
Bletzinger
,
K.-U.
, 2011, “
3D Simulation of Wind Turbine Rotors at Full Scale. Part II: Fluid-Structure Interaction Modeling With Composite Blades
,”
Int. J. Numer. Methods Fluids
,
65
, pp.
236
253
.
57.
Takizawa
,
K.
,
Moorman
,
C.
,
Wright
,
S.
,
Spielman
,
T.
, and
Tezduyar
,
T. E.
, 2011, “
Fluid-Structure Interaction Modeling and Performance Analysis of the Orion Spacecraft Parachutes
,”
Int. J. Numer. Methods Fluids
,
65
, pp.
271
285
.
58.
Takizawa
,
K.
,
Wright
,
S.
,
Moorman
,
C.
, and
Tezduyar
,
T. E.
, 2011, “
Fluid-Structure Interaction Modeling of Parachute Clusters
,”
Int. J. Numer. Methods Fluids
,
65
, pp.
286
307
.
59.
Manguoglu
,
M.
,
Takizawa
,
K.
,
Sameh
,
A. H.
, and
Tezduyar
,
T. E.
, 2011, “
Nested and Parallel Sparse Algorithms for Arterial Fluid Mechanics Computations With Boundary Layer Mesh Refinement
,”
Int. J. Numer. Methods Fluids
,
65
, pp.
135
149
.
60.
Tezduyar
,
T. E.
,
Takizawa
,
K.
,
Brummer
,
T.
, and
Chen
,
P. R.
, 2011, “
Space-Time Fluid-Structure Interaction Modeling of Patient-Specific Cerebral Aneurysms
,”
Int. J. Numer. Methods Biomedical Eng.
,
48
, pp.
247
267
.
61.
Takizawa
,
K.
, and
Tezduyar
,
T. E.
, 2011, “
Multiscale Space-Time Fluid-Structure Interaction Techniques
,”
Comput. Mech.
,
48
, pp.
247
267
.
62.
Stein
,
K.
,
Benney
,
R.
,
Tezduyar
,
T.
, and
Potvin
,
J.
, 2001, “
Fluid-Structure Interactions of a Cross Parachute: Numerical Simulation
,”
Comput. Methods Appl. Mech. Eng.
,
191
, pp.
673
687
.
63.
Stein
,
K. R.
,
Benney
,
R. J.
,
Tezduyar
,
T. E.
,
Leonard
,
J. W.
, and
Accorsi
,
M. L.
, 2001, “
Fluid-Structure Interactions of a Round Parachute: Modeling and Simulation Techniques
,”
J. Aircr.
,
38
, pp.
800
808
.
64.
Stein
,
K.
,
Tezduyar
,
T.
,
Kumar
,
V.
,
Sathe
,
S.
,
Benney
,
R.
,
Thornburg
,
E.
,
Kyle
,
C.
, and
Nonoshita
,
T.
, 2003, “
Aerodynamic Interactions Between Parachute Canopies
,”
J. Appl. Mech.
,
70
, pp.
50
57
.
65.
Stein
,
K.
,
Tezduyar
,
T.
, and
Benney
,
R.
, 2003, “
Computational Methods for Modeling Parachute Systems
,”
Comput. Sci. Eng.
,
5
, pp.
39
46
.
66.
Takizawa
,
K.
,
Moorman
,
C.
,
Wright
,
S.
,
Purdue
,
J.
,
McPhail
,
T.
,
Chen
,
P. R.
,
Warren
,
J.
, and
Tezduyar
,
T. E.
, 2011, “
Patient-Specific Arterial Fluid-Structure Interaction Modeling of Cerebral Aneurysms
,”
Int. J. Numer. Methods Fluids
,
65
, pp.
308
323
.
67.
Tezduyar
,
T. E.
, 1992, “
Stabilized Finite Element Formulations for Incompressible Flow Computations
,”
Adv. Appl. Mech.
,
28
, pp.
1
44
.
68.
Tezduyar
,
T. E.
,
Behr
,
M.
, and
Liou
,
J.
, 1992, “
A New Strategy for Finite Element Computations Involving Moving Boundaries and Interfaces—The Deforming-Spatial-Domain/Space-Time Procedure: I. The Concept and the Preliminary Numerical Tests
,”
Comput. Methods Appl. Mech. Eng.
,
94
, pp.
339
351
.
69.
Tezduyar
,
T. E.
,
Behr
,
M.
,
Mittal
,
S.
, and
Liou
,
J.
, 1992, “
A New Strategy for Finite Element Computations Involving Moving Boundaries and Interfaces—The Deforming-Spatial-Domain/Space-Time Procedure: II. Computation of Free-Surface Flows, Two-Liquid Flows, and Flows With Drifting Cylinders
,”
Comput. Methods Appl. Mech. Eng.
,
94
, pp.
353
371
.
70.
Tezduyar
,
T. E.
, 2003, “
Computation of Moving Boundaries and Interfaces and Stabilization Parameters
,”
Int. J. Numer. Methods Fluids
,
43
, pp.
555
575
.
71.
Hughes
,
T. J. R.
, and
Brooks
,
A. N.
, 1979, “
A Multi-Dimensional Upwind Scheme With No Crosswind Diffusion
,”
Finite Element Methods for Convection Dominated Flows
,
T. J. R.
Hughes
, ed., AMD-Vol.
34
,
ASME
,
New York
, pp.
19
35
.
72.
Brooks
,
A. N.
, and
Hughes
,
T. J. R.
, 1982, “
Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows With Particular Emphasis on the Incompressible Navier-Stokes Equations
,”
Comput. Methods Appl. Mech. Eng.
,
32
, pp.
199
259
.
73.
Tezduyar
,
T. E.
,
Mittal
,
S.
,
Ray
,
S. E.
, and
Shih
,
R.
, 1992, “
Incompressible Flow Computations With Stabilized Bilinear and Linear Equal-Order-Interpolation Velocity-Pressure Elements
,”
Comput. Methods Appl. Mech. Eng.
,
95
, pp.
221
242
.
74.
Tezduyar
,
T. E.
,
Behr
,
M.
,
Mittal
,
S.
, and
Johnson
,
A. A.
, 1992, “
Computation of Unsteady Incompressible Flows With the Finite Element Methods—Space-Time Formulations, Iterative Strategies and Massively Parallel Implementations
,”
New Methods in Transient Analysis
, PVP-Vol. 246/AMD-Vol.
143
,
ASME
,
New York
, pp.
7
24
.
75.
Johnson
,
A. A.
, and
Tezduyar
,
T. E.
, 1994, “
Mesh Update Strategies in Parallel Finite Element Computations of Flow Problems With Moving Boundaries and Interfaces
,”
Comput. Methods Appl. Mech. Eng.
,
119
, pp.
73
94
.
76.
Saad
,
Y.
, and
Schultz
,
M.
, 1986, “
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
,”
SIAM J. Sci. Stat. Comput.
,
7
, pp.
856
869
.
77.
Karypis
,
G.
, and
Kumar
,
V.
, 1998, “
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
,”
SIAM J. Sci. Comput. (USA)
,
20
, pp.
359
392
.
You do not currently have access to this content.