In the conventional approach for fluid-structure-interaction problems, the fluid and solid components are treated separately and information is exchanged across their interface. According to the conventional terminology, the current numerical methods can be grouped in two major categories: partitioned methods and monolithic methods. Both methods use separate sets of equations for fluid and solid that have different unknown variables. A unified solution method has been presented in the previous work of Giannopapa and Papadakis (2004, “A New Formulation for Solids Suitable for a Unified Solution Method for Fluid-Structure Interaction Problems,” ASME PVP 2004, San Diego, CA, July, PVP Vol. 491–1, pp. 111–117), which is different from these methods. The new approach treats both fluid and solid as a single continuum; thus, the whole computational domain is treated as one entity discretized on a single grid. Its behavior is described by a single set of equations, which are solved fully implicitly. In this paper, the elastodynamic equations are reformulated so that they contain the same unknowns as the Navier–Stokes equations, namely, velocities and pressure. Two time marching and one spatial discretization scheme, widely used for fluid equations, are applied for the solution of the reformulated equations for solids. Using linear stability analysis, the accuracy and dissipation characteristics of the resulting difference equations are examined. The aforementioned schemes are applied to a transient structural problem (beam bending) and the results compare favorably with available analytic solutions and are consistent with the conclusions of the stability analysis. A parametric investigation using different meshes, time steps, and beam dimensions is also presented. For all cases examined, the numerical solution was stable and robust and therefore is suitable for the next stage of application to full fluid-structure-interaction problems.

1.
Wiggert
,
D. C.
, and
Tijsseling
,
A. S.
, 2001, “
Fluid Transients and Fluid-Structure Interaction in Flexible Liquid-Filled Piping
,”
Appl. Mech. Rev.
0003-6900,
54
, pp.
455
481
.
2.
Korterweg
,
D. J.
, 1878, “
Uber die Fortplanzungs-Geschwindigkeit des Schalles in Elastischen Rohen
,”
Ann. Phys. Chem.
0003-3804,
5
, pp.
525
542
.
3.
Felippa
,
C. A.
,
Park
,
K. C.
, and
Farhat
,
F.
, 2001, “
Partitioned Analysis of Coupled Mechanical Systems
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
190
, pp.
3247
3270
.
4.
Hübner
,
B.
,
Walhorn
,
E.
, and
Dinkler
,
D.
, 2004, “
A Monolithic Approach to Fluid-Structure Interaction Using Space-Time Finite Elements
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
193
, pp.
2087
2104
.
5.
van Brummelen
,
E. H.
, and
Koren
,
B.
, 2003, “
A Pressure-Invariant Conservation Godonov-Type Method for Barotropic Two-Fluid Flows
,”
J. Comput. Phys.
0021-9991,
185
, pp.
289
308
.
6.
Giannopapa
,
C. G.
, and
Papadakis
,
G.
, 2004, “
A New Formulation for Solids Suitable for a Unified Solution Method for Fluid-Structure Interaction Problems
,”
ASME PVP 2004
,
San Diego, CA
, July PVP Vol.
491-1
, pp.
111
117
.
7.
Giannopapa
,
C. G.
, 2004, “
Fluid-Structure Interaction in Flexible Vessels
,” Ph.D. thesis, University of London, London.
8.
Eringen
,
A. C.
, 1980,
Mechanics of Continua
,
2nd ed.
,
Robert E. Krieger Publishing Company
,
Huntington, USA
.
9.
Bathe
,
K. J.
, 1997,
Finite Element Procedures
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
10.
Vardy
,
A. E.
, and
Alsarraj
,
A. T.
, 1989, “
Method of Characteristics Analysis of One-Dimensional Members
,”
J. Sound Vib.
0022-460X,
129
(
3
), pp.
477
487
.
11.
Vardy
,
A. E.
, and
Alsarraj
,
A. T.
, 1991, “
Coupled Axial and Flexural Vibration of 1-D Members
,”
J. Sound Vib.
0022-460X,
148
(
1
), pp.
25
39
.
12.
Lin
,
X.
, and
Ballman
,
J.
, 1995, “
A Numerical Scheme for Axisymmetric Elastic Waves in Solids
,”
Wave Motion
0165-2125,
21
, pp.
115
126
.
13.
Giese
,
G.
, and
Fey
,
M.
, 2002, “
A Genuinely Multidimensional High-Resolution Scheme for the Elastic-Plastic Wave Equation
,”
J. Comput. Phys.
0021-9991,
181
, pp
338
353
.
14.
Voinovich
,
P.
,
Merlen
,
A.
,
Timofeev
,
E.
, and
Takayama
,
K.
, 2003, “
A Godunov-Type Finite Volume Scheme for Unified Solid-Liquid Elastodynamics on Arbitrary Two-Dimensional Grids
,”
Shock Waves
0938-1287,
13
, pp.
221
230
.
15.
Voinovich
,
P.
, and
Merlen
,
A.
, 2003, “
Two-Dimensional Unstructured Elastic Model for Acoustic Pulse Scattering at Solid-Liquid Interfaces
,”
Shock Waves
0938-1287,
13
, pp.
421
429
.
16.
Leveque
,
R. J.
, 2002,
Finite Volume Methods for Hyperbolic Problems
,
Cambridge University Press
,
Cambridge
.
17.
Issa
,
R. I.
, 1986, “
Solution of the Implicit Discretised Fluid Flow Equation by Operator-Splitting
,”
J. Comput. Phys.
0021-9991,
62
, pp.
40
65
.
18.
Gresho
,
P. H.
, and
Sani
,
R. L.
, 1987, “
On Pressure Boundary Conditions for the Incompressible Navier–Stokes Equations
,”
Int. J. Numer. Methods Fluids
0271-2091,
7
, pp.
1111
1145
.
19.
Ferziger
,
J. H.
, and
Peric
,
M.
, 1996,
Computational Methods of Fluid Dynamics
,
Springer-Verlag
,
Berlin, Germany
.
20.
Mattheij
,
R. M. M.
,
Rienstra
,
S. W.
, and
tenTije Boonkkamp
,
J. H. M.
, 2005,
Partial Differential Equations: Modeling, Analysis and Computing
,
SIAM
,
Philadelphia, PA
.
21.
Case
,
J.
,
Chilver
,
A. H.
, and
Ross
,
C. T. F.
, 1999,
Strength of Materials and Structures
,
Arnold
,
New York
.
22.
Geradin
,
M.
, and
Rixen
,
D.
, 1997,
Mechanical Vibrations
,
Wiley
,
Chichester, UK
.
23.
Sloan
,
A. K.
,
Bailey
,
C.
, and
Cross
,
M.
, 2003, “
Dynamic Solid Mechanics Using the Finite Volume Methods
,”
Appl. Math. Model.
0307-904X,
27
, pp.
69
87
.
24.
Bath
,
M.
, and
Kamm
,
R. D.
, 1999, “
A Fluid Structure Interaction Finite Element Analysis of Pulsatile Blood Flow in Arterial Structures
,”
ASME J. Biomech. Eng.
0148-0731,
121
, pp.
361
369
.
You do not currently have access to this content.