The dynamic interaction between an elastic structure and a fluid has been the subject of intensive investigations in recent years, e.g. ([45], [57], [67], [73], [161], [163], [232], [255]). Since analytical solutions procedures are available only for very simple problems, numerical approaches, which can be formulated in the time or frequency domain, had to be employed, e.g. ([54], [131], [150], [210], [241], [242]). Vonestorff et al [285] investigated the coupled fluid-structure systems subjected to dynamic loads using the finite element and boundary element methods. Feng et al [58] analyzed the nonlinear three dimensional fluid-structure interaction by using the finite element method for the structure and the finite-infinite element method for the fluid. The infinite element was also used by Olson [211] to present a procedure to analyze fluid-structure interaction.
Since the variational principles are employed to derive numerical solutions, many researchers have attempted to derive variational principles for different classes of the fluid-structure interaction problems. Pinsky and Abboud [224] proposed two mixed variational principles for transient and harmonic analyses of non-conservative coupled exterior fluid-structure interaction systems. Kock and Olson [141] presented a finite element formulation directly derived from a variational indicator based on Hamilton's principle. Liu [157] presented a general variational principle for fluid-structure interaction problems with sloshing. Abboud [1] proposed a mixed variational principle for transient and harmonic analysis of nonconservative coupled structure-exterior fluid systems. The formulation provided a basis for finite element approximation which is applicable to the analysis of radiation from shell structures with viscoelastic compliant coatings. In addition to the variational principles, energy methods have been employed to investigate the problem. Zeng et al [298] developed an energy-based symmetric coupled finite element and boundary integral method which is valid for all frequencies.
The size of the coupled fluid-structure interaction problem is generally large. Many researchers have attempted to reduce the problem size in different ways. Seybert [247] employed Ritz Vectors and Eigenvectors along with a combination of finite element and boundary element methods to reduce the problem size. Haroun [106] employed a boundary integral technique to evaluate the added mass matrix to the tank shell due to the fluid. Rajasankar et al [227] presented the results of investigations conducted to evaluate the added mass to represent the fluid effect in 3-D problems.
Out of all the work done in the area of developing a finite element method for fluid-structure interaction problems, two approaches predominate. In the first approach, the displacement based method, the displacements are the nodal variables in both the fluid and the structure. Belytschko and Kennedy ([20], [21], [22]), Bathe and Hahn [18], Nitikitpaiboon and Bathe [202] and Chopra et al [34] described the method in detail. This approach is not well suited for problems with large fluid displacements. Another difficulty with this method is that special care must be taken to prevent zero-energy rotational modes from arising. In the second approach, the potential-based method, displacements remain the nodal variables in the structure, while velocity potentials or pressures are the unknowns in the fluid. Zienkiewicz and Newton [301], Morand and Ohayon [185], Everstine et al ([55], [56]), Olson and Bathe ([212], [213]) and others ([77], [206], [300]) demonstrated techniques for formulating finite elements using potential-based methods. In all these works, only a linearized version of the problem has been considered.
Several finite element studies have considered the gravity and free surface effects along with the fluid structure interaction. Wilson and Khalvati [291] incorporated the gravity and the free surface effects in a displacement-based method with rotational constraints. They demonstrated results for both a static and a dynamic floating body problem. Their method necessitates the use of a reduced integration scheme to prevent element locking. Aslam [10] incorporated the linearized dynamic free surface condition into a velocity potential-based finite element fluid formulation, but did not consider the fluid-structure interaction problem. Ohayon [205] included gravity terms in a displacement-based finite element method for fluid-structure modal analysis.
The search for variational principles resembling Hamilton's principle for fluid mechanics problems has concerned many researchers including Seliger and Whitham [245], Miles [180], Serrin [246] and Luke [162]. While Seliger and Whitham have pointed out that the Lagrangian density for the fluid variational principle is the pressure, they did not consider the implication of a variable boundary. Luke has incorporated a variable boundary in his variational principle in order to generate the governing equations for the special case of gravity waves in an incompressible fluid. The role of the Bernoulli constant as a Lagrange multiplier constraining global conservation of mass was not noted by any of the aforementioned researchers. Kock and Olson [141] were the first to use Bernoulli constant in there formulation to conserve the total mass. Ikegawa and Washizu [123] introduced a variational principle, utilizing the stream function, in order to model incompressible flow with a free surface under gravity using the finite element method but did not extend their method to the general compressible fluid-structure problem. More recently, Ecer and his coworkers ([50], [286], [49]) suggested variational approaches for use in modelling incompressible, viscous fluid flows but have not extended their approach to include fluid-structure interaction. Liu and Uras [157] using a variational principle which is not based on Hamilton's principle, developed a mixed variational formulation and demonstrated that several of the fluid-structure interaction formulations already in use can be obtained from it.
Birk et al [24] investigated the influence of fluid-containing appendages on the dynamic response of multi-degree-of-freedom system subjected to stochastis environmental loads. They expressed the modal properties of the system comprising of a fluid-containing appendage attached to a multi-degree-of-freedom system in terms of the individual dynamic properties of the primary and the secondary systems. They obtained the peak response value at any level on the structure by following the evolutionary distribution of the extreme values and reported that an important feature of the combined system was that the response of the primary system was suppressed when one of the sloshing modes of the secondary fluid appendage is tuned to the fundamental mode of the primary system. They used a building with a water tank situated at any floor and excited by an earthquake to illustrate the methodology.