Hydroelastic analysis of VLFS

A typical very large floating structure (VLFS) has large horizontal dimensions ranging from several hundred meters to several kilometers. On the other hand, the depth of the VLFS is only several meters. For example, the MegaFloat Phase 1 model has a length of 300 m, a width of 60 m and a depth of 2 m, while the Mega-Float Phase 2 model (Figure 3.1) has a length of 1,000 m, a width of 121 m (at the widest part) and a depth of 3 m. Given such a small depth to length ratio, the VLFS would behave almost like an elastic plate when considering the vertical responses. In other words, the VLFS response in the vertical direction cannot be assessed only by its rigidbody motions. It is therefore crucial to estimate the flexural response of a VLFS when designing the structure. However, for the design of the mooring system, the VLFS may be considered as a rigid-body since it has a large flexural rigidity in the horizontal directions. When the dynamic response of a floating body due to waves is analyzed,

the pressure of the fluid acting on the floating body must be evaluated at the same time. If a floating body moves, the fluid surrounding the floating body also moves. Thus, the pressure surrounding the floating body changes in order to satisfy the Bernoulli’s equation. On the other hand, if the pressure changes, the motion of the floating body is affected. This kind of mutual relationship is referred to as a fluid-structure interaction. If the motion of the floating body consists of elastic deformations, the fluid-structure interaction is called hydroelasticity. The vertical response of a VLFS typically shows hydroelasticity between its flexural behavior and the fluid motions. The hydroelastic analysis is thus necessary in order to assess the vertical response of a VLFS. Many papers on hydroelastic analysis of VLFSs have been published

to date. They may be found in the review papers by Kashiwagi (2000a), Watanabe et al. (2004a,b), Newman (2005), Ohmatsu (2005) and Suzuki et al. (2006). To name a few, Mamidipudi and Webster (1994) undertook pioneering work on the hydroelastic analysis of a mat-like floating

airport by combining the finite-difference method for plate problem and the Green’s function method for fluid problem. Wu et al. (1995) solved the two dimensional (2-D) hydroelastic problem by the analytical method using eigenfunctions. Yago and Endo (1996) analyzed a zero-draft VLFS using the direct method and also compared this with their experimental results. Ohkusu and Nanba (1996) analyzed an infinite-length VLFS analytically. Kashiwagi (1998a) applied the B-spline panels for the analysis of a zerodraft VLFS using the pressure distribution method. Nagata et al. (1998) and Ohmatsu (1998) analyzed a rectangular VLFS by using a semi-analytical approach based on eigenfunction expansions in the depth direction. Seto and Ochi (1998) developed a hybrid element method (as a combination of the Finite ElementMethod (FEM) and infinite element) for hydroelastic analysis of a VLFS in stepped-depth configuration. Iijima et al. (1998) analyzed the hydroelastic behavior of semi-submersible type VLFSs using their program named VODAC. Takagi et al. (2000) analyzed the performance of an anti-motion device for a VLFS using the eigenfunction expansion method.

The transient response of a VLFS due to airplane landings and takeoffs was analyzed by Kim and Webster (1998), Watanabe and Utsunomiya (1996), and then for a moving load by Watanabe et al. (1998), Endo (2000) and Kashiwagi (2000b). In the hydroelastic analysis of a VLFS, there are two major issues to be

tackled. The first issue is concerned with the development of an accurate method for fluid-structure interaction analysis of an elastic floating vessel. In other words, how to model and analyze accurately the hydroelastic behavior of an elastic floating structure? The second issue deals with the applicability of themethod to aVLFS. This is an important point of view because relatively large computational resources are required when analyzing a “very large” floating structure. Any established method for the hydroelastic analysis of VLFS must adequately satisfy these two requirements. One way to tackle this problem is to use an analytical approach. If the

problem has been solved analytically, the computational requirement for a VLFS is a non-issue. However, the drawback of the analytical approach is that only simple geometries are tractable. For example, modeled as a 2-D problem a floating plate, a circular plate, a ring-shaped plate or a rectangular plate can be solved analytically/semi-analytically. In these analytical/semi-analytical approaches, the constant sea depth is assumed. These analytical/semi-analytical approaches can be powerful tools, when the fundamental behavior of a VLFS is examined for scientific purposes in order to obtain a deeper knowledge or for preliminary design purposes. More importantly, the analytical/semi-analytical method can provide benchmark solutions for the verification of numerical solutions. In order to explain this analytical/semi-analytical approach in greater detail, the eigenfunction expansion matching method for a floating elastic plate modeled as a 2-D problem, which was first solved by Wu et al. (1995), is presented in the next section. Another way to solve the hydroelastic problem of a VLFS is by using a

numerical approach. In a numerical approach, there can be several combinations of the methods for solving the structure part and the fluid part. In principle, any method that can be applied to solve the vibration problem of a plate or a 3-D structure can be used for solving the structure part. Among these various methods, FEM may be considered as the most promising method because of its versatility in handling complicated geometries of real structures. For a linear hydroelastic analysis, commercial FEM codes may be used for modeling the VLFS’s vessel structure. In some cases, the entire VLFS can be modeled as a floating plate. At the preliminary design stage, this simple modeling is very efficient and is frequently employed in the actual design procedure. For solving the fluid part, the often used method is the Green’s function method. If one adopts Green’s function which satisfies the boundary condition on the free-surface, the sea bottom and that at infinite distance from the floating structure, the unknown parameters to be

determined for the fluid part can be minimized to be only those associated with the wetted surface of the floating body. At the same time, general geometry of the floating body and general topography of the seabed (including breakwater effect) can be considered. Alternatively, FEM can also be used for solving the fluid part although there are some difficulties encountered in satisfying accurately the infinite boundary condition and the whole fluid domain within the fictitious infinite boundary must be discretized into finite elements. In order to couple the problem between the structure part and the fluid

part, there exist two competing approaches. These two approaches are sometimes referred to as the modal method and the direct method. In the modal method, the flexural response of the structure is represented by a combination of the global modal responses of the structure. If each modal response corresponds to the natural mode for the vibration of the structure in air, the method may be called the dry-mode superposition method. On the other hand, if each modal response corresponds to the natural mode for vibration of the structure in fluid, the method may be referred to as the wet-mode superposition method. In actual applications, the dry-mode superposition method is frequently used since the dry-modes can be obtained in a straightforward manner by any standard FEM package. In some cases, the direct method is used. In the direct method, the response of the structure is not represented through a superposition of the global modal responses. Instead, the response of the structure is directly represented by, for example, the nodal responses if the structure is modeled by FEM. In most cases, the direct method is computationally more demanding than the modal method. Thus, nowadays the dry-mode superposition method seems more frequently used than the direct method for the actual design of a VLFS. In Section 3.3, the modal method is explained in detail as an example of

the numerical approach, where the classical thin-plate theory is employed for modeling the structural behavior and Green’s function method is used for solving the fluid problem. This method can be used for a relatively large problem, if a computer equipped with large storage is available. However, a “very large” floating structure may be still difficult to solve because of its large requirements for storage and computational time. Fortunately, there exist several methods to overcome this difficulty. These methods will be explained briefly in the same section. Finally, some numerical results of a VLFS in a variable-depth sea are presented.