An Eigenfunction-Expansion Method for Hydroelastic ...

Proceedings of The Thirteenth (2003) International Offshore and Polar Engineering Conference Honolulu, Hawaii, USA, May 25 –30, 2003 Copyright © 2003 by The International Society of Offshore and Polar Engineers ISBN 1 –880653 -60 –5 (Set); ISSN 1098 –6189 (Set)

An Eigenfunction-Expansion Method for Hydroelastic Analysis of a Floating Runway S.Y. Hong*, J.W. Kim**, R.C. Ertekin*** and Y.S. Shin** *Korea Research Institute of Ships and Ocean Engineering, KORDI Daejeon, Korea **American Bureau of Shipping Houston, TX, USA *** Univ. of Hawaii at Manoa, Dept. of Ocean and Resources Engineering Honolulu, USA

in the initial, conceptual design stage before determining design parameters such as the size, overall structural stiffness, depth, draft and freeboard, performance evaluation of breakwater depending on its size, and structural arrangements. It was previously addressed that the global structural behavior of a VLFS is more important than its local behavior in design (Basu, 2002).

ABSTRACT A mat-type very large floating structure (VLFS) is one of the most popular conceptual designs for future floating plants and airports. With its high length-to-draft and breadth-to-draft ratios, a mat-type structure is quite flexible and is vulnerable to dynamic motions caused by waves. As a result, such structures would usually be located near the shore and be protected by a breakwater. Hydroelastic analyses of VLFS have mostly been done under the assumption of 'zero-draft' to achieve numerical efficiency. Recently, a higher-order boundary element method (HOBEM) analysis of MEGA-FLOAT by Hong et al. (2001) showed that the non-zero draft effect may not be negligible in the practical range of incoming wave lengths. The two-dimensional analysis of Kim & Ertekin (2000) also provided a similar conclusion. However, the HOBEM approach could not easily be applied to a fullscale VLFS because of its high demanding computational resources. In the present paper, we extend Kim & Ertekin's eigenfunction expansion method to three dimensions, and provide a more efficient numerical model for the hydroelastic analysis of a mat-type VLFS. The presence of a breakwater and the proximity of a shoreline will also be considered. Validation of the new model is done by comparison with the HOBEM results. The new model is expected to provide a useful method for site selection and preliminary or conceptual design of a mat-type VLFS.

It appears, therefore, that a design procedure based on a direct-analysis method is a prerequisite in the design of VLFS. To determine several design parameters as reliable as possible, a series of intensive performance evaluation of hydroelastic behavior of the structure should be done prior to its basic design. Hydroelasticity analysis can provide answers to a number of useful design factors, such as the wave loads, global and local structural responses that depend on changes in the size of the VLFS, global stiffness of the structure, and the location of breakwaters if needed. In principle, conventional boundary-element method (BEM) can be applied to this purpose, but a more numerically efficient method is desirable because the BEM is not always computationally affordable since it requires tremendously large computer memory and very long turn-around time in obtaining the solution. For this reason, some numerically efficient methods have been devised, for example, the ‘B-spline Galerkin method’ proposed by Kashiwagi (1998a), and the ‘eigenfunction expansion method’ of Ohmatsu (1998). These methods are based on the generalized-mode approach (see also Wu (1984) and Newman (1994)).

KEY WORDS: hydroelasticity, eigenfunction-expansion method, draft effect, mat-type VLFS, breakwater, drift force.

As another line of approach to hydroelastic analysis of VLFS, Ohkusu & Namba (1996) showed that the modified free-surface condition due to the presence of the thin plate structure, as in the capillary-gravity wave problem, can be applied to solve the elastic behavior of pontoontype VLFS. In this approach, the floating plate and fluid beneath are treated as uniform media of hydroelastic waves on the horizontal plane. Since the waves on the plate are identified by only one wavelength for a given frequency, this enables us to describe the deformation of the plate much more efficiently than the other approaches in which large number of modes is required. This approach has been widely used in the study of the elastic deformation of ice floes in the arctic region (see, Stoker (1957), Evans & Davies (1968), Meylan & Squire (1994), Fox & Squire (1994)). Similarly, Kim & Ertekin (1998) introduced an eigenfunction-expansion method, where the velocity potential under the plate is expanded by eigenfunctions. This method has proven to be very

INTRODUCTION A mat-type VLFS is one of the most popular conceptual designs for planned floating plants and airports due to its cost effectiveness compared with a footing-type VLFS. In spite of the existence of the 1km long Mega-Float structure that had been successfully constructed and installed in Japan, the concept of VLFS is new as a huge floating structure since proven design technology is not fully established due to lack of sufficient experience with such structures. Furthermore, unlike popular floating structures such as FPSO, TLPs and semisubmersibles, widely accepted design guidelines for VLFS are not yet available because of insufficient experience with the construction and operations of VLFS. This is the main reason why intensive efforts should be made

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the thin-plate theory − mω 2ζ + D∆2ζ = p f ,

efficient in the parametric study of the hydroelastic response of a pontoon-type VLFS because the discretization of the structure is needed only along its edges, unlike in the usual panel methods that require the discretization of the entire mat. Consequently, the timeconsuming evaluation of the surface integrals is replaced by the evaluation of line integrals with much more efficiency.

by

assumption

(Rayleigh (1894)): (1)

where p f = p f ( x, y ) is the spatial part of the time-harmonic pressure on the lower surface of the plate and ∆ = ∂ 2 / ∂x 2 + ∂ 2 / ∂y 2 is the twodimensional Laplacian on the horizontal plane.

Kim and Ertekin (2002), for example, introduced Green-Naghdi theory to hydroelasticity of an infinitely long plate to investigate the shallowwater wave effect. Most of the numerically efficient approaches to hydroelasticity of pontoon-type VLFS have adopted the zero-draft approximation, but it was pointed out that neglect of draft effects may lead to the underestimation of hydroelastic response in the practical range of wave frequency (see e.g., Hong et al. (2001); Kim & Ertekin (2000)). So it is desirable that we consider the effect of non-zero draft on the hydroelastic response, as well as the presence of breakwaters and proximity of shorelines so as to include a more realistic environment that can be considered in a parametric study and during the preliminary or conceptual design stage.

shoreline

y

n

J

s O

Region II

.

z (up)

B

x

L

S

Region I

breawater Incident wave β

In the present paper, we extend Kim & Ertekin's (1998) eigenfunction expansion method when the draft is non-zero, and provide a computationally efficient and practical numerical model applicable to pontoon-type VLFS. A similar model has been successfully applied to the zero-draft model of Kim & Ertekin (1998). In this study, the presence of the breakwater and the proximity of the shoreline is also considered. Validation of the new model is presented in comparison with the HOBEM results. The new model should provide a useful method in estimating design parameters such as the main dimensions, structural stiffness, and site selection of a mat-type VLFS and breakwaters.

(a) plan view z

Incident wave y

Region I

h

J

Region II

J

Region I

(b) side view Fig. 1 Definition sketch of the problem

FORMULATION Since the plate is freely floating, the bending moment and shear force should vanish at the edges of the plate, i.e., ∂ 2ζ ∂ 2ζ ∂ 3ζ ∂ 3ζ ν ν + = 0 , + ( 2 − ) = 0 on J, , (2a) ∂n 2 ∂s 2 ∂n 3 ∂s 2∂n which can alternatively be written on J as

An elastic mat of rectangular plan geometry, with length, L, beam, B, and draft d, is considered, see Fig. 1. The mat is freely floating on an inviscid fluid-layer of constant density ρ and depth h, and is under the action of linear incoming waves of angular frequency ω and direction β . The origin of the Cartesian coordinate system Oxyz is placed at the center of the runway. The breakwater is defined by a finite length, Lb , at y = − B / 2 − S , and constant thickness Bb , at a distance of S from the breakwater-side edge of the mat, and it extends to the sea floor having vertical sides. Two regions describing the fluid Region (I) and the fluid-plate Region (II) are considered, and these regions are separated by the juncture boundary J.

∂ ζ ∂  ∂ 2ζ =0, (2b). ∆ζ + (1 − ν ) 2  = 0, 2 ∂s ∂n  ∂s  where n and s denote, respectively, the normal and tangential directions as seen in Fig. 1, and ν is Poisson's ratio. ∆ζ − (1 − ν )

2

For the fluid motion, we assume that the fluid is inviscid and incompressible and its motion is irrotational. Then we can introduce the velocity potential, φ ( x, y, z ) by which the velocity field is defined as u ( x, y, z ) = ∇φ . The velocity potential must satisfy the Laplace equation in the fluid domain:

We assume that the wave amplitude, A, of the incoming waves is much smaller than the wavelength, λ = 2π / k , i.e., kA