AN OPTIMIZATION OF EIGENFUNCTION EXPANSION METHOD FOR THE INTERACTION OF WATER WAVES WITH AN ELASTIC

526Available ontline at www.sciencedirect.comScienceDirecttIHDJoumal of HydrodynamicsELSEVIER2009,21(4):526-530www.sciencedirect.com/DOI: 10.1016/S1001-6058(08)60180-8science/jourmal/10016058AN OPTIMIZATION OF EIGENFUNCTION EXPANSION METHOD FORTHE INTERACTION OF WATER WA VES WITH AN ELASTIC PLATE*XU Feng, LU Dong-qiangShanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China,E-mail: xufeng 163192@shu.edu.cnState Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, China(Received November 27, 2008, Revised January 18, 2009)Abstract: The hydroelastic interaction of an incident wave with a semi-infinite horizontal elastic plate floating on a homogenousfluid of fininite depth is analyzed using the eigenfunctionn method. The fluid is assumed to be inviscid and incompressibleand the wave amplitudes are assumed to be small. A two-dimensional problem is formulated within the framework of linear potential, regions, namely an open water region and a plate-covered region. In this paper, theorthogonality property of eigenfunctions in the open water region is used to obtain the set of simultaneous equations for theexpansion coefficients of the velocity potentials and the edge conditions are included as a part of the equation system. The resultsindicate that the thickness and the density of plate have almost no influence on the reflection and transmission coefficients.Numerical analysis shows that the method proposed here is effctive and has higher convergence than the previous results.Key words: wave scattering, eigenfunction expansion method, orthogonality1. Introductionconcern for the dynamics of the ice sheet in the polarThe models of theinteraction between waterocean. Recently, Squire et al.summarized thewaves and an elastic plate are derived from two kindsstudies related to the interaction between water wavesof physical problems. One is the design of Very Largeand ice sheets.Floating Structures (VLFS) in offshore zones. ItBecause the horizontal scales of VLFS and icerecent years, countries such as Japan and Singapore,sheet are very large in comparison with the incidenthave constructed many VLFS， which comprisewavelength and their thicknesses, researchers usuallyfloating airports, mobile offshore base, floating pier,idealize VLFS or ice sheet as a thin elastic platel-Is.floating plants, floating emergency bases, floatingVarious approaches for the interaction of water wavesstorage facilities, etc. Attention must be paid to thewith an elastic plate were summarized by Kashiwagihlhydrodynamic characteristics of VLFS because theand Squire et al.4. Among them, the eigenfunctioninteraction of water waves with VLFS on the sea isexpansion method is often used. It is a direct method,very important for the safety of VLFS. So there areas it . combines the kinematic and dynamic surfacelots of literatures on this subject". The other is theconditions，which gives the dispersion relationsatisfied by the wave numbers. Fox and Squire5l used* Project supported by the Innovation Program ofthe eigenfunction expansion method to study theinteraction of surface waves with an ice-coveredShanghaiMunicipal Education Commissionirant No.09YZ04), the State Key Laboratory of Ocean Enginceringsurface and obtained the solution by the conjugate(Grant No. 0803) and the Shanghai Rising-Star Program (Grantgradient method. Fox and Squirel5l observed that theNo. 07QA14022).eige:red region are notBiography: XU Feng (1985- ), Male, Master Candidateorth中国煤化工cnventa ineCorresponding author: LU Dong qiang,prodYHC N M H Gs are complete. ForE-mail: dqlu@ shu.edu.cnhe set oI eigentuncuons oI une place-covered region,527Sahoo et al.4) defined a different inner product withinwhere E1= Ed /[12(1-0)] is the flexural rigiditywhich the original eigenfunctions were orthogonal.of the plate, E the effective Young's modulus of theBhattacharjee and Sahool5l further applied this methodelastic plate, d the constant thickness of plate, Uinto the case of two-layer fluids. Teng et al.0lPoisson's ratio, m. =p,d，p。the density of theimproved Fox and Squire' s' method by removing thearbitrarily defined tuning parameter, and optimizedelastic plate， ρ the fluid density, and g thethe method of Sahoo et al.4 to make one of thegravitational acceleration. The boundary conditionscoefficient matrices diagonal and hence the solver forand matching relations for a free edge arethe associated linear system particularly is simplified.In the present article, we will revisit theinteraction of a plane incident surface wave with ax(5a)semi-infinite elastic plate floating on an inviscid fluidof finite depth. The beauty of the present method isthat it can directly deal with the problem by using the(=0(5b)conventional inner product, and the new inner productproposed by Sahoo et al.4 is not necessary. Moreover,it has higher convergence than previous methods sincewe can use lesser terms in the expansion for. theat (x,z)=(0,0) andvelocity potentials suggested by Fox and Squire'sl toattain the same accuracy.8中(0-,z)_ a中(0+,2)(6a)ax2. FormulationA two-dimensional problem is considered here.中(0-,z)= q(0+,z)(6b)The mathematical formulation can be found in Refs.[4]and [6]， and is repeated here for the sake offor -h