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CHAPTER THREE Method of Solution for Coupled Water-Plate Problem 3.1 Overview Recall in the previous chapter that we first described the equation of motion of the water using the linear wave theory while the equation of motion for the plate by using the Mindlin plate theory in a coupled form We then employed the modal expansion method to decouple the plate deflections (Eq 2.14) and water motions (Eq 2.15) so that the boundary value problem for each of the unit-amplitude radiation and diffracted potentials can be represented in an uncoupled form The problem at hand is to devise numerical techniques to solve these decoupled equations of motion for the plate deflections and water motions (i.e the velocity potential) There are several methods in solving the equations of motions To name a few, Mamidupudi and Webster (1994) have used the finite difference method for solving the governing equations of the plate (modelling a pontoon-type floating airport) and the Green function for solving the fluid’s Laplace equation The eigenfunctions method (analytical method) has also been employed by researchers such as Wu et al (1995), Nagata et al (1998) and Ohmatsu (1998) for determining 44 Method of Solution for Coupled Water-Plate Problem the hydroelastic response of a VLFS However, such an analytical method is only applicable for floating structures with simple geometries such as circular and rectangular shapes In this chapter, we present a hybrid finite element-boundary element (FE-BE) method for solving the equations of motion for the water and plate Although the method is used here for tackling the hydroelastic problems of rectangular shaped VLFS, it can be applied for any arbitrarily shaped VLFS The details on the finite element method could be found in books by Petyt (1990) and Cook et al (2002) whereas details on boundary element method in books by Brebbia (1984) and Becker (1992) The flow chart describing the algorithm of this numerical method is given in Fig 3.1 The finite element method is used to solve the governing equations of motion for the plate because of its versatility in handling complicated plate geometries In the finite element method, the 8-node NC-QS Mindlin plate element is used for the first time in such floating plate problem This plate element is known for its high accuracy in predicting the stress resultants as explained in the literature (see Section 1.2.2) Details of the discretisation technique for the Mindlin plate are given in Appendix B On the other hand, for determining the fluid motion, the boundary element method is adopted to transform the Laplace equation (2.16) together with the boundary conditions (given by Eqs 2.17-2.21) into a boundary integral equation This means that only the boundaries of the computational fluid domain are needed to be discretised The free-surface Green function derived by Linton (1999) that satisfies 45 Method of Solution for Coupled Water-Plate Problem automatically the boundary conditions of the fluid domain at the seabed, free surface and the Sommerfeld condition is used So the remaining unknown parameters to be determined for the fluid part are only those associated with the wetted surface of the floating body The boundary element method thus significantly reduces the computational time needed for the hydroelastic analysis The velocity potential of the water is obtained by solving the boundary integral equation The computed velocity potentials are used to derive the exciting force and added mass for the water-plate equations (Eq 2.11) By solving Eq (2.11) using the finite element method, one obtains the modal amplitudes ς l Together with the modal ˆ functions, these modal amplitudes define the plate deflections w (see Eq 3.5) In the sequel, we will present the method of solutions for the plate deflection, the flow field surrounding the floating structures and the steady drift force 46 Method of Solution for Coupled Water-Plate Problem START Input Data Plate dimensions: L, d, h Plate properties: E, Sea states: T, H, Defining GE and BCs for plate displacement w, x, y (Eqs 2.11 and 2.13) Decoupling plate deflection w and water motion using modal expansion Discretise plate equation using NC‐ QS Mindlin plate element (Appendix B) wj vi ˆ (d ) M ( ˆ xj + ∑ N i θ xid ) = ∑ N j ψ j =1 ψ (d ) i =1 θ (d ) ˆ yj yi (d ) w (d ) (d ) Obtain plate stiffness and mass matrices using Hamiltonian principle (Appendix C) Defining GE and BCs for water motion (Eqs 2.2 to 2.6) w = ∑ clwς l , φ = φ D + ∑ ς lφl φl M M l =1 l =1 Transforming GE (Laplace equation) with BCs into a BIE using free surface Green’s function and solve for (Appendix D) Perform free vibration analysis to obtain modal functions cl by solving the eigenproblem − 2πφl (x ) + ∫ s l l ∂G (x,x' )φl (x' )dS ∂n = − ∫ G (x,x' )iωclw (x' ) dS S ˆ ˆ ([K ] + [K ]){c } = ω [M ]{c } f Obtain GE and BCs for water in uncoupled form (Eqs 2.16 to 2.21) l S Assembly discretised plate stiffness and mass matrices into plate equation in decoupled form ˆ ˆ ˆ {c }([K ] + [K ]+ [K ]- ω [M] + ω [M ]){c }{ς } = {c }{F} T l s f rf a Solve for modal amplitudes ς l l T l l Solve for plate deflection Abbreviations GE: Governing equation BC: Boundary condition BIE: Boundary integral equation ˆ ˆ {w} ={cl }{ςl } END Fig 3.1 Flow chart describing algorithm for hybrid FE-BE method 47 Method of Solution for Coupled Water-Plate Problem 3.2 Hydroelastic Analysis After obtaining the discretised version of the plate equation and the boundary value problem of the water, the method of solution for the hydroelastic problem involves the following four steps: Step Solving for velocity potential using boundary integral equation Substitute the computed modal functions clw ( x, y ) into Eq (2.18) Solve the Laplace equation (2.16) with the boundary conditions given by Eqs (2.17) to (2.21) for the velocity potential components φ l This step can be done by transforming these Laplace equation and boundary conditions into a boundary integral equation using the boundary element method (BEM) The BEM easily handles the Sommerfeld radiation condition by using the free-surface Green function G For details on the transformation of the Laplace equation into a boundary integral equation using the free-surface Green function, please refer to Appendix D The boundary integral equation for the velocity potential is given as ∂G (x; x' )φl (x' )dS S HS ∪ S HB ∂n w − iω ∫SHB G (x; x' ) cl (x' )dS = 4πφ In − 2πφl (x ) + ∫ for l = 1,2, , N (3.1) for l = D where x = (x, y, z ) are the source points and x' = ( x' , y ' , z ') the field points By rearranging Eq (3.1), the radiated and diffracted potentials [ φ R (x ) and φ D (x ) ] are can then be re-written in the matrix form as 48 Method of Solution for Coupled Water-Plate Problem ~ {φ R } = {φ l }{ς l } = −iω φ l clw {ς l } = −iω [I ] + ∂G 2π ∂n { }{ } {φ D } = 2[I ] + ∂G 2π ∂n −1 {φ In }, −1 [G ]{clw }{ς l }, (3.2a) (3.2b) where [G ] is the global matrix for the free surface Green function, [I ] is the identity ˆ matrix, {clw } the mode shapes (eigenvectors) obtained by performing a free vibration analysis on the plate and ∂G ~ φ l = [I ] + 2π ∂n {} Step −1 [G ] (3.3) Assembly the plate equation in global form [ ] With the computed flexural stiffness K f , shear stiffness [K s ] and mass [M ] given in Appendix C, the equation of motion for the plate can be written in the global form as ˆ ([K ] + [K ] + [K ] − ω [M ] + [M ]){w} = {F} f s rf a (3.4) [ ] where K rf , [M a ] and {F} are the global restoring force matrix, global added mass ˆ matrix and global exciting force matrix, respectively The displacement vector {w} ˆ may be expanded in an appropriate set of modes {cl } as ˆ ˆ {w} = {cl }{ς l } (3.5) 49 Method of Solution for Coupled Water-Plate Problem ˆ Note that {cl } could be obtained by performing a free vibration test on the Mindlin { ˆ ˆ plate where {cl } = clw { } ˆψ cl y ˆl cψ x { } } {cˆ } is the eigenvectors correspond to the plate T w l ˆl ˆψ deflection, cψ x and cl y the eigenvectors correspond to the rotation about y and ˆ x-axis Due to the orthogonal properties of the free vibration modal function cl , we ˆ could multiply the obtained plate equation with clT (where T denotes the transpose of the matrix) and arrive at ~ ˆ ˆ ˆ {c }([K ] + [K ] + [K ] − ω [M ] + [M ]){c }{ς } = {c }{F} = {F}, T l f s rf a l l T l (3.6) {} { } ~ ˆ where l = 1,2, K , M and F = clT {F} is the generalized exciting force By using the computed velocity potentials (Eqs 3.2a-b), the global matrices for the added mass ~ [M a ] and exciting force {F} can be calculated from [M a ] = ∫ S {φ~ }dS , {~} = iω {cˆ }∫ F HB T l l S HB {φ D }dS (3.7a) (3.7b) The generalised added mass is given as a complex matrix, and thus it includes the effect of radiation damping (Wang et al., 2008) 50 Method of Solution for Coupled Water-Plate Problem Step Solving for plate deflection Equation (3.6) is solved for the complex amplitudes {ς l } These complex amplitudes are the back-substituted into Eq (3.5) to get the plate deflection The stress resultants may be readily obtained from taking appropriate derivatives of the deflection and bending rotations as given in Eqs (2.13a-c) 3.3 Flow Field Analysis The wave flow field can be determined based on the velocity potential obtained from Step of Section 3.2 This flow field provides designers a good understanding of the wave elevations surrounding the floating structures Below, the method of solution for the flow field is described The diffracted and radiated potentials in the fluid domain [ φ R (x ) and φ D (x ) ] can be derived by rearranging the boundary integral equation given in Eq (3.2) (see Appendix D for the details) as follows w ∂G ~ G cl {ς l } + φR , 4π 4π ∂n {φ R } = iω { } {φ D } = −{φ In } + ∂G {~D }, φ ∂n { } { } (3.8a) (3.8b) { } ~ ~ where φ R and φ D are the radiated and diffracted velocity potentials distributed on the wetted surface of the floating structure obtained in Step of Section 3.2 By 51 Method of Solution for Coupled Water-Plate Problem summing the velocity potentials obtained from Eq (3.8), the wave elevation η e can then be derived from the dynamic free surface boundary condition given as ∂ {φ }, g ∂t {η e } = − 3.4 on S F (3.9) Steady Drift Force Analysis Based on the computed velocity potential and plate deflection, one can obtain the steady drift force of the floating plate The drift forces are important for the design of the mooring system Below, the method of solution for the steady drift force is presented The steady drift force acting on the floating plate is evaluated using the near-field method (or the pressure distribution method) In the near-field approach, all the second-order quantities in terms of the wave amplitudes have to be considered The wave forces acting on the floating body have to be evaluated at the instantaneous position of the floating body, as opposed to the mean position in the linear wave theory (Utsunomiya et al., 2001) We follow the numerical scheme given by Utsunomiya et al (2001) in evaluating the steady drift force The wave force F acting on the floating body is given by F=∫ SH ' P ' n' dS ' , (3.10) 52 Method of Solution for Coupled Water-Plate Problem where n = (n x , n y , n z ) is the unit normal vector pointing inward from the fluid domain into the floating body and prime (') indicates that the quantities are being evaluated at the instantaneous position of the body P ' is the pressure distribution on the wetted surface of the floating body S H ' derived from the Bernoulli equation given as follow 2 ∂φ ∂φ ∂φ P(x, y,−d ; t ) = ρωφ − ρgw − ρ + + ∂x ∂y ∂z (3.11) By applying the perturbation theory to Eq (3.10) and expanding the pressure P ' using Taylor’s series expansion, Utsunomiya et al (2001) obtained the expression for the total horizontal steady drift force Fd( ) as a sum of four drift force components _ _ (2 ) Fd = Fd(2 ) + Fd(2 ) + Fd(2 ) + Fd(2 ) , _ _ _ _ (3.12) where the overbar denotes the time-average of the steady drift forces during one wave period The steady drift force in Eq (3.12) is based on the assumption that the motion of the floating body is constrained in the horizontal direction The four steady drift force components in Eq (3.12) are given as follows (Pinkster, 1979; Utsunomiya et al., 2001) 53 Method of Solution for Coupled Water-Plate Problem • Component 1: Steady drift force due to the first order hydrostatic and hydrodynamic pressure, i.e (1) _ _ _ ∂φ (1) − ∇w (1) dS = − ρ ∫ gw + S HB ∂t _ ( ) Fd • Component 2: Steady drift force due to the velocity of the fluid _ (2 ) Fd • (3.13) _ n x = − ρ ∫ ∇φ dS n S HS y (3.14) Component 3: Steady drift force due to the product of the first order motion and the gradient of the first order pressure _ (2 ) d F • _ _ ∂ ∂φ (1) n x = − ρ ∫ w (1) dS S HS ∂z ∂t n y (3.15) Component 4: Steady drift force due to the relative wave height = ρg ∫ ζ r(1) CH _ (2 ) Fd nx dl n y (3.16) 54 Method of Solution for Coupled Water-Plate Problem where n x and n y are the unit normal in the x and y directions, respectively C H is the intersection of the mean wetted surface of the floating body and the plane z = () ζ r1 is the relative wave elevation defined as ζr = − ∂φ (1) − w (1) , g ∂t (3.17) where w (1) is the first order deflection of the plate obtained from Step of Section 3.2, φ (1) the first order velocity potential obtained from Step of Section 3.2 3.5 Concluding Remarks We have presented the solution technique to solve for the decoupled equations of motion for the plate and water by using the hybrid finite element-boundary element (FE-BE) method The governing equation of motion for the plate is solved using the finite element method where a modified non-conforming quadratic-serendipity (NCQS) Mindlin plate element is used for the first time in modelling the floating structure On the other hand, the governing equation for the water is solved using the boundary element method which involves the transformation of the Laplace equation together with the boundary conditions into a boundary integral equation (BIE) The BIE could then be solved for the velocity potential φ by using the constant panel method The computed potential φ is substituted into the equation of motion of the floating plate (Eq 2.11) as the hydrodynamic force and the equation is solved using the finite element method for the complex amplitudes ς l of the plate The 55 Method of Solution for Coupled Water-Plate Problem deflection w of the plate can then be obtained by taking the product of the modal functions clw of the freely vibrating plate and the complex amplitudes ς l The flow field surrounding the floating structure and the steady drift forces could then be computed once we have the deflection w and velocity potential φ obtained from the hydroelastic analysis The solution techniques presented in this chapter could be used to investigate the hydroelastic interactions of the two floating storage modules placed side-by-side towards the structural responses, wave runup along the channel and the steady drift forces acting on the floating modules Before we proceed to perform the hydroelastic analysis on the FFSF, the accuracy of the modified NC-QS Mindlin plate element (that is used to model the plate) in predicting the plate deflection and stress resultants will first be verified in the next chapter 56 ... the hydroelastic interactions of the two floating storage modules placed side-by-side towards the structural responses, wave runup along the channel and the steady drift forces acting on the floating. .. } { } (3. 8a) (3. 8b) { } ~ ~ where φ R and φ D are the radiated and diffracted velocity potentials distributed on the wetted surface of the floating structure obtained in Step of Section 3. 2 By... velocity of the fluid _ (2 ) Fd • (3. 13) _ n x = − ρ ∫ ∇φ dS n S HS y (3. 14) Component 3: Steady drift force due to the product of the first order motion and the