Hydroelastic responses and interactions of mega floating fuel storage modules 2

CHAPTER TWO Problem Definition and Mathematical Formulation 2.1 Overview In this chapter, we will present the numerical model, assumptions and governing equations used for solving the hydroelastic analysis of a floating fuel storage facility The hydroelastic analysis could be solved by using the time domain or frequency domain approach The time domain approach is useful for solving aircraft landing or wave impact problems on VLFS For instance, Kim and Webster (1998), Watanabe et al (1998) and Kashiwagi (2000, 2004) have investigated the transient response using the time domain approach but the analysis is time consuming as the response of the structure needs to be evaluated at every time-step On the other hand, the frequency domain approach is computationally efficient and proven to predict good results particularly for long term response (Chakarbarti, 1988) Usually the frequency domain approach is used instead of the time domain approach when determining the hydroelastic response of the floating structure because of its simplicity and ability to capture the pertinent response parameters in a steady state condition Hence, we will employ the frequency domain approach in our hydroelastic model 29 Problem Definition and Mathematical Formulation We consider the water and the structure to oscillate in a steady-state harmonic motion with the circular frequency ω The water is modeled using the linear potential theory and the floating structure is modeled as solid plate by using the Mindlin plate theory The detail numerical model and mathematical formulations for the equation of motions of the fluid and plate are shown in the subsequent sections 2.2 Problem Definition Consider a floating fuel storage facility (FFSF) comprising of two box-like floating fuel storage modules (with each module having the dimensions of L2 × L1 × hm ) being placed side-by-side at a distance s apart as shown in Fig 2.1 The storage modules are protected by floating breakwater with dimensions L3 × L4 × hb placed in the front The FFSF is subjected to an incoming wave of wave period T and a wave height A that impacts the structure at a wave angle θ with respect to the negative x-axis The seabed is assumed to be dredged and leveled and the water depth is H The top, bottom and side wall thicknesses t s of the floating storage module are assumed to be the same The storage modules may have different drafts d m depending on the amount of fuel stored in them 30 Problem Definition and Mathematical Formulation y Floating storage module Floating storage module 2L2 Incoming wave 2L4 x s 2L1 2L1 Floating breakwater 2L3 (a) (b) Fig 2.1 Figure depicting the box-like floating fuel storage modules and breakwater in (a) plan view (b) side view 2.3 Pictorial Description of Numerical Model The floating storage module is modelled as an equivalent solid plate for the hydroelastic analysis The cross section of the equivalent solid plate is shown in Fig 2.2 The symbols ∆ , ∆ and ∆ denote the plate domain for the storage module 1, 31 Problem Definition and Mathematical Formulation module and the floating breakwater, respectively The plates are assumed to be perfectly flat with free edges The water domain is denoted by The symbols S F , S B , S HB , S HS and S ∞ represent the free surface, the seabed, the wetted bottom surface of the floating body, the wetted side surface of the floating body and the artificial boundary at infinity, respectively The free and undisturbed water surface is at z = Fig 2.2 Schematic diagram depicting cross sectional of equivalent solid plate In order to simplify and parameterise the governing equations of the fluid-structure interaction problems, we shall non-dimensionalise the spatial and time variables in the mathematical formulations of the fluid-structure interaction problems presented next 2.4 Assumptions, Modelling and Formulation of Governing Equations for Water Motion The water is assumed to be a perfect fluid with no viscosity and incompressible and the fluid motion to be irrotational Based on these assumptions, the fluid motion 32 Problem Definition and Mathematical Formulation may be represented by a velocity potential φ (x, y, z ) , and the velocity of the fluid could be obtained from φ (x, y, z ) as follows Vx = Vy = Vz = ∂φ ( x, y, z ) ∂x (2.1a) ∂φ ( x , y , z ) ∂y (2.1b) ∂φ ( x, y, z ) ∂z (2.1c) where Vx , Vy and Vz are respectively the velocity in the x , y , and z directions It is also assumed that the fluid motion is small so that the linear potential theory can be applied for formulating the fluid-motion problem The single frequency velocity potential φ (x, y, z ) of the water must satisfy the Laplace equation (Lamb, 1932; Wehausen and Laitone, 1960) ∇ 2φ ( x, y, z ) = 0, in (2.2) where ∇ = ∂ (•) ∂x + ∂ (•) ∂y + ∂ (•) ∂z is the three dimensional Laplace operator The velocity potential φ (x, y, z ) must also satisfy the following boundary conditions: i Boundary condition on the seabed S B , which expresses impermeability (i.e no fluid enters or leaves the seabed and hence the velocity component normal to the seabed is zero) 33 Problem Definition and Mathematical Formulation ∂φ ( x, y, z ) = 0, ∂z (2.3) on S B ii Boundary condition on the hull surface S H , which denotes that the velocity of the wave particles at the wetted surface of the floating body must be equivalent to the velocity of the structure motion Vn , where n denotes the unit normal to the wetted surface This means that the floating structure must always be in contact with the water (i.e no air gap exists between the floating structure and the water) − iω w(x, y ) on S HB ∂φ (x, y, z ) = Vn =  , on S HS ∂n  (2.4) iii Boundary condition on the free surface S F , which denotes that the velocity of the fluid particle must be equal to the velocity of the free surface and the pressure on the free surface must always be constant ∂φ ( x, y, z ) = ω 2φ ( x, y, z ), ∂z on S F (2.5) iv The Sommerfeld radiation condition at the artificial fluid boundary at infinity S ∞ , which denotes that the wave must radiated away from the floating bodies and behaves like the incident wave φ In ( x, y, z ) at S ∞ (Sarpkaya and Issacson, 1981) 34 Problem Definition and Mathematical Formulation lim x →∞  ∂  x − ik [φ ( x, y, z ) − φ In ( x, y, z )] = ∂x    on S ∞ , (2.6) where x = (x, y ) and i = − The wave number k satisfies the dispersion relationship (Lamb, 1932) k tanh(kH ) = ω (2.7) and φ In (x, y, z ) is the incident velocity potential given by (Sarpkaya and Isaacson, 1981) φ In (x, y, z ) = A cosh (k (z + H )) ik ( x cosθ + y sin θ ) e , ω cosh kH (2.8) where A is the wave amplitude 2.5 Assumptions, Modelling and Formulation of Governing Equations for Structure Motion The box-like floating structure is commonly modelled as an equivalent solid plate (see Section 2.3, Fig 2.2) The sloshing effect due to the fuel in the storage modules is neglected in the study, but the fuel load is taken into consideration The lengths and the height of the plate are kept the same as the actual structure (Fig 2.1) but the Young modulus E and the Poisson ratio ν of the equivalent solid plate are tweaked to match the natural frequencies and vibration modes of the actual 35 Problem Definition and Mathematical Formulation structure The equivalent plate is assumed to be perfectly flat with free edges The plate material is commonly assumed to be isotropic or orthotropic and obeys Hooke’s law For the hydroelastic analysis, most researchers modelled the plate according to the classical thin plate theory More recently, the adoption of the Mindlin plate theory for equivalent floating plate is getting popular due to the theory’s ability to provide a better prediction of the stress resultants and its allowance for the effects of transverse shear deformation and rotary inertia Unlike the classical thin plate theory where the equations of motion are described solely by the deflection w(x, y ) variable, the motion of the Mindlin plate is represented by the vertical deflection w(x, y ) , the rotation about the y-axis ψ x (x, y ) and the rotation about the x-axis ψ y ( x, y ) The additional rotation variables allow for the effect of transverse shear deformation and also reduce the order of derivatives needed for computations of the stress resultants In the hydroelastic analysis, the plate is assumed to be restrained in the x-y plane by the station keeping system and the plate can only deform in the vertical direction (i.e z-direction) Thus, when one uses the finite element method for the plate analysis, the nodes of the elements are constrained from moving in the x-y direction and are only allowed to move in the vertical z-direction The vertical deflection w is measured from the free and undisturbed water surface as shown in Fig 2.3 36 Problem Definition and Mathematical Formulation Fig 2.3 Schematic diagram depicting deflection of plate with respect to water surface Mindlin (1951) used the variational approach for deriving the governing equation of motion for the first-order shear deformable plates (or commonly known as Mindlin plates) which includes the effect of rotary inertia as well The detail mathematical formulation of the governing equation is given in Appendix A The governing equations in time-harmonic motion for an isotropic plate are given by (Liew et al., 1998; Reddy, 2007)  ∂ w ∂ w   ∂ψ x ∂ψ y κ Gh  +  +  +  ∂y ∂y   ∂x  ∂x       + ρ ' hω w = − P ,    D(1 − ν )  ∂ 2ψ x ∂ 2ψ x   ∂x + ∂y 2   D(1 + ν )  ∂ 2ψ x ∂ ψ y  + +   ∂x 2 ∂x∂y   2 D(1 − ν )  ∂ ψ y ∂ ψ y  +  ∂x 2 ∂y   D(1 + ν )  ∂ 2ψ x ∂ 2ψ y +  +   ∂y∂x ∂y   (2.9a)   − κ Ghψ x + ∂w     ∂x    , (2.9b) ρ'h =− ω ψx 12   − κ Ghψ y + ∂w      ∂y    (2.9c)  ρ' h3 =− ω ψy 12 [ ( where (x, y ) ∈ ∆ , G = E / [2(1 + ν )] is the shear modulus, D = Eh / 12 − ν )] the flexural rigidity, ρ ' the mass density of the plate The right hand sides of Eqs (2.9b-c) 37 Problem Definition and Mathematical Formulation are the rotary inertia terms whereas the shear correction factor κ (usually taken as 5/6) is introduced to compensate for the error due to the assumption of a constant shear strain (and thus constant shear stress) through the plate thickness that violates the zero shear stress condition at the free surface (Liew et al., 1998; Reddy, 2007) The hydrostatic and hydynamic pressure P acting on the bottom of the structure (i.e z = ) are given by the linearised Bernoulli equation, i.e P(x, y,−d ) = iωφ (x, y,−d ) − w(x, y ), (2.10) Equation (2.9) may be expressed in the following compact matrix equation {} ˆ ˆ ˆ β ([B1 ] + [B2 ]){w} + γω [B3 ]{w} = − f , (2.11) where β = D /( ρgL4 ) is the stiffness constant, γ the specific gravity of the plate, ˆ {w} = {w ψ x ψ y } the generalised displacement vector, the forcing vector {fˆ}= {iωφ - w T 0} , [B1 ] and [B2 ] are the differential operators T  ∂2 ∂2 +  ∂x ∂y  6κ (1 −ν )  ∂ [B1 ] = −  h ∂x  ∂  − ∂y   ∂ ∂y   −1 ,   − 1   ∂ ∂x (2.12a) 38 Problem Definition and Mathematical Formulation  0 0  2  +ν  ∂        [B2 ] = 0 −ν  ∂ + ∂  + (1 +ν )  ∂  ∂y   ∂x  ∂x   ∂x∂y           ∂2   ∂2  +ν ∂ +ν −ν   + + 0  ∂x∂y    ∂x   ∂y         ,   ∂     ∂y    (2.12b) and finally [B3 ] is the matrix of constant multipliers  h  [B3 ] = 0 h  12  0    0    3 h  12   (2.12c) Based on the strain-displacement relations and assuming a plane stress distribution in accordance with Hooke’s law, the boundary conditions at the free edges are given by (Liew et al., 1998) ∂ψ s   ∂ψ = 0, Bending moment M nn = D  n + ν ∂s   ∂n   −ν Twisting moment M ns = D    ∂ψ n ∂ψ s  + =0,  ∂n    ∂s   ∂w  Shear force Qn = κ Gh  + ψ n  = ,  ∂n  (2.13a) (2.13b) (2.13c) 39 Problem Definition and Mathematical Formulation where s and n denote the tangential and normal directions to the section of the plate Decoupling of Governing Equations using Modal Expansion Method 2.6 The plate equation (2.11) indicates that the response of the plate w(x, y ) is coupled with the fluid motions (or velocity potential φ (x, y, z ) ) On the other hand, the fluid motion can only be obtained when the plate deflection w(x, y ) is specified in the boundary condition (Eq 2.4) In order to decouple this interaction problem into a hydrodynamic problem in terms of the velocity potential and a plate vibration problem in terms of the generalised displacement, we shall adopt the modal expansion method as proposed by Newman (1994) In this method, the deflection of the plate w(x, y ) is expanded by a series of the products of the modal functions clw and their corresponding complex amplitudes ς lw (x, y ) , w(x, y ) = ∑ clwς lw (x, y ) , M (2.14) l =1 where M denotes the total number of modes taken in the plate analysis As the problem is linear, the total velocity potential can be represented by a linear superposition of the diffracted part φ D (x, y, z ) and the radiated part φ R ( x, y, z ) , where φ D (x, y, z ) is computed from the sum of the incident wave φ In (x, y, z ) and scattered wave φ S ( x, y, z ) By using the modal expansion method, the total velocity potential φ (x, y, z ) may be expressed as (Eatock Taylor and Waite, 1978) 40 Problem Definition and Mathematical Formulation φ ( x, y, z ) = φ D (x, y, z ) + φ R (x, y, z ) = φ D (x, y, z ) + ∑ ς lφl φ l (x, y, z ) , M (2.15) l =1 where φ l =1, 2, ,M is the radiation potential corresponding to the unit-amplitude motion of the l-th modal function Note that the complex amplitudes ς lφl in Eq (2.15) are assumed to be the same values as ς lw in Eq (2.14) (Newman, 1994) By substituting Eqs (2.14) and (2.15) into the Laplace equation (2.2) and the fluid boundary conditions (Eqs 2.3-2.6), we arrive at the following decoupled governing equation and boundary conditions for each of the unit-amplitude radiation potentials (i.e for l = 1,2,…,M) and the diffraction potential (i.e for l = D) ∇ 2φl = 0, in , (2.16) ∂φl = 0, ∂z on S B , (2.17) ∂φl − iω clw (x, y ) = ∂n  ∂φl = 0, ∂z x →∞ on S HB , (2.18) on S HS , ∂φl = ω 2φl ( x, y, z ), ∂z lim for l = 1,2, K, M , for l = D (2.19) (2.20) on S F ,  ∂  x − ik (φ − φ In ) = 0, ∂x    on S ∞ (2.21) The boundary value problem for each of the unit-amplitude radiation potential and diffracted potential are defined by Eqs (2.16) to (2.21) in an uncoupled form 41 Problem Definition and Mathematical Formulation This boundary value problem could be solved by using the boundary element method in order to compute the velocity potential The water-plate equation (2.11) can then be solved by using the finite element method once we have the velocity potential 2.7 Concluding Remarks We have presented the numerical model, assumptions and governing equations used for solving the hydroelastic analysis of a floating fuel storage facility We assumed that the water to be a perfect fluid with no viscosity and incompressible and the fluid motion is irrotational, so that the velocity potential φ (x, y, z ) exists The velocity potential φ (x, y, z ) must satisfy the Laplace equation (2.2) and the boundary conditions on the computational boundaries (Eqs 2.3 to 2.6) On the other hand, the floating structures are modeled as solid plates with free edges by using the Mindlin plate theory (Eq 2.11) that takes into account the transverse shear deformation and rotary inertia To obtain the deflection w of the floating plate, we need to solve the equation of motion for the plate (Eq 2.11) that unfortunately involves the fluid velocity potential φ (x, y, z ) On the other hand, in order to determine the velocity potential φ (x, y, z ) , we need to solve the Laplace equation (2.2) which in turn has a surface boundary condition in contact with the floating plate that contains the unknown deflection w(x, y ) Hence, we are faced with a coupled water-plate problem To decouple these coupled differential equations, we employed the modal expansion method By using the modal expansion method, we 42 Problem Definition and Mathematical Formulation take a series of modal functions clw ( x, y ) from a free vibration analysis of the plate and substitute them into the boundary condition associates with the water surface and plate surface interface (Eq 2.4), thereby removing the unknown deflection w Hence, we obtained the decoupled boundary value problem for each of the unitamplitude potential defined by Eqs (2.16) to (2.21) The boundary value problem could be solved by using the boundary element method to obtain the velocity potential φ (x, y, z ) Upon solving the velocity potential, the plate deflection w could be obtained by solving the water-plate equation using the finite element method In the next chapter, we will present the detail solution techniques of the decoupled water-plate equation (Eq 2.11) and the Laplace equation (2.16) together with its boundary conditions (Eqs 2.17 to 2.21) by using the hybrid finite elementboundary element (FE-BE) method A flow chart is given to show the algorithm of the hydroelastic analysis program 43 ... amount of fuel stored in them 30 Problem Definition and Mathematical Formulation y Floating storage module Floating storage module 2L2 Incoming wave 2L4 x s 2L1 2L1 Floating breakwater 2L3 (a)... sections 2. 2 Problem Definition Consider a floating fuel storage facility (FFSF) comprising of two box-like floating fuel storage modules (with each module having the dimensions of L2 × L1 ×... (a) (b) Fig 2. 1 Figure depicting the box-like floating fuel storage modules and breakwater in (a) plan view (b) side view 2. 3 Pictorial Description of Numerical Model The floating storage module