Proceedings of the ASME th International Conference on Ocean, Offshore and Arctic Engineering OMAE2013 June 9-14, 2013, Nantes, France

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1 Proceedings of the ASME th International Conference on Ocean, Offshore and Arctic Engineering OMAE2013 June 9-14, 2013, Nantes, France OMAE APPLYING STRIP THEORY BASED LINEAR SEAKEEPING LOADS TO 3D FULL SHIP FINITE ELEMENT MODELS Chengbi Zhao Naval Architecture and Ocean Engineering, South China University of Technology, Guang Zhou, P.R.China Ming Ma Advanced Marine Technology Center, DRS Technologies Stevensville, MD, USA Owen Hughes Aerospace and Ocean Engineering Virginia Polytechnic Institute and State University Blacksburg, VA, USA, ABSTRACT Panel based hydrodynamic analyses are well suited for transferring seakeeping loads to 3D FEM structural models. However, 3D panel based hydrodynamic analyses are computationally expensive. For monohull ships, methods based on strip theory have been successfully used in industry for many years. They are computationally efficient, and they provide good prediction for motions and hull girder loads. However, many strip theory methods provide only hull girder sectional forces and moments, such as vertical bending moment and vertical shear force, which are difficult to apply to 3D finite element structural models. For the few codes which do output panel pressure, transferring the pressure map from a hydrodynamic model to the corresponding 3D finite element model often results in an unbalanced structural model because of the pressure interpolation discrepancy. To obtain equilibrium of an imbalanced structural model, a common practice is to use the inertia relief approach to rebalance the model. However, this type of balancing causes a change in the hull girder load distribution, which in turn could cause inaccuracies in an extreme load analysis (ELA) and a spectral fatigue analysis (SFA). This paper presents a method of applying strip theory based linear seakeeping pressure loads to balance 3D finite element models without using inertia relief. The velocity potential of strip sections is first calculated based on hydrodynamic strip theories. The velocity potential of a finite element panel is obtained from the interpolation of the velocity potential of the strip sections. The potential derivative along x-direction is computed using the approach proposed by Salvesen, Tuck and Faltinsen (1972). The hydrodynamic forces and moments are computed using direct panel pressure integration from the finite element structural panel. For forces and moments which cannot be directly converted from pressure, such as hydrostatic restoring force and diffraction force, element nodal forces are generated using Quadratic Programing. The equations of motions are then formulated based on the finite element wetted panels. The method results in a perfectly balanced structural model. An example is given to compare the ordinary strip theory to the proposed direct pressure integration method. The accuracy proves the validity of this new method. 1. INTRODUCTION With the advent of new ship types, designs of increasingly larger scale, combined with their respective commercial service and/or military mission requirements, interest in using seakeeping loads for ship structural design has increased dramatically in recent years. Tools ranging from simple 2D strip theories to complex 3D-CFD numerical simulation methods have been used for practical designs. For seakeeping analyses, the main objective is to predict ship motions, the global hull girder loads, such as sectional bending moments and shear forces, are a by-product of the seakeeping analyses and have not been given enough attention. As a three-dimensional finite element model has become a de facto standard for ship structural design, accurately transferring seakeeping loads becomes 1 Copyright 2013 by ASME

2 more important. For three-dimensional finite element models, the basic load vectors are the panel pressure, inertia forces and element nodal forces. The global hull girder loads reported in a structural analysis, such as vertical bending moment, vertical shear force, horizontal bending moment, horizontal shear force and longitudinal torsional moment, are derived from the integration of those basic loads. Ideally, the hull girder load distributions reported from a structural model should be in good agreement with the ones obtained from the seakeeping analysis, and be closed on both ends of the ship because the structural model is also in an equilibrium state. However, because meshes for hydrodynamic analyses are often much coarser than the corresponding finite element model meshes, the loads mapped to the structural models usually result in imbalanced models, even for tools using threedimensional panel methods which can output panel pressure directly. Often, the inertia relief method (ABS, 2010) has to be used for the final adjustment to balance the structural model. The inertia relief technique is simply to change the accelerations such that the structural model is forced to satisfy Newton s second law. The method is very powerful and can correct any imbalanced model. However, there are two shortcomings to this approach. First, the additional inertial forces cause a change in the hull girder response (such as bending moment). Second, the change of the accelerations has to be relatively small to ensure the fictitious inertial forces do not significantly distort the original structural response. This often requires a visual inspection and engineering judgment. For the extreme load analysis (ELA), where the number of load cases is limited, visually inspecting each load case is possible. But for the spectral fatigue analysis (SFA), where there are thousands of load cases, it is not practical. Zhao, Ma and Danese (2012) presented an approach to apply frequency domain linear seakeeping loads to 3D finite element models. To ensure a balanced structural model, two techniques were employed. 1) The equations of motion were formulated based on the structural mesh, rather than on the hydrodynamic mesh (Malenica et al., 2008). 2) The unrealistic hydrostatic restoring forces due to direct pressure integration were corrected using quadratic programming. For linear seakeeping analyses using strip theories, it is even more challenging to transfer seakeeping loads to structural models because usually only sectional forces and moments, but not the panel pressure, are available. To transfer the sectional loads to 3D finite element models, Ma, Hughes and Zhao (2012) have recently proposed a method using quadratic programming to convert the hull girder sectional loads to 3D finite element nodal forces. The approach assigns nodal forces to all wettable nodes of the structural model such that the integration of all nodal forces results in identical hull girder sectional loads. The method can adjust multiple sectional loads, such as vertical bending moment, vertical shear force, horizontal bending moment, horizontal shear force, and torsional moment simultaneously. While most strip theory methods use 2D sectional integration, some strip theory variations, such as VERES (2012), offer direct pressure integration approaches. The basic idea is to derive 3D panel potentials by interpolating 2D sectional velocity potentials. Once the panel potentials are obtained, the pressure can be calculated using Bernoulli s equation, and the forces and moments can be calculated by integrating panel pressures. The calculation of hydrodynamic forces and moments involves evaluating the potential derivative in the lengthwise x-direction, i.e.. One approach is to evaluate the derivative term using numerical differentiation. This approach, often called strip theory pressure integration method, seems to lack a strong theoretical basis because strip theory is essentially a 2D method. However, the advantage of using this approach is that it is easy to obtain panel pressures similar to 3D panel methods and therefore the hydrodynamic loads can be interfaced to a structural model. To avoid the evaluation of the derivative, Salvesen, Tuck and Faltinsen (1972) used Stokes theorem to convert the surface derivative integral to a line integral. With this approach, the wave diffraction force can be expressed as a function of radiation force and incident wave force. This approach, often called the STF method or ordinary strip theory method, is more rigorous in terms of mathematics, but it is difficult to apply loads to a structural model because it only outputs hull girder sectional forces and moments. Using different approaches often results in quite different response, especially on hull girder loads. To illustrate the problem, a container ship, S175, was computed using two methods provided in VERES (2012). One is the ordinary strip theory, i.e. sectional integration method. The other is the direct pressure integration method. The surface model and section model of S175 are shown in Figure 1 and 2 respectively. Fig. 1. S175 surface model 2 Copyright 2013 by ASME

3 Fig.2 S175 section model The heave motion RAO is shown in Figure 3. The longitudinal vertical bending moment distribution at wave period 11.5 second is shown in Figure 4. As can be seen, the results of both motion and load depend on the method chosen. The bending moment distribution with the pressure integration method is not closed at both ends, which means the model is not in equilibrium. The difference of the maximum bending moments of the two methods is over 20%. Fig. 3. S175 Heave RAO Fig. 4. Vertical BM longitudinal distribution In this paper, a hybrid approach is proposed to apply strip theory based frequency domain linear seakeeping loads to three-dimensional finite element structural models. The method combines the best of the ordinary strip theory and the pressure integration method. The potential derivative along the lengthwise x-direction is computed using the approach proposed by Salvesen, Tuck and Faltinsen (1972), who presented a thorough formulation of the theory, referred to as STF. The STF method and its variations have been the most widely used strip theories for the past 40 years. The hydrodynamic forces due to incident wave potential and radiation wave potentials are represented by panel pressure. The hydrodynamic forces due to diffraction wave potential and hydrostatic restoring force are given as both element panel pressure and elemental nodal forces using quadratic programming. The method not only has excellent agreement with the STF method, but also is able to apply panel pressure and element nodal forces to a finite element model. Numerical examples are given for the validation. 2. THEORETICAL BACKGROUND We assume that the fluid is homogeneous, inviscid and incompressible and the fluid motion is irrotational. It is also assumed that the motions are small. We consider a body that is submerged or floating on the surface of the fluid. As such, the ship motion problem may be formulated in terms of potential flow theory, thus the fluid velocity vector may be represented by the gradient of a total velocity potential, ψ, which is separated into two parts assuming a slender hull at slow forward speed: ψ(x, y, z, t) = Φ s (x, y, z) + φ u (x, y, z)e iωt (1) where Φ is a steady contribution due to forward speed U of the ship, and is an unsteady part associated with the incident wave system and the unsteady body motion. = -, is the frequency of encounter, which is related to the ship s speed U, the incident wave frequency, the wave number, and relative heading to the incident wave direction. With the assumption of small oscillations, the total unsteady velocity potential (, which varies during one oscillation) around the ship can be divided into a series of independent velocity potentials: an incident wave (,,, ) component, the diffracted wave (,,, ) component and the six radiation wave = (,,, ) components due to the six-degrees of ship motions. φ u e iωt = [ξ 0 (φ 0 + φ 7 ) + 6 =1 ξ φ ] e iωt (2) In the case of long crested, harmonic progressive waves the incident potential for infinitely deep water is defined as, φ 0 = i g k(z i cos β iy sin β) e ω 0 where = ω 0 g is the wave number. The steady velocity can be separated into different terms as well. It is common practice to assume that there is a base flow, either the double-body flow or the uniform flow, and a small steady disturbance potential which is linear in forward speed. Φ s (x, y, z) = x + φ(x, y, z) (3) 3 Copyright 2013 by ASME

4 The radiation and diffraction potentials must each satisfy the Laplace equation in the fluid domain and the following linear boundary conditions. 1. Steady perturbation potential φ φ + g x = (4) ( x + φ) = (5) 2. Incident and diffracted potential [(iω x ) + g ] (φ 0, φ 7 ) = (6) φ 7 = φ 0 (7) 3. Radiation potential (iω x ) φ + g φ = (8) φ = iω + m (9) where j=1,,6; n j is the generalized directional cosine with n = (n 1, n, n 3 ) T and r n = (n 4, n 5, n 6 ) T ; n is a unit normal vector outward to the wetted body surface and r= (,, ) T is a position vector of a point on the wetted body surface. The m-terms provide a coupling between the basis flow Φ s and the time dependent potential (m 1, m, m 3 ) T = (n ) Φ s (m 4, m 5, m 6 ) T = (n ) [r Φ s ] =r (m 1, m, m 3 ) T n Φ s For a slender body, the steady perturbation potential due to forward motion is negligible in the unsteady flow, i.e., Φ =-. We have (m 1, m, m 3, m 4, m 5, m 6 ) T = (,,,, n 3, n ) T The radiation potentials can be written into two parts φ = φ 0 + iω φ U Substituting the potential into equation (9) results in the two hull conditions φ 0 = iω φ U = iωm Now since both φ 0 and φ U must satisfy the Laplace equation and the same boundary conditions, it follows that φ U = for j=1,2,3,4 and that φ U 0 5 = φ 3 φ U 6 = φ 0 Thus, the radiation potential components can be expressed in terms of the speed-independent part of the potential, φ 0, as φ = φ 0 for j=1,2,3,4 φ 5 = φ iω φ 3 0 φ 6 = φ 6 0 iω φ 0 where φ 0 (j=1,,6) must satisfy the conditions j 0 n = iω on the mean hull position. The solution of the linearized unsteady forward motion is then obtained by means of the Green function for solving the boundary integral equation (BIE). Once the potentials are obtained, the pressure of the body surface on wave can be calculated by Bernoulli s equation. p = ρ ( ψ + gz) t + 2 ψ ψ 2 Ignoring higher order terms, the pressure at a position of the body can be expanded into a Taylor series around the mean position p u = ρ (iω x ) φ ue iωt ρg(ξ 3 + ξ 4 y ξ 5 x) (10) where within the accuracy of the linearization the pressure can be conveniently evaluated at the undisturbed position of the hull. The last term in equation (10) gives the ordinary buoyancy restoring force and moment. Integration of the pressure of the first term of (10) over the hull surface yields the hydrodynamic force and moment amplitudes: H = ρ (iω ) φ u j=1,,6 (11) The force and moment can be divided into two parts as H = X + G where X j is the exciting force and moment due to incident and diffracted potentials: X = ρ (iω x ) (φ 0 + φ 7 ) (12) and G j is the force and moment due to radiation potentials G = ρ (iω x ) φ k 6 = T k ξ k (13) k=1 Here T k denotes the hydrodynamic force and moment in the jth direction per unit oscillatory displacement in the kth mode: T k = ρ (iω ) φ k = ω a k iωb k (14) Equations of motion that govern the steady-state time-harmonic response of the body follow from the application of Newton s second law. H (t) = ω e iωt 6 M k k=1 ξ k j =,, (15) where ξ = ξ R + i ξ I is the motion RAO. M k is the body mass matrix. Combining equation (12) and (13), the response amplitude operator RAO can be determined by the linear equation system, 4 Copyright 2013 by ASME

5 6 [ ω (M k + a k) + iωb k + c k] ξ k = X (16) k=1 where j=1,..,6, and a k and b k are the added mass and damping coefficient which originate from the radiation potential. X is the exciting force due to the incident and diffracted wave potential. a k, b k a X are derived from the first term of (10) and depend on both the forward speed and the frequency of oscillation. The restoring coefficient, c k, is derived from the second term of (10). 2.1 STRIP THEORY ASSUMPTION Strip theory for the seakeeping of surface ships has been developed over the past 50 years and used with great success. It is a quasi two-dimensional theory in which the ship s hull is represented with a series of two dimensional cross-sectional stations or strips. All the potential flow calculations are computed independently in two dimensions, and the final results include the three-dimensional influence of forward speed. The method, often referenced as the ordinary strip theory, was first introduced by by Korvin-Kroukovski and Jacobs (1957) who presented a method for calculating heave and pitch motions of ship in regular waves. Two forces, one due to the hull motion and one due to the diffraction of the incident wave, can be represented using relative motion descriptions. The method was extended by Salvesen, Tuck and Faltinsen (1970; STF) who provided a thorough derivation of the method. From the 3D potential flow theory presented previously, assume: x y, z 1, 3 ω x Thus the free surface boundary condition used for the radiation potentials is ω φ + g φ = The two-dimensional radiation potential at each section, Ψ D k, is equal to the three dimensional potential φ 0 k for sway, heave and roll: φ 0 D k = Ψ k for k=2,3,4 In addition, from hull condition, we have for pitch and yaw: φ 0 D 5 = Ψ 3 φ 0 6 = Ψ D and φ 0 1 φ 0 k for k=2,,6 each cross section, the velocity potential at a point p(y,z) can be represented as: 2πΨ D D k = (Ψ G k G Ψ D k ) (17) s where G is the Green function. Straight line segments are used to approximate the control surfaces, and constant values of the velocity potential and its normal derivatives are assumed at each segment. It can be interpreted as the response of a system at a field point p(y, z) due to a delta function input at the source point q. Two types of Green functions, the free surface Green function and the simple Rankine source function, can be used to solve the problem. It can be shown that the free surface Green function satisfies the Laplace equation, the free surface boundary conditions and the condition at infinity. Only the body boundary conditions need to be solved. The fluid domain and the control surfaces are sketched in Figure 5. Figure 5: The fluid domain and control surfaces For the simple Rankine source function, there are five sets of boundary conditions to satisfy: these are the body surface, the free surface, the left control surface, the right control surface, and the bottom surface. The fluid domain and the control surfaces are sketched in Figure 6. Figure 6: The fluid domain and control surfaces 3.0 HYDRODYNAMIC FORCE AND MOMENT USING HYBRID PANEL PRESSURE INTEGRATION The 3D panel potentials can be obtained by using interpolation from 2D sectional potentials, as illustrated in Figure 7. Applying Green s second identity on twodimensional velocity potential on the boundary of 5 Copyright 2013 by ASME

6 Fig. 7 3D panel potentials The problem left is to determine the added-mass and damping coefficients, which involves evaluating φ k x In the STF approach, φ k = m φ x k C x φ k l Applying the above relationship to the equation ( 14) we have T k = ρiω φ k ρ + ρ m φ k C A φ k l where CA refers to the aftermost cross section of the ship. The added mass and damping coefficients are M a k = ρ ω Re ( niωφ kn S n M L M m nφ kn S n + n φ kn L n ) b k = ρ ω Im ( niωφ kn S n L M m nφ kn S n + n φ kn L n ) where M is the number of the wetted 3D panels, and L is the number of the line segment of the aftermost cross section. Discretized hydrodynamic force and moment due to the incident potential are h I = ρiω 0 φ 0 M = ρiω 0 n φ 0n S In the ordinary strip theory, the Froude-Krylov force before and after the section cuts is often ignored. With the panel pressure integration approach, the force and moment can be easily obtained by directly integrating over all 3D wetted panels, as illustrated in Figure 8. Fig. 8 Panels with only Froude-Krylov force and moment Discretized hydrodynamic force and moment due to the diffraction potential are h D = ρ (φ 0 iω φ U ) φ 0 + ρ iω φ φ 0 0 C A The first term can be obtained by integrating the potentials over all 3D wetted panels. The forces and moments introduced from the second term are first computed using sectional integration, then evenly distributed as the nodal forces to all 3D wetted panels. Discretized hydrostatic restoring force and moment are computed using Quadratic Programming (Zhao Ma, and Danese 2012). The added mass, damping coefficient, hydrostatic restoring force and exciting force and moment are constructed from the wetted panels of a three-dimensional finite element model. The response amplitude operator RAO can be determined by the equations of motion (16). With this approach, the equilibrium of the structural model is automatically satisfied because the equations of motion are constructed on the structural mesh. 4.0 NUMERICAL VALIDATION The S-175 container ship is well known because it was used by the ITTC (2010) to carry out a comparative numerical study of linear waveinduced motions and structural loads. The database that resulted from that study includes numerical results from many institutions, and also some experimental data. Table 1 lists the main particulars. Table 1: Main particulars of the S175 container ship Length between perpendiculars 175 m Breadth 25.4 m Depth 15.4 m Draught 9.5 m Displacement ton LCB (from AP) m Block coefficient Midship section coefficient 0.98 Total mass ton l 6 Copyright 2013 by ASME

7 XCG(from AP) m YCG(from centerline) 0 m ZCG(from baseline) 9.5m R xx R yy R zz Three hydrodynamic analyses are conducted for the comparison. For the first analysis, which is labeled as MAESTRO-Wave-2D-Green Function in the following result figures, the free surface Green function is used. For the second analysis, labeled as MAESTRO-Wave-2D-RK, the simple Rankine source function is used. The ship is advancing with constant speed v= knots with 4 different headings, =0, =120 and =180. All results are presented in a non-dimensional way using wave amplitude (A), wave number (k), encounter frequency ( ), water density ( ρ), gravitational acceleration (g), ship beam(b) and ship length between perpendiculars (L pp ) as given in Table 2: Fig. 10 Pitch RAO (β = 8, F r =.2 5) Fig. 11 Heave RAO (β =, F r =.2 5) Table 2: Non-dimensional parameter definition Translational motions (heave, sway and surge) x = x A Rotational motions (roll, pitch and yaw) θ = θ A The motion RAO results, along with the available experimental data from ITTC, are shown in Figures 9 to 14. Fig. 12 Pitch RAO (β =, F r =.2 5) Fig. 9 Heave RAO (β = 8, F r =.2 5) Fig. 13 Heave RAO (β = 2, F r =.2 5) 7 Copyright 2013 by ASME

8 5.0 A CASE STUDY Fig. 14 Pitch RAO (β = 2, F r =.2 5) The hydrodynamic pressure distribution at a wave period of 11 seconds is shown in Figure 15. The vertical bending moment distribution envelope for the head seas is given in Figure 16. Each cross section curve represents the bending moment RAO of the station. Figure 17 gives a comparison of vertical bending moment RAO at midship. To illustrate the complete procedure of applying strip theory based linear seakeeping pressure load for a structural analysis, including generating design waves, a full ship example is given in this section. A finite element model was provided by NAPA Ltd of a nominal frigate 150 meters long and displacing 4000 tons. The model was created using NAPA-Steel as a molded form structural model, as shown in Figure 18. The NAPA-Steel/MAESTRO interface program can generate a full ship finite element mesh in MAESTRO format from a molded form structural model with one click of a button. The generated finite element model has over 61,000 nodes and 125,000 elements, as shown in Figure 19. This interface program also automatically translates the compartment and wetted surface definitions as MAESTRO groups, as shown in Figure 20. In addition, the weight distribution, tank loads and floating condition defined in the NAPA hydrostatic module are also translated into MAESTRO. With a complete finite element mesh and load definition, the generated finite element model is ready for a linear static analysis without any additional manual editing. Fig. 15 Hydrodynamic pressure distribution at wave period=11 seconds Fig. 18 NAPA-Steel molded form model Fig. 16 Vertical bending moment RAO envelope Fig. 19 NAPA-Steel generated finite element model Fig. 20 Wetted surface and internal tank definition Fig. 17 Vertical bending moment RAO (F r =.2 5) Once the finite element model and the weight distribution are imported (or constructed), MAESTRO will first perform a hydrostatic balance to obtain the mean draft plane and to identify the wetted surface elements. Next, a strip theory based seakeeping analysis is performed by executing MAESTRO-Wave. The calculations are done for a forward speed of 20 knots, 7 different headings, and 8 Copyright 2013 by ASME

9 30 frequencies. A unit wave response database is generated. The database includes wave-induced accelerations, panel pressures, nodal forces and the cross sectional hull girder loads. The unit wave pressure distribution, ship motion RAOs, and hull girder load RAOs can be displayed for sanity checks. Figure 21 shows the hull girder vertical bending moment RAO of all sections. The closure of the bending moment for all frequencies verifies that the model is indeed in equilibrium. Note that the hull girder loads reported in MAESTRO, such as vertical bending moment, vertical shear force, horizontal bending moment, horizontal shear force and longitudinal torsional moment, are derived from the basic loads such as panel pressure, nodal force and inertia force, and they are computed solely for the purpose of verifying the integrity of individual load components. A design wave is generated based on a desired dominant load parameter (DLP), a sea state diagram and the return period. The dynamic design wave, which has a perfect equilibrium based on the presented method, combined with the static loads, becomes a regular static analysis load case. Figures 22 to 27 show the combined dynamic and static panel pressure, hull girder bending moment and shear force distribution, and the deformation responses of the model under sagging and hogging loads. Fig. 24 Structural response on hogging wave Fig. 25 Sagging wave pressure distribution Fig. 26 Vertical bending moment distribution on sagging wave Fig. 21 Vertical bending moment RAO and unit wave pressure distribution Fig. 22 Hogging wave pressure distribution Fig. 23 Vertical bending moment distribution on hogging wave Fig. 27 Structural response on hogging wave 6.0 CONCLUDING REMARKS This paper presents a method to apply strip theory based linear seakeeping pressure loads to 3D finite element models. The velocity potentials of each cross section are first computed using 2D boundary integral equations. The 3D panel potentials are obtained by linear interpolation between cross sections. The potential derivative along the x- direction is computed using the STF method. The hydrodynamic pressure and forces are calculated at the panel level. For forces which do not have direct pressure conversion, such as force and moment generated by hydrostatic restoring component and wave diffraction component, additional nodal forces are generated using Quadratic Programming. The equations of motions are constructed in the structural mesh and equilibrium is guaranteed. The proposed direct pressure integration method is in excellent agreement with the STF sectional approach (VERES, 2012). 9 Copyright 2013 by ASME

10 8. ACKNOWLEDGMENTS The authors thank Tomi Holmberg of NAPA Group for providing the finite element model presented in this paper. REFERENCES AMERICAN BUREAU OF SHIPPING, (2010), Guide for Dynamic Loading Approach for floating production, storage and offloading (FPSO) installations ITTC (2010), Comparative study on linear and nonlinear ship motion and loads, ITTC workshop on seakeeping, Seoul, Korea. Korvin-Kroukovsky, B.V and Jacobs, W.R, 'Pitching and heaving motions of a ship in regular waves', Trans. SNAME, Vol.65, MA M., HUGHES O., ZHAO CB. (2012), Applying sectional seakeeping loads to full ship structural models using quadratic programming, Proc. of ICMT2012, Harbin, China MAESTRO Version (2012). Program documentation, Advanced Marine Technology Center, DRS Technologies Inc., Stevensville, MD, USA. MALENICA S., STUMPF E., SIRETA F.X. & CHEN X.B. (2008), Consistent hydro-structure interface for evaluation of global structural responses in linear seakeeping., Proc. of OMAE08, Estoril, Portugal Salvesen, N., Tuck, E.O. and Faltinsen, O., 1970, Ship Motions and Sea Loads, Trans. SNAME, Vol. 78, ZHAO CB., MA M., DANESE, N. (2012) A method of applying linear seakeeping panel pressure to full ship structural models, Proc. of COMPIT2012, Liege, Belgium VERES Version (2012), ShipX Vessel Responses, MARINTEK, Trondheim, Norway 10 Copyright 2013 by ASME