The weakly nonlinear approach is inconsistent but effective to reduce computational cost because nonlinear radiation and diffraction forces are missing. The nonlinear Froude–Krylov pressure is calculated by Taylor expanding of the incident wave potential about the calm water level as follows: equation(15) z<0ϕI=gAωekzsin(k(x+Ut)cosβ+kysinβ−ωt)0

nonlinear Froude–Krylov pressure works with an extension of restoring pressure, which is negative above the calm water level. The nonlinear pressure is integrated over the instantaneously wetted surface. The linear part of the dynamic pressure is obtained by dropping the terms related with the incident wave selleck kinase inhibitor potential from Eq. (14) as equation(17) pLD=−ρ(∂∂t−U¯⋅∇)(Φ+ϕd)+∇Φ⋅∇(12Φ+ϕd)The linear part is integrated over the mean body surface. For calculation of slamming forces, the ship is discretized into 2-D sections along the longitudinal axis, which covers the whole ship from stern to bow. The sections are perpendicular to the free surface of the calm water in Fig. 2. Longitudinal

mesh for each section is used to integrate slamming loads. Symmetric slamming forces acting on the sections are considered by either wedge approximation or GWM. Only water entry problem is considered. Asymmetric slamming forces for torsion and horizontal bending are not considered. learn more Wedge approximation is based on momentum conservation, which is expressed O-methylated flavonoid as equation(18) F=ddtMaḣ=Mah¨+∂Ma∂tḣThe relative displacement and velocity are calculated as follows: equation(19) ḣ=−∂u→∂t⋅(0,0,1)+∂ζI∂t

equation(20) h=−u→⋅(0,0,1)+ζI+DWedge approximation follows von Karman׳s solution with simplified wedge shapes. Once the surrounding flow is assumed as a potential flow, the infinite frequency added mass of the wedge is calculated as equation(21) Ma=π2ρb2(t)(1−γ2π) In case of GWM, the body geometry enters water with a vertical velocity shown in Fig. 3. Slamming pressure is limited to the water entry problem without flow separation. The space-fixed coordinate system is used, the origin of which is located at the intersection of the vertical axis of symmetry and the free surface of the calm water. The set of the initial value problem is expressed as follows (Zhao and Faltinsen, 1993, Korobkin, 2010 and Khabakhpasheva et al., 2014): equation(22) 2∇φ=0∇2φ=0 equation(23) φ=0(y=H(t)) equation(24) S(x,t)=φy(x,H(t),t)(|x|>c(t)) equation(25) φy=f′(x)φx−ḣ(t)(y=f(x)−h(t),|x|