PSI - Issue 48

Nurman Firdaus et al. / Procedia Structural Integrity 48 (2023) 58–64 Firdaus et al. / Structural Integrity Procedia 00 (2023) 000–000

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∇ . ∇= 0

(2) In solving Laplace's equation several boundary conditions are needed. Boundary conditions must match the physical environment of the oceans to describe realistic results. The relevant boundary conditions for solving fluid problems with potential theory are the boundary conditions on the free-surface elevation  . The kinematic and dyanmic free-surface boundary conditions can be formulated in the following equation (Xu et al., 2019),    ݐ −    ݖ += 0 for z=η, the kinematic free-surface (3) g η+ ∂ ∂ ϕ t + 1 2 ∇ϕ 2 = 0 for z=η, the dynamic free-surface (4) ∂ ∂ ϕ z = 0 for z=η, condition on the seabed (5) The conditions for the position of body boundary  and initial time domain problem are as follows,   ϕ  = (6) ϕ= 0, ( t < 0 )    ݐ = 0 ݐ( = 0, ݖ =η) (7) (Pan, 2022) described regarding on the mathematical equation of a BEM model to accomplish potential flow matter, which the calculating of the added mass coefficients  , the damping coefficients b , and the wave excitation force  can be summarized based on potential velocity and body boundary conditions as follows  ݆ − i   ݆ =     ݆  ׭ +     Δ݆  ׭ , ( ,݆ = 1~6 ) (8) ݆ =−   ݆   + 7   ׭ −   ݆ Δ 7   ׭ , (݆ = 1~6 ) (9) where  is the water density,  are the incident wave frequency,   and   are the undisturbed position and the thin shell surface of the body boundary condition and  is the position vector in Cartesian coordinate system, for the source point. 3. Mathematical equation of motion A simplified analysis of frequency-domain method adopted to review a characteristic of platform spar-FOWT are summarized below. The application of this approach method is used to solve non-linear problems, all aspects of non-linearity are changed in a linear form (Low & Langley, 2006) . The frequency-domain derived by the linear differential motion equation based on the assumption that the excursion of floating body is harmonic with small amplitude around the equilibrium position. The general equation of motion in the analysis of the frequency-domain method can be written as follows (Firdaus, Budi Djatmiko, et al., 2021): +() .  � + () .  � + . = (10) where  is the body mass and inertia and  is the hydrostatic stiffness of system. To obtain the transfer function or so-called Response Amplitude Operator from the frequency-domain analysis, we perform to analyze the behavior of dynamic motion for platform spar-FOWT due to harmonic incident wave. According to (Journee & Massie, 2001) , the formulation of RAO body motion is derived with assumption that the wave-induced forces are sinusoidal and the response motion of the floating body only come from wave-induced exciting. In the following the complex notation   ݐ and wave forces in the time domain and the body response in various directions are written as follows, =   ݋  ݐ + ሬԦ  ,  =    .   ሬԦ  ,  .   ݐ (11)