Effect of Mesh Number on Accuracy of Semi Submersible Motion Prediction

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1 RAO (motion) Jurnal Teknologi Full paper Effect of Meh Number on Accuracy of Semi Submerible Motion Prediction Haan Abyn a*, Adi Maimun b, M. Rafiqul Ilam c, Allan Magee d, Jawar a, Behnam Bodaghi a, Mohamad Pauzi b, C. L. Siow a a Department of Aeronautic, Automotive and Ocean Engineering, Faculty of Mechanical Engineering, Univeriti Teknologi Malayia, 8131 UTM Johor Bahru, Johor, Malayia b Marin Technology Center, Univeriti Technologi Malayia, 8131 UTM Johor Bahru, Johor, Malayia c Bangladeh Univerity of Engineering and Technology, Dhaka, Bangladeh d Technip Malayia *Correponding author: abynhaan@gmail.com Article hitory Received :1 Augut 213 Received in revied form : 1 November 213 Accepted :28 November 213 Graphical abtract Abtract Generally, floating tructure play an important role for exploring the oil and ga from the ea. The force and motion prediction of offhore tructure may be carried out by uing time domain or frequency domain or model tet. In thi paper the frequency domain analyi ued becaue it i the implified and linearized form of the equation of motion. Motly in numerical calculation the number of mehe play an important role in the accuracy of reult, time of calculation and facing to computer memory limitation. The 3D ource ditribution panel method i hown to be enitive to meh near the reonance frequencie of the floating body. So, it i important to etablih bet practice and determine the meh requirement for a given level of accuracy. The reult obtained from numerical commercial oftware HydroSTAR (Htar) on emi ubmerible prove it. Keyword: Semi ubmerible; floating tructure; 3D ource denity ditribution technique; potential theory; meh generation 214 Penerbit UTM Pre. All right reerved. 1. INTRODUCTION All of fixed, floating and moored tructure uch a hip, emiubmerible, FPSO, TLP and other are ubjected to wave, wind and current at ea. They have ix-coupled degree of freedom of motion. Namely, linear and angular motion are urge, way, heave, roll, pitch and yaw. Ocillating of floating tructure affect the loading and offloading operation ytem. They may experience reonant motion, which hould be avoided a much a poible under intallation, operation and urvival condition. In particular, the vertical plane motion induced by heave, roll and pitch of a floating tructure hould be kept adequately low to guarantee the afety of rier and umbilical pipe a the mot important component in the equipment of oil production. There are different theorie for tudying motion of floating tructure uch a trip theory and potential theory. 3D ource denity ditribution technique i ued to get the potential over the floating tructure by many reearcher and oftware. Having flow velocity potential on and off the panel, hydrodynamic coefficient of floating tructure can be determined. Uing Bernoulli equation lead to calculation of preure ditribution and force over the floating tructure. A numerical model i a mathematical tructure which can be ued to decribe and tudy a real ituation. A econd-order linear differential equation for coupled ix degree of freedom can decribe the hydrodynamic of floating tructure; conit of added ma, damping coefficient, tiffne coefficient, force and motion in ix direction. He and Smith 1 tudied on non-lifting potential flow calculation about arbitrary 3D object. They utilized a ource denity ditribution on the urface of the tructure and olved for ditribution neceary to lake the normal component of the fluid velocity zero on the boundary. Plane quadrilateral ource element were ued to approximate the tructure urface, and the integral equation for the ource denity i replaced by a et of linear algebraic equation for the value of the ource denity on the quadrilateral element. By olving thi et of equation, the flow velocity both on and off the urface wa calculated. Wu, et al. 2 tudied on the motion of a moored emi ubmerible in regular wave and wave induced internal force numerically and experimentally. In their mathematical formulation, the moored emi ubmerible wa modeled a an externally contrained floating body in wave, and derived the linearized equation of motion. Yilmaz 3 analyzed the exceive motion of moored emi ubmerible. They developed and employed two different time domain technique a due to mooring tiffne, vicou drag force 66:2 (214) eissn

2 68 Haan Abyn et al. / Jurnal Teknologi (Science & Engineering) 66:2 (214), and damping; there are trong nonlinearitie in the ytem. In the firt technique, firt-order wave force acting on tructure conidered a a olitary excitation force and evaluated according Morion equation. In their econd technique, they ued mean drift force to calculate lowly varying wave force and imulation of lowly varying and teady motion Söylemez 4 developed a technique for prediction of damaged emi ubmerible motion under the wind, current and wave. He ued Newton econd law for reolving equation of motion and developed numerical technique of nonlinear equation for the intact and damaged condition in time domain. Clau, et al. 5 analyzed numerically and experimentally the ea-keeping behavior of a emi ubmerible in rough wave in the North Sea. They ued panel method TiMIT (Time-domain invetigation, developed at the Maachuett Intitute of Technology) for wave/tructure interaction in time domain. The theory behind TiMIT i trictly linear and thu applicable to moderate ea condition only. Newman 6 carried out convergence tudie uing WAMIT in the frequency domain for repreentative floating bodie uing different dicretization cheme. An important requirement for a unit with drilling capabilitie i the low level of motion in the vertical plane (motion induced by heave, roll and pitch. Mato, et al. 7 numerically and experimentally invetigated Second-order reonant of a deep-draft emi-ubmerible heave, roll and pitch motion. One of the manner to improve the hydrodynamic behavior of a emi-ubmerible i to increae the draft. The low frequency force computation ha been performed in the frequency domain by WAMIT a commercial Boundary Element Method (BEM) code. They generated a different number of mehe on the tructure and calculated pitch force ( Figure 7). Thi tudy focue on vertical motion of GVA 4 emi ubmerible which i characterized by favorable ea-keeping behavior and calculate motion of a body at Head and Beam Sea for different number of mehe. 2. TEORY AND MATHEMATICAL MODELING The individual emi ubmerible i treated a a rigid body with ix degree of freedom ( Figure 1). It i ubjected to hydrodynamic force due to ea incident wave, radiated wave by itelf and diffracted wave due to itelf a well a bodie. Two right hand coordinate ytem are defined. One i fixed to the pace on water urface and the other one i fixed to the center of gravity. The fluid i aumed to be incompreible, invicid and irrotational and the veel i aumed to be floating in open water. Then velocity potential with boundary condition of the tructure, water free urface, ea bottom and far filed are conidered to atify the Laplace equation. Time dependence of fluid motion i limited to imple harmonic motion, o the flow field can be characterized by the following velocity potential: Φ = Re[φ(x, y, z)e iωt Figure 1 Semi ubmerible wetted urface φ = iω [(φ + φ 7 ). ζ a + 6 j=1 (X j ϕ j )] (2) φ = igζ a ω (1) coh[k(z + h)] e ik(xcoα+yinα) (3) coh kh The differential equation governing the fluid motion follow from the application of the continuity equation which yield the Laplace equation. The individual potential are the olution of the following Laplace equation: 2 φ = (4) 2.1 Boundary Condition In the fluid domain bounded by the mean wetted urface area of body S, the above linear velocity potential mut atify the Laplace equation and alo the following boundary condition: -linearized free urface condition: φ + ω2 z g φ z φ =, at z= (5) -boundary condition on the ea floor: = on z = h (6) Another boundary condition i the wetted urface of the floating tructure. Becaue of linearization, mentioned boundary condition may be applied on the wetted urface of the floating tructure in it equilibrium poition φ o + φ 7 n n =, n S (7) φ j n = iωn j, on S (8) In which nj i the direction coine on the urface of the body in the j-th mode of motion and ha the following form: n 1 = co(n, x), n 2 = co(n, y), n 3 = co(n, z) n 4 = (y y G )n 1 (z z G )n 2 n 5 = (z z G )n 1 (x x G )n 3

3 69 Haan Abyn et al. / Jurnal Teknologi (Science & Engineering) 66:2 (214), n 6 = (x x G )n 2 (y yz G )n 1 The radiation condition of the potential φ j, in which in polar co-ordinate: lim r (r1 2( φ r iω2 φ) = (9) g 2.2 Velocity Potential However, there i no analytical olution for φ 7 and φ i, o the problem hould be olved numerically. According to the 3-D ink ource method, the potential φ 7 and φ i can be expreed in term of well known Green function that can be expreed by the following equation 8. φ j (x, y, z) = 1 6 4π j=1 σ(ξ, η, ζ) G(x, y, z, ξ, η, ζ)d (1) The integral i to be carried out over complete immered urface of the object. The Green function G (ource potential) mut in order of the repreentation in (Equation (1) to be valid, atify all the boundary condition of the problem with the exception of the body boundary condition and have a ource like behavior. A a reult, boundary condition are reduced only to on wetted urface of the bodie. So, the wetted urface hould be ubdivided into panel to tranform integral equation to a ytem of algebraic equation to determine unknown ource denity over each panel. The appropriate Green function ued in thi paper to the boundary value problem poed i given by Wehauen, et al. 8 After getting the ource denity, the velocity potential on each panel can be obtained uing the (Equation (1). 2.3 Force and Moment Once the velocity potential i obtained, the hydrodynamic preure at any point on the body can be obtained from the linearized Bernoulli equation and can be written a: Φ t ( Φ)2 + P + gz = (11) ρ repectively. Thee coefficient which are ued in the equation of motion can be expreed from the equation a: a kj = ρ. Re [ φ j n k. d] (16) b kj = ρω. Im [ φ j n k. d] (17) The uffixek, j = 1, 2, 3, 4, 5, 6 repreent urge, way, heave, roll, pitch and yaw mode, repectively. 2.4 Equation of Motion in Frequency Domain By having exciting force, added ma and damping coefficient, the motion of emi ubmerible can be calculated by the following coupled equation. The equation of motion will be coupled dynamically becaue of hydrodynamic interaction between the element. So the equation can be conidered by uing the following matrix relationhip: 6 j=1 (M kj + a kj )X j + b kj X j + CX j = F k k = 1, 2 6 & j = 1, RESULTS AND DISCUSSION (18) One of the obtacle in numerical method i meh number, computer ram limitation and finally calculate time conumption. To obtain the motion repone of a floating emi ubmerible calculation ha been carried out at Head and Beam Sea by uing HydroSTAR 9 which i commercial oftware baed on potential theory. The principal particular of the elected Semi Submerible i howed in Table 1. A different number of flat quadrilateral mehe 436, 552, 896, 1192 and 2248 have been generated on the wetted urface of emi ubmerible. It wa tried to chooe mall mehe at the edge and near water urface for more accuracy (Figure 2). For more viibility of figure only half of the body wa hown. The HydroSTAR itelf wa ued to generate the mehe. Now after putting the value of Φ in the (Equation (11), the following expreion i obtained, P ρ = iωφ ( φ)2 + gz (12) By neglecting the higher order term, we can write: P = ρgz + iρωφ (13) A firt part of (Equation 13) i aociated with the hydrotatic and teady force, o neglecting thi part, the firt order wave exciting and ocillatory force caued by the dynamic fluid preure acting on the body can be obtained from the following integral: F k e iωt = iρωe iωt {φ + φ 7 }n k d (14) F kj e iωt = ρωe iωt {ω 2 X j φ j }n k. d (15) Moreover, it i uual to decompoe the hydrodynamic force reulting from the movement of the bodie into component in phae with the acceleration and velocity of the rigid body motion. It lead to the added ma and damping coefficient

4 7 Haan Abyn et al. / Jurnal Teknologi (Science & Engineering) 66:2 (214), Figure 2 Wetted urface meh Table 1 Principal particular of the emi ubmerible Character Size Unit Length m Width m Draft m Diplacement m 3 Water Plan Area m 2 Number of Column 4 Pontoon depth 6.3 m Pontoon beam 13.3 m Pontoon centerline eparation m Column longitudinal pacing (centre) m Column diameter 1.59 m GM T 2.87 m GM L 4.6 m K XX m K YY m K ZZ 35 m CG X m CG Y m CG Z -.28 m ω n3.32 rad/ ω n4.9 rad/ ω n5.12 rad/ Computation of firt order problem of wave diffraction and radiation have been carried out for motion repone of a emi ubmerible at Head Sea and Beam Sea and plotted againt wave frequency in Figure 3-Figure 6 at a water depth of 175 meter. Figure 3 and Figure 5 and how non-dimenional heave motion at heading 9 and 18 repectively. The RAO plateau around one until around.2 rad/ and after that get a maximum peak at reonant frequency.35 rad/. In the range of natural frequency reonance take place, o the calculation i difficult and trongly depend on the damping. Then they fell down dramatically to reach a nadir at the o-called cancellation frequency.38 rad/. At the cancellation period, the vertical excitation on top and bottom of the pontoon reduce the exciting force to near zero. The RAO then rie to reach the econd peak about.5rad/ and finally lope down to zero at high frequencie. In the roll motion RAO (Figure 4) the firt peak occur at around natural frequency.1 rad/ and econd one at.65 rad/. The RAO decreae from.61 to.2 at cancelling frequency 1.1

5 RAO (motion) RAO (motion) RAO (motion) RAO (motion) 71 Haan Abyn et al. / Jurnal Teknologi (Science & Engineering) 66:2 (214), rad/. The firt peak take place at 1.2 rad/ and after that the RAO lope down. Figure 4 and 7 how that calculation in very low or high frequencie hould be done at different mehe number. In Figure 6 pitch motion RAO reach a peak at natural frequency.12 rad/, then drop at cancelling frequency.19 rad/. There are rie of.4 at.6 rad/ and after that they decreae dramatically to reach.15 at 1.5 rad/. A the Figure 3-Figure 6 how, calculation reult for different number of mehe are nearly identical (not enitive to meh), except at natural frequencie. Normally calculation accuracy i low at reonance frequency. The RAO are highly dependent on damping and potential damping i low. So, more panel are required to give a converged reult. For verification Mato, et al. 7 wave exciting pitch reult which were calculated by uing WAMIT hown in Figure 7. A it how the reult in different meh number excluding 241 panel, are almot ame. In thi tudy calculation carrie out by uing Dell laptop, CPU 1.83 GHz, RAM 3 GB, Window 7, 64 bit operating ytem. Figure 8 how the calculation time by HydroSTAR for different meh number of emi ubmerible. The mentioned time ha been hown in percentage form in Figure meh and related calculation time conidered a a bae and the other computed repect for them. A it i hown by increaing meh number almot 2% (2 time), calculation time increae about 5% (5 time) and increaing 4% caue 2% of calculation time Figure 4 Roll motion (RAO) at beam ea 4. CONCLUSION Motion of floating tructure ha ignificant influence on loading and unloading operation. In thi paper, a emi ubmerible modeled in well-known commercial oftware HydroSTAR. A calculation ha been carried out at the different number of mehe on the wetted urface of the tructure and computation time recorded. Calculation time increae dramatically by increaing meh number. For larger mehe, improved calculation time may be achieved uing larger RAM to avoid wapping data to the hard dik Figure 5 Heave motion (RAO) at head ea Figure 6 Pitch motion (RAO) at head ea Figure 3 Heave motion (RAO) at beam ea

6 Time increaing (%) Time (min) 72 Haan Abyn et al. / Jurnal Teknologi (Science & Engineering) 66:2 (214), Nomenclature Figure 7 Pitch wave exciting force 7 a kj b kj Added ma coefficient matrix of kj Damping coefficient matrix of kj BUET Bangladeh Univerity of Engineering and Technology C Hydrotatic retoring force coefficient matrix of kj F k denote the k-th component of wave exciting force F kj denote the k-th component of force ariing from the j- th component of motion of the body. Fn Froude number g Gravitational acceleration M kj Inertia matrix in k mode due to the motion in j mode RAO Repone Amplitude Operation UTM Univeriti Teknologi Malayia x G, y G, z G Co-ordinate of the centre of gravity of the body x, y, z Invetigating point on the wetted urface of the body X j Vector containing the three tranlational and three rotational ocillation about the coordinate axe in j mode. α Wave heading angle from X -axi ω Circular frequency of incident wave ζ a Incident wave amplitude, φ Incident wave potential φ 7 Diffraction wave potential on body φ j Potential due to motion of the body in j-th mode, (ξ, η, ζ) denote a point on urface S σ(ξ, η, ζ) denote the unknown ource ditribution Meh No. Figure 8 Calculation time for different meh number Meh increaing (%) Reference [1] He, J. L. and A. M. O. Smith, Calculation of Nonlifting Potential Flow About Arbitrary 3D Bodie. Journal of Ship Reearch. 1(1): [2] Wu, S., J. J. Murray, and G. S. Virk, The Motion and Internal Force of a Moored Semi-ubmerible in Regular Wave. Ocean Engineering. 24(7): [3] Yilmaz, O. and A. Incecik, Extreme Motion Repone Analyi of Moored Semi-ubmerible. Ocean Engineering. 23(6): [4] Söylemez, M., Motion Tet of a Twin-hulled Semi-ubmerible. Ocean Engineering. 22(6): [5] Clau, G. F., C. Schmittner, and K. Stutz. 22. Time-domain Invetigation of a Semi Submerible in Rogue Wave. in 21t International Conference on Offhore Mechanic and Arctic Engineering (OMAE22). Olo, Norway: ASME. [6] Newman, J. N. Progre in Wave Load Computation on Offhore Structure. In OMAE Vancouver. [7] Mato, V. L. F., A. N. Simo, and S. H. Sphaier, 211. Second-order Reonant Heave, Roll and Pitch Motion of a Deep-draft Semiubmerible: Theoretical and Experimental Reult. Ocean Engineering. 38(17 18): [8] Wehauen, J. V. and E. V. Laitone. 22. Surface Wave. Berlin: Springer. [9] Chen, X.-b. 21. HydeoSTAR for export - Uer Manual. Figure 9 Calculation time in percentage form