Numerical Study of Solitary Wave Slamming on a 3-D Flexible Plate by MPS-FEM Coupled Method

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1 Proceedngs of the Twenty-eghth (018) Internatonal Ocean and Polar Engneerng Conference Sapporo, Japan, June 10-15, 018 Copyrght 018 by the Internatonal Socety of Offshore and Polar Engneers (ISOPE) ISBN ; ISSN Numercal Study of Soltary Wave Slammng on a 3-D Flexble Plate by MPS-FEM Coupled Method Guanyu Zhang, Chengpng Rao, Decheng Wan * Collaboratve Innovaton Center for Advanced Shp and Deep-Sea Exploraton, State Key Laboratory of Ocean Engneerng, School of Naval Archtecture, Ocean and Cvl Engneerng, Shangha Jao Tong Unversty, Shangha, Chna * Correspondng author ABSTRACT In ths paper, the smulaton of the nteracton between three-dmenson soltary wave and horzontal plate s nvestgated usng the movng partcle sem-mplct and fnte element coupled method (MPS-FEM). The MPS method s used to calculate the flud doman, whle the FEM s adopted to solve the structure doman. The smulaton of soltary wave slammng onto the flexble plate s ntally conducted, the effects of wave ampltude and plate elevaton on the wave-nduced mpact force are nvestgated. The nteracton between wave and flexble plate s fnally compared wth that regardng the rgd plate to study the contrbuton of the structural flexblty to the wave-nduced force and energy dsspaton. KEY WORDS: Movng Partcle Sem-Implct; MPS-FEM coupled method; flud structure nteracton (FSI); wave-nduced force. INTRODUCTION Plate-lke offshore structures, such as the per, jetty and very large floatng structure (VLFS), are vulnerable to flud mpact loads. Especally the VLFS, whose stffness s relatvely small, presents the smlar behavor to that of a flexble plate above the water surface. When encounterng extreme wave, the plate may produce elastc vbraton and consderable deformaton whch would brng the new challenges for the structural safety. The study of Flud-Structure nteracton (FSI) problems due to the flud mpact loads onto the structures, becomes of paramount mportance n the feld of naval archtecture and ocean engneerng. Expermental studes n FSI problems are scarce and there are only a few papers nvolved these problems, e.g. Cox et al. (00), Guomo et al. (007) and Nell et al. (017). In addton, varous numercal methods have been conducted n order to smulate FSI problems. By usng fnte element method, Irahpanah (1983) smulated the mpact of wave on horzontal platform, analyzed the wave-nduced load at the bottom of the deck. Seffert et al. (014), based on fnte volume method, smulated the soltary wave mpactng on a horzontal plate wth the help of an open source CFD software OpenFOAM. The nfluence of the depth of the numercal flume, the plate elevaton and the wave ampltude are nvestgated, the obtaned results agreed well wth the experment data. However, there are much fewer FSI analyss about flexble plates. Lu et al. (00) combned the boundary element method (BEM) and FEM to nvestgate the hydroelastc response of two-dmensonal elastc plate under wave-nduced force. Lao and Hu (013) appled a coupled FDM FEM method to nvestgate the nteracton between surface flow and thn elastc plate, and obtaned good results. Nell (017) conducted the experment of the wave mpactng a thn flexble plate, to analyze the reflecton and transmsson of regular ncdent water waves. It can draw a concluson that the ampltude was attenuated due to the green water and wave breakng, and the attenuaton s small when the plate s more flexble. Lu et al. (018) conducts the quas-statc and dynamc tensle tests and plate mpact experments, demonstrates that the stran rate effect vares wth the plastc stran and provdes a practcal advce for shp collson assessments. Despte the effectveness, these mesh-based methods may suffer from the dffcultes such as the adjustment and regeneraton of mesh whle coordnatng the nterface between flud and sold doman. Therefore, some newly mesh-free partcle methods have drawn a great deal of attenton. In contrast wth the mesh-based method, these mesh-free methods are nherently Lagrangan methods, by whch a contnuum s dscretzed nto movng partcles, so that the calculaton of the numercal dsspaton of the convecton term s avoded. In addton, the meshfree partcle methods avert the treatment of mesh or the capture of free surface. Thus, meshfree partcle methods can deal wth the large deformaton and strong nonlnear phenomenon of free surfaces wth relatve ease, as well as the movng boundares. SPH (Smoothed Partcle Hydrodynamcs) proposed by Lucy (1977), s a tradtonal meshfree method, whch was frst appled n astrophyscs. Subsequently, SPH has been developed more actvely, and has been appled to ncompressble flow and FSI problems n Antoc et al. (007). In addton, some scholars managed to combne SPH wth other methods 46

2 to solve the FSI problems. The SPH FEM couplng method has been frst proposed by Attaway et al. (1994) to nvestgated structurestructure nteracton, but t was used n FSI problems soon afterwards (Fourey et al. 01; Yang et al. 01; Long et al. 016). The MPS (Movng Partcle Sem-mplct) proposed by Koshzuka and Oka (1996), s another typcal partcle-based meshfree method. Compared wth the SPH, the pressure of the partcle s obtaned by solvng the Posson's pressure equaton (PPE) n the MPS method. Thus, the obtaned pressure feld through MPS method s relatvely smoother. MPS was later appled nto the feld of ocean engneerng (Gotoh and Khayyer. 016). Some prelmnary researches correspondng to FSI has been conducted on the bass of the MPS method. The FEM method s a general choose to combne, n order to address complcated FSI problem. Lee et al. (007) successfully smulated the nteracton between dam-break and sloshng flow through the coupled MPS-FEM method. Some other research performng the MPS-FEM model (Mtsume et al. 014; Hwang et al. 014; Hwang et al. 016; Zhang et al. 016a) also dsplayed far agreement wth avalable expermental results. The objectve of the present paper s to nvestgate the slammng on a 3-D flexble plate nduced by soltary wave usng the MPS-FEM coupled method. A number of cases should be conducted because of the uncertanty regardng ther occurrence and scale. The outlne of the present paper s shown as follows. The theores of MPS, FEM and couplng strategy are brefly ntroduced. Subsequently, the smulaton of 3-D soltary wave slammng onto the flexble plate s conducted, the effects of wave ampltude and plate elevaton on the wave-nduced mpact force are nvestgated. Fnally, the nteracton between wave and flexble plate s fnally compared wth the rgd plate to study the contrbuton of the structural flexblty to the wave-nduced force and energy dsspaton. NUMERICAL METHOD In ths study, the MPS-FEM coupled method s adopted to nvestgate the wave-plate nteracton problem. The MPS method s used to calculate the flud doman, whle the FEM s adopted to solve the structure doman. Specfcally, the 4-node thn plate element s employed n the FEM calculaton, and a parttoned couplng strategy s adopted to unte the two methods. The theory for the MPS and FEM have been presented wth detals n our prevous papers (Zhang and Wan, 01; Zhang et al., 014; Tang et al., 015; Tang et al., 016; Zhang et al., 016b; Zhang and Wan. 017; Zhang and Wan. 018). These are ntroduced brefly n ths secton. MPS formulaton for flud dynamcs The governng equatons of the MPS method for vscous ncompressble flud can be expressed n Lagrangan form as followng: V= 0 (1) DV 1 () = P + ν V + g Dt ρ where V, ρ, P, ν and g denote the velocty vector, the flud densty, the pressure, the knematc vscosty and the gravtatonal acceleraton. The kernel functon n present paper can be formulated as: re 1 0 r < re W( r) = 0.85r+ 0.15re (3) 0 re r where r = rj r s the dstance between partcle and j, and re denotes the nfluence radus of the target partcle. In MPS, the models of partcle nteracton nvolve the gradent, dvergence, and Laplacan models. They are wrtten as Eq. (4), Eq. (5) and Eq. (6). D φj + φ φ = ( ) ( ) 0 r j r W rj r n r r j j ( V V ) ( r r ) D V = W ( ) 0 j j r j r n r r j j D ( ) ( r r ) (6) φ = φ 0 j φ W j n λ j λ = j W ( ) W ( rj r ) j j j r r r r where D s the dmenson number, r s the poston vector, and n 0 s the ntal densty of the partcle number and defned as: n = W( rj r ) (8) j The pressure felds are obtaned through solvng the PPE. In present paper, the mxed source term method s employed combned wth a velocty dvergence-free condton and a constant partcle number densty (Tanaka et al., 010 and Lee et al., 011) as followng: k 0 k 1 ρ * ρ n n P + = (1 γ) V γ < > (9) 0 Δt Δt n where t denotes the calculaton tme step, k and k+1 ndcate the physcal quantty n the k th and k+1 th tme steps, and γ s the weght of the partcle number densty term between 0 and 1. In ths paper, γ =0.01 s selected for all numercal experments. FEM formulaton for structure dynamcs Based on FEM theory, the spatally dscretzed structural dynamc equatons, whch govern the moton of structural nodes, can be formulated as: M y+ Cy + K y= F ( t) (10) C= α1m+ αk (11) where M, C, K denote the mass matrx, the Raylegh dampng matrx, and the stffness matrx of the structure, respectvely. F s the external force vector that acts on the structure and vares wth computatonal tme. y s the dsplacement vector of the structure element node. α1 and α are coeffcents related to the natural frequency and the dampng ratos of the structure. Accordng to Newmark (1959), the structural node dsplacement at t=t+ t can be solved wth the help of Taylor s expansons of velocty and dsplacement: y t+δ t = y t + (1 γ) ytδ t+ γ y t+δtδ t, 0 < γ < 1 (1) 1 β yt+δ t = yt + y tδ t+ ytδ t + β yt+δ tδt, (13) 0< β < 1 where β and γ are paramount parameters n the Newmark-β method and are set as β=0.5, γ=0.5 for all smulatons. Then the dsplacement at t=t+ t s proposed by Hsao et al. (1999): K y = F (14) t+δ t t+δt K = K + a0m+ a1c (15) F = t+δ t F + t M( a0y t ay t a + 3yt) C( a 1yt a y + a y ) (16) 4 t 5 t (4) (5) (7) 47

3 1 γ 1 1 a0 =, a 1 =, a =, a3 = 1, βδt βδt βδt β (17) γ Δt γ a4 = 1, a5 = ( ), a6 =Δt(1 γ), a7 = γδt β β where K and F denote the effectve stffness matrx and effectve force vector. Subsequently, the acceleratons and veloctes related to the next tme step are updated as follows: yt+δ t = a0( yt+δ t yt) ay t a 3y t (18) y = y + a y + a y (19) t+δ t t 6 t 7 t+δt Couplng strategy for MPS-FEM coupled method In ths study, the weak couplng between MPS and the FEM method s mplemented. The tme step szes for structure and flud analyses are ts and tf, respectvely. Due to the stablty of Newmark-β scheme, ts can be longer than tf, so that ts s k multples of tf, where k s an nteger. The nteracton procedure can be summarzed as follows: 1) The pressure of the flud wall boundary partcle s calculated at each flud tme step. Then the pressure should be averaged durng ts, to obtan the external force on the element node, as follows: p 1 k n+ 1 pn+ k = 1 = (0) where p n + 1 s the pressure of the flud partcle on the wall boundary at the nstant t + Δ t, and p n + 1 s the average pressure of the flud partcle wthn Δ. ts f ) The values of structural nodal poston y t, velocty y t, and acceleraton y t can be determned based on the prevous tme step. 3) The external force vector F t+δt s of the structural boundary partcles s calculated by multplyng the average pressure n 1 and the nfluental area, whch equals the square of the ntal partcle spacng dp. F t+δ t = p s n+ 1 dp (1) 4) The structural nodal dsplacements and veloctes at next structural tme step can be calculated based on the Newmark-β scheme. 5) Update the velocty and poston of the structural boundary partcles at each structural tme step and the flud partcles at each flud tme step. p + η = Asech ( k( x ct)) () k A H 3 = 3 / 4 (3) c = g( A + H ) (4) where A s the water heght, also s the wave ampltude for soltary wave. H, x and c denote the water depth, the horzontal coordnate and the wave speed, respectvely. In ths paper, the soltary wave s generated by a pston-type wavemaker, whose moton was descrbed by Gorng (1978). The speed of the wavemaker s formulated as: d X( t) casech ( k( X ct)) Ut () = = (5) dt H+ Asech ( k( X ct)) Thus, the poston of the wavemaker at tme t can be expressed as: A Xt () = tanh( kct ( X)) (6) kh The stroke length s calculated by the dfference value between the wavemaker poston at t = + and t = : 16AH S = (7) 3 The wave perod s approxmately: A T (3.8 ) kc + H (8) After one wave perod, the wavemaker reaches ts maxmum poston and then becomes stll. NUMERICAL SIMULATION In ths secton, the nteracton of 3D soltary wave nteractng wth a flexble horzontal plate s nvestgated based on aforementoned MPS- FEM coupled method. Fgure shows the model of the numercal wave tank together wth the plate. The tank s.00 m n length and the water depth (H) s 0.114m. As s shown, the left sde of the tank s a pstontype wavemaker that s employed to generate the soltary wave. A fxed support horzontal plate s nstalled n the mddle of the tank. t n t n + 1 t n+ k 1 t n+ k pn p n + 1 n+ 1 Flud solver Structure solver Fg. 1 Schematc dagram of the parttoned couplng strategy between the flud and structure domans Numercal wave generaton p n+ k 1 n + k = p k Only update structure poston Accordng to the potental flow theory, a soltary wave conssts of a sngle crest of nfnte length. The profle of the soltary wave can be expressed as follows (Korteweg and De Vres, 1895): p n + k yn, y n, yn yn+ 1, y n+ 1, y n+ 1 y, y, y n+ k n+ k n+ k Fg. Geometrc model of the numercal wave tank Numercal wave generatons Before studyng the wave-structure nteracton, t s of mportance to examne the accuracy of the generated soltary wave, the verfcaton s conducted n the numercal wave tank wthout the plate. The soltary wave, wth dfferent wave ampltudes (A), ncludng A/H=0., 0.3, 0.4 and 0.5, adopted n the smulatons. The computatonal parameters are lsted n Table 1. Table 1. The computatonal parameters of MPS Parameter Value Water densty 1000 kg/m 3 48

4 Knematc vscosty m /s Gravtatonal acceleraton 9.81 m/s Partcle spacng m Flud number 8614 Total number The surface elevaton at wave ampltudes of A/H= 0.~0.5 s depcted n Fg.3. The comparson of the wave profle between the numercal smulaton and the theoretcal soluton shows that the wave crest of the smulaton agrees well wth those presented by Gorng (1978). Wave mpactng onto flexble plate In ths subsecton, the nteracton between the three-dmenson soltary wave and a flexble plate s smulated wth the help of the n-house MPSFEM-SJTU solver. Two dmensonal FSI problems has been smulated usng MPSFEM-SJTU solver by Zhang et al. (018) and Rao et al. (017), and the obtaned results are well relable. The structural parameter s shown n Table. The elastc modulus of the plate s 50MPa. In addton, the dstance between the bottom of the plate and the stll water lne (SWL) s defned as plate elevaton (D), whch can be altered by movng the plate vertcally. In the smulatons, the plate elevaton (D) and wave ampltude (A) vary from case to case n order to nvestgate ther effects on the wave-nduced force on the plate. The dmensonless parameters of all the cases are shown n Table 3. Fgure 4 shows the partcular snapshots of the nteracton between soltary wave and flexble plate under the condton of A/H=0.3, D/H=0.1. The calculated wave-nduced force on the plate s shown n Fg. 5 and the dsplacement hstory n the mddle of the plate s shown n Fg.6. It can be seen that the plate possesses a slght upward deformaton mmedately as the wave contacts the plate around t=1.6s. Then the wave crest hts the leadng edge of the plate at around 1.8s, and the vertcal force reaches maxmum. The maxmum vertcal force lasts about one second wth slght oscllaton, untl 1.9s. Subsequently, the deformaton reaches the maxmum, and the force begns to descend wthout evdent oscllaton. Fnally, the plate suffered a negatve vertcal force after t=.08s owng to the green water on the plate. It can be easly observed that the horzontal force s far less than vertcal force, whch ndcates that the nteracton between wave and plate manly reflects n vertcal slammng. Table. The computatonal parameters of FEM Parameters Values Structural densty 1800 kg/m 3 Elastc modulus 50 MPa Thckness m Posson's rato 0.3 Dampng coeffcent α1 0 Dampng coeffcent α 0 Element type 4-node thn plate element Element number 800 Table 3. Confguratons of the cases Case No. Ampltude (A/H) Elevaton (D/H) (a) A/H=0. (b) A/H=0.3 (c) A/H=0.4 Fg. 3 The wave elevaton on the wave gauge (d) A/H=0.5 49

5 (a) t=1.6s (b) t=1.76s (c) t=1.8s (d) t=1.9s (e) t=1.98s Fg. 4 Snapshots of wave-plate nteracton (flexble plate) (f) t=.04s Fg. 5 wave-nduced forces on the plate Fg. 6 dsplacement hstory on the mddle of plate To nvestgate the effects of the wave ampltude and plate elevaton on the wave-nduced force, especally the vertcal slammng, the maxmum value of vertcal force hstory n each case s collected. It can be nferred n the Fg. 7 that the maxmum vertcal force on the plate s approxmatvely n proporton to the wave ampltude under the same plate elevatons, whle the maxmum vertcal force decreases wth the plate elevaton ncreasng. Smlar to the results from Lu (016), the force ncreases at frst wth the ascendng of elevaton, then goes down. The dscrepancy n smaller elevaton s owng to the fnte depth of tank. In addton, the comparson of vertcal force hstory under dfferent plate elevatons s depcted n Fg. 8, the loadng duraton ncreases wth the ascendng of the ampltude and the descendng of the elevaton. Besdes, evdent oscllaton durng the rse of force can be observed n a greater elevaton. A/H=0. A/H=0.3 A/H=0.4 A/H=0.5 Fg. 8 Hstory of the vertcal force on the plate Fg. 7 The maxmum vertcal force on the plate 50

6 (a) t=1.6s (b) t=1.66s (c) t=1.76s (d) t=1.80s (e) t=1.9s (f) t=.04s Fg. 9 Snapshots of wave-plate nteracton (rgd plate) The nfluence of the structure flexblty on slammng In ths subsecton, the nteracton between the soltary wave and a rgd plate s smulated wth A/H=0.3, D/H=0.1. The result s compared wth the flexble case n order to nvestgate the contrbuton of the structural flexblty to the wave-nduced force and energy dsspaton. Some partcular snapshots n the selected smulaton are gven n Fg. 9. The hstory of wave-nduced force on plate s shown n Fg. 10. It can be seen that at around t=1.6s the wave contacts the leadng edge of plate and mpacts the plate. When the wave crest hts the leadng edge of plate at around 1.76s, the vertcal force reaches maxmum. At 1.8s, evdent spray and wave breakng can be observed behnd the plate, whch dsspate energy from the system, resultng to a drastc drop n the curve of vertcal force. mportant: An evdent spray behnd the aft end of the plate durng the post stage s an evdent proof to the energy dsspate, as well as the smaller velocty transmttng from the aft end of plate. Consder the above reasons, durng the mpact, the energy s shown to dsspate, partcularly n rgd plate. It llustrates that wave attenuaton s more sgnfcant n the rgd case than the flexble case. It s smlar to the results of the experment of Nell et al (017). Compared wth the flexble plate s counterpart, evdent dfferences can be observed n the vertcal force. For the rgd plate, t takes only 0.16 second to reach the maxmum vertcal force, then the vertcal force descends drastcally wth evdent oscllaton. However, n the flexble plate, the maxmum vertcal force lags slghtly and lasts for one second wth sght oscllaton, besdes, the maxmum vertcal force s lower than the rgd case. Then there s no obvous oscllaton durng the descendng of the vertcal force. The reason for these dfferences probably s that the upward dsplacement due to the mpact provdes a cushonng for wave to spend more tme on suffusng the bottom of flexble plate, so t leads to the dfference n loadng tme of vertcal force. Moreover, t also results to a lower peak value for flexble plate. In addton, the comparson of horzontal force between rgd and flexble plate shows a far agreement, whle there are some evdent oscllatons n the flexble case. The velocty dstrbuton of the flud feld s depcted n Fg.11. When the soltary wave has no contact wth the plate, the velocty vector of the water partcle s acclvtous, and when the wave contacts the plate, the velocty vector of water partcle near the leadng edge of the plate s almost perpendcular to the horzontal plate, n both two cases. However, the flud under the plate presents evdent dfference. In the rgd plate, the velocty vector s bascally parallel to the plate durng the whole mpact. Whle n the flexble plate, the velocty vector s nclned to the plate durng the slammng stage, then the velocty vector s downward to the plate durng the post stage, shown n Fg.11 (b ~ c). In addton, durng the post stage, evdent spray can be observed behnd the rgd plate, whch does not exst n the flexble plate. Besdes, the velocty transmttng from the aft end of rgd plate s obvously smaller than that of flexble plate. The dscrepancy of energy s the reason for the above dfference. For the flexble case, energy transfer s manly taken nto account: Durng the slammng stage, the plate absorbs some of the wave energy due to the upward deformaton, then t returns most energy to the flud under the plate. Whle n the rgd case, energy dsspate s prmary Fg. 10 Hstory of wave-nduced forces on plate 51

7 Rgd plate Elastc plate (a) t=1.60s (b) t=1.76s (c) t=.04s Fg. 11 The velocty dstrbuton of the flud feld (Top panels are rgd plate; Bottom panels are flexble plate) CONCLUSIONS In present paper, the nteracton between the soltary wave and the horzontal flexble plate s numercally nvestgated based on aforementoned MPS-FEM coupled method. Based on the results of smulatons, the followng conclusons can be summarzed as follows: The wave ampltude (A/H=0., 0.3, 0.4 and 0.5) and plate elevaton (D/H=0.06, 0.1 and 0.18) are frst taken nto account to study ther effects on the nteracton. The results ndcate that the maxmum vertcal force s n proporton to the wave ampltude, whle the maxmum vertcal force decreases wth the ascendng of the elevaton. In addton, the loadng duraton ncreases wth the ascendng of the ampltude, as well the descendng of the elevaton. Evdent oscllaton durng the rse of force can be observed n a greater elevaton. Subsequently, the wave-nduced force s prmarly focused n ths research to study the effects of flexblty on wave-plate nteracton. The nteracton between the soltary wave and a flexble plate s smulated under the condton of A/H=0.3 and D/H=0.1, compared wth the rgd plate. The dscrepances n vertcal force help us clearly observe the contrbuton of the structural flexblty. In the flexble case, the upward dsplacement due to the mpact provdes a cushonng for wave to spend more tme on suffusng the bottom of the plate, so t leads to the dfference n loadng tme of vertcal force. Moreover, t also results to a lower peak value for flexble plate. However, the comparson of horzontal force between rgd and flexble plate shows a far agreement, whch ndcates that the nteracton between wave and plate manly reflects n vertcal slummng. Smultaneously, the analyss of flow feld for rgd and flexble plate shows the contrbuton of the structural flexblty to the energy. For the flexble case, energy transfer s manly taken nto account, whle n the rgd case, energy dsspate s prmary mportant. It llustrates that, the energy s shown to dsspate durng the mpact, partcularly n rgd plate. So, wave attenuaton s more sgnfcant n the rgd case than the flexble case. The concluson s smlar to the results of the experment of Nell et al (017). ACKNOWLEDGEMENTS Ths work s supported by the Natonal Natural Scence Foundaton of Chna ( , , ), Chang Jang Scholars Program (T014099), Shangha Excellent Academc Leaders Program (17XD140300), Program for Professor of Specal Appontment (Eastern Scholar) at Shangha Insttutons of Hgher Learnng (0130), Innovatve Specal Project of Numercal Tank of Mnstry of Industry and Informaton Technology of Chna (016-3/09) and Lloyd s Regster Foundaton for doctoral student, to whch the authors are most grateful. REFERENCES Antoc, C, Gallat, M, and Sblla, S (007). Numercal Smulaton of Flud structure Interacton by SPH, Computers and Structures, 85(11), Attaway, SW, Hensten, MW, and Swegle, JW (1994). Couplng of Smooth Partcle Hydrodynamcs wth the Fnte Element Method, Nuclear engneerng and desgn, 150(-3), Cox, DT, and Ortega, JA (00). Laboratory observatons of green water overtoppng a fxed deck, Ocean Engneerng, 9(14): Cuomo, G, Trndell, M, and Allsop, W (007). Wave-n-deck loads on exposed jettes, Coastal Engneerng, 54(9), Fourey, G, Oger, G, Touzé, DL, and Alessandrn, B (010). Volent Flud-Structure Interacton Smulatons usng a Coupled SPH/FEM method, Iop Conference Seres Materals Scence & Engneerng, 10, Gorng, DG (1978). Tsunams-the Propagaton of Long Waves onto a Shelf, PhD thess, Calforna Insttute of Technology, Pasadena, Calforna, USA. Gotoh, H, and Khayyer, A (016). Current achevements and future perspectves for projecton-based partcle methods wth applcatons n ocean engneerng, Journal of Ocean Engneerng and Marne Energy, (3), Hwang, SC, Khayyer, A, Gotoh, H, and Park, JC (014). Development of a Fully Lagrangan MPS-based Coupled Method for Smulaton of Flud-structure Interacton Problems, Journal of Fluds & Structures, 50(), Hwang, SC, Park, JC, Gotoh, H, Khayyer, A, and Kang, KJ (016). Numercal Smulatons of Sloshng Flows wth Elastc Baffles by usng a Partcle-based Flud-structure Interacton Analyss Method, Ocean Engneerng, 118, Hsao, KM, Ln, JY, and Ln WY (1999). A Consstent Corotatonal Fnte Element Formulaton for Geometrcally Nonlnear Dynamc Analyss of 3-D Beams, Comput. Methods Appl. Mech. Engrg, 169, Irajpanah K (1983). Wave uplft force on horzontal platform, PhD thess, Unversty of Southern Calforna, Los Angeles, USA. Korteweg DJ, and De Vres, G (1895). On the change of form of long waves advancng n a rectangular canal, and on a new type of long statonary waves, Phlosophcal Magazne, 39(40),

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