Numerical Simulation on Hydroelastic Response of Structure under Impact Load from Water Using Eulerian Scheme with Lagrangian Particles

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1 Journal of Shn and Ocean Enneern 2 ( D DAVD PUBLSHG umercal Smulaton on Hydroelastc Resonse of Structure under mact Load from Water Usn Euleran Scheme wth Laranan Partcles Hdem Mutsuda 1, Suandar Baso 2 and Yasuak Do 1 1. Dvson of Enery and Envronmental Enneern, Hroshma Unversty, Hash-Hroshma , Jaan 2. Deartment of aval Archtecture, Faculty of Enneern, Hasanuddn Unversty, Makassar 90245, ndonesa Abstract: Hydroelastcty caused by water mact s of concern n many alcatons of ocean enneern/naval archtect and s a comlcated hyscal henomenon. The authors have develoed a couled Euleran scheme wth Laranan artcles to combne advantaes and to comensate dsadvantaes n both rd based method and artcle based method. n ths study, the develoed numercal model was aled to hydroelastc roblems due to mact ressure such as water entry of an elastc cylnder and elastc tanker moton n wave. The authors showed the numercal results whch s overall areement wth exermental results. The roosed numercal scheme can be useful and effectveness to evaluate hydroelastcty and sh-wave nteracton n nonlnear wave moton wth breakn. Key words: Hydroelastcty, slammn, mact ressure, Euleran-Laranan scheme, sh-wave nteracton. 1. ntroducton Hydroelastcty caused by water mact s of concern n many alcatons of ocean enneern/naval archtect and s a comlcated hyscal henomenon. The mact load due to a slammn can result n substantal damae n an ocean structure and a sh. Therefore, several works have been erformed to redct the mact ressure actn on an ocean structure and a sh. Moreover, a sh s not a really rd constructon and ths means that a sh has elastc behavors where t exerences strans and stresses because of ts structural flexblty. Ths can not be nelected that the hydroelastc behavors of a sh contrbutes some effects to sh erformances. Therefore, the hydroelastc behavors of a sh have to be consdered n redctn sh motons, ressure, bendn moment and torque as resulted by the stronly nteracton between wave-sh assocated wth Corresondn author: Hdem Mutsuda, assocate rofessor, research felds: flud enneern, comutatonal mechancs, naval archtect, and ocean envronment. E-mal: mutsuda@naoe.hroshma-u.ac.. hydroelastc effects toward roer sh desn and sh safety. Hauen [1] suested that ar-cushon effects may be mortant when there are several domnant natural erods of structural vbratons. Meyerhoff et al. [2] have already studed wave mact on elastc beams. Faltnsen [3] studed an aroxmate three-dmensonal theoretcal nvestaton of hydroelastc wet-deck slammn. The hydroelastc slammn roblem must be hydrodynamcally studed from a structural ont of vew [4]. Senanovc et al. [5] analyzed the hydroelastc effect on a flexble semented bare moton n waves and dstorton. The slam events [6] were also characterzed exermentally by usn a hydroelastc semented model. Recently many onon researches n marne enneern/naval archtect have been attemted to yeld CFD (comutatonal flud dynamcs tool toward accurate tool wth consdern CFD requrements. These can redct wave mact as hydrodynamc effects due to stronly nonlnear sh-wave nteractons, however, nvolvement of hydroelastc

2 364 umercal Smulaton on Hydroelastc Resonse of Structure under mact Load from Water Usn Euleran Scheme wth Laranan Partcles effects assocated wth caturn nonlnear free surface flows on a sh moton under severe wave condtons s stll rarely devoted. The results have been also enerally concerned on water entry roblems wth caturn technque of free surface henomena or wth a weakly nteracton between wave and an elastc sh. Why nonlnear free surface flows are dffcult? That s because t s a comlex roblem to kee the sharness of the ar-water nterface tolerable and to handle movn free surface and elastc sh boundary. Therefore, the develoments of CFD technques to redct accurately hydrodynamc and hydroelastc effects on sh moton n severe wave condton need tremendous efforts. Under ths backround, the authors have develoed a couled Euleran scheme wth Laranan artcles to combne advantaes and to comensate dsadvantaes n both rd based method and artcle based method. The model has two knds of Laranan artcles,.e., SPH (smoothed artcle hydrodynamcs [7] and free surface artcle on Euleran rds to correct nterface trackn error. The develoed model has already aled to several knds of flud-structure roblems [8-11] for comutn stronly nteracton between water and body. n ths study, the develoed numercal model s aled to hydroelastc roblems due to mact ressure such as water entry of an elastc cylnder, a floatn moton of elastc structure n headn wave. The authors nvestate the valdaton and verfcaton of the develoed model n hydroelastc roblems and show that the roosed scheme s useful and effectveness to evaluate hydroelastcty and sh-wave nteracton n nonlnear wave moton wth breakn. 2. Comutatonal Method 2.1 Caturn Technques for Multhase Laranan Partcles The schematc llustraton of the model s shown n F. 1. As shown n F. 2, ths scheme uses a staered rd system and has two knds of Larane artcles, F. 1 Grd arranement and artcle dstrbuton ( : SPH artcles, : Free surface artcles. F. 2 Defnton of hyscal values on a rd and Laranan artcles..e., SPH artcles denoted by black dots and free surface artcles denoted by whte dots havn hyscal roertes, e.., densty and velocty. A sold model s reresented by the SPH artcles, whch are characterzed by densty and ther radus n order to comute sh moton and deformaton. Elastc deformaton and fracture of sh moton can be comuted by usn the SPH artcles. The free surface artcles defned on Euleran rd are located near free surface to cature accurately an nterface between dfferent hases, e.., water surface. The free surface artcles have densty functon to characterze hyscal roertes at each hase. Densty functon s defned on node ont of the staered rd and t lays an mortant role of trackn the nterface between dfferent hases, such as ar, water and sold. Althouh the artcles n both sdes of the nterface are located n artcle level set method [12], the free surface artcles n ths model are arorately laced

3 umercal Smulaton on Hydroelastc Resonse of Structure under mact Load from Water Usn Euleran Scheme wth Laranan Partcles 365 n only one sde of free surface to reduce comutatonal cost and tme n redstrbuton rocess of the free surface artcles Tme Evoluton of Laranan Partcles and nterolaton of Physcal Value Partcle locaton on free surface s nterated by usn evoluton of the fourth order accurate Rune-Kutta method, follown the evoluton equaton: dx u( x (1 dt where, x s the artcle locaton, u x s the ( artcle velocty calculated by nterolatn velocty on nehborn cell faces of rd. Meanwhle, SPH artcles are laced n the area of sold hase and are advanced from the equaton of rd moton by usn SPH method [7] descrbed n the follown secton. Velocty comonents are defned on cell faces of rds and ressure n all hases s defned at the center as shown n F. 2. The velocty of the free surface artcle s nterolated from veloctes on the nehborn cell faces of the rds by usn blnear n 2D or trlnear nterolaton n 3D deendn on the requred accuracy and effcency. For ths smlfyn reason, the number of artcles can be corrected n one rd. n addton, the Laranan artcles (SPH artcles and free surface artcles are advected to mantan densty functon of the artcle durn calculaton Redstrbuton of Free Surface Partcles Usn the roosed model, to cature nterface accurately durn a calculaton, redstrbuton of free surface artcles s erodcally needed to add and delete the artcles usn the technque develoed by Enrht et al. [12]. F. 3 shows one examle of the artcle dstrbuton for water entry roblem of a crcular obstacle n 2D. The artcles are located near the free surface characterzed by densty functons as mentoned n the next secton. Usn the redstrbuton rocess, both comutatonal effcency and stablty are enhanced as well. Gas Sold F. 3 One examle of artcle dstrbuton n water entry roblem of a tranular wede Correcton of Densty Functon on Grds n the roosed model, densty functon defned on a rd node s corrected by usn densty functon on free surface artcles wthn referenced area wth radus h. The densty functon reduces a numercal error enerated n advecton rocess usn the follown equaton: m max, W x x, h (2 1 where, s the densty functon on the node of the rds after the error correcton, m / s the artcle volume, V s characterzed by the artcle radus r, W s the kernel functon defned as a cubc slne functon, x and x are reresented as satal ostons on a rd and a artcle, resectvely. 2.2 Governn Equatons and Comutatonal Scheme n the roosed model, the overnn equatons for flud hase consst of the mass conservaton equaton, ncomressble aver-stokes equaton and the equaton of contnuty, -hase densty functon ( 0 1 and ther advecton equatons. The overnn equatons are exressed as follow: u 0 (3 x 2 u 1 u P u u t x x x x x (4 F fs u 0 t x (5 Lqud

4 366 umercal Smulaton on Hydroelastc Resonse of Structure under mact Load from Water Usn Euleran Scheme wth Laranan Partcles where, -hases means as-hase ( = 1, lqud-hase ( = 2 and sold-hase ( = 3, s the ravty, F fs s the flud structure nteracton term, s the SGS (sub rd scale stress term. To reduce the model arameter, the SGS stress term s solved by usn the LES (lare eddy smulaton model. n addton, the densty functon s used to dentfy hyscal roertes of dfferent hase and the densty and the vscosty can be derved by usn the densty functon as the follown equatons: (6 1 The densty functon 3 (sold hase s corrected by usn Eq. (2 wth SPH artcles defned n sold hase. The overnn equatons can be solved usn the slttn method as a well-known conventonal multhase technque. n the roosed model, the authors emloyed the C-CUP (constrant nterolaton rofle/cp combned, unfed rocedure method develoed by Yabe et al. [13] as the slttn method. The overnn equatons for sold hase, whch are dscretzed usn the SPH method [7] n the model, are the contnuty equaton and momentum equaton as follows: D u 0 (7 Dt x Du F fs (8 Dt x where, s the densty, u s the velocty, x s the oston vector of vector comonents, s the stress tensor of the sold hase, and F fs s the flud structure nteracton term. The stress tensor s n Eq. (8 s ven by s P S (9 where, S s the devatorc stress tensor, P kk / 3 the ressure solved by the Posson equaton. The model consders a lare deformaton of an elastc body. The stress of a sold body chanes at every calculaton ste by usn the follown equaton: { ds e } [ D ]{ d } (10 e where, D s the elastc-lastc matrx, d the tme ncrement of the stran, and ds the tme ncrement of the devatorc stress. To solve rotaton of the sold hase durn a deformaton, the Jaumann dervatve s used to ensure materal frame ndfference wth resect to the rotaton as follow: ds 1 k k k k 2 S S dt (11 3 where, s the stran rate tensor and Ω the sn tensor. The flud structure nteracton F fs s solved by acceleraton obtaned from the ressure on SPH artcles nterolated usn the ressure on rds solved by the Posson equaton. n the roosed model, the flud structure nteracton F fs n Eqs. (4 and (8 can be ven by the follown equaton: 1 P( rb F fs ( ra mb aw ( ra rb, h (12 ( ra b ( rb To kee comutatonal effcency and stablty, the tme ncrement n the sold hase s aroxmately 1/10 to 1/50 of that n flud hase. The boundary condton for velocty on a body s mosed usn the follown equaton: u b 3u s ( 1 3 u (13 where, u b s the velocty on the Euleran rd where SPH artcle on a sold nterface s located, 3 the densty functon of sold hase, u s the velocty on SPH artcle n each rd, u the velocty at the face of a rd. Eq. (13 means that no-sl boundary condton can be mosed on the surface of the sold nterface. 2.3 Sold Moton n 3D n the roosed model, a sold body conssts of a lare number of SPH artcles to cature motons and deformaton of a sh n 3D. Therefore, the 3D moton of a sh hull s reresented by descrbn translaton and rotaton of the center of ravty of a sold usn the follown equatons:

5 umercal Smulaton on Hydroelastc Resonse of Structure under mact Load from Water Usn Euleran Scheme wth Laranan Partcles x s, k Fs, k 2 t m F fs (14 T (15 t (16 t where, s the rotatonal anle, s the anular velocty, T s the torque, s the nerta moment, and F fs s the flud structure nteracton n Eq. (4. n addton, the center of ravty of a sh hull can be obtaned by calculatn the nerta moment of the SPH artcles, and ths s roosed by usn Baraff theory [14]. Based on ths theory, n the roosed model, the k+1 equatons for a 3D moton are ven by r n 1 2 r r m r r ( where, s the coordnate of the center of ravty of a sold body. r s the oston of the ravty center, s the nerta moment, r s the oston of the the SPH artcle and m s the mass of the artcle. The authors can reresent tme nteraton of the locaton of the sold body by the follown equatons: '* k1 k r (18 r r k1 1 k1 r ˆ r ( r ˆ r ( k k1 dt mrˆ (21 ( r r 1 1 k k1 r r R ( r r (22 k 1 r u (23 t k1 k r (24 r r where, the R s a rotaton matrx. The nerta moment s set at ntal condton. Therefore, the coordnates of velocty of each SPH artcle n every tme ste can be tracked by usn the rotaton matrx and the amount of the anle rotaton of the center of ravty to avod the mbal lock henomenon. Therefore, the quaternon s used nstead of the rotaton matrx n Euler anles. 3. umercal Results 3.1 Water Entry Problem of Elastc Cylnder The authors aled to a water entry roblem of an elastc cylnder made of olyvnylchlorde lastc as a benchmark test to valdate the develoed numercal model. As shown n F. 4, the dameter s 150 mm and the wdth s 13.5 mm and the thckness of the cylnder s 15 mm, whch s a very thn structure. The Youn modulus s 3,400 MPa and the oson rato s 0.38 and the densty s 1,700 k/m 3. The nner strans of the elastc cylnder at three dfferent onts (Pont 1, 2 and 3 were measured by PVDF (olyvnyldene dfluorde flm sensors. The ntal entry seed s 2.8 m/s. The oston of the elastc cylnder was catured by hh seed camera (1,000 flame er second. F. 5 shows snashots of the water entry rocess catured by the hh seed camera. F. 6 shows tme hstory of the voltae of stran on the nner face of the elastc cylnder. All of the tme hstores of stran are damn oscllaton whch corresonds to the natural frequency of the submered elastc cylnder. The frequency sectrum of the stran durn the entry rocess s shown n F. 7. n ths case, the domnant frequency s about 175 Hz. To valdate the develoed numercal model, the entry rocess n 2D was comuted wth the ntal condtons as show n F. 8. The elastc cylnder conssts of the SPH artcles. The total number of the SPH artcles s 1,885 n 2D and ts radus s 0.31 mm. The free surface artcles are located near the free surface and ts radus s 0.75 mm. The total number of the free surface artcles s 7,224. The rd sze s 3 mm and the total rd number s 120,000 n 2D. The montorn onts (Pt.1 to 12 are located at the nner face of the cylnder to comare the stran of the elastc cylnder wth the exermental result. F. 9 shows the entry rocess of the elastc cylnder nto the stll water. t can be seen that the stronly slashn was enerated from the thn layer between the cylnder and the water. F. 10 shows comarson of

6 368 umercal Smulaton on Hydroelastc Resonse of Structure under mact Load from Water Usn Euleran Scheme wth Laranan Partcles Pont 1 Pont 2 Pont 3 Pont 3 Pont 2 Pont 1 F. 4 Elastc cylnder model and stran measurn onts at the nner surface. F. 6 Tme hstores of voltae of stran aue at the nner face of the elastc cylnder durn the entry rocess n the exerment. Pont 1 Pont 2 Pont 3 F. 7 Domnant frequency of the stran n the exerment. domnant frequency s shown n F. 12. The domnant frequency s about 180 Hz whch concdes wth the exermental results as shown n F. 7. The authors nvestated the nternal stran of the elastc cylnder durn the entry rocess as shown n F. 13. The nternal stran durn the entry s radly exchaned n tme and sace. 3.2 Elastc Sh Body n Wave F. 5 Snashots of the elastc cylnder durn the water entry rocess n the exerment. the vertcal oston of the cylnder between the exerment and the numercal result. The numercal result s overall areement wth the exermental one. F. 11 shows the tme hstory of the stran at the nner face of the cylnder durn the water entry rocess. The The authors aled to numercal smulaton of an elastc sh moton n headn wave. F. 14 shows the brd s-eye vew of the sh hull as a tanker tye. As shown n the revous secton, n the roosed model, the tanker s reresented by a lare number of the SPH artcles traced by artcle based method, SPH. The ncdent wave heht s H w / L = 0.06, where L s sh lenth and the wave lenth s / L = 1.0. The Froude number F r s at the ntal condton. The rd sze s L PP and the radus of free surface

7 umercal Smulaton on Hydroelastc Resonse of Structure under mact Load from Water Usn Euleran Scheme wth Laranan Partcles 369 v F. 8 ntal condtons of water entry roblem and artcle dstrbuton n the elastc cylnder. F. 12 Domnant frequency of the stran n numercal result. ε xx ε xy ε yy t = s t = s F. 9 Water entry rocess of the elastc cylnder. Vertcal oston (cm F. 10 Comarson of vertcal oston of the elastc cylnder durn the entry rocess (dots: exerment, sold lne: numercal result. F. 11 Tme hstores of stran at the nner face of the elastc cylnder durn the water entry rocess comuted by the resent model. F. 13 nternal stran feld of the elastc cylnder durn the entry rocess. Bow F. 14 Dstrbuton of SPH artcle on a tanker. Stern artcle s L PP. The total number of the free surface artcle located near the free surface s 1,300,000 n 3D. The radus of the SPH artcle s L PP and the total number s 26,000 for the tanker. F. 15 shows snashots of the freely elastc tanker moton n headn wave. These results show that heave and tch motons of the tanker n the headn wave can be seen.

8 370 umercal Smulaton on Hydroelastc Resonse of Structure under mact Load from Water Usn Euleran Scheme wth Laranan Partcles F. 15 Snashots of freely elastc tanker moton n headn wave. There are the stronly nonlnear henomena such as slashn, breakn, slammn and reen water wth the tanker moton. F. 16 shows the stran dstrbuton of the elastc tanker n the headn wave. t can be seen that the hon and san motons were occurred under the mact load due to the stronly nonlnear henomena such as slammn and reen water. F. 17 shows the tme hstory of the stran at the three dfferent onts on the deck (T1, T2 and T3 and bottom (B1, B2 and B3. The hh frequency stran was enerated by the hon and san motons on both faces. 4. Conclusons To nvestate hydroelastc henomena caused by water mact n nonlnear wave moton, the authors have develoed a couled Euleran scheme wth Laranan artcles to combne advantaes and to comensate dsadvantaes n both rd based method and artcle based method. The develoed numercal model was aled to hydroelastc roblems due to mact ressure such as water entry of the elastc cylnder and the elastc tanker moton n headn wave. The numercal results are n ood areement wth the exermental results. The resent model can kee the sharness of the sh-wave nterface and the resent numercal scheme can handle movn free surface moton and elastc sh boundary. t makes clear that the resent model can redct wave mact as hydrodynamc effects due to stronly nonlnear F. 16 Deformaton and stran dstrbuton of the elastc tanker under the hon and san motons. [10-4 ] 1.2 xx Tme[s] (a Deck [10-5 ] 8.0 B1 B2 4.0 B3 xx Tme[s] (bbottom F. 17 Tme hstory of stran on the deck and bottom under the hon and san motons. T1 T2 T3

9 umercal Smulaton on Hydroelastc Resonse of Structure under mact Load from Water Usn Euleran Scheme wth Laranan Partcles 371 sh-wave nteractons. The hydroelastc effects assocated wth caturn nonlnear free surface flows on a sh moton under severe wave condton can be examned usn the resent model. Ths numercal model can be useful and effectveness to evaluate hydroelastcty and moton of a sh body n a structure desn rocess. More detaled valdaton wll be necessary as future work. Acknowledments The authors arecate that the numercal comutatons and dscusson were artly erformed by Mr. Kenta Kawakam and Mr. Koch Hashhra n ths aer. The research was artly suorted by Tsunesh Shbuldn Co., Ltd. and the Fundamental Research Develon Assocaton for Shbuldn and Offshore, REDAS. References [1] E.M. Hauen, Hydroelastc analyss of slammn on stffened lates wth alcaton to catamaran wet-deck, Ph.D. Thess, Det. Marne Hydrodynamcs, orwean Unversty of Scence and Technoloy, 1999, [2] W.K. Meyerhoff, Added mass of thn rectanular lates calculated from otental, Journal of Sh Research 14 ( [3] O.M. Faltnsen, Water entry of a wede by hydroelastc orthotroc late theory, Journal of Sh Research 43 ( [4] O.M. Faltnsen, Hydroelastc slammn, Journal of Marne Scence and Technoloy 5 ( [5]. Senanovc, S. Malenca, S. Tomasevc, nvestaton of sh hydroelastcty, Ocean Enneern 35 ( [6] G. Thomas, S. Wnkler, M. Davs, D. Holloway, S. Matsubara, J. Lavroff et al., Slam events of hh-seed catamarans n rreular waves, Journal of Marne and Scence and Technoloy 16 ( [7] R.A. Gnold, J.J. Monahan, Smoothed artcle hydrodynamcs, theory and alcaton to non-shercal stars, Mon. ot. Roy. Astr. Soc. 181 ( [8] H. Mutsuda, Y. Shnkura, Y. Do, An Euleran scheme wth Laranan artcles for solvn mact ressure caused by wave breakn, n: Proc. of the 18th nternatonal Socety of Offshore and Polar Enneers Conference, Canada, 2008, [9] H. Mutsuda, Y. Shmzu, Y. Do, umercal Study on nteracton Between Volent Wave and Structure Usn SPH, Partcle-Based Methods, Fundamentals and Alcatons, Barcelona, 2009, [10] H. Mutsuda, T. Kurokawa, S. Baso, Y. Do, umercal Smulaton of nteracton Between Wave and Floatn Body Usn Euleran Scheme wth Laranan Partcles, n: Proc. of 4th Euroean Conference on Comutatonal Mechancs (ECCM 2010 [CD-ROM], Pars, [11] S. Baso, H. Mutsuda, T. Kurokakawa, T. Kurokawa, Y, Do, J. Sh, An Euleran scheme wth Laranan artcle for evaluaton of seakeen erformance of sh n nonlnear wave, nternatonal Journal of Offshore and Polar Enneern 21 (2 ( [12] D. Enrht, R. Fedkw, J. Ferzer,. Mtchell, A hybrd artcle level set method for mroved nterface caturn, J. Comut. Phys. 183 (1 ( [13] T. Yabe, P.Y. Wan, Unfed numercal rocedure for comressble and ncomressble flud, Journal of the Physcal Socety of Jaan 60 (7 ( [14] D. Baraff, An ntroducton to Physcally Based Modeln, Rd Body Smulaton Unconstraned Rd Body Dynamcs, Carnee Mellon Unversty, USA, 1997.