Implement of the MPS-FEM Coupled Method for the FSI Smulaton of the 3-D Dam-break Problem Youln Zhang State Key Laboratory of Ocean Engneerng, School of Naval Archtecture, Ocean and Cvl Engneerng, Shangha Jao Tong Unversty, Collaboratve Innovaton Center for Advanced Shp and Deep-Sea Exploraton Shangha 0040, Chna Decheng Wan * State Key Laboratory of Ocean Engneerng, School of Naval Archtecture, Ocean and Cvl Engneerng, Shangha Jao Tong Unversty, Collaboratve Innovaton Center for Advanced Shp and Deep-Sea Exploraton Shangha 0040, Chna * Correspondng author: dcwan@sjtu.edu.cn Abstract In the present study, the coupled movng partcle sem-mplct (MPS) method and fnte element method (FEM) s developed for the 3-D flud structure nteracton (FSI) problems. Heren, the MPS method s employed for the smulaton of flud doman whle the FEM approach s used for the analyss of structural doman. For the mplementaton of the coupled approach, we proposed a mappng algorthm to transfer quantty values between the partcles of flow feld and the elements of structural feld. In ths mappng algorthm, the nonmatchng refnement levels of both domans are permtted, whch mples that the much larger sze of element can be used n the FSI smulaton and the computatonal effcency can be mproved. Wth the beneft of the proposed MPS-FEM coupled method, the 3-D FSI problem of dam-break flow mpactng onto the flexble wall s numercally nvestgated. The evolutons of free surface and the mpactng loads on the wall are compared aganst those regardng rgd tank. In addton, the deformaton and the strength behavors of the flexble wall are exhbted. I. INTRODUTION The flud structure nteracton (FSI) problems wth volent free surface have ganed great attentons snce they are often encountered n many engneerng applcatons, such as the lqud sloshng n an ol tanker, very large floatng structure nteractng wth waves, flexble structures experencng dambreak flows, etc. In the past decades, the grd-based methods are much popular n the contrbutons regardng the smulaton of FSI problems. However, t s qute a challenge for such methods to model phenomena that nvolve complex free surface flows, large deformatons of flexble structures. By contrast, the mesh-less methods are free from these dffcultes. Therefore, the mesh-less methods, cooperatng wth the fnte element method (FEM), are promsng for the FSI problems nvolvng flexble structures and free surface flows. In the recent decade, several representatve mesh-less methods have been proposed for the FSI problems. For nstance, the smoothed partcle hydrodynamcs (SPH) method has been extensvely studed and extended to the FSI problems by couplng wth the FEM method. Snce the SPH method s flexble n descrbng the volent evoluton of the flud free surface, the movement and the deformaton of elastc structures, Rafee and Thagarajan [1] proposed a SPH-based solver for smulaton of the dam-break flow nteractng wth an elastc gate. Lu et al. [] employed an mproved SPH method for modelng the hydro-elastc problems of the water mpactng onto a forefront elastc plate. Besdes, the partcle fnte element method (PFEM) s another good canddate for the FSI smulatons. In ths method, the same Lagrangan formulaton s used for both the flud and the sold analyss, whch ndcates that a monolthc system of equatons could be created for the smultaneous soluton of the flud and structural response. Based on ths monolthc approach, Zhu and Scott [3] smulated the process of sloshng wave nteractng wth a soft beam. In the recent few years, the movng partcle sem-mplct (MPS) method, whch s orgnally proposed by Koshzuka and Oka for ncompressble flow [4], has also been ntroduced nto the flud doman analyss of the FSI problems. Hwang et al. [5] employed the MPS method to nvestgate the sloshng phenomenon n partally flled rectangular tanks wth elastc baffles. Zha et al. [6] developed an mproved MPS method to solve the hydro-elastc response of a wedge enterng calm water. Accordng to these results, the mesh-less methods are of great prospect n the smulaton of -D FSI problems. However, there are very few works about the applcaton of the mesh-less methods on the 3-D FSI problems. In the present study, we devote to extend the mesh-less method for 3-D FSI problems. The MPS s employed for the smulaton of flud doman and the FEM method s used for the analyss of structural dynamc response. In the MPS-FEM coupled method, the spaces of flud doman and structural doman wll be dspersed by partcles and grds, respectvely. Hence, the nterface between the flud boundary partcles and structural grds s somerous and a specal technque for the data communcaton crossng the nterface s necessary. Here, a kernel functon based nterpolaton (KFBI) technque s proposed to meet the requrement. The accuracy of the KFBI technque for the force and dsplacement nterpolatons between the flud and the structural domans s valdated. Then, the MPS-FEM coupled method s appled to the FSI problem of 3-D dam-break flow nteractng wth an elastc tank wall, and the nfluence of structural elastcty on the evoluton of volent free surface s comparatvely nvestgated. 43
SPHERIC Bejng 017 Bejng, Chna, October 17-0, 017 II. NUMERICAL METHODS In the present study, the MPS-FEM coupled method whch s a parttoned coupled approach for the FSI problems s developed. Here, the formulas of MPS method, FEM method and the data nterpolaton on the nterface between the flud and structure domans are brefly ntroduced. A. MPS Method for Flud Analyss In the MPS method for ncompressble vscous flow, the governng equatons whch nclude the contnuty equatons and the Naver-Stokes equatons, should be expressed by the partcle nteracton models based on the kernel functon. Here, the kernel functon presented by Zhang et al. [7] s employed. re Wr () 0.85r 0.15re 0 1 0 r r r e r where r s dstance between partcles and r e s the effect radus. The partcle nteracton models, ncludng the dfferental operators of gradent, dvergence and Laplacan, are defned as j r r r r 0 j W j n j rj r e (1) dm ( ) ( ) () Φj Φ rj r dm Φ ( r ) j r 0 W (3) n r r j j dm ( ) ( ) (4) r r 0 j W j n j n 1 * n n P (1 ) V t t n * 0 0 where γ s a blendng parameter wth a value between 0 and 1. The range of 0.01 γ 0.05 s better accordng to numercal experments conducted by Lee et al.[8] In ths paper, γ=0.01 s adopted for all the smulatons. B. Structure Solver Based on the FEM Method In the FEM method, the deformaton of structure s governed by the dynamc equatons expressed as (6) M y C y K y F( t) (7) C M K (8) 1 where M, C, K are the mass matrx, the Raylegh dampng matrx, the stffness matrx of the structure, respectvely. F s the external force vector actng on structure, and vares wth computatonal tme. y s the dsplacement vector of structure. α 1 and α are coeffcents whch are related wth natural frequences and dampng ratos of structure. To solve the structural dynamc equatons, another two group functons should be supplemented to set up a closedform equaton system. Here, Taylor s expansons of velocty and dsplacement developed by Newmark [9] are employed: y y (1 ) y t y t, 0 1 (9) tt t t tt 1 yt t yt ytt ytt yt tt, 0 1 (10) where β and γ are mportant parameters of the Newmark method, and selected as β=0.5, γ=0.5 for all smulatons n present paper. The nodal dsplacements at t=t+ t can be solved by the followng formulas [10]: where ϕ s an arbtrary scalar functon, Ф s an arbtrary vector, dm s the number of space dmensons, n 0 s the ntal partcle number densty for ncompressble flow, λ s a parameter defned as K y F (11) tt tt K K a M ac (1) 0 1 j W ( r r ) r r j j j W ( r r ) whch s ntroduced to keep the varance ncrease equal to that of the analytcal soluton [4]. In the present MPS method, the pressure Posson equaton (PPE) wth the mxed source term s employed to satsfy the ncompressble condton of flud doman and defned as j (5) F F tt t M( a0 y t a y t a3 yt ) C( a y a y a y ) 1 t 4 t 5 t 1 1 a0, a 1, a, t t t 1 a3 1, a4 1, t a5 ( ), a6 t(1 ), a7 t (13) (14) 44
SPHERIC Bejng 017 Bejng, Chna, October 17-0, 017 where K and F are the so-called effectve stffness matrx and effectve force vector, respectvely. Fnally, the acceleratons and veloctes correspondng to the next tme step are updated as follows. F n P l0 W ( r rn ) W ( r r ) n (17) y a ( y y ) a y a y (15) tt 0 tt t t 3 t y y a y a y (16) tt t 6 t 7 tt C. KFBI Technque for Data Interpolaton For the smulaton of 3D FSI problems based on aforementoned MPS-FEM coupled method, the space of flud doman wll be dspersed by partcles whle the space of structural doman wll be dspersed by grds. In general, the fne partcles should be arranged wthn the flud doman to keep a satsfactory precson for the flud analyss. By contrast, the much coarser grds could be accurate enough for the structure analyss, whch ndcates that the flud partcles are usually not concded wth the structural nodes on the nterface between the flud and structure doman, as shown as Fgure 1. Hence, the somerous nterface between the two domans may result n the challenge of data exchange n the process of FSI smulaton. In the present study, the kernel functon based nterpolaton technque s proposed to apply the external force carred by the flud partcles onto the structural nodes and update the postons of boundary partcles correspondng to the dsplacements of structural nodes. where P s the pressure of the boundary partcle obtaned from the flud doman, l 0 s the ntal dstance between the neghbour partcles. The schematc dagram of the technque for the deformaton of the flud structure nterface s shown n Fgure 3. The flud boundary whch s conssted of partcles wll deform correspondng to the deformaton of structural boundary. The deflecton value of boundary partcle w m can be obtaned by the nterpolaton based on the kernel functons W( r -r m ) and the nodal dsplacement δ. w m W ( r r ) m W ( r r ) m (18) Fgure. Schematc dagram of the force nterpolaton Fgure 3. Schematc dagram of the dsplacement nterpolaton Fgure 1. Interface between the flud and structure domans The schematc dagram of the KFBI technque for the force transformaton from the flud doman to the structural boundary s shown n Fgure. In the KFBI technque, the boundary partcle of the flud doman wll be denoted as the neghbour partcle of the structure node whle the dstance between the partcle and the node s smaller than the effect radus r e of nterpolaton. The weghted value of the flud force of the neghbour partcle W( r -r n ) s calculated based on (1). Then, the equvalent nodal force F n correspondng to the node n s obtaned by the summaton of force components regardng to the neghbour partcles. III. NUMERICAL VALIDATION In ths paper, a FSI solver, whch ncludes the flud doman calculaton module, the structural doman calculaton module and the nterface data nterpolaton module, s developed based on the parttoned couplng strategy. For the flud smulaton, the accuracy of the flud doman calculaton module has been valdated by a seres of benchmarks n the prevous works of our study group [7][11] [1][13][14]. Here, the objectve of the study n the present secton s to valdate the relablty of the other two modules. A. Valdaton of Structural Doman Calculaton Module For the valdaton of the structural doman calculaton module, the numercal test of the response of a square sheet s carred out. The schematc dagram of geometrc dmensons of the square sheet s shown as the Fgure 4. The sde length and the thckness of the sheet are m, 0.001 m, respectvely. The sheet s clamped at all the four edges and dspersed by quadrlateral elements wth sze of 0.1 m. The concentrated force F(t)=10 N s appled at the geometrc center A of the plate n the normal drecton. The detaled calculaton parameters can be found n the Table I. 45
SPHERIC Bejng 017 Bejng, Chna, October 17-0, 017 TABLE I. PARAMETERS FOR STRUCTURAL RESPONSE TEST Structural parameters Values Structure densty (kg/m3) 1800 Young's modulus (GPa) 40 Posson's rato 0.3 Element sze (m) 0.1 Dampng coeffcents α1 0.0 Dampng coeffcents α 0.0 Tme step sze (s) 1 10-3 B. Valdaton of Data Interpolaton Module To valdate the nterpolaton accuracy of the nterface data transformaton module, two numercal tests are carred out n the present secton. The accuracy of the flud force transformaton from the flud doman to the structural doman wll be nvestgated by the frst test. In ths test, the rectangular sheet, whch s wth the length of m and wdth of 0.6 m, s dspersed by the flud partcles and the structural elements, respectvely. The dstance between neghbour partcles s 0.0 m and the sze of element s 0.05 m. The trangular dstrbuted force n the normal drecton s appled onto the partcle model of the rectangular sheet, as shown n the Fgure 7. The value of force carred by the partcles s calculated by Dsplacement (s) Fgure 4. Schematc dagram of the square plane 0.15 0.1 0.09 0.06 0.03 0.00 ANSYS Present F=1000-500x (19) Wth the help of the nterface data nterpolaton module, the force on the partcle model can be transferred to the nodes of the element model. Fgure 8 shows the force dstrbuton on the element model. It can be notced that the present maxmum.493 N approxmates to the theoretcal value.5 N. -0.03 0.0 0.5 1.0 1.5.0 Tme (s) Fgure 5. Tme hstory of the dsplacement at the center of the square plane Fgure 7. Dagrammatc sketch of force dstrbuton on the partcle model Fgure 8. Force dstrbuton on the element model Fgure 6. Deformatons of the square plane at t=0.75 s (Left: ANSYS, rght: present result) Fgure 5 shows the tme hstores of structural vbraton at the geometrc center A of the plate. Good agreement between the results from the present structural doman calculaton module and the ANSYS software can be observed. Furthermore, the structural deformaton contour at the tme 0.75 s s compared aganst that calculated by the ANSYS software, as shown n the Fgure 6. It can be notced that the consstent deformaton form s obtaned by the two structure solver, whch ndcates that the structural doman calculaton module of the present solver s relable and can be appled to the structural dynamc response analyss of the FSI problems. Fgure 9. Dagrammatc sketch of deformaton of the element model 46
SPHERIC Bejng 017 Bejng, Chna, October 17-0, 017 IV. NUMERICAL SIMULATIONS Fgure 10. Dsplacement of the geometrc center pont of the partcle model A. Numercal Setup In ths secton, the numercal smulaton of the flud structure nteracton between the dam-break slammng wave and the elastc wall s carred out. The applcablty of the MPS-FEM coupled method n the 3-D flud structure nteracton problem s nvestgated. Fgure 1 shows the schematc dagram of the computatonal model. All the walls of the tank are rgd except the rght one. Elastc materal s used for the rght wall of the tank and the four edges of the wall are clamped on the adjacent walls. A pressure measurng pont s mounted at the mdpont P of the bottom of the elastc wall, and fve dsplacement measurng ponts (A-E) are arranged above the pont P wth the spacng of 0.1 m. Detaled parameters of the materal property and the numercal condton are shown n Table II. Fgure 11. Deformaton of the partcle model (t=0.1 s) In the second test, the accuracy of the deformaton of the flud boundary n accordance wth that of the structural boundary s nvestgated. The rectangular sheet s dspersed to the same partcle and element models. The structural boundary s forced to deform by settng the nodal velocty of the element model. As shown n Fgure 9, the structural node wll move n the normal drecton and the dstrbuted nodal velocty performs the parabolc shape n both x and y drectons. The nodal velocty can be calculated by TABLE II. Flud Structure FLUID AND STRUCTURAL PARAMETERS OF SIMULATION Parameters Values Flud densty (kg/m 3 ) 1000 Knematc vscosty (m /s) 1 10-6 Gravtatonal acceleraton (s/m ) 9.81 Partcle spacng (m) 0.03 Number of flud partcles 1500 Total number of partcles 44371 Tme step sze (s) 1 10-4 Structure densty (kg/m 3 ) 1800 Young's modulus (GPa) 10 Posson's rato 0.3 Element sze (m) 0.01 Dampng coeffcents α 1 0.05 Dampng coeffcents α 0.0005 Tme step sze (s) 1 10-4 V f ( x) f ( y)cos( t) 100 where f ( x) x x, f ( y) 1 y 9 (0) By the nterface data nterpolaton module, the deformaton values of the boundary partcles can be obtaned. Fgure 10 shows the present nterpolaton value of the deflecton on the geometrc center pont of the partcle model and the theoretcal value of the structure nodal dsplacement. Good agreement can be acheved between the two results. Furthermore, the deformed partcle model and element model are compared at tme 0.1 s, as shown n Fgure 11. It can be found that the shapes of the two models are concdent wth each other. Accordng the results of the two tests, we deem that the nterface data nterpolaton module s accurate n force and deformaton value nterpolaton between the flud and the structure domans. Fgure 1. Schematc dagram of the computatonal model (Unt:m) B. Numercal Results To nvestgate the effect of structural elastcty to the evoluton of free surface, the dam-break flows n both rgd tank and elastc tank are numercal smulated. Fgure 13 shows the comparson of free surfaces n the rgd tank and the elastc tank at four nstants. At the tme 0.8 s, the upward jet flow s generated after the dam-break flow mpactng onto the rght wall of the tank. The heght of jet flow n the elastc tank s slghtly lower than that n the rgd tank. At the nstant 1.4 s, the water front of jet flow falls down and the water surface wth the rollng shape s formed. It can be notced that the shape of rollng water surface approxmates crcular arc n the rgd tank, whle that s closed to an 47
SPHERIC Bejng 017 Bejng, Chna, October 17-0, 017 ellptcal arc n the elastc tank. In addton, more the splashed water partcles n the rgd tank are observed than those n the elastc tank. It may nduced by the energy dsspaton durng the deformaton of the lateral wall. At the nstant 1.6 s, an ar bubble s generated after the rollng of water front. As shown n the sde vew, the bubble n the elastc tank presents rregular three-dmensonal shape whle that n the rgd tank presents the two-dmensonal characterstc. At the tme 1.9 s, the water partcles near the elastc wall are bounced back nto the tank durng the vbraton of the structure, and the separaton between the water and the wall s formed. mpact phenomenon, the frst peak value of pressure on the lateral wall of the elastc tank s 95 Pa, whch s 1% smaller than that regardng the rgd tank (11673 Pa). And at the nstant t 1.56 s, whch s close to the tme that the largest ampltude of structural deformaton occurs, the second peak of pressure s observed. In the followng stage, the pressure hstory actng on the elastc wall presents sgnfcant fluctuaton whch s smlar wth the trend of structural vbraton. It could be nferred that the varyng of mpact load actng on the wall s more complex snce the vbraton of the tank wall. Dsplacement (m) 0.0 0.15 0.10 0.05 A (h=0.1 m) B (h=0. m) C (h=0.3 m) D (h=0.4 m) E (h=0.5 m) 0.00 0.0 0.5 1.0 1.5.0.5 Tme (s) Fgure 14. Tme hstores of vbratons on the elastc wall 1000 10000 Rgd wall Elastc wall 8000 Pressure (Pa) 6000 4000 000 Fgure 13. Evoluton of free surface n sde vew (from top to bottom: t=0.8, 1.4, 1.6 1.9 s) Fgure 14 shows the tme hstores of the structural vbraton at the fve measurng ponts on the elastc wall. It can be notced that the trends of the curves are smlar to each other except the peak values of the vbraton. At the tme 0.6 s, the dam-break wave reaches the rght end of the tank and the elastc wall starts to deform due to the mpact load. As more and more water partcles actng onto the elastc wall, the ampltudes of the deformaton ncrease and reach to the maxmums at the nstant 1.6 s. In the followng stage, the elastc wall vbrates wth the decreasng ampltudes snce the combned acton of the flud load and the structural dampng force. Besdes, the elastc wall vbrates wth the maxmum ampltude n the transverse drecton at the measurng pont C whch s 0.3 m above the bottom of the tank. In the Fgure 15, the tme hstores of mpact loads at the measurng pont P are compared between the elastc tank and the rgd tank. At the ntal nstant (t=0.6 s) of the 0 0.0 0.5 1.0 1.5.0.5 Tme (s) Fgure 15. Tme hstores of mpact loads at the measurng pont P V. CONCLUSIONS In the present study, the MPS-FEM coupled method s developed for 3-D FSI problems. The kernel functon based nterpolaton (KFBI) technque s proposed for the data transformaton on the nterface between the flud and structural domans. To valdate the relablty of the structural doman calculaton module, the numercal test of the vbraton response of a square sheet under the constant force s carred out. Both the tme hstory of structural vbraton and the deformaton form of the sheet are n good agreements wth the results calculated by the ANSYS software. Then, the nterpolaton accuracy of the KFBI technque s valdated by two numercal tests. Concdent quanttes of force and structural deformaton can be acheved between the flud boundary and the structural boundary. Fnally, the MPS-FEM coupled solver s used to smulate the nteracton between the dam-break flow and the 48
SPHERIC Bejng 017 Bejng, Chna, October 17-0, 017 elastc wall. Dfferent phenomena of the dam-break flow can be observed n the elastc tank. For nstance, the qualtatve results show that the evolutons of free surface n the elastc tank present obvously three dmensonal characters n comparson wth those of rgd tank. The quanttatve results show that the mpulse pressure nduced by the dam-break wave mpactng onto the elastc wall s 1% smaller than that regardng the rgd tank. In summary, as a prelmnary attempt, present study develops a fully Lagrangan FSI approach and shows the capablty of the nhouse solver n smulaton the 3-D FSI problems wth volent free surface. ACKNOWLEDGEMENT Ths work s supported by the Natonal Natural Scence Foundaton of Chna (5137915, 51490675, 1143009, 51579145), 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 [1] A. Rafee and K. P. Thagarajan, An SPH projecton method for smulatng flud hypoelastc structure nteracton, Comput. Methods Appl. Mech. Eng., vol. 198, pp. 785 795, 009. [] M. B. Lu, J. R. Shao, and H. Q. L., Numercal smulaton of hydro-elastc problems wth smoothed partcle hydrodynamcs method, Journal of Hydrodynamcs, Ser. B, vol. 5(5), pp.673 68, 013. [3] M. J. Zhu and M. H. Scott, Drect dfferentaton of the quasncompressble flud formulaton of flud structure nteracton usng the PFEM, Computatonal Partcle Mechancs, vol. 4(3), pp. 1 13, 016. [4] S. Koshzuka and Y. Oka, Movng partcle sem-mplct method for fragmentaton of ncompressble flud, Nuclear Scence and Engneerng, vol. 13, pp. 41 434, 1996. [5] S. C. Hwang, J. C. Park, H. Gotoh, A. Khayyer, and K. J. Kang, Numercal smulatons of sloshng flows wth elastc baffles by usng a partcle-based flud structure nteracton analyss method, Ocean Engneerng, vol. 118, pp. 7 41, 016. [6] R. S. Zha, H. Peng, and W. Qu, Solvng D coupled water entry problem by an mproved MPS method, The 3nd Internatonal Workshop on Water Waves and Floatng Bodes, Dalan, Chna, pp. 3 6, Aprl, 017. [7] Y. X. Zhang, D. C. Wan, and T. Hno, Comparatve study of MPS method and level-set method for sloshng flows, Journal of hydrodynamcs, vol. 6(4), pp. 577 585, 014. [8] B. H. Lee, J. C. Park, M. H. Km, S. J. Jung, M. C. Ryu, and Y. S. Km, Numercal smulaton of mpact loads usng a partcle method, Ocean Engneerng, vol. 37, pp. 164 173, 010. [9] N. M. Newmark, A method of computaton for structural dynamcs, Journal of the engneerng mechancs dvson, vol. 85(3), pp. 67 94, 1959. [10] K. M. Hsao, J. Y. Ln, and W. Y. Ln, A consstent co-rotatonal fnte element formulaton for geometrcally nonlnear dynamc analyss of 3-D beams, Comput. Methods Appl. Mech. Engrg, vol. 169, pp. 1 18, 1999. [11] Y. L. Zhang, X. Chen, and D. C. Wan, MPS-FEM coupled method for the comparson study of lqud sloshng flows nteractng wth rgd and elastc baffles, Appled Mathematcs and Mechancs, vol. 37(1), pp. 1359-1377, 016. [1] Y. L. Zhang, Z. Y. Tang, and D. C. Wan, Numercal nvestgatons of waves nteractng wth free rollng body by modfed MPS method, Internatonal Journal of Computatonal Methods, vol. 13(4), pp. 1641013-1 1641013-14, 016. [13] Z. Y. Tang, Y. L. Zhang, and D. C. Wan, Mult-resoluton MPS method for free surface flows, Internatonal Journal of Computatonal Methods, vol. 13 (4), pp. 1641018-1 1641018-17, 016. [14] Y. L. Zhang and D. C. Wan, Numercal study of nteractons between waves and free rollng body by IMPS method, Computers and Fluds, vol. 155, pp.14 133, 017. 49