Brittle Fracture and Hydroelastic Simulations based on Moving Particle Simulation

Transcription

1 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 Brttle Fracture and Hydroelastc Smulatons based on Movng Partcle Smulaton R.A. Amaro Junor 1 and L.Y. Cheng 1 Abstract: In ths paper smulatons of brttle fracture and hydroelastc problems are carred out by usng a numercal approach based on the Movng Partcle Smulaton (MPS) method. It s a meshless method used to model both flud and elastc sold, and all the computatonal doman s dscretzed n Lagrangan partcles. A hgher order accuracy gradent operator s used heren by adoptng a correcton matrx. Also, n order to correctly smulate the collson of the fragments, a contact detecton algorthm that takes nto account the presence of the sold surfaces generated by brttle fracture s proposed. In case of flud-structure nteracton, a parttoned couplng between flud and sotropc elastc sold s adopted. Explct and sem-mplct tme ntegraton algorthms are used for elastc sold and flud domans, respectvely. Matchng of the tme steps n both domans s done by subcyclng technque to mprove the computatonal effcency. The performance of method s evaluated by analyzng the results of several cases of study. At frst, the dynamcs of slender cantlever beam s analyzed to check the convergence of the method. After that, a collson between two elastc solds wth brttle fracture and further collson between the fragments s smulated to show the mprovement acheved. Fnally, to valdate the flud-structure nteracton smulaton approach, comparson wth avalable numercal results of a dam breakng on an elastc plate s performed, as well as the comparson wth avalable expermental measurements of the nteracton between lqud sloshng and an elastc plate. Keywords: Flud-structure nteracton, brttle fracture, elastc sold, movng partcle smulaton, partcle method, hydroelastcty. 1 Introducton Over the last 50 years, numercal smulaton enabled the study of complex physcal systems n several areas of scences and engneerng. The most commons meth- 1 Department of Constructon Engneerng, Polytechnc School, Unversty of São Paulo, São Paulo, SP, Brazl.

2 88 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 ods are the Fnte Dfference Method (FDM), Fnte Volume Method (FVM), Fnte Element Method (FEM) and Boundary Element Method (BEM). However, generally these mesh-based methods face several restrctons when the problems nvolve large deformaton or dsplacement of the boundares, such as free surface, fragmentaton, mergng or multbody nteractons, and ngenous boundary trackng or remeshng technques are requred, ncreasng the computatonal complexty. Due the easy mplementaton and flexblty, the meshless methods has attracted much attenton n recent years. An mportant class of meshless method are the partcle-based methods, where the behavor of a physcal problem s represented by a collecton of ponts (partcles). In a partcle method each partcle moves accordngly wth ts own mass and the nternal/external forces evaluated by the nteracton wth the neghborng partcles [Idelsohn and Onate (2006)]. Examples of partcle methods utlzed n dfferent feld of scences and engneerng are: Partcle n Cell (PIC) [Harlow (1964)], Smooth Partcle Hydrodynamcs (SPH) [Lucy (1977); Gngold and Monagham (1977)], Materal Pont Method (MPM) [Sulsky, Chen, and Schreyer (1994)], Movng Partcle Sem-mplct (MPS) [Koshzuka, Tamako, and Oka (1995)] and Partcle Fnte Element Method (PFEM) [Idelsohn, Onate, and Pn (2004)]. In the recent years, problems nvolvng dynamc flexble bodes and ther nteracton wth free surface flow have receved great attenton n several feld of engneerng and scence. Nevertheless, ther computatonal modelng remans as a great challenge. Regardng the fracture phenomenon, t s governed by nteracton of mcro and macro-vods, mcrostructural defects and ntal flaws, and nvolve multple physcal processes occurrng at dfferent tme and length scales [Das and Cleary (2010)]. Generally, when nvestgatng fracture problems usng mesh-based methods, several shortcomngs, such as the mpossblty of crack propagaton along the element s edge and mesh dstortng may occur. To overcome the constrants mposed by mesh, partcle methods have been proposed to model the fracture phenomenon. For example, Guo and Narn (2006) descrbed algorthms for three-dmensonal dynamc stress and fracture analyss usng MPM. Das and Cleary (2010) used SPH to modelng breakage of rocks under mpact, common n many ndustral processes. Xao, Han, and Hu (2011) presented a FEM-SPH couplng algorthm and proposed an adaptve couplng technque to smulate mpact problems nvolvng large deformatons and fractures. Chen, Wang, Xe, and Qn (2013) presented a hybrd approach based on SPH for rapd crack smulaton of brttle materal n physcs-based anmaton area. On the other hand, n the case of flud-structure nteracton (FSI) problems, a couplng strategy to satsfy both the geometrcal compatblty and the equlbrum con-

3 Brttle Fracture and Hydroelastc Smulatons 89 dtons on the nterface s a key ssue [Ishhara and Yoshmura (2005)]. The man approaches employed n the soluton of FSI problems are roughly dvded nto the monolthc methods, n whch fully coupled flud-structure nteracton n the nterface s handled synchronously, and the parttoned methods, n whch the equatons of flud and structure are alternately ntegrated n tme, and nterface condtons are handled asynchronously. For numercal smulatons of hydroelastc problems nvolvng free surface, the Lagrangan formulaton have been wdely used, especally numercal approach based on partcle methods. Idelsohn, Onate, and Pn (2004) modeled nteractons between floatng and submerged bodes and free surface flows by usng PFEM. Antoc, Gallat, and Sblla (2007) studed expermentally and numercally based on SPH method, the deformaton of an elastc plate under the effect of a rapdly varyng flud flow. Campbell, Vgnjevc, Patel, and Mlsavljevc (2009) presented a explct SPH-fnte element approach to smulate floatng body, and showed the ablty of the numercal approach to represent large structural deformaton due to water mpact. Wthn ths context and to take advantage of partcle methods, a practcal computatonal technque for modelng and smulaton of brttle fracture and hydroelastc problems s presented n ths paper. The numercal approach s based on the Movng Partcle Smulaton (MPS) method, n whch all the computatonal doman, ncludng sold and flud, s dscretzed n Lagrangan partcles. Due to fully Lagrangan descrpton, numercal dffuson caused by advecton term does not arse. As a meshless method, t s very effectve for the smulaton of problems nvolvng complex deformatons of the boundares and has been wdely appled n several engneerng problems, for example: Nonlnear hydrodynamcs such as green water [Shbata, Koshzuka, Saka, and Tanzawa (2012); Bellez, Cheng, and Nshmoto (2013)], sloshng [Lee, Jeong, Hwang, Park, and Km (2013)], coupled moton of sloshng and vessel [Lee, Park, Km, and Hwang (2011); Km, Lee, Km, and Park (2011)], and sloshng moton coupled to a movng suppresson devce [Tsukamoto, Cheng, and Nshmoto (2011)]; Multphase flows [Koshzuka, Ikeda, and Oka (1999); Park and Jeun (2011); Cheng, Gomes, Yoshno, and Nshmoto (2011)]; Elastc sold [Chkazawa, Koshzuka, and Oka (2001); and other ndustral problems [Kakuda, Ushyama, Obara, Toyotan, Matsuda, Tanaka, and Katagr (2010); Cheng, Olvera, Favero, Olvera, and Gonçalves (2013); Motezuk and Cheng (2013)] and bomedcal engneerng applcatons [Chhatkul, Koshzuka, and Uesaka (2009); Nagayama and Honda (2012)]. In the present paper, a hgher order accuracy gradent operator s used as an attempt to mprove the accuracy. Also, n order to correctly smulate the collson of the fragments, a smple condton of brttle fracture assocated to a more generc contact detecton algorthm that takes nto account the presence of the sold surfaces

4 90 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 generated by brttle fracture s proposed. To solve hydroelastc problems nvolvng free surface, a parttoned couplng between flud and sotropc elastc sold s adopted. The sold surface partcles are treated lke a flud partcle and the pressures of the sold surface partcles are computed by solvng Posson equaton for the pressure, together wth the flud partcles. Therefore, the couplng between sold and flud s done at frst by usng the dsplacement and velocty of elastc sold as the boundary condtons for the flud doman. Then the pressure on the sold surface partcles s obtaned by solvng the flud moton. After that the calculated pressure feld s taken nto account n the calculaton of the moton of the elastc sold. Explct and sem-mplct tme ntegraton algorthms are used for elastc sold and flud domans, respectvely. Matchng of the tme steps n both domans s done by sub-cyclng technque to mprove the computatonal effcency. In the next sectons, a bref descrpton of the MPS model for both flud and elastc sold s presented, as well as the brttle fracture condton, the contact detecton algorthm and the couplng technque between flud and elastc sold. Fnally, the performance of method s evaluated by smulatons of several cases of study. At frst, the dynamcs of slender cantlever beam s analyzed to check the convergence of the method. After that, a collson between two elastc solds wth brttle fracture and further collson between the fragments s smulated to show the mprovement acheved. Fnally, to valdate the flud-structure nteracton smulaton approach, comparson wth avalable numercal results of a dam breakng on an elastc plate s performed, as well as the comparson wth avalable expermental measurements of the nteracton between lqud sloshng and an elastc plate. 2 Governng equatons 2.1 Flud The governng equatons of ncompressble vscous flow are expressed by the conservaton laws of mass and momentum: Dρ Dt + ρ( v) = 0 (1) Dv Dt = 1 ρ P + ν 2 v + f (2) where ρ s the densty, v s the velocty vector, P s the pressure, ν s the knematc vscosty, f s the external force vector.

5 Brttle Fracture and Hydroelastc Smulatons Sold For the dynamcs analyss of elastc solds, the governng equaton of moton can be wrtten as: ρ Dv Dt = (2µε + λtr(ε)i) + b (3) where ε s the stran tensor and b s the body force vector. The Lamé s constants µ and λ are gven by: µ = E 2(1 + v) λ = Ev (1 + v)(1 2v) (4) where E s the Young s modulus and v s the Posson s rato. The Eq. 3, can be rewrtten by ntroducng stress tensor σ and sotropc pressure p as: ρ Dv Dt = (σ pi) + b (5) where stress tensor and sotropc pressure are calculated as: σ = 2µε p = λtr(ε ) (6) 3 Numercal model In MPS method, the dfferental operators of the governng equatons are replaced by dscrete dfferental operators on rregular nodes [Isshk (2011)], whch are derved from a model of nteracton between partcles. For a gven partcle, the nfluence of a neghbor partcle j s defned by weght functon w( r j ) gven n Eq. 7: { re r w( r j ) = j 1, r j r e (7) 0, r j > r e where r e s the effectve radus that lmts the range of nfluence and r j s the dstance between and j. In the other words, r e defnes the neghborhood of the partcle. The summaton of the weght of all the partcles n the neghborhood of the partcle s defned as ts partcle number densty n : n = w( r j ) j (8)

6 92 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 As a result, for a scalar functon φ and a vector functon φ, the gradent, dvergence, rotaton and Laplacan operators can be defned by Eq. 9, Eq. 10, Eq. 11 and Eq. 12, respectvely: φ = d (φ j φ ) n 0 r j r 2 (r j r )w( r j ) (9) j φ = d (φ j φ ) (r j r ) n 0 j r j r 2 w( r j ) (10) φ = d n 0 j (φ j φ ) s j w( r j ) (11) r j r 2 φ = 2d λ l n 0 (φ j φ )w( r j ) (12) j where d s the number of spatal dmensons, s j s a versor perpendcular to r j and n 0 s the ntal value of n. In case of flud, the ntal value of n s ndependent to the partcle so that n 0 can be used nstead of n 0. Fnally, λ l s a correcton parameter so that the varance ncrease s equal to that of the analytcal soluton, and s calculated by: λ l = j r j 2 w( r j ) j w( r j ) (13) Adoptng correcton technque, ntally proposed for the SPH method [Bonet and Lok (1999)], the gradent operator (Eq. 9) can be mproved by ntroducng a correcton matrx A: φ = A A = [ ] 1 (φ j φ ) (r j r ) n j r j r r j r w( r j ) (14) [ ] 1 1 (r j r ) n j r j r (r j r ) r j r w( r j ) (15) For two-dmensonal cases analyzed heren, r e s set to 2.1l 0 to calculate gradent, dvergence and rotaton operators and r e s set to 4.0l 0 to calculate Laplacan operator, where l 0 s the ntal dstance between two adjacent partcles.

7 Brttle Fracture and Hydroelastc Smulatons 93 4 Soluton algorthm 4.1 Algorthm for ncompressble vscous flow To solve the ncompressble vscous flow, a sem-mplct algorthm s used n the MPS method. At frst, predctons of the partcle s velocty v and poston r are carred out explctly by usng vscosty and external forces terms of the momentum conservaton (Eq. 2) [Ikeda, Koshzuka, Oka, Park, and Sugmoto (2001)]: v r = v n + t ( ν 2 v n + f ) = r n + tv (16) (17) As the flud densty ρ s proportonal to the partcle number densty n, by applyng the conservaton law of mass (Eq. 1), we have: v = 1 t ρ n+1 ρ 0 ρ = 1 t n n+1 n n 0 (18) where n s the partcle number densty calculated after the movement of partcles n the frst predcton step and the devaton of the velocty v s owng to the mplct pressure term n the momentum conservaton equaton: v = t ρ P n+1 By substtutng Eq. 18 nto Eq. 19, and consdered ncompressble flow,.e. n n+1 = n 0, the Posson equaton for the pressure can be wrtten as: 2 P n+1 (19) = ρ n 0 n t 2 n 0 (20) In ths work, t s assumed that densty s a lnear functon of pressure: n n+1 n 0 = ρn+1 ρ 0 = 1 + P n+1 ρc 2 (21) where c s the sound velocty. By substtutng Eq. 21 nto Eq. 20, we have: 2 P n+1 = ρ ( n 0 n ) t 2 n 0 + αp n+1 where α = 1/(ρc 2 ). Assumng a weakly compressble model of the Eq. 22, the numercal soluton of the system of equaton s faster than the Eq. 20 n each tme step, mprovng the computatonal effcency. Compared wth ncompressble results, the weakly compressble (22)

8 94 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 condton has a neglectable effect on the accuracy of the results [Shakbaena and Jn (2010)]. Also, as the pressure computatonal s relatvely senstve to small varaton of n, a relaxaton coeffcent κ s used for mprovng stablty of a computaton method and Eq. 22 s rewrtten as: 2 P n+1 αp n+1 = κ ρ t 2 n 0 n n 0 (23) The value of κ = 0.01 s used n ths study. From Eq. 19, the velocty v n+1 of a partcle s updated and the poston r n+1 can be obtaned by: r n+1 = r n + tv n Algorthm for elastc sold For elastc sold, an explct algorthm s used n the MPS method [Chkazawa, Koshzuka, and Oka (2001)]; [Song, Koshzuka, and Oka (2003)]. Frst the dsplacement vector u j between partcles and ts neghbor j s calculated by: (24) u j = r j R j r 0 j (25) where R j s the rotaton matrx and r 0 j the ntal poston vector. The dsplacement vector can be dvded nto normal (u j ) n and shear (u j ) s components: (u j ) n = (u j r j ) r j r j r j (26) (u j ) s = u j (u j ) n (27) Thus the normal ε j and shear γ j stran vectors can be calculated by: ε j = (u j ) n r 0 j (28) γ j = (u j ) s r 0 j (29) The volumetrc deformaton tr(ε) s descrbed usng the dvergence of dsplacement vector: tr(ε) = (u j ) = d u j r j n 0 r 0 j r j w( r 0 j ) (30) j

9 Brttle Fracture and Hydroelastc Smulatons 95 From the stran vectors, the normal stress vector σ j, shear stress vector τ j and sotropc pressure p are calculated as: σ j = 2µε j (31) τ j = 2µγ j p = λtr(ε) For elastc sold, a modfed dfferental operators s used n governng equaton. A functon φ s calculated n the ntermedate poston between two partcles and j, wth the varable r j replaced by (r + r j )/2, and the gradent operator (Eq. 9) s rewrtng as: φ = 2d n 0 j φ j (32) (33) r j r 2 (r j r )w( r j ) (34) where φ j s the functon calculated n the ntermedate poston. Smlar of the gradent operator, the dvergence operator can be calculated by: φ = 2d n 0 (φ j ) (r j r ) j r j r 2 w( r j ) (35) Applyng Eq. 5, translaton of partcles can be obtaned from dvergence of the dsplacement (normal and shear stress vector) and sotropc pressure gradent: ( ) v ρ = 2d σ j t 0 n n 0 r 0 j w( r j ) (36) ρ ρ ( ) v = 2d t s n 0 ) ( v t p = 2d n 0 j j j τ j 0 r 0 j w( r j ) (37) p j r j r 0 j r j w( r 0 j ) (38) where p j = (p + p j )/2 s the average pressure. Velocty v n+1 and poston r n+1 of a partcle can be explctly calculated as: [ ( v ) v n+1 = v n + t t + n ( ) v + t s ( ) ] n v (39) t p r n+1 = r n + tv n+1 (40)

10 96 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 To ensure the conservaton of angular moment, the force due the shear stress vector s taken n account for the rotaton of partcles, n order to cancel the torque between each par of partcles. The force can be calculated as Eq. 41 and the moment can be wrtten as Eq. 42. F j = 2d ld 0 n 0 τ j 0 r 0 j w( r j ) (41) M j = (r j r ) F j (42) If the moment of nerta I s constant along the tme, the angular acceleraton vector of partcles can be calculated as: I ω t = 1 2 j M j (43) where the moment of nerta I s calculated as: I = m l2 0 6 = ρl2+d 0 6 (44) Fnally, angular velocty ω n+1 and rotaton θ n+1 of a partcle are explctly calculated as: ( ) n ω n+1 = ω n ω + t (45) t θ n+1 = θ n + tω n+1 (46) 4.3 Boundary Condtons In order to dentfy free surface partcles, the partcle number densty and the number of neghborng partcles are used as checkng parameters. A partcle s defned as free surface partcle and ts pressure s set to zero when ts partcle number densty n s smaller than β 1 n 0 and ts number of neghborng partcles s smaller than β 2 N 0, where N 0 s the number of neghborng partcles nsde the effectve radus r e n the ntal dstrbuton. The value of β 1 used n ths study s 0.97 and β 2 s 0.85, based on Lee, Park, and Km (2010). Ths double check technque mproved the effectveness of the free surface partcle detecton, whch results n more stable computaton of the flud pressure. Sold wall boundary condton s represented by three layers of fxed partcles. The partcles that form the layer n contact to the flud are denomnated wall partcles,

11 Brttle Fracture and Hydroelastc Smulatons 97 of whch the pressure s computed by solvng Posson equaton for the pressure (Eq. 23), together wth the flud partcles. The partcles that forms two other layers are denomnated dummy partcles. Dummy partcles are use to assure the correct calculaton of the partcle number densty of the wall partcles. Pressure s not calculated n the dummy partcles. As the boundary condton for a fxed end of elastc sold, the motons of the partcles are constraned by settng the dsplacements to zero. 4.4 Brttle fracture A smplfed condton of brttle fracture s mplemented n order to smulate multbody dynamcs wth rupture of brttle solds. When the stran ε j between partcles and j s greater than a crtcal value ε max, the weght functon between the partcles and j subjected to fracture s set to zero, Eq. 47. Thus the connecton between the partcles s lost, characterzng the brttle fracture pont of the sold. ε j > ε max = w( r j ) = 0 (47) To avod problems of overlappng among fractured partcles, that subsequently can collde each other, a more generc contact detecton algorthm s proposed heren. Intally the partcles that forms the solds are dvded nto surface and nternal partcles. Smlar to the condton of free surface for flud [Koshzuka and Oka (1996)], the partcle number densty n s used as crteron. If the value of n s lower than a certan value, Eq. 48, the partcle s classfed as a sold surface partcle. n < n max n c (48) where n max s the maxmum value of partcle number densty and n c s a constant value. In the present work n c = 0.5 s adopted. Ths reclassfcaton of the sold partcle n surface partcle and nternal partcle s carred out n every tme step. In practce, for each partcle, the weght functons between the partcle and ts ntal neghborng partcles are stored n a matrx, together the ndex of neghbor partcles. In case of fracture of a nternal sold partcle and ts neghbor j, snce the dstance between them ncreases, ther partcle number densty decrease. As a result, they are reclassfed as surface partcles, as shown n Fg. 1, and the weght functon between the partcles and j s set to zero. As rebndng of fractures surfaces s not consdered n the present study, once a sold partcle s classfed as surface one, t s not reversble. In order to check the occurrence of collson between a par of surface partcles, the relatonshp and the weght functon between the partcles are checked frst. If

12 98 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 Fgure 1: Surface partcles (red) and nternal partcles (orange) durng fracture. the par of surface partcles are neghbors and the weght functon s dfferent of zero, t means that they are adjacent partcles on the surface of a sold. Then the par of partcles are treated lke nternal partcles and no specal treatment for the collson s performed. In case the partcles are neghbors and the weght between them s zero, the nteracton between the par of partcles s taken nto account when the dstance between them s less than the ntal dstance l 0,.e., when they are n compresson. Ths results n repulsve forces actng between the partcles. By applyng ths collson check and treatment, new surface partcles assocated to the newly formed sold surface due fracture can be detected dynamcally, so that overlappng of surface partcles belongng to dfferent surfaces can be avoded, as well as overcomng the problem of msdetecton of a par of adjacent partcles of a sold surface as partcles that belong to two dstnct surfaces n collson. 4.5 Flud-structure nteracton In case of flud-structure nteracton, a parttoned weak couplng s adopted. When solvng the flud moton, the sold surface partcles are treated lke a flud partcle and ther pressures are computed by solvng Posson equaton for the pressure (Eq. 23), together wth the flud partcles. After that, the calculated pressure feld of the sold surface partcles and ts neghborng flud partcles s taken nto account to determne the moton of elastc sold by rewrtten the momentum (Eq. 5) for sold surface partcles as: ρ Dv Dt = (σ pi) + b + F f s (49) where the body force vector F f s s calculated usng the average pressure of the partcles and j as: F f s = 2d ñ j P j r j r 2 (r j r )w( r j ) (50)

13 Brttle Fracture and Hydroelastc Smulatons 99 where P j = (P + P j )/2 and j represents flud and sold surface partcles n the neghborhood of the partcle, excludng the dummy partcles. ñ s the partcle number densty computed consderng only the neghborhood flud and sold surface partcles. For a sold surface partcle, ñ can be obtaned by subtractng ts ntal partcle number densty calculated consderng only sold partcles (n 0 ) sold from the partcle number densty calculated consderng all of ts neghbor partcles (n ) total : ñ = (n ) total (n 0 ) sold (51) Fg. 2 shows the couplng between the flud and sold partcles. Only flud and sold surface partcles (blue + brown) wthn the neghborhood range r e are consdered for the calculaton of the force F f s on the sold surface partcles (brown) whle the sold partcles (orange) wthn the neghborhood range r e are neglected. Fgure 2: The nteracton among flud, sold surface and sold partcles. On the other hand, to complete the couplng between sold and flud, the dsplacement and velocty of elastc sold are used as the boundary condtons of the flud. Ths procedure s repeated for each tme step. 4.6 Numercal stablty In order to avod numercal nstabltes, stablty crtera should be satsfed n the smulatons. An mportant stablty crteron utlzed n Computatonal Flud Dynamcs s the CFL condton [Courant, Fredrchs, and Levy (1967)]. The CFL condton adopted n ths work s gven by the followng equaton: t < l 0C v max, 0 < C 1 (52) where, t s the tme step, v max the maxmum velocty and C the Courant number.

14 100 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 A relevant aspect of the couplng algorthm s the tme steps of flud t f and sold t s. The transent response for each materal has a dfferent tme scales and n most cases the CFL condton requres a much smaller tme step for elastc sold than ncompressble flow. To avod the use of very small tme step for the flud doman, whch s much more tme consumng due to the soluton of the system of Posson equaton for the pressure (Eq. 22), and ncreasng computatonal cost, a sub-cyclng algorthm of elastc sold s adopted durng the smulaton. 5 Results and dscussons In order to evaluate the performance of the method, several example smulatons are carred out. Frst, a cantlever beam s smulated and the maxmum dsplacement and the natural frequency obtaned by the numercal smulaton are compared to analytcal results, allowng to verfy the convergence of the method. After that, the smulatons of a collson between two elastc solds are carred out wthout and wth the proposed contact detecton algorthm to show the mprovement when nvolvng brttle fracture. Fnally, two FSI problems are analyzed. The frst case presents a dam-break problem, where the collapsng water column hts a fxed elastc plate and horzontal dsplacements obtaned by the numercal smulatons are compared wth results avalable n the lterature, obtaned by other numercal methods. The second case conssts of lqud sloshng nsde a rectangular tank wth a fxed elastc plate. For ths case two fllng levels are consdered. The horzontal dsplacements of the elastc plate obtaned by numercal smulatons are compared to the results provded by the experment performed by Idelsohn, Mart, Souto- Iglesas, and Onate (2008b). 5.1 Cantlever beam As a smple dynamc case, the frst mode shape of a cantlever beam of length l = 2.0 m wth square cross secton b = 0.1 m, s consdered. The materal propertes are densty ρ = 1000 kg/m 3, Young s modulus E = 100 MPa and Posson s rato ν = 0.3. In order to nvestgate the convergence of the method, dfferent dstance of partcles (dp) are consdered d p = 20.00, 10.00, 5.00, 2.50 and 1.25 mm, wth rato b/d p = 5, 10, 20, 40 and 80. All cases are smulated wth tme step t = 10 6 s. The computed results are compared to analytcal results of maxmum ampltude max and natural frequency f, approxmated by Eq. 53 and Eq. 54, respectvely. max = ρgal4 8EI (53) f = EI 2π ρal 4 (54)

15 Brttle Fracture and Hydroelastc Smulatons 101 where A s the cross secton area. The plotted results of dsplacement can be seen n the graphc presented n Fg. 3 and the results of natural frequency, maxmum ampltude and dscrepancy between computed results and analytcal results of maxmum ampltude, for the frst mode shape, are shown n Tab. 1. Fgure 3: Maxmum deflecton of the cantlever beam. Table 1: Results of the convergence test for the cantlever beam. Model b/d p f (Hz) max (cm) Dscrepancy (%) MPS (d p = mm) MPS (d p = mm) MPS (d p = 5.00 mm) MPS (d p = 2.50 mm) MPS (d p = 1.25 mm) Analytcal soluton Results of natural frequency obtaned by the smulatons show excellent agreement compared wth the analytcal soluton. The computed maxmum ampltude shows a good agreement wth analytcal solutons, especally for d p = 1.25 mm, rato b/d p = 80, wth relatve dscrepancy about 1.36%. Also, the results show convergence of the numercal approach for the dynamc smulaton. Even for the roughest case of the relatvely long elastc beam consdered heren, wth b/d p = 5, the dscrepancy of the maxmum deflecton s about 5.00%.

16 102 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , Brttle fracture smulaton Fg. 4 shows ntal condtons of the case of collson between two elastc solds nvolvng brttle fracture. A cube (magenta) 0.5 x 0.5 x 0.5 m wth ntal velocty v y = 15 m/s colldes a block (blue) 1.0 x 0.25 x 2.0 m ntally wthout moton. The materal propertes of the cube are densty ρ = 1000 kg/m 3, Young s modulus E = 6 MPa and Posson s rato ν = 0.3 and the materal propertes of the block are densty ρ = 1000 kg/m 3, Young s modulus E = 10 MPa and Posson s rato ν = 0.3. The smulaton parameters are partcle dstance d p = 0.05 m and tme step t = 10 6 s. The crtcal dstance of fracture s ε max = 0.2. Fgure 4: Intal condtons of the case of collson. Fg. 5 gves the snapshots of the relevant nstants of the smulatons. The left and the rght columns show, respectvely, fracture wthout and wth the proposed contact detecton. The collson of two elastc solds occur at 0.07 s. After the fracture occurs n the left sde of the block at 0.60 s, the collson between the newly formed fracture surfaces s detected at 0.70 s. The repulson due to collson between partcles of fractured surface s vsble n the case wth contact detecton at 1.00 s. As a result, the parts of block move away from each other. Meanwhle as shown n the left column of Fg. 5, f collson detecton between the surface created by fracture s not performed, a new and stronger collson between the fractures surfaces occurs

17 Brttle Fracture and Hydroelastc Smulatons 103 at 1.70 s. Due of the absence of collson detecton and treatment of the fracture surfaces, the smulaton dverged owng to the overlap of the partcles s 0.60 s 0.70 s 1.00 s 1.70 s 1.90 s Fgure 5: Snapshots of the man nstants of the fracture due to collson between a cube and a block. Left: smulaton carred out wthout the contact detecton; Rght: smulaton carred out wth the contact detecton.

18 104 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , Dam-break on elastc plate A dam-break problem, smlar to the case nvestgated by Koshzuka, Tamako, and Oka (1995) usng a rgd obstacle, s smulate wth a fxed elastc plate, allowng the nvestgaton of nteracton between flud and elastc sold. The ntal confguraton of the problem s shown n Fg. 6. The dmensons of elastc plate, whch s hghlghted n orange, are 12.0 x 80.0 mm. The physcal propertes of the elastc sold are: densty ρ = 2500 kg/m 3, Young s modulus E = 1 MPa and Posson s rato v = 0.0. The physcal propertes of the flud are: densty ρ = 1000 kg/m 3 and knematc vscosty ν = 10 6 m 2 /s. As smulaton parameters, the values n Tab. 2 are consdered. The smulatons are performed for 2 s. Table 2: Dam-break on elastc plate. Smulaton parameters. d p (mm) t f lud (s) t sold (s) Flud partcles Sold partcles Fgure 6: Dmensons and ntal condtons of dam-break on elastc plate. Fg. 7, Fg. 8 and Fg. 9 show the tme hstory of the horzontal dsplacement of the top of the elastc plate obtaned by the present smulaton wth d p = 3.00, 1.50 and 0.50 mm, respectvely, and the results of the others methods avalable n the lterature [Walhorn, Kolke, Hubner, and Dnkler (2005); Mart, Idelsohn, Lmache, Calvo, and D Ela (2006); Idelsohn, Mart, Lmache, and Onate (2008a); Amanfard, Hesan, and Rahbar (2011)]. Comparng the result obtaned by the present

19 Brttle Fracture and Hydroelastc Smulatons 105 smulaton wth those from other methods, t may be noted that ntally all of the results has the same tendency. The maxmum dsplacement of 4.1 cm of the present smulaton s almost dentcal to that obtaned by FEM and SPH, but slghtly lower than the results from PFEM computatons. For d p = 3.00 mm, after the nstant 0.4 s the dsplacement obtaned by the present smulaton remans n 2.1 cm, hgher than the results of the others methods. From 0.6 s the oscllatng behavor of the curve resembles the curve of SPH, but wth dfferences n dsplacement ampltudes. For d p = 1.50 mm, after the nstant 0.4 s the dsplacement obtaned by the present smulaton remans n 1.7 cm whch s smlar to the results from SPH and PFEM smulatons. From 0.6 s, the curve shows less vbraton cycles and the obtaned result s relatvely close to the SPH result. Fnally, the smulaton wth d p = 0.50 mm presents dstnct decreasng curve from the other two smulatons wthout concavty between the nstants 0.3 s and 0.6 s, reachng a value of 1.7 cm at the nstant 0.6 s. After the nstant 0.6 s, two peaks of 1.1 cm followed by a peak of 1.1 cm and another peak 0.9 cm are computed. Each numercal method exhbt slghtly dstnct oscllatng behavor and the dsplacement calculated by the present smulaton based on the MPS method shows less vbraton cycles. Also, the obtaned result s relatvely close to the SPH result. In addton to ths, the oscllatons tend to a pont of equlbrum between flud and structure due to the dampng of elastc plate on the flud. Fgure 7: Horzontal dsplacement of the elastc plate d p = 3.00 mm. Fgure 8: Horzontal dsplacement of the elastc plate d p = 1.50 mm. Fg. 10 shows a sequence of frames from the smulaton obtaned by SPH [Rafee and Thagarajan (2009)], PFEM [Idelsohn, Mart, Lmache, and Onate (2008a)] and the smulatons carred n the present study. It can be observed a good agreement between the methods. As there are no expermental results of the present case, takng nto account the physcs of the problem, the results show that, compared

20 106 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 Fgure 9: Horzontal dsplacement of the elastc plate d p = 0.50 mm. wth the methods currently avalable, the present approach s also able to reproduce the man behavors of the problems nvolvng free surface flow and elastc solds. 5.4 Sloshng wth an elastc plate The case conssts of a rectangular tank partally flled wth ol, and an elastc plate fxed at the mdpont of the tank bottom. It s a 2D experment performed by Idelsohn, Mart, Souto-Iglesas, and Onate (2008b). The dmensons and ntal confguraton of the experment are llustrated n Fg. 11. The tank s subjected to a oscllatory moton of ampltude α = 4, around the mdpont of the bottom, and perod T obtaned by the followng equaton: ( ( ) ) 1 πg T = 2π L tanh πhl L where L s the tank length and H L s the fllng level. Two fllng levels are consdered and the horzontal dsplacements of the elastc plate obtaned by numercal smulatons are compared to the expermental measurements. The horzontal dsplacement (x) of the elastc plate s relatve to the local reference frame fxed to the tank, Fg. 12. It should be noted the ntal dfferences between the experment and numercal smulaton due to the nerta of the tank. In the experment, the transton from ntal statc state to harmonc moton occurs gradually, whle n the numercal smulaton, ths transton s nstantaneous. Thus, n the early stages, the expermental measurement and the numercal results are dfferent. In order to compare the results dsregardng the transent responses, the results obtaned by the present smulaton (55)

21 Brttle Fracture and Hydroelastc Smulatons 107 Fgure 10: Dam-break on elastc plate: comparson between SPH [Rafee and Thagarajan (2009)], PFEM [Idelsohn, Mart, Lmache, and Onate (2008a)] and present MPS (0.14 s, 0.16 s, 0.26 s, 0.34 s, 0.42 s, 0.62 s, 0.80 s, 1.48 s).

22 108 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 Fgure 11: Dmensons and ntal condtons of sloshng. Fgure 12: Local reference frame fxed to the tank. are shfted n tme to match the frst peak of the expermental and numercal results after achevng steady state Elastc plate mmersed n low fllng The tank s flled to H L = 57.4 mm and subjected to a oscllatory moton of perod T = s. The dmensons of elastc plate, whch s hghlghted n orange color, Fg. 11, are 4.0 x 57.4 mm and the physcal propertes are: densty ρ = 1100 kg/m 3, Young s modulus E = 6 MPa and Posson s rato v = The physcal propertes of the flud are: densty ρ = 917 kg/m 3 and knematc vscosty ν = 5x10 5 m 2 /s. As smulaton parameters, the values n Tab. 3 are consdered. The smulatons are performed for 10 s. Table 3: Elastc plate mmersed n low fllng. Smulaton parameters. d p (mm) t f lud (s) t sold (s) Flud partcles Sold partcles Fg. 13 gves the computed tme hstory and the measured results of the horzontal dsplacement of the top of the elastc plate. Due to the dfferences n exctaton n the early stages of the experment and numercal smulaton, as mentoned above, the expermental values have a dsplacement of 0.12 cm at 0.48 s and 0.47 cm at 0.92 s, whle the numercal result presents a dsplacement of 0.8 cm at 1.00 s. After the nstant 1.00 s, the perod of oscllaton of the elastc plate s, approxmately, 1.70 s for both expermental values and numercal results. Between the nstants 1.00 s and 7.00 s, the measured magntude of the crest and valley of the dsplace-

23 Brttle Fracture and Hydroelastc Smulatons 109 ment are dfferent wth respect to undeformed condton of the elastc plate. On the other hand, for the computed result, ths symmetry s observed. Fgure 13: Horzontal dsplacement of the elastc plate mmersed n low fllng. Results of present MPS smulatons and the expermental measurements [Idelsohn, Mart, Souto-Iglesas, and Onate (2008b)]. Table 4: Elastc plate mmersed n low fllng. Results of present MPS smulatons and the expermental measurements [Idelsohn, Mart, Souto-Iglesas, and Onate (2008b)]. d p (mm) Expermental (cm) Present MPS (cm) Error (%) The error of the computed s evaluated between 1.72 s and 6.76 s, whch s the nterval when experment reached the steady state and remans stable. The error s defned as the rato between the modulus of the dfference between average ampltudes of the expermental measurement and numercal smulaton by average ampltudes obtaned expermentally. Tab. 4 shows the errors for partcle dstance d p = 1.00, 0.80 and 0.50 mm. When partcle dstance are 1.00 and 0.80mm, the

24 110 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 Fgure 14: Elastc plate mmersed n low fllng. Comparson between present MPS smulatons and experment [Idelsohn, Mart, Souto-Iglesas, and Onate (2008b)] (0.90 s, 1.20 s, 1.67 s, 2.07 s, 2.47 s, 2.87 s, 3.33 s, 3.76 s).

25 Brttle Fracture and Hydroelastc Smulatons 111 error s relatvely large achevng 27%. However, for partcle dstance 0.50mm, the error reduces to 18.75%. Fg. 14 shows a sequence of frames from the present MPS smulatons and experment. It can be observed a smlar behavor between the experment and the present MPS smulatons Elastc plate mmersed n hgh fllng The tank s flled to H L = mm and subjected to a oscllatory moton of perod T = s. The dmensons of elastc plate, whch s hghlghted n orange, Fg. 11, are 4.0 x mm and the physcal propertes are: densty ρ = 1100 kg/m 3, Young s modulus E = 6 MPa and Posson s rato v = The physcal propertes of the flud are: densty ρ = 917 kg/m 3 and knematc vscosty ν = 5x10 5 m 2 /s. As smulaton parameters, the values n Tab. 5 are consdered. The smulatons are performed for 10 s. Table 5: Elastc plate mmersed n hgh fllng. Smulaton parameters. d p (mm) t f lud (s) t sold (s) Flud partcles Sold partcles Fgure 15: Horzontal dsplacement of the elastc plate mmersed n hgh fllng. Results of present MPS smulatons and the expermental measurements [Idelsohn, Mart, Souto-Iglesas, and Onate (2008b)].

26 112 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 Fgure 16: Elastc plate mmersed n hgh fllng. Comparson between present MPS smulatons and experment [Idelsohn, Mart, Souto-Iglesas, and Onate (2008b)] (1.10 s, 1.34 s, 1.74 s, 1.98 s, 2.30 s, 2.54 s, 2.94 s, 3.18 s).

27 Brttle Fracture and Hydroelastc Smulatons 113 Table 6: Elastc plate mmersed n hgh fllng. Results of present MPS smulatons and the expermental measurements [Idelsohn, Mart, Souto-Iglesas, and Onate (2008b)]. d p (mm) Expermental (cm) Present MPS (cm) Error (%) Fg. 15 shows the computed tme hstory and the measured results of the horzontal dsplacement of the top of the elastc plate. Smlar to the former low fllng case, computed results show transent behavor dfferent from the expermental measurement because the nerta of the system s not consdered n the numercal smulaton. After the nstant 1.00 s, the oscllaton of the elastc plate becomes stable wth a perod of approxmately 1.20 s for both expermental measurement and numercal results. The expermental measurement shows peak values around 7.52 cm and 7.38 cm, whle the numercal result presents peak values around 8.70 cm and 8.70 cm. Both experment and numercal results show that the plate oscllates symmetrcally wth respect to ntal undeformed poston of the elastc plate. The errors evaluated from 2.44 s to 9.08 s are gven n Tab. 6, for partcle dstance d p = 1.00, 0.80 and 0.50 mm. In relaton to the former low fllng case, the error reduced sgnfcantly and the convergence of the numercal results can also observed. Fg. 16 shows a sequence of frames from the present MPS smulatons and experment. As n the prevous case, t s agan observed a smlar behavor between the experment and the present MPS smulatons. 6 Concludng remarks A computer code for the modelng and smulaton of brttle fracture and hydroelastc problems has been mplemented n the present work. A more generc contact detecton algorthm that takes nto account the presence of the sold surfaces generated by brttle fracture s proposed. To solve hydroelastc problems nvolvng free surface, a parttoned couplng between flud and sotropc elastc sold s adopted. Explct and sem-mplct tme ntegraton algorthms are used for elastc sold and flud domans, respectvely. Matchng of the tme steps n both domans s done by sub-cyclng technque to mprove the computatonal effcency. A dynamc case of slender cantlever beam s analyzed and the convergence of the method s verfed. After that, a collson between two elastc solds wth brttle fracture and further collson between the fragments s smulated showng the mprovement

28 114 Copyrght 2013 Tech Scence Press CMES, vol.95, no.2, pp , 2013 acheved. Fnally, to valdate the flud-structure nteracton smulaton approach, two FSI problems are smulated and compared wth other computatonal methods or expermental results found n lterature. Comparsons wth avalable numercal results of a dam breakng on an elastc plate are performed and t can be observed a good agreement between the results of present method and another methods, takng nto account the physcs of the problem. In the second case, avalable expermental measurements of the nteractons between lqud sloshng, and an elastc plate are compared wth numercal results. For the low fllng case, t was noted sgnfcant errors n dsplacements, whle n the hgh fllng case the mprovement on the accuracy can be observed. On the other hand, the convergence of the numercal results can be observed n the both fllng cases. The comparsons of the results showed the effectveness of the present approach to reproduce the man behavors of the problems nvolvng fracture of brttle materals and nteracton between free surface flow and elastc solds. Acknowledgement: Ths work had fnancal support from CAPES and the authors are thankful to Petrobras for fnancal support on the development of the smulaton system based on MPS method. References Amanfard, N.; Hesan, M.; Rahbar, B. (2011): An SPH approach for fludhypoelastc structure nteractons wth free surfaces. In Proceedngs of the World Congress on Engneerng. Antoc, C.; Gallat, M.; Sblla, S. (2007): Numercal smulaton of fludstructure nteracton by SPH. Computers & Structures, vol. 85, no , pp Bellez, C. A.; Cheng, L. Y.; Nshmoto, K. (2013): A numercal study of the effects of bow shape on green water phenomenon. In Proceedngs of the Twentythrd (2013) Internatonal Offshore and Polar Engneerng - ISOPE2013. Bonet, J.; Lok, T.-S. L. (1999): Varatonal and momentum preservaton aspects of smooth partcle hydrodynamc formulatons. Computer Methods n Appled Mechancs and Engneerng, vol. 180, no. 1 2, pp Campbell, J. C.; Vgnjevc, R.; Patel, M.; Mlsavljevc, S. (2009): Smulaton of water loadng on deformable structures usng SPH. Computer Modelng n Engneerng & Scences, vol. 49, no. 1, pp Chen, F.; Wang, C.; Xe, B.; Qn, H. (2013): Flexble and rapd anmaton of brttle fracture usng the smoothed partcle hydrodynamcs formulaton. Computer Anmaton and Vrtual Worlds, vol. 24, no. 3 4, pp

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