Flud Structure Interacton Usng the Partcle Fnte Element Method SR Idelsohn(1,2), E Oñate (2), F Del Pn(1) andnestor Calvo(1) (1)Internatonal Center for Computatonal Methods n Engneerng (CIMEC) Unversdad Naconal del Ltoral and CONICET, Santa Fe, Argentna e mal: sergo@cerdegovar (2)Internatonal Center for Numercal Methods n Engneerng (CIMNE) Unversdad Poltécnca de Cataluña, Barcelona, Span e mal: onate@cmneupces Abstract In the present work a new approach to solve flud structure nteracton problems s descrbed Both, the equatons of moton for fluds and for solds have been approxmated usng a materal (lagrangan) formulaton To approxmate the partal dfferental equatons representng the flud moton, the shape functons ntroduced by the Meshless Fnte Element Method (MFEM) have been used Thus, the contnuum s dscretzed nto partcles that move under body forces (gravty) and surface forces (due to the nteracton wth neghborng partcles) All the physcal propertes such as densty, vscosty, conductvty, etc, as well as the varables that defne the temporal state such as velocty and poston and also other varables lke temperature are assgned to the partcles and are transported wth the partcle moton The so called Partcle Fnte Element Method (PFEM) provdes a very advantageous and effcent way for solvng contact and free surface problems, hghly smplfyng the treatment of flud structure nteractons Key words: Flud Structure nteracton, Partcle methods, Lagrange formulatons, Incompressble Flud Flows, Meshless Methods, Fnte Element Method 1 Introducton Many classfcatons have been proposed to enclose the numercal formulatons that approxmate the contnuum equatons that govern ncompressble flud flows In partcular the one descrbng the way that convecton s treated dvdes the numercal formulatons nto two classes, namely materal (or lagrangan) formulatons and spatal (or euleran) formulatons The frst one descrbes convecton by placng a set of axes over the materal partcles that move accordngly to the equatons of moton In the
euleran case the axes are set fxed n space and convecton terms are ncluded n the equatons descrbng the transport of the flud flow The present work wll descrbe a method that uses a materal formulaton The equatons of moton for both, the sold and flud do not present convecton terms, mplyng that the convecton effect s drectly obtaned by movng the dscrete doman Many authors have taken advantage of lagrangan formulatons to descrbe dfferent types of problems The Smooth Partcle Hydrodynamcs (SPH) method developed by Monaghan(1977)[mon81, mon97] should be mentoned as a poneer method of ths knd Many other methods have been derved from SPH One that has shown remarkable results s the Movng Partcle Sem Implct method (MPS) ntroduced by Koshzuka and Oka (1996)[kos96] These methods use a kernel functon to nterpolate the unknowns SPH uses a weak formulaton whle MPS uses a strong form of the governng equatons Ramaswamy (1986)[ram87] proposed a lagrangan fnte element formulaton for a 2 D ncompressble flud flow In that paper the mesh was convected accordng to the equatons of moton but wthout change of topology, makng t rather lmtng when the elements got hghly dstorted The equatons of moton were dscretzed n space by usng the fnte element method wth lnear shape functons Another possble classfcaton for numercal formulatons may be the one that separates the methods that make use of a standard fnte element mesh (lke those made of tetrahedra or hexahedra), and the methods that do not need a standard mesh, namely the meshless methods The formulaton descrbed n ths paper can be consdered a partcular class of meshless method Agan, SPH mght be cted as one the frst meshless methods Indeed, after Monaghans work and n partcular n the past 20 years, many have been the attempts to develop a robust meshless method that could approxmate PDE s n 2 D and 3 D wth acceptable accuracy, convergence and speed Among others, the methods based on Movng Least Square nterpolatons [nay92, bel94], Partton of Unty [dua95], and the ones based on the natural neghbor nterpolaton functons [bra95, suk98] may be lsted In ths work the nterpolaton functon used by the Meshless Fnte Element Method (MFEM) [de03a] wll be mplemented Ths functon uses the oronoï dagram of the cloud of ponts to construct the nterpolant The extended Delaunay tessellaton (EDT) [de03b] s appled to connect the neghborng partcles The EDT provdes polyhedral elements that are slver free n 3 D, avodng nstabltes of the Delaunay tessellaton due to dstorted tetrahedra The MFEM shape functons adapt automatcally to the polyhedra and n the case that the polyhedron s a smplex, the shape functon behaves exactly as the lnear fnte element shape functon
Flud structure nteracton (FSI) problems have been of specal nterest for desgners and engneers n the past 20 years Ths explans why more robust and stable formulatons have been developed to assst the approxmaton of contact problems Embedded methods have been developed by Löhner et al [loh03] where a sngle mesh s used to partton the flud as well as the structure Also Arbtrary Lagrangan Euleran (ALE) formulatons [Sou00] have gven acceptable results when the dsplacements or the geometry deformatons are not excessvely large The approxmaton for the FSI problem depends bascally on the couplng of the flud and structure equatons Based on ths couplng FSI problems may be dvded nto problems wth weak nteracton and problems wth strong nteracton The later are found when elastc deformaton of the sold takes place The weak nterpolaton case happens when large rgd dsplacements are present Ths stuaton s typcal n shp hydrodynamcs, when a rgd body moves accordng to the forces gven by the pressure feld obtaned from the flud dynamc problem These forces appled to the rgd body wll accelerate t, changng ts velocty and therefore, ts poston FSI problems have been classcally solved n a parttoned manner solvng teratvely the dscretzed equatons for the flow and the sold doman separately The soluton of both, flud flow and sold, wth the same materal formulaton, open the door to solve the global coupled problem n a monolthc fashon Nevertheless, n ths paper the rgd sold wll stll be solved separately from the flud A parttoned method [pp95, mok01] or teratve method [rug00, rug01, zha01 s chosen to solve the couplng between the flud and sold The advantage to use a materal formulaton for both, sold and flud parts wll be used here only to better reproduce breakng waves or separated drops n the flud, whch are phenomena mpossble to reproduce usng a spatal formulaton The layout of the paper s the followng: n the next secton the basc lagrangan equatons of moton for the flud and sold domans are gven Next the dscretzaton method chosen to solve the ncompressble flud flow equatons and the sold dynamcs n tme equatons are detaled The algorthm for the recognton of the boundary nodes and the treatment of the free surface n the flud s explaned Fnally the effcency of the Partcle Fnte Element Method for solvng a varety of flud structure nteracton problems nvolvng large moton of the free surface n the flud s shown 2Equatons of moton 21 Flud dynamc problem: updatng the flud partcle postons The flud partcle postons wll be updated va solvng the lagrangan form of the Naver Stokes equatons
Let X the ntal poston of a partcle a tme t=t0 and let x the fnal poston Been u x j, t =u the velocty of the partcle n the fnal poston the followng approxmate relaton can be wrtten: x =X f u, t, Du / (1) Conservaton of momentum and mass for ncompressble Newtonan fluds n the lagrangan frame of reference are represented by the Naver Stokes equatons and the contnuty equaton n the fnal x poston, as follows: Mass conservaton: u D + = 0 x Momentum conservaton: Du = p + τ + f, x x j j (2) (3) where s the densty, p the pressure, τj the devatorc stress tensor, f the source D term (usually the gravty) and represents the total or materal tme dervatve For Newtonan fluds the stress tensor τj may be expressed as a functon of the velocty feld through the vscosty μ by u u j 2 ul τj =μ δ (4) x j x 3 x l j For near ncompressble flows 2 μ u 0, 3 x u x << uk xl (5) and t may be neglected from Eq 4 Then: u u j τj μ x j x the term: In the same way, the term ncompressble flows as: (6) τ j n the momentum equatons may be smplfed for near x j
u u j u uj τj = μ =μ μ = xj xj x j x xj xj x j x u u j u μ μ μ xj xj x x j xj xj Usng eq (7), the momentum equatons can be fnally wrtten as: Du u = p τj f p μ f x xj x xj xj (7) (8) Note: eq(3) or the equvalent for ncompressble flud flow eq(8) are non lnear In euleran formulatons the non lnearty s explctly present n the convectve terms In ths lagrangan formulaton, the non lnearty s due to the fact that eqs (3) and (8) are wrtten n the fnal postons of the partcles, whch are unknown There are others way to wrte lagrangan formulatons, for nstance stayng n the ntal poston [aub04] In all cases, the equatons are non lnear Boundary condtons On the boundares, the standard boundary condtons for the Naver Stokes equatons are: τj ν j pν =σ n on Γσ on uν = un Γn uζ = ut on Γ t where ν and ζ are the components of the normal and tangent vectors to the boundary 22 Sold dynamcs problem: updatng the rgd body poston In ths paper, the structure wll be consdered as a rgd sold Then, the equatons of moton for a rgd body are: DU (9) m =F where F are the resultant of the external forces (surface forces, gravty force, etc), whose lne of acton passes through the mass center of the body, U s the velocty of the mass center and m the total mass of the sold The actual moton of the rgd body conssts n the superposton of the translaton produced by the resultant force F and the rotaton produced by the couple T satsfyng:
DM =T, (10) where M s the angular momentum about the mass center It must be noted that n (10) the tme dervatve s expressed as the rate of change wth respect to any non rotatng system of axs It may be also expressed as the dervatve wth respect to the body fxed axes by: DM D M εjk e j M k =T, (11) where Ω denotes the angular velocty of the body, e are orthogonal unt bass vectors, ε the permutaton symbol and D/ D t s the dervatve wth respect to the body fxed axes Let now the body fxed axes be the prncpal axes of nerta of the body, wth ts orgn at the center of mass, then: M =I, (wthout summaton n the ndex ) (12) where I are the prncpal moments of nerta and then: D M D =I (13) Fnally, the equatons of moton of the body mght be summarzed as: DU m =F, D I εjk e j I k k =T, (14) (15) Callng a and α the lnear and the angular acceleraton of the mass center of the body: m a =F, (16) I α εjk e j I k k =T, (17) Ths s a non lnear system of partal dfferental equatons that has to be lnearzed for ts numercal approxmaton The fnal rgd body velocty of an arbtrary pont s a combnaton of both, the lnear velocty of the center of mass U and the angular velocty accordng to: u =U εjk e j r k (18) where r s the dstance from the orgn of the body axes to an arbtrary pont attached to the body The velocty u wll be used later as a boundary condton for the flud dynamcs problem A very large number of problems nvolve plane moton In ths case, equaton (15) reduces to:
I D I α=t, (19) where Ω, I, α and T are the planar angular velocty, the moment of nerta, the planar angular acceleraton and the external couple respectvely 3The dscrete flud dynamcs problem The Naver Stokes equatons present three man dffcultes: The equatons are tme dependent and thus a temporal ntegraton needs to be carred out A spatal dependency s also present and thus the space wll be dscretzed Fnally, Eq 3 presents a non lnearty, whch must be solved teratvely Each of the above tems wll be explaned and a soluton algorthm wll be ntroduced to obtan a fnal accurate and robust numercal scheme 31 Implct explct tme ntegraton Let t n and t n 1 be the ntal and fnal tme step Let =t n 1 t n be the tme ncrement Eq (8) s ntegrated mplctly n tme as: Du u x, t n 1 u X, t n = un 1 un [ ] u = p μ f x xj xj n θ (20), where [ φ x, t ] n θ means θφ x, t n 1 1 θ φ x, t n =θφn 1 1 θ φ n and φ n =φ x, t n represents the value of the functon at tme t n but at the fnal poston x For smplcty φn wll be used nstead of φ n Only the case of θ =1 (fully mplct scheme) wll be consdered next Other values, as for nstance θ =1/2, may be consdered wthout major changes The tme ntegrated equatons become: n 1 n 1 un 1 un u = p μ f (21) x x x [ ] [ ] j j The mass conservaton s also ntegrated mplctly by: un 1 D n 1 n n 1 = x (22) The tme ntegraton of Eq (20) presents some dffcultes: t s a fully coupled equaton nvolvng four degrees of freedom by node When the flud s ncompressble or nearly ncompressble advantages can be taken from the fact that n Eq (20) the three
components of the velocty are only coupled va the pressure The fractonal step method proposed n [cod01] wll be used for the tme soluton Ths bascally conssts n splttng each tme step n two pseudo tme steps In the frst step the mplct part of the pressure s avoded n order to have a decoupled equaton n each of the velocty components The mplct part of the pressure s added at a second step The fractonal step algorthm for eqs (21) and (22) s the followng: Splt of the momentum equatons Du un 1 un = un 1 u u un n θ 1 n 1 1 τj = p f, x xj where u are fcttous varables termed fractonal veloctes defned by the splt: n θ u =u n f γp n τ, x x j j u n 1 =u p n 1 γp n, x (23) (24) (25) where p n = p x, t n s the value of the pressure at tme t n evaluated at the fnal poston and f s consdered constant n tme In Eqs (24) and (25) γ s a parameter gvng the amount of pressure splttng, varyng between 0 and 1 A larger value of γ means a small pressure splt In ths paper γ wll be fxed to 0 n order to have the larger pressure splt and hence, a better pressure stablzaton Other values as, for nstance γ=1, may be used to derve hgh order schemes n tme[cod01] Takng nto account Eq (7), the last term n Eq (24) may be wrtten as: n 1 n θ n n θ u u u τ =μ = μ 1 θ μθ x j j xj xj xj xj xj xj The followng approxmatons have been ntroduced [cod01]: n θ n u u u μ μ 1 θ μθ xj xj xj xj xj xj (27) Ths allows to wrte Eq (24) as: n u u n u =u n f γ p μ 1 θ μθ x xj xj xj xj For γ=1 and θ =1 the equatons for the fractonal veloctes becomes: u u μ =u n f xj xj (26) (28) (29)
Splt of the mass conservaton equatons un 1 u u D n 1 n n 1 n, = = x (30) where s a fcttous varable defned by the splt u n =, x (31) u n 1 u n 1 = x (32) Coupled equatons From eqs (25), (31) and (32) the coupled mass momentum equaton becomes: 2 n 1 2 = 2 p n 1 x (33) Takng nto account Eq (31) the above expresson can be wrtten as: n 1 n u 2 = 2 p n 1 2 x x (34) It s mportant to note that n eq (34) the ncompressblty condton has not be ntroduced yet The smplest way to ntroduce the ncompressblty condton n a lagrangan formulaton s to wrte: (35) n 1 = n = 0 = Then, the frst term of Eq (34) dsappears, gvng: u 2 = 2 p n 1 x x (36) The three steps of the fractonal method used here can be summarzed by: u u, u μ =u n f xj xj u 2 = 2 p n 1 x x u n 1 =u p n 1 x p n 1, un 1 (37) (38) (39)
32 The spatal dscretzaton provded by the MFEM One of the key to solve a flud mechancs problem usng a lagrangan formulaton s to generate effcently the shape functons to approxmate the spatal unknown In the Fnte Element context, ths means to generate permanently, at each tme step, a new mesh In ths work the nterpolaton functon used by the Meshless Fnte Element Method (MFEM) [de03a] wll be mplemented Ths functon uses the oronoï dagram of the cloud of ponts to construct the nterpolant The extended Delaunay tessellaton (EDT) [de03b] s appled to connect the neghborng partcles The EDT provdes polyhedral elements that are slver free n 3 D, avodng nstabltes of the Delaunay tessellaton due to dstorted tetrahedral EDT provde a way to generate meshes at each tme step very effcently n a computng tme whch s largely smaller than the computng tme needed to solve the lnearzed system of equaton EDT together wth the MFEM are the man key to make the PFEM presented n ths paper a useful tool The unknown functons are approxmated usng an equal order nterpolaton for all varables n the fnal confguraton: u = N l X, t U l, (40) l p= N l X, t P l (41) l In matrx form: u =N T X, t U p=n T X, t P, or n compact form: NT u =N T U = NT [ (42) (43) N T ] U, (44) where N T are the MFEM shape functons and U, P the nodal values of the three components of the unknown velocty and the pressure respectvely It must be noted that the shape functons N X, t are functons of the partcle coordnates only Then, the shape functons may change n tme followng the partcle postons Durng the tme step, a mesh update may ntroduce a change n the shape functon defnton, whch must be taken nto account Durng the tme ntegraton there are two tmes nvolved: t n and t n 1 The followng notaton wll be used to dstngush between N X, t n and N X, t n 1 : (45) N X, t n =N n and N X, t n 1 =N n 1
In ths work, the followng hypothess wll be ntroduced: There s not mesh update durng each tme step Ths means that f a mesh update s ntroduced at the begnnng of a tme step, the same mesh (but deformed) wll be kept untl the end of the tme step Mathematcally ths means: N X, t n =N X, t n 1 (46) Unfortunately, ths hypothess s not always true and ths ntroduces small errors n the computaton, whch are neglected n ths paper Usng the Galerkn weghted resdual method to solve the spltted equatons, the followng ntegrals are obtaned: N u d N un d N f d N x γpn d (47) n θ u n d N σ n τ n θ ν j γp ν dγ =0, N μ x x j Γ j j [ [ σ ] ] u 2 N x 2 pn 1 γpn d N u n 1 ν un 1 ν dγ =0, x Γu [ N un 1 u x p n 1 γpn ] d N p n 1 γ p n ν dγ = 0, Γ (48) (49) σ where the boundary condtons have been also splced and s the volume at tme t n 1 Integratng by parts some of the terms, the above equatons become: n θ N u n n (50) N γp d μ d N σ n γp ν dγ =0, x Γσ xj xj N N p n 1 γp n u d d x x x Γu p n 1 γp n }d N p n 1 γp n dγ = 0 Γ σ The essental and natural boundary condtons of Eq (51) are: p=0 on Γσ u n 1 ν=u n 1 s ν on Γu where u n +1 s (51) N { un 1 u x N un n 1 dγ =0, s the rgd body velocty obtaned from Eq (19) (52)
Dscrete equatons Usng the approxmatons gven be Eqs (44), (45) and (46) the dscrete equatons become: N N T d U = N N T d U n N f d T γ N μ N d P n x In compact form: MU =MU n F N N T x j xj d U n θ N σ n γp n dγ (53) Γσ γ T n μ n θ B P KU (54) Makng use of the approxmaton descrbed before for U n ϑ : μθ γ T n μ 1 θ n M K U =MU n F B P KU, (55) For θ =1 and γ=0 : μ M K U =MU n F (56) In the same way: N T N d U x N un n 1 dγ = Γ u N NT d P n 1 γp n x x (57) In compact form: S P n 1 = B U U S γ P n, (58) For θ =1 and γ=0 : S P n 1= B U U (59) Fnally: N N T d U n 1= N N T d U In compact form: N NT d P n 1 γp n N N T dγ P n 1 γp n x Γ σ (60)
T n 1 B P γp n For θ =1 and γ=0 : MU n 1=MU MU n 1=MU (61) T n 1 B P (62) Where the matrces are: Mp 0 0 M= 0 Mp 0, [ 0 0 Mp M p = NN d, [ T ] (63) B= ] N T N T N T N d ; N d ; N d, x y z N NT N NT N NT d, x x y y z z S= (64) (65) (66) U = N u n 1 dγ, n Γ [ ] S K= 0 0 [ 0 S 0 0 0 S, F T = N T f x d ; 1 [ (67) u (68) N T f y d ; N T f z d ] N T σ nx dγ ; N T σ ny dγ ; N T σ nz dγ Γσ Γσ Γσ ] (69) Stablzaton of the ncompressblty condton In the euleran form of the momentum equatons, the dscrete form must be stablzed n order to avod numercal wggles n the velocty and pressure results Ths s not the case n the lagrangan formulaton where no stablzaton terms must be added to eqs (62) Nevertheless, the ncompressblty condton must be stablzed n equal order approxmatons to avod pressure oscllatons n some partcular cases For nstance for small pressure splt ( γ 0 ) or for small tme step ncrements (Courant number much less than one) t s well known that the fractonal step does not stablze the pressure waves [cod01] In those partcular cases, a stablzaton term must be ntroduced n Eqs (62) n order to elmnate pressure oscllatons
A smple and effectve procedure to derve a stablzed formulaton for ncompressble flows s based n the so called Fnte Calculus (FIC) formulatons [ona98, ona00, ona02] In all the examples presented n ths paper the FIC formulaton was used to stablze the pressure oscllatons 33 Non lnearty of the lagrangan formulaton Many algorthms are avalable to lnearze the equatons of moton The Newton Raphson scheme s probably the most popular because of ts robustness and fast convergence It has been appled wth success n ths type of formulaton n [rad98] Nevertheless we consder that t mght not be the most approprate opton for the type of equatons we are ntendng to solve as t requres large memory storage Instead, the successve teraton algorthm has been chosen for the present analyss In ths case, only the varables that nduces the non lnearty need to be stored n successve teratons Let us now descrbe the process that may take place untl convergence: I) II) III) I) ) Approxmate u n 1 (For the frst teraton u n 1=0 For the subsequent teratons the value of u n 1 correspondng to the last teraton s taken) Move the partcles to the x n 1 poston and perform an EDT polyhedrzaton Evaluate the fractonal velocty u from (56) It must be noted that the matrces M and K are separated n 3 blocks Then, these equatons may be solved separately for U x, U y and U z For θ 0 (mplct scheme) nvolves the soluton of 3 symmetrc lnear systems of equatons For θ =0 (explct scheme) the M matrx may be lumped and nverted drectly Evaluate the pressure p n 1 by solvng the laplacan equaton (58) Evaluate u n 1 usng (61) 4 Tme ntegraton of the sold dynamcs problem Eqs (14) and (15) that govern the movement of rgd bodes are ntegrated n tme by the explct Newmark algorthm It conssts n evaluatng the velocty by lnearzng the acceleraton between two tme steps: 1 γ α γα U n 1=U n 1 γ an γan 1 n 1 = n n n 1 (70) The pont poston for the explct verson of the Newmark algorthm s evaluated by: x n 1 =x n U n 2 an / 2 (71) To ntegrate the angular acceleraton n a 3D system by Newmark algorthm two steps are needed, namely: predctor step:
= n 1 γ αn (72) then, the acceleratons are predcted by usng Eq (17): αn 1 =T n 1 / I εjk e j I k k / I corrector step n 1 = γ αn 1 The lnear veloctes are ntegrated drectly usng Eqs(16) and (70): an 1 =F n 1 / m U n 1=U n 1 γ an γan 1 (73) (74) (75) For the present analyss γ=1/2 wll be consdered Both veloctes, U n 1 and n 1 are used n (18) to evaluate the new velocty of all the ponts of the body In the explct verson of the Newmark algorthm, the new poston of the rgd body s evaluated by : x n 1 =x n U n 2 an /2 (76) φn 1 = n 2 αn / 2 where x s the new poston of the center of mass and φ are the angular rotatons of the body It must be noted that for planar moton, the predctor step s unnecessary and αn 1 may be evaluated drectly usng (19) 41 Τη ε Χου π λ ε δ Πο β λ ε µ On the couplng boundary the flud velocty and the sold velocty should converge to the same value Ths could be expressed as: u f Γ1 =u s Γ2 (77) Thus, two subsystems need to be solved, namely the flud system: N u n 1,p n 1,x n 1,u n 1,x n 1 =0 f f s s and the sold system: S u n 1,x n 1,u n 1,p n 1,x n 1 =0 s s f f In the equatons above only the varables to be solved at tme step n+ 1 are shown An teratve procedure has to be mplemented to couple both systems A fxed pont algorthm may be mplemented and thus the system could be wrtten as:
u u n 1 n 1 n 1 f,k+1,p k+1,x f,k+1 =F u n 1 n 1 s,k+1,x s,k+ 1 n 1 n 1 n 1 n 1 u n 1,p,x f s,k,x s,k,u f =G n 1 n 1 n 1 n 1 n 1,x f,k,u s,x s f,k,p k (78) The frst equaton n Eqs (78) denotes the flud subsystem and the second equaton the sold subsystem The subscrpt k s the teraton counter In the present analyss a Gauss Sedel process has been chosen [cod96] In ths way, the teratve procedure means to solve frst one of the subsystems, for nstance the flud system Next the sold system s solved usng the nformaton from the flud computaton Eqs (78) should be modfed and the fnal expresson used for the computaton s as follow: n 1 n 1 n 1 n 1 n 1 n 1 n 1 u n 1,x f,k,u s,k,x s,k f,k+1,p k+ 1,x f,k+1 =F u f,k,p k u n 1 n 1 s,k+1,x s,k+ 1 =G n 1 n 1 n 1 n 1 u n 1 s,k,x s,k,u f,k+ 1,p k+ 1,x s,k+ 1 Convergence occurs when the dfference between veloctes of successve teraton steps s less than the acceptable error 5 Free surface and boundary recognton The soluton of partal dfferental equatons (PDE) requres to prescrbe boundary condtons as a necessary step to a well posed problem When the PDEs are approxmated n space and the doman s parttoned nto dscrete elements (fnte elements, partcles, balls, nodes, etc) the boundary elements should be provded at the ntal tme step, such that, at run tme the algorthm knows where to mpose or fx the varables of the analyss (pressure, velocty and ther dervatves for nstance) Ths would be the case of a statc doman, where the geometry does not change n tme and the boundares reman constant In ths work the nterest s focused on problems where the soluton doman s hghly dstorted, and boundary elements can change between tme steps In ths case an effcent boundary recognton algorthm s mandatory n order to mpose boundary condtons over the rght elements, thus avodng possble error accumulaton over the tme When applyng the MFEM [Ide03a] to the dscrete space problem, the EDT [Ide03b] s computed to connect the partcles that dscretze the doman, thus, all the empty oronoï spheres are found and stored These spheres wll be used to compute the boundary usng the alpha shape technque [ede94] The partcles wll follow a gven h(x) dstrbuton accordng to the maxmum error allowed for the dscrete space problem, where h(x) s the expected dstance among neghborng partcles Then, havng all the empty oronoï spheres and h(x) the boundary partcles are regarded as: all the partcles whch are on an empty sphere wth a radus r bgger than α h In ths crteron, α s a parameter close to one, typcally α=1 2 and h s the mean value taken from the defnng partcles of the sphere under nspecton
Once a decson has been made concernng whch of the nodes are on the boundares, the boundary surface must be defned It s well known that, n 3 D problems, the surface fttng a number of nodes s not unque For nstance, four boundary nodes on the same sphere may defne two dfferent boundary surfaces, one concave and the other convex In ths work, the boundary surface s defned by all the polyhedron faces havng all ther nodes on the boundary and belongng to just one polyhedron See [de03b] The correct boundary surface may be mportant to defne the correct normal external to the surface Furthermore; n weak forms (Galerkn) t s also mportant a correct evaluaton of the volume doman It must however be noted that n the crteron proposed above, the error n the boundary surface defnton s of order h Ths s the standard error of the boundary surface defnton n a meshless method for a gven node dstrbuton Another mportant feature of the alpha shape technque related to the contact problem s shown n Fg51 The mage shows two dfferent tme steps of a sold cube fallng nto water after the alpha shape algorthm for the boundary recognton has been appled The partcles, as well as the connectons provded by the EDT are depcted At the frst tme step all the rad of the empty crcles constructed wth the nodes of the cube and the nodes regarded as belongng to the free surface are larger than α h x and thus the elements that they defne are elmnated from the tessellaton The second pcture shows a more evolved state, wth the cube reachng the water surface Thus, at ths state the crcle rad are less than α h x and so the connectons between the cube and the flud take part of the computaton In ths way free surface and contacts are solved at once Fg 51: The alpha shape technque used for contact recognton 6 Jonng and breakng partcles
The dea of h varable mesh s rather dfferent n partcle methods than n classcal euleran formulatons In partcle methods, each partcle s followed n tme and the same partcle can cross domans n whch the soluton need small h n order to represent hgh gradents or can cross a regon wth large h where the soluton s smooth The concept of varable h s ntroduce n partcle methods by jonng two partcles when they are too close to each other or breakng a partcle n two when all the neghborng partcles are too far and the soluton needs a hgher gradent In the example presented n ths paper the followng crteron has been used: 1) Durng the EDT algorthm to buld the polyhedral mesh a partcle s not added f there s a prevous pont at a dstance d < λ h(x), beng λ = 05 a constant parameter 2) on the contrary, when there s an empty sphere whose radus r > Λ h(x) a pont s added to ts center and assgned wth values nterpolated from the sphere defnng partcles The parameter s currently taken as Λ = 11*sqrt(dm)/2 n order to accept, wth a 10% of tolerance, near square (dm = 2, Λ 78) or near cube (dm = 3, Λ 95) local arrays as connected nner ponts Ths parameter must be less than the alpha shape parameter α h x n order to avod nterference wth the boundary recognton algorthm 7 aldaton Examples PFEM was developed as a general purpose method for solvng dfferent knd of problems on whch large free surface or nterface boundares changes are nvolved The method s well suted to solve a large varety of mechancal problems ncludng mxng flud and sold materals, wave moton problems, mould fllng, coupled thermal mechancal problems and flud sold nteracton as well In ths secton some problems are ncluded n order to show the valdaton of the present approach towards expermental and numercal tests In the followng secton, some more specfc examples on flud structure nteracton wll be shown 71 Sloshng The smple problem of the free oscllaton of an ncompressble lqud n a contaner s consdered frst Numercal solutons for ths problem can be found n several references [rad98] Ths problem s nterestng because there s an analytcal soluton for small ampltudes For larger ampltudes the wave breaks and also some partcles can be separated from the flud doman due to ther large velocty PFEM can solve very large ampltudes, even n a 3D doman [de04] However, n ths secton only the small ampltude, two dmensonal example s shown to valdate the method Fgure 71 shows the varaton n tme of the ampltude compared wth the analytcal results for the near nvscd case Lttle numercal vscosty s observed on the phase wave and ampltude n spte of the relatve poor pont dstrbuton
Fg 71 Sloshng: Comparson of the numercal and analytcal soluton 72 Wave on a channel: comparson wth expermental results Ths example was performed n order to compare and valdate the method wth expermental results A wave s runnng from the left to the rght arrvng to a shallow doman were the wave breaks The example s represented on Fg 72 were the calculated partcle postons are shown at dfferent tme steps The wave was produce by a partcular movement of the left wall Ths example was reproduced expermentally n the CIEM (Martme Expermental and Research Channel) of the Escuela Técnca de Ingeneros de Canales Camnos y Puertos n the Unversty of Catalunya The channel s 100 meters length, 3 meters wde and 5 meters hgh A pressure sensor was placed on the rght wall at 02 meters from the bottom
Fg 72 Wave on a channel: partcle dstrbuton for dfferent tme steps
Expermental and numercal pressure results are compared at dfferent tme step n Fg 73 Both results, expermental and numercal, were smoothed n order to gnore the artfcal oscllatons from hgh order waves present n the problem The comparson of the results n the pressure values shows a reasonably agreement 20000 pressure (Pa) 16000 12000 Expermental pressure 8000 Numercal pressure 4000 0 17,4 17,7 18 18,3 18,6 18,9 19,2 tme (s) Fg 73 Wave on a channel: expermental and numercal comparson of the pressure 73 Dam collapse Fg 74 Dam Collapse Intal poston Left: expermental [Kos96] Rght: 3D smulaton
The dam collapse problem represented n Fgure 74 was solved by Koshzu and Oka [Kos96] both expermentally and numercally n a 2D doman It became a classcal example to test the valdaton of the Lagrangan formulaton n flud flows The water s ntally located on the left supported by a removable board The collapse starts at tme t = 0, when the removable board s sld up scosty and surface tenson are neglected The water s runnng on the bottom wall untl, near 03 sec, t mpnges on the rght vertcal wall Breakng waves appear at 06 sec Around t=1 sec the man water wave reaches the left wall agan In [Ide04] the results obtaned usng the method proposed n 2D and 3D domans are presented and compared wth expermental results Agreement wth the expermental results of [kos96] both n the shape of the free surface and n the tme development are excellent In ths example the power of the method to represent breakng waves and flow separaton for a very complcated and random problem s verfed and compared to expermental results Ths example s further exploted here to compare results obtaned wth three dfferent node denstes at some tme steps n order to check the convergence of the method Fg 75 Dam Collapse Comparson between the experment and numercal results obtaned n dfferent tme steps, wth dfferent refnement levels
Fgure 75 shows the doman profle at dfferent tme steps and wth dfferent refnement levels At the top there are some photos taken on the expermental setup correspondng to 2, 6 and 1 second from the startng pont From top to bottom grds wth 22, 44 and 88mm n typcal dstance between neghborng partcles are shown The excellent agreement already obtaned wth low refnement s even better wth hgher level of refnement 8 Further Examples on flud structure nteracton 81 Shp profle ht by a wave In the example of Fgure 81 the moton of a fcttous rgd shp ht by an ncomng wave s analyzed Ths s the frst example n whch the rgd body s moved by the flud forces n a couplng problem as was explaned n the prevous chapters The dynamc moton of the shp s nduced by the resultant of the pressure and the vscous forces actng on the shp boundares The horzontal dsplacement of the mass centre of the shp was fxed to zero In ths way, the shp moves only vertcally although t can freely rotate The poston of the shp boundary at each tme step s evaluated usng eq (76) and the velocty of the body by usng eq (18) Ths defnes a movng boundary condton for the free surface partcles n the flud as ntroduced n eq (59) Fgure 77 shows nstants of the moton of the shp and the surroundng flud It s nterestng to see the breakng of a wave on the shp prow at t= 091 sec as well as on the stern at t= 205 sec when the wave goes back Note that some water drops slp over the shp deck at t= 13 sec and 295 sec 82 Ol shp tank under a lateral wave The present example depcts the flexblty of the algorthm ntroduced n ths paper to solve some complcated confguratons as the one shown n Fg 82 The traversal cut of an ol shp tank s ht by a wave The structure of the shp does not only nteract wth the external water but t also moves due to the flud forces nduced by the flud n the tank Fg 82 shows the temporal development of the problem The blue lnes over each partcle represent the magntude of the velocty feld Intally (t = 00) the shp s released from a fxed poston and the equlbrum confguraton found s consstent wth Arqumdes prncple Durng the followng tme steps the external wave hts the shp and both the nternal and the external fluds nteract wth the shp boundares At tmes t = 510 sec and 600 sec breakng waves and some water drops slppng along the shp deck can be observed Fgure 83 shows the pressure pattern at two tme steps
83 Tanker Snkng Ths example represents the snkng of a tanker, whch s beng flooded by the shp prow The shp has many connected compartments that are serally flooded
Fg 81 Shp profle ht by a wave: partcle postons for dfferent tme steps
Fg 82: Ol shp tank under a lateral wave: Partcle dstrbuton and velocty feld Fg 83: Ol shp tank under a lateral wave: Pressure profle at two dfferent tme steps
Fg 84: Snkng tanker Partcle dstrbuton at three dfferent tme steps
Fg 85: Snkng tanker elocty feld In ths example proper flud structure nteracton s dsplayed as the buoyancy, pressure and drag forces from the flud are actng over the shp and, on the other hand, the shp dsplacement moves nternal and external free surfaces on the flud In Fgure 84 there are three tme steps shown wth the partcle postons and the tanker n three dfferent snkng stages Fgure 85 dsplays velocty profle on a zoom of the frst and last tme steps form prevous fgure In ths fgure, the vortcty s also easly shown
Ths s an nterestng example usng a varable dstance between partcles to enhance the soluton near the shp and free surfaces Ths varable dstrbuton was obtaned followng the method outlned n secton 6 above The large cyan dots are representng the free surface recognzed by the method as explaned n secton 5 9 Conclusons The partcle fnte element method (PFEM) seems deal to treat problems nvolvng fluds wth free surface and submerged or floatng structures wthn a unfed lagrangan fnte element framework Problems such as the analyss of flud structure nteractons, large moton of flud or sold partcles, surface waves, water splashng, separaton of water drops, etc can be easly solved wth the PFEM The success of the method lays n the accurate and effcent soluton at each tme step of the equatons of an ncompressble flud and the nteractng sold Essental ngredents of the numercal soluton are the effcent regeneraton of the polyhedral mesh usng an extended Delaunay tessellaton, the polyhedral fnte element nterpolaton va the MFEM and the dentfcaton of the boundary nodes usng an Alpha Shape type technque The examples presented have shown the potental of the PFEM for solvng a wde class of practcal FSI problems Other examples of applcaton of the PFEM can be found n [ona04] Acknowledgements Thanks are gven to Mr Mguel Angel Calena for computng the numercal results usng the PFEM of the wave on a channel presented n 72 Thanks also to the Martme Expermental and Research Channel (CIEM) of the Escuela Técnca de Ingeneros de Canales Camnos y Puertos, Unversty of Catalunya for the expermental results of the same problem References [aub04] Aubry R, Idelsohn SR and Oñate The Partcle Fnte Element Method n flud mechancs ncludng thermal convecton dfuson Submtted to Computer & Structures (2004) [bel94] T Belytschko, L Gu and Y Y Lu, Fracture and crack growth by element free Galergn methods, Modellng Smul Mater Sc Eng, vol 2, pp 519 534, 1994 [cod01] RCodna, Pressure Stablty n fractonal step fnte element methods for ncompressble flows, Journal of Comput Phsycs, 170,(2001) 112 140 [cod96] Codna R, Cervera M Block teratve algorthms for nonlnear coupled problems In: Papadrakaks M, Bugeda G, edtors, Advanced Computatonal Methods n Structural mechancs Barcelona: CIMNE, 1996
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