The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves

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1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2004; 61: (DOI: /nme.1096) The partcle fnte element method: a powerful tool to solve ncompressble flows wth free-surfaces and breakng waves S. R. Idelsohn 1,2,,, E. Oñate 2 and F. Del Pn 1 1 Internatonal Center for Computatonal Methods n Engneerng (CIMEC), Unversdad Naconal del Ltoral and CONICET, Santa Fe, Argentna 2 Internatonal Center for Numercal Methods n Engneerng (CIMNE), Unversdad Poltécnca de Cataluña, Barcelona, Span SUMMARY Partcle Methods are those n whch the problem s represented by a dscrete number of partcles. Each partcle moves accordngly wth ts own mass and the external/nternal forces appled to t. Partcle Methods may be used for both, dscrete and contnuous problems. In ths paper, a Partcle Method s used to solve the contnuous flud mechancs equatons. To evaluate the external appled forces on each partcle, the ncompressble Naver Stokes equatons usng a Lagrangan formulaton are solved at each tme step. The nterpolaton functons are those used n the Meshless Fnte Element Method and the tme ntegraton s ntroduced by an mplct fractonal-step method. In ths manner classcal stablzaton terms used n the momentum equatons are unnecessary due to lack of convectve terms n the Lagrangan formulaton. Once the forces are evaluated, the partcles move ndependently of the mesh. All the nformaton s transmtted by the partcles. Flud structure nteracton problems ncludng free-flud-surfaces, breakng waves and flud partcle separaton may be easly solved wth ths methodology. Copyrght 2004 John Wley & Sons, Ltd. KEY WORDS: partcle methods; fnte element methods; fractonal step; lagrange formulatons; ncompressble Naver Stokes equatons; mplct tme ntegraton; flud structure nteractons; free-surfaces; breakng waves 1. INTRODUCTION Over the last 20 years, computer smulaton of ncompressble flud flow has been based on the Euleran formulaton of the flud mechancs equatons on contnuous domans. However, t s stll dffcult to analyse problems n whch the shape of the nterface changes contnuously Correspondence to: S. R. Idelsohn, Internatonal Center for Computatonal Methods n Engneerng (CIMEC), Unversdad Naconal del Ltoral and CONICET, Santa Fe, Argentna. E-mal: sergo@cerde.gov.ar Receved 11 July 2003 Revsed 8 January 2004 Copyrght 2004 John Wley & Sons, Ltd. Accepted 13 February 2004

2 THE PARTICLE FINITE ELEMENT METHOD 965 or n flud structure nteractons wth free-surfaces where complcated contact problems are nvolved. More recently, Partcle Methods n whch each flud partcle s followed n a Lagrangan manner have been used [1 4]. The frst deas on ths approach were proposed by Monaghan [1] for the treatment of astrophyscal hydrodynamc problems wth the so-called Smooth Partcle Hydrodynamcs Method (SPH). Ths method was later generalzed to flud mechanc problems [2 4]. Kernel approxmatons are used n the SPH method to nterpolate the unknowns. Wthn the famly of Lagrangan formulatons the Free Lagrange Method (FLM) [5, 6] receved a lot of attenton throughout 1980s. Bascally the FLM s an adaptaton of the fnte volume method n a Lagrangan scheme that uses the orono dagram of freely movng ponts to partton the doman. As a drawback, we mght say that poor aspect-rato orono cells results n poor resoluton of the fnal results. On the other hand, a famly of methods called Meshless Methods have been developed both for structural [7 9] and flud mechancs problems [10 13]. All these methods use the dea of a polynomal nterpolant that fts a number of ponts mnmzng the dstance between the nterpolated functon and the value of the unknown pont. These deas were proposed frst by Nayroles et al. [9] whch were later used n structural mechancs by Belytschko et al. [7] and n flud mechancs problems by Oñate et al. [10 13]. In a prevous paper, the authors presented the numercal soluton for the flud mechancs equatons usng a Lagrangan formulaton and a meshless method called the Fnte Pont Method (FPM) [10]. Lately, the meshless deas were generalzed to take nto account the fnte element type approxmatons n order to obtan the same computng tme n mesh generaton as n the evaluaton of the meshless connectvtes [13]. Ths method was called the Meshless Fnte Element Method (MFEM) and uses the extended Delaunay tessellaton [14] to buld a mesh combnng elements of dfferent polygonal (or polyhedral n 3D) shapes n a computng tme whch s lnear wth the number of nodal ponts. It must be noted that partcle methods may be used wth ether mesh-based or meshless shape functons. The only practcal lmtaton s that the connectvtes n meshless methods or the mesh generaton n mesh-based methods need to be evaluated at each tme step. In ths paper, a partcle method wll be used together wth a partcular form of the FEM. The new method wll be called the Partcle Fnte Element Method (PFEM). To evaluate the forces on each partcle the ncompressble Naver Stokes equatons on a contnuous doman wll be solved usng the MFEM shape functons [13] n space. Those functons are generated n a computng tme order n where n beng the number of partcles. From the computng tme pont of vew, ths s the same (or even better) than the computng tme to evaluate the connectvtes n a meshless method. Furthermore, the shape functons proposed by the MFEM have bg advantages compared wth those obtaned va any other meshless method: all the classcal advantages of the FEM for the evaluaton of the ntegrals of the unknown functons and ther dervatves are preserved, ncludng the facltes to mpose the boundary condtons and the use of symmetrc Galerkn approxmatons. The Lagrangan flud flow equatons for the Naver Stokes approxmaton wll be revsed n the next sectons ncludng an mplct fractonal-step method for the tme ntegraton. Then, the partcle method proposed wll be used to solve some FSI problems wth rgd solds and flud flows ncludng free-surfaces and breakng waves. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

3 966 S. R. IDELSOHN, E. OÑATE AND F. D. PIN 2. PARTICLE METHODS Partcle Methods am to represent the behavour of a physcal problem by a collecton of partcles. Each partcle moves accordngly wth ts own mass and the nternal/external forces appled on t. External forces are evaluated by the nteracton wth the neghbour partcles by smple rules. A partcle may be a physcal part of the doman (spheres, rocks, powder, etc.) or a specfc part of the contnuous doman prevously defned. Another characterstc of Partcle Methods s that all the physcal and mathematcal propertes are attached to the partcle tself and not to the elements as n the FEM. For nstance, physcal propertes lke vscosty or densty, physcal varables lke velocty, temperature or pressure and also mathematcal varables lke gradents or volumetrc deformatons are assgned to each partcle and they represent an average of the property around the partcle poston. Partcle methods are advantageous to treat dscrete problems lke granular materals but also to treat contnuous problems n whch there are possbltes of nternal separatons, contact problems or free-surfaces wth breakng waves. Accordngly to the way to evaluate the forces appled to each partcle, the method may be dvded nto two categores: those n whch the nteractng forces between the partcles are evaluated by a local contact problem [15] and those n whch the forces are evaluated by solvng a contnuous dfferental equaton n the entre doman [16]. Ths paper concerns wth the last category. Fnally, the most crucal characterstc of a Partcle Method s that there s not a specfed soluton doman. The problem doman s defned by the partcle postons and hence, there s not a boundary surface or lne. Ths s the reason why, when a dfferental equaton s to be solved n order to evaluate the forces, the boundary surface needs to be dentfed n order to mpose the boundary condtons. In addton, the partcles can be used to generate a dscrete doman wthn whch the ntegral form of the governng dfferental equatons s solved (see Fgure 1). In ths paper, a Partcle Fnte Element Method s proposed to deal wth the ncompressble Naver Stokes equatons. Then, the true materal wll be contnuous and ncompressble when t s submtted to compresson forces, but wth the possblty to separate under tracton forces. Ths s the case of most physcal fluds, lke water, ols and other fluds wth low rate of surface tractons. Both, flud and sold materals wll be modelled by an arbtrary number of partcles. On each partcle the actng forces wll be the gravty force (nternal force of the partcle) and the nteractng forces wth the neghbour partcles (external force to the partcle). The external forces wll be evaluated solvng the Naver Stokes equatons. For ths reason a doman needs to be defned at each tme step wth a defned boundary surface where the boundary condtons wll be mposed. Also at each tme step a new mesh s generated n order to defne shape functons to solve the dfferental equatons. Ths mesh s only useful for the defnton of the nteractng forces and vanshes once the forces are evaluated (see Fgure 1). The nterpolaton functons to be used are a partcular case of the Fnte Element Method shape functons. The boundary surface s defned usng the Alpha-Shape Method explaned n Secton 4. The evaluaton of the nteractng forces between partcles s descrbed next. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

4 THE PARTICLE FINITE ELEMENT METHOD 967 Fgure 1. Recognton of the boundary of the analyss doman and mesh update for successve pont dstrbutons. 3. PARTICLE POSITION UPDATE The partcle postons wll be updated va solvng the Lagrangan form of the Naver Stokes equatons. Let X be the ntal poston of a partcle at tme t n. Let x the fnal poston of a partcle at tme t n+1 and the tme ncrement Δt = t n+1 t n and u (x, t n+1 ) = u n+1 beng the velocty of the partcle at tme t n+1, the fnal poston can be approxmated by x = X + u n+1 Δt In the same way the dsplacement of the partcle d (x, t n+1 ) = d n+1 = u n+1 Δt Governng Lagrangan equatons n a vscous flud flow In the fnal x poston, the mass and momentum conservaton equatons can be wrtten as Mass conservaton: D Dt + u = 0 (1) x Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

5 968 S. R. IDELSOHN, E. OÑATE AND F. D. PIN Momentum conservaton: Du = p + τ j + f (2) Dt x where s the densty u are the Cartesan components of the velocty feld, p the pressure, τ j the devator stress tensor, f the source term (normally the gravty) and D/Dt represents the total or materal tme dervatve of a functon. For Newtonan fluds the stress tensor τ j may be expressed as a functon of the velocty feld through the vscosty μ by ( u τ j = μ + u j 2 ) u l δ j (3) x 3 x l For near ncompressble flows (u /x >u k /x l ) the term 2μ 3 and t may be neglected n Equaton (3). Then ( u τ j μ u x 0 (4) + u ) j x (5) In the same way, the term (/ )τ j near ncompressble flows as n the momentum equatons may be smplfed for τ j = ( ( u μ + u )) j = μ ( ) u + μ ( ) uj x x = μ ( ) u + μ ( ) uj μ x ( ) u (6) Then, the momentum equatons can be fnally wrtten as Du Dt = x p + τ j + f p + μ x ( u ) + f (7) Boundary condtons: On the boundares, the standard boundary condtons for the Naver Stokes equatons are τ j ν j pν = σ n u ν =ū n u ζ =ū t on Γ σ on Γ n on Γ t where ν and ζ are the components of the normal and tangent vector to the boundary. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

6 THE PARTICLE FINITE ELEMENT METHOD Implct explct tme ntegraton Equaton (7) wll be ntegrated mplctly n tme as Du Dt u (x,t n+1 ) u (X,t n ) Δt = un+1 u n Δt = [ x p + μ ( u ) + f ] n+θ (8) where [(x, t)] n+θ means θ(x, t n+1 )+(1 θ)(x, t n ) = θ n+1 +(1 θ) ˆ n and ˆ n = (x, t n ) represents the value of the functon at tme t n but at the fnal poston x. For smplcty n wll be used nstead of ˆ n. Only the case of θ = 1 (full mplct) wll be consdered next. Other values, as for nstance θ = 2 1, may be consdered wthout major changes. The tme ntegrated equatons become un+1 u n [ = ] n+1 [ p + μ ( u Δt x The mass conservaton s also ntegrated mplctly by ) + f ] n+1 (9) D Dt n+1 n Δt n+1 (un+1 ) = (10) x 3.3. The tme splttng The tme ntegraton of Equatons (9) presents some dffcultes because 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 Equatons (9) the three components of the velocty are only coupled va the pressure. The fractonal-step method proposed n Reference [17] wll be used. Ths bascally conssts n splttng each tme step nto 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 durng a second step. The fractonal-step algorthm for Equatons (9) and (10) s the followng: Splt of the momentum equatons Du Dt un+1 u n Δt = un+1 u + u un Δt = 1 p n τ n+θ j + f (11) x where u are fcttous varables termed fractonal veloctes defned by the splt (A) u = un + f Δt Δt γp n + Δt x τ n+θ j (12) (C) u n+1 = u Δt x (p n+1 γp n ) (13) Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

7 970 S. R. IDELSOHN, E. OÑATE AND F. D. PIN n whch p n = p(x, t n ) s the value of the pressure at tme t n but evaluated at the fnal poston and f s consdered constant n tme. In Equatons (12) and (13) γ s a parameter gvng the amount of pressure splttng, varyng between 0 and 1. A larger value of γ means 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 [17]. Takng nto account (6), the last term n (12) may be wrtten as τ n+θ j = μ ( u n+θ ) ( û n = μ(1 θ) ) + μθ The followng approxmatons have been ntroduced [17]: ( ) μ u n+θ ( û n ) μ(1 θ) + μθ ( u ) Ths allows to wrte Equaton (12) as u = u n + f Δt Δt γ ˆp n + Δt μ(1 θ) x For γ = 1 and θ = 1 u Δt μ ( u Splt of the mass conservaton equatons D Dt n+1 n Δt = n+1 + n Δt where s a fcttous varable defned by the splt ( û n ) + Δt μθ ( ) u n+1 ( u ) ) = u n + f Δt (14) = (un+1 u + u ) x (15) n Δt n+1 Δt = u x = (un+1 u ) x (16a) (16b) Coupled equatons From Equatons (13) and (16) the coupled mass momentum equaton becomes n+1 (B) Δt 2 = 2 x 2 (pn+1 ) (17) Takng nto account Equaton (16a), the above expresson can be wrtten as n+1 n Δt 2 + Δt u x = 2 x 2 (p n+1 ) (18) Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

8 THE PARTICLE FINITE ELEMENT METHOD 971 In Equaton (18) the ncompressblty condton has not been ntroduced yet. The smplest way to ntroduce the ncompressblty condton n a Lagrangan formulaton s to wrte Then, the frst term of Equaton (18) dsappears, gvng n+1 = n = 0 = (19) Δt u x = 2 x 2 (p n+1 ) The three step fractonal method used here can be summarzed by (A) u Δt μ ( u ) = u n x + f Δt u j (B) Δt u x = 2 x 2 (p n+1 ) p n+1 (20) (C) u n+1 = u Δt (p n+1 ) u n+1 x 3.4. Generaton of a new mesh One of the key ponts for the success of the Lagrangan flow formulaton descrbed here s the fast regeneraton of a mesh at every tme step on the bass of the poston of the nodes n the space doman. In ths work, the mesh s generated usng the so-called extended Delaunay tesselaton (EDT) presented n Reference [14]. The EDT allows to generate meshes of elements wth arbtrary polyhedrcal shapes (combnng trangles, quadrlaterals and other polygons n 2D and tetrahedra, hexahedra and arbtrary polyhedra n 3D) n a computng tme of order n, n beng the total number of nodes n the mesh. The shape functons for arbtrary polyhedral elements can be smply obtaned usng the so-called non-sbsonan nterpolatons [18]. Detals of the mesh generaton procedure and the shape functons for arbtrary polyhedra can be found n References [13, 14]. Once the new mesh has been generated at each tme step the numercal soluton s found usng the fnte element algorthm descrbed n the paper. The combnaton of elements wth dfferent geometrcal shapes n the same mesh s one of the nnovatve aspects of the Lagrangan formulaton presented here Spatal dscretzaton va the Meshless Fnte Element Method (MFEM) The unknown functons are approxmated usng an equal order nterpolaton for all varables n the fnal confguraton u = l N l (X, t)u l p = l N l (X, t)p l Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

9 972 S. R. IDELSOHN, E. OÑATE AND F. D. PIN In matrx form u = N T (X, t)u p = N T (X, t)p (21) or n compact form u = N T U = N T N T N T U (22) 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 co-ordnates. Then, the shape functons may change n tme followng the partcles poston. Durng the tme step a mesh update may ntroduce 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 ): N(X, t n ) = N n and N(X, t n+1 ) = N n+1 (23) Nevertheless, the followng hypothess wll be ntroduced: There s no 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 ) (24) Unfortunately, ths hypothess s not always possble to satsfy for all meshes and thus ntroduces small errors n the computaton whch are neglected n ths paper. Usng the Galerkn weghted resdual method to solve the splt equatons the followng ntegrals must be wrtten: (A) N u d Δt + N μ N u n d Δt {( )} u n+θ d N f d + N x γp n d Γ σ N ( σ n (τ n+θ j ν j γp n ν )) dγ = 0 (25) (B) { ( u ) } N 2 Δt x x 2 (p n+1 γp n ) d + N(ū n+1 ν u n+1 ν ) dγ = 0 (26) Δt Γu Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

10 THE PARTICLE FINITE ELEMENT METHOD 973 (C) { N (u n+1 u ) Δt + } (p n+1 γp n ) d x N (p n+1 γ ˆp n )ν dγ = 0 (27) Γ σ where the boundary condtons have also been splt and s the volume at tme t n+1. Integratng by parts some of the terms, the above equatons become (A) N (u f Δt) Δt d + μ N u n+θ d N u n Δt d + N γp n d x N ( σ n + γp n ν ) dγ = 0 (28) Γσ (C) (B) Δt N u x d N (p n+1 γp n ) d + x x Δt Γ u Nū n+1 n dγ = 0 { N (u n+1 u ) Δt + } (p n+1 γp n ) d N (p n+1 γp n )dγ = 0 x Γ σ It must be noted that the essental and natural boundary condtons of Equatons (29) are (29) (30) p = 0 on Γ σ (31) ū n+1 ν = 0 on Γ u (32) Dscrete equatons. Usng approxmatons (22) (24) the dscrete equatons become (A) In compact form N N T d U = γδt + Δt N N T x N N T d Un + Δt d P n Δtμ N N f d N T d U n+θ Γ σ N (σ n + γp n ) dγ (33) MU = MU n + ΔtF γδt BT P n Δtμ KUn+θ Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

11 974 S. R. IDELSOHN, E. OÑATE AND F. D. PIN and makng use of the approxmaton descrbed before for U n+ϑ ( M + Δtμθ ) K U = MU n + ΔtF γδt BT P n and for θ = 1 and γ = 0 In the same way Δt In compact form and for θ = 1 and γ = 0 Fnally (A) Δtμ(1 θ) KU n ( M + Δtμ ) K U = MU n + ΔtF (34) ( ) N N T d U + ( N Nū n+1 N T ) n dγ = d(p n+1 γp n ) x Δt Γ u x x (35) SP n+1 = Δt (BU Û) + SγP n (B) SP n+1 = Δt (BU Û) (36) N N T d Un+1 = N N T d U Δt N T N d(p n+1 γp n ) x + N N T dγ(p n+1 γp n ) (37) Γ σ In compact form and for θ = 1 and γ = 0 where the matrces are (C) MU n+1 = MU Δt BT (P n+1 γp n ) MU n+1 = MU Δt BT P n+1 (38) M p 0 0 M = 0 M p M p (39) Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

12 THE PARTICLE FINITE ELEMENT METHOD 975 M p = NN T d (40) [ ( ) ( ) ( ) ] N N N B = x NT d ; y NT d ; z NT d (41) ( N S = x N T x + N y N T y + N z N T z ) d (42) Û = Nū n+1 n dγ (43) Γ u S 0 0 K = 0 S 0 (44) 0 0 S [ F T = N T f x d ; N T f y d ; + 1 [ N T σ nx dγ; Γ σ Γ σ N T σ ny dγ; ] N T f z d ] N T σ nz dγ Γ σ (45) 3.6. Summary of a full teratve tme step A full tme step may be descrbed as follows: startng wth the known values u n and p n n each partcle, the computaton of the new partcle poston nvolves the followng steps: (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). (II) Move the partcles to the x n+1 poston and generate a mesh. (III) Evaluate the u velocty from (34). (It must be noted that the matrces M and K are separated n 3 blocks. Then, ths equatons may be solved separately for Ux,U y and U z.forθ = 0 (mplct) nvolves the soluton of 3 Laplacan equatons. For θ = 0 (explct) the M matrx may be lumped and nverted drectly). (I) Evaluate the pressure p n+1 by solvng the Laplacan Equaton (36). () Evaluate the velocty u n+1 usng (38). Go to (I) untl convergence. The Lagrangan splt scheme descrbed has two mportant advantages: (1) Step III s lnear and may be explct (θ = 0) or mplct (θ = 0). The use of a Lagrangan formulaton elmnates the standard convecton terms present n Euleran formulatons. The convecton terms are responsble for non-lnearty, non-symmetry and non-self-adjont operators whch requre the ntroducton of hgh-order stablzaton terms to avod numercal Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

13 976 S. R. IDELSOHN, E. OÑATE AND F. D. PIN oscllatons. All these problems are not present n ths formulaton. Only the non-lnearty remans due to the unknown of the fnal partcle poston. (2) In all the steps, the system of equatons to be solved are the evaluaton of the velocty components (step III) and the evaluaton of the pressure (step I). Those systems are scalar (only one degree of freedom by node), symmetrc and postve defnte. Then, t s very easy to solve them usng a symmetrc teratve scheme (such as the conjugate gradent method) 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 parameter must be added n Equatons (34) and (38). Nevertheless, the ncompressblty condton must be stablzed n equal-order approxmatons to avod possble 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. In those partcular cases, a stablzaton term must be ntroduced n Equatons (B) n order to elmnate pressure oscllatons. A smple and effectve procedure to derve a stablzed formulaton for ncompressble flows s based on the so-called Fnte Calculus formulatons [19 21]. In all the examples presented n ths paper, the γ parameter was always fxed equal to zero and the tme ncrements were fxed to a gven value of the Courant number 1, avodng n ths way all the stablzaton problems. 4. BOUNDARY SURFACES RECOGNITION One of the man problems n mesh generaton s the correct defnton of the boundary doman. Sometmes, boundary nodes are explctly defned as specal nodes, whch are dfferent from nternal nodes. In other cases, the total set of nodes s the only nformaton avalable and the algorthm must recognze the boundary nodes. Such s the case n Partcle Methods n whch, at each tme step, a new partcle poston s obtaned and the boundary-surface must be recognzed usng the new partcle postons. The use of the MFEM wth the extended Delaunay partton makes t easer to recognze boundary nodes. Consderng that the partcles follow a varable h(x) dstrbuton, where h(x) s the mnmum dstance between two partcles, the followng crteron has been used: All partcles on an empty sphere wth a radus r(x) bgger than αh(x) are consdered as boundary partcles (see Fgure 2). Thus, α s a parameter close to, but greater than one. Note that ths crteron s concdent wth the Alpha Shape concept [22]. Once a decson has been made concernng whch of the partcles are on the boundares, the boundary surface must be defned. It s well known that n 3D problems the surface fttng a number of partcles s not unque. For nstance, four boundary partcles on the same sphere may defne two dfferent boundary surfaces, a concave one and convex one. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

14 THE PARTICLE FINITE ELEMENT METHOD 977 Fgure 2. Contour recognton: Empty crcles wth radus α h(x) defne the boundary partcles. In ths work, the boundary surface s defned wth all the polyhedral surfaces havng all ther partcles on the boundary and belongng to just one polyhedron. See Reference [13]. The correct boundary surface may be mportant to defne the correct normal external to the surface. Furthermore, n weak forms (Galerkn) a correct evaluaton of the volume doman s also mportant. Nevertheless, t must be noted that n the crteron proposed above, the error n the boundary surface defnton s proportonal to h. Ths s the error order accepted n a numercal method for a gven node dstrbuton. The only way to obtan more accurate boundary surface defnton s by decreasng the dstance between the partcles. 5. NUMERICAL RESULTS A number of free-surface flow and flud structure nteracton problems wll be presented. In a frst group of examples the nteractng sold wll be consdered nfntely rgd and fxed. Those cases are useful to compare the results wth expermental and analytcal ones. The nteractng sold wll also be represented wth partcles but wth mposed velocty equal to zero. In a second group of examples movng rgd sold motons wll be consdered. In all cases, the elastc strans wll be neglected. The sold wll be consdered n two dfferent ways: (a) As a partcular materal wth a hgh vscosty parameter, much hgher than the flud doman. For practcal purposes a μ value wll be consdered. Ths value s enough to represent a sold wthout ntroducng numercal problems. (b) The sold wll be consdered as a boundary contour wth an mposed velocty. After each tme step, the flud forces on the sold due to the pressure and the vscous terms wll be evaluated. In the next step, the sold wll move rgdly usng Newton law. Tme steppng and teratve process: The tme step length Δt was mposed to a varable value and evaluated at the begnnng of each tme step. The crteron to calculate the tme step Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

15 978 S. R. IDELSOHN, E. OÑATE AND F. D. PIN Table I. Average tme (n s) for a standard PC consderng 3 teratons by tme step. n Monolthc Fractonal step Mesh generaton s 30 s 7 s mn 20 mn 1.5 mn mn 796 mn 18 mn was: durng the teratve mplct process (Secton 6.3), the tme step may be as bg as possble wth the lmtaton that at the end of the teraton any element cannot have a negatve or zero volume. In ths way the mesh s preserved durng the entre tme step. Ths crteron s less restrctve than mposng a Courant number less than one. In all the examples performed, a maxmum of 3 teratons n the teratve process (see table on Secton 3.6) was needed to reach a reasonable convergence. Computng tme: The computng tme of each tme step s of the same order as a standard ncompressble flud mechancs problem solved va a fractonal step method, addng the computng tme needed to generate the polyhedral mesh and the boundary recognton. One of the key ponts for the success of the Lagrangan flow formulaton descrbed here s the fast generaton of a mesh. In Reference [14] t s shown that the EDT and the Alpha-Shape method solve ths problem n order n 1.1. In partcular, Reference [14] presents the computng tme evaluatons for dfferent sze problems performed n a standard PC of 1 GHz. The computng tme n second for the mesh generaton and boundary recognton s t(s) = n 1.1 Ths tme must be compared wth the computng tme needed to solve a Laplacan equaton. Ths s very problem dependent, but for a standard 3D problem usng a conjugate gradent teratve method, an optmst number of operaton to acheve a reasonable convergence error s of order n 1.6. For the monolthc case, (the entre unknown solved together) ths means to solve a system of 4n degrees of freedom (d.o.f). Supposng 3 teratons by tme step, ths means for a 1 GHz PC: t(s) = (4n) 1.6 For the fractonal step method, ths means to solve 4 Laplacan of n d.o.f. at each teraton. Consderng also 3 teratons by tme step n the same PC means t(s) = n 1.6 Table I shows a comparson of the computng tme for dfferent number of partcles. It s clear that for large d.o.f problems, the computng tme needed to evaluate a new mesh at each tme step s not mportant compared wth computng tme nvolved to solve the non-lnear system Sloshng problems 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 [23]. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

16 THE PARTICLE FINITE ELEMENT METHOD 979 Fgure 3. Sloshng. Intal pont dstrbuton. Ths problem s nterestng because there s an analytcal soluton for small ampltudes. Fgure 3 shows a schematc vew of the problem, and the pont dstrbuton n the ntal poston. The dark ponts represent the fxed ponts where the velocty s fxed to zero. It s worth mentonng that n ths problem the wall has been represented by two layers of nodes but the elements constructed between layers are omtted from the ntegraton process. Thus, the nodes on the external layer do not take part n the computaton and are ncluded n the fgure only for vsualzaton purposes. Fgure 4 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. The analytcal soluton s only acceptable for small wave ampltudes. For larger ampltudes, addtonal waves are overlappng and fnally, the wave breaks and also some partcles can be separated from the flud doman due to ther large velocty. Fgure 5 shows the numercal results obtaned wth the method presented n ths paper for larger sloshng ampltudes. Breakng waves as well as separaton effects can be seen on the free-surface. Ths partcular and very complcated effect s apparently well represented by ths model. In order to test the potentalty of the method n a 3D doman, the same sloshng problem was solved as a 3D problem. Fgure 6 shows the dfferent pont poston at two tme steps. Each pont poston was represented by a sphere and only a half of the fxed recpent s represented on the fgure. The sphere representaton s used only to mprove the vsualzaton of the flud movement Dam collapse Ths problem was solved by Koshzuka and Oka [4] both expermentally and numercally n a 2D doman. It became a classcal example to test the valdaton of the Lagrangan formulaton n flud flows. In ths paper, the results obtaned usng the method proposed n 2D and 3D domans are presented. The water s ntally located on the left supported by a removable board. See Fgure 7. The collapse starts at tme t=0, when the removable board s sld-up. scosty and surface tenson are neglected. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

17 980 S. R. IDELSOHN, E. OÑATE AND F. D. PIN Fgure 4. Sloshng: Comparson of the numercal and analytcal soluton. Fgure 5. Sloshng: Dfferent tme step for large ampltudes. Fgures 8(a) (d) show the pont postons at dfferent tme steps. The blue ponts represent the free-surface detected wth the alpha-shape algorthm wth an alpha parameter α = 1.1. The nternal ponts are sky-blue and the fxed wall s yellow n the 3D and brown n the 2D example. The water s runnng on the bottom wall untl, near 0.3 s, t mpnges on the rght vertcal wall. Breakng waves appear at 0.6 s. Around t = 1 s the man water wave reaches the left wall agan Agreement wth the expermental results of Reference [4] both n the shape of the free surface and tme development s excellent. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

18 THE PARTICLE FINITE ELEMENT METHOD 981 Fgure 6. Sloshng: Dfferent tme step for 3D domans. Fgure 7. Dam Collapse. Intal poston. Left: expermental [6]. Rght: 3D smulaton. 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 wth expermental results Wave breakng on a beach A smulaton of the propagaton of a water wave and ts breakng due to shoalng over a plane slope s presented next. Ths example was numercally studed n Reference [23] wth a Lagrangan formulaton usng drectly the standard Fnte Element Method wth remeshng. There s also an analytcal soluton for a smplfed approxmaton that s used for comparson [24]. Fgure 9 shows the ntal pont dstrbuton and Fgure 10(a) comparson wth the analytcal free-surface at a dfferent tme step. The geometry of the problem as well as a dscusson of the analytcal soluton may be found n Reference [23]. Intally (Fgures 10(a) and (b) the wave travels over a constant depth bottom towards the slope wth no ostensble change of shape. Strongly non-lnear effects appear when the wave hts the slope (Fgure 10(c)). The crest of the wave accelerates whle the rest lags behnd (Fgure 10(d)). At ths tme the comparsons wth the analytcal soluton are n agreement only n the wave poston. The shape of the wave obtaned wth the numercal soluton s totally dfferent. The reason s that the analytcal soluton gves symmetrcal shape waves, whch are not physcal, before the breakng process. Subsequently, a water jet s formed at the crest Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

19 982 S. R. IDELSOHN, E. OÑATE AND F. D. PIN Fgure 8. Dam Collapse. Comparson wth expermental results of Reference [6]: (a) expermental, 2D and 3D numercal soluton at t = 0.2 s; (b) expermental, 2D and 3D numercal soluton at t = 0.4 s; (c) expermental, 2D and 3D numercal soluton at t = 0.6 s; and (d) expermental, 2D and 3D numercal soluton at t = 0.8s. plunge makng the breakng wave (Fgures 10(e) and (f)) and comng n contact wth the nearly stll surface of the water ahead. In Reference [23] the evaluaton s stopped before ths contact pont. Usng the methodology proposed n ths paper, the analyss may be contnued Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

20 THE PARTICLE FINITE ELEMENT METHOD 983 Fgure 9. Wave breakng on a beach. Intal geometry and pont poston. Fgure 10. Wave breakng on a beach. Comparson wth analytcal results at dfferent tme steps. Top: Numercal soluton. Bottom: Analytcal soluton: (a) t = 0 sec; (b) t = 4 sec; (c) t = 8 s; (d) t = 9.8s; and (e) t = 11.2 sec, (f) t = 14.2s. untl the end. In Fgures 10(g) and (h), the wave fnally hts a lateral wall (ntroduced n the model to stop the lateral effects) producng drop separatons, and then comng back towards the left as a new wave. The ablty of the model to accurately smulate the varous stages of the wave breakng s noteworthy. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

21 984 S. R. IDELSOHN, E. OÑATE AND F. D. PIN Fgure 11. Breakng wave on a beach: Oblque wave on a 3D doman. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

22 THE PARTICLE FINITE ELEMENT METHOD 985 Fgure 12. Sold floatng on a free-surface. Intal geometry and pont dstrbuton. Nevertheless, a 2D doman s an easy case and may be solved acceptably wth any mesh generator. The true problems appear n a 3D doman, where the mesh generaton s complcated wth the presence of slvers and other geometrc mesh generaton problems. In order to show the power of the tool presented, the same problem was solved n a 3D doman. To transform the wave breakng descrbed before n a true 3D problem, the ntal poston of the wave was ntroduced havng an oblque angle wth the beach lne. In ths way, a 3D effect appears. When the wave hts the slope, the crest of the wave accelerates dfferently n accordance wth the depth, nducng the wave to correct ts oblque poston and break parallel to the beach. The results may be seen n Fgure 11 for dfferent tme steps Sold floatng on a free-surface The followng example, shown schematcally n Fgure 12, represents a very nterestng problem of flud structure nteracton when there s a weak nteracton between the flud and a large rgd deformaton of the structure. In ths case, there s also a free-surface problem, representng a schematc case of sea-keepng n shp hydrodynamcs. The example shows a recpent wth a floatng pece of wood n whch a wave s produced on the left sde. The wave ntercepts the wood pece producng a breakng wave and movng the floatng wood. In ths example the sold was represented by very vscous flows wth a vscosty parameter order ten tmes greater than the water vscosty. Fgure 13 shows the pressure contours and the free-surface poston for dfferent tme steps. Ths example, as well as the next example to be presented n Secton 5.5, has no analytcal or expermental result to use as comparson. The reason to present t n ths paper s to show the possblty of the method to carry out flud structure nteracton problems. The behavour of the sold seems to be correct and the flow movng s acceptably realstc Sold cube fallng n a recpent wth water Ths last example s also a case of flud structure nteracton. The sold s ntally totally free and s fallng down nto a recpent wth a flud. Fgure 14 shows the ntal poston and the ntal mesh. In ths example, the sold was modelled as a boundary condton for the flud. Once the pressure and the vscous forces have been evaluated n the flud, the sold s accelerated usng Newton law. The sold has a mass and a gravty force concentrate n ts gravty centre. The sold s consdered to be lght compared to the lqud weght. At the begnnng the sold falls free due to the gravty forces. Once n contact wth the water free-surface (t = 0.31 s) the alpha-shape method recognzes the dfferent boundary contours. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

23 986 S. R. IDELSOHN, E. OÑATE AND F. D. PIN Fgure 13. Sold floatng on a free-surface. Pressure contours and free-surface postons for dfferent tme steps: (a) t = 0 s; (b) t = 0.29 s; (c) t = 0.49 s; (d) t = 0.71 s; and (e) t = 1.23 s. For nstance, the red ponts on the sold cube are dry partcles whle the blue ponts on the sold cube are wet partcles. The sky-blue ponts are free-surface ponts. The pressure and the vscous forces are evaluated n the entre doman and n partcular on the sold cube. Ths flow forces ntroduce a negatve acceleraton to the vertcal velocty untl, once the sold s completely nsde the water, the fallng velocty becomes zero. Then, Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

24 THE PARTICLE FINITE ELEMENT METHOD 987 Fgure 14. Sold cube fallng nto a recpent wth water. Intal mesh. Arqumdes prncple makes the sold to go up to the free-surface. Fgure 15 shows dfferent tme steps. It s nterestng to observe that there s a rotaton of the sold. Ths s due to the fact that the centre of the floatng forces s hgher n the rotated poston than n the ntal ones. 6. CONCLUSIONS Partcle Methods combned wth a Fnte Element Method n whch the meshes are generated lnearly wth the number of partcles are an excellent tool to solve flud mechanc problems, especally flud structure nteractons wth movng free-surfaces. The Meshless Fnte Element Method seems to be the best adapted FEM to ths knd of combnaton. In fact, the MFEM has the advantages of a meshless method concernng the easy ntroducton of the nodes connectvty n a bounded tme of order n. The method also preserve the classcal advantages of the FEM such as: (a) the smplcty of the shape functons, (b) C 0 contnuty between elements, (c) an easy ntroducton of the boundary condtons, and (d) symmetrc matrces. The fractonal step approach presented here has proved to be an effcent procedure for solvng accurately the Lagrangan flow equatons. Both Partcle Methods and the MFEM are the key ngredents to the Partcle Fnte Element Method, a very sutable method to solve flud structure nteracton problems ncludng freesurface, breakng waves, flow separatons, contact problems and collapse stuatons. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

25 988 S. R. IDELSOHN, E. OÑATE AND F. D. PIN Fgure 15. Sold cube fallng nto a recpent wth water. Dfferent tme steps. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:

26 THE PARTICLE FINITE ELEMENT METHOD 989 REFERENCES 1. Gngold RA, Monaghan JJ. Smoothed partcle hydrodynamcs, theory and applcaton to non-sphercal stars. Monthly Notces of the Royal Astronomcal Socety 1997; 181: Bonet J, Kulasegaram S. Correcton and stablzaton of smooth partcle hydrodynamcs methods wth applcatons n metal formng smulaton. Internatonal Journal for Numercal Methods n Engneerng 2000;47: Dlts GA. Movng least squares partcle hydrodynamcs. I. Consstency and stablty. Internatonal Journal for Numercal Methods n Engneerng 1999; 44: Koshzuka S, Oka Y. Movng partcle sem-mplct method for fragmentaton of ncompressble flud. Nuclear Engneerng Scence 1996; 123: Crowley WP. Lecture Notes n Physcs. Sprnger: Berln, 1970; Frtts MJ, Crowley WP, Trease HE. The Free Lagrange Method, Lecture Notes n Physcs, vol Sprnger: New York, Belytschko T, Lu Y, Gu L. Element free Galerkn methods. Internatonal Journal for Numercal Methods n Engneerng 1994; 37: De S, Bathe KJ. The method of fnte spheres wth mproved numercal ntegraton. Computers and Structures 2001; 79: Nayroles B, Touzot G, llon P. Generalzng the fem: dffuse approxmaton and dffuse elements. Computatonal Mechancs 1992; 10: Oñate E, Idelsohn SR, Zenkewcz OC, Taylor RL. A fnte pont method n computatonal mechancs. Applcatons to convectve transport and flud flow. Internatonal Journal for Numercal Methods n Engneerng 1996; 39(22): Oñate E, Idelsohn SR, Zenkewcz OC, Taylor RL, Sacco C. A stablzed fnte pont method for analyss of flud mechancs problems. Computer Methods n Appled Mechancs and Engneerng 1996; 39: Idelsohn SR, Stort MA, Oñate E. Lagrangan formulatons to solve free surface ncompressble nvscd flud flows. Computer Methods n Appled Mechancs and Engneerng 2001; 191: Idelsohn SR, Oñate E, Calvo N, Del Pn F. The meshless fnte element method. Internatonal Journal for Numercal Methods n Engneerng 2003; 58(6): Calvo N, Idelsohn SR, Oñate E. Polyhedrzaton of an arbtrary 3D pont set. Computer Method n Appled Mechancs and Engneerng 2003; 192: Rojek J, Oñate E, Zarate F, Mquel J. Modellng of rock, sol and granular materals usng sphercal elements. Proceedngs of the European Conference on Computer Mechancs (ECCM 2001), Cracow, Poland, June Idelsohn SR, Oñate E, Del Pn F. A Lagrangan meshless fnte element method appled to flud structure nteracton problems. Computer and Structures 2003; 81: Codna R. Pressure stablty n fractonal step fnte element methods for ncompressble flows. Journal of Computatonal Physcs 2001; 170: Belkov, Semenov A. Non-sbsonan nterpolaton on arbtrary system of ponts n Eucldean space and adaptve generatng solnes algorthm. Numercal Grd Generaton n Computatonal Feld Smulaton, Proceedngs of the 6th Internatonal Conference. Greenwch Unversty, July Oñate E. Dervaton of stablzed equatons for advectve dffusve transport and flud flow problems. Computer Methods n Appled Mechancs and Engneerng 1998; 151(1 2): Oñate E. A stablzed fnte element method for ncompressble vscous flows usng a fnte ncrement calculus formulaton. Computer Methods n Appled Mechancs and Engneerng 2002; 182(1 2): Oñate E. Possbltes of fnte calculus n computatonal mechancs. Internatonal Journal for Numercal Methods n Engneerng 2004; 60: Edelsbrunner H, Mucke EP. Three-dmensonal alpha-shape. ACM Transactons on Graphcs 1994; 3: Radovtzky R, Ortz M. Lagrangan fnte element analyss of a Newtonan flows. Internatonal Journal for Numercal Methods n Engneerng 1998; 43: Latone E. The second approxmaton to cnodal waves. Journal of Flud Mechancs 1960; 9:430. Copyrght 2004 John Wley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61: