SPH and ALE formulations for sloshing tank analysis

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1 Int. Jnl. of Multphyscs Volume 9 Number SPH and ALE formulatons for sloshng tank analyss Jngxao Xu 1, Jason Wang 1 and Mhamed Soul*, 2 1 LSTC, Lvermore Software Technology Corp. Lvermore CA 94550, USA 2 Unversté de Llle Laboratore de Mécanque de Llle, UMR CNRS 8107, France ABSTRACT Desgn of fuel tanks requres the knowledge of hydrodynamc pressure dstrbuton on the structure. These can be very useful for engneers and desgners to defne approprate materal propertes and shell thckness of the structure to be resstant under sloshng or hydrodynamc loadng. Data presented n current tank desgn codes as Eurocode, are based on smplfed assumptons for the geometry and materal tank propertes. For complex materal data and complex tank geometry, numercal smulatons need to performed n order to reduce expermental tests that are costly and take longer tme to setup. Dfferent formulatons have been used for sloshng tank analyss, ncludng ALE (Arbtrary Lagrangan Euleran) and SPH (Smooth Partcle Hydrodynamc). The ALE formulaton uses a movng mesh wth a mesh velocty defned trough the structure moton. In ths paper the mathematcal and numercal mplementaton of the FEM and SPH formulatons for sloshng problem are descrbed. From dfferent smulatons, t has been observed that for the SPH method to provde smlar results as ALE formulaton, the SPH meshng, or SPH partcle spacng needs to be fner than ALE mesh. To valdate the statement, we perform a smulaton of a sloshng analyss nsde a partally flled tank. For ths smple, the partcle spacng of SPH method needs to be at least two tmes fner than ALE mesh. A contact algorthm s performed at the flud structure nterface and SPH partcles. In the paper the effcency and usefulness of two methods, often used n numercal smulatons, are compared. Keywords: SPH, ALE, Sloshng tank 1. INTRODUCTION In computatonal mechancs there s a growng nterest n developng meshless methods and partcle methods as alternatves to tradtonal FEM (Fnte Element Method). Among the varous meshfree and partcle methods, Smoothed Partcle Hydrodynamcs (SPH) s the longest establshed and s approachng a mature stage. SPH s a Lagrangan meshless method n whch the problem to be solved s dscretzed usng partcles that are free to move rather than element ted by classcal mesh connectvty. SPH method has been extensvely used for hgh mpact velocty applcatons, n aerospace and defense ndustry for problems where classcal FEM methods fal due to hgh meshes dstorton. For small deformaton, FEM Lagrangan *Correspondng Author: E-mal: mhamed.soul@unv-llle1.fr

2 210 SPH and ALE formulatons for sloshng tank analyss formulaton can solve structure nterface and materal boundary accurately, the man lmtaton of the formulaton s hgh mesh dstorton for large deformaton and movng structure. One of the commonly used approach to solve these problems s the ALE formulaton whch has been used wth success for smulaton of flud structure nteracton wth large structure moton such as sloshng fuel tank n automotve ndustry and brd mpact n aeronautc ndustry. It s well known from prevous analyss, see Aquelet et al. [1], that the classcal FEM Lagrangan method s not sutable for most of the FSI problems due to hgh mesh dstorton n the flud doman. To overcome dffcultes due to large mesh dstorton, ALE formulaton has been the only alternatve to solve flud structure nteracton for engneerng problems. For the last decade, SPH and DEM (Dscrete Element Method), have been used usefully for engneerng problems to smulate hgh velocty mpact problems, hgh explosve detonaton n sol, underwater exploson phenomena, and brd strke n aerospace ndustry, see Han et al. [2] for detals descrpton of DEM method. SPH s a mesh free Lagrangan descrpton of moton that can provde many advantages n flud mechancs and also for modellng large deformaton n sold mechancs. For some applcatons, ncludng underwater exploson and hydrodynamc mpact on deformable structures, engneers have swtched from ALE to SPH method to reduce CPU tme and save memory allocaton. Unlke ALE method, and because of the absence of the mesh, SPH method suffers from a lack of consstency than can lead to poor accuracy, as descrbed n Randles et al. [14] and Vgnevc et al. [16]. In ths paper, devoted to ALE and SPH formulatons for sloshng tank problems, the mathematcal and numercal mplementaton of FEM and SPH formulatons are descrbed. From dfferent smulatons, t has been observed that for the SPH method to provde smlar results as FEM Lagrangan formulatons, the SPH meshng, or SPH spacng partcles needs to be fner than the ALE mesh, see Messahel et al. [15] for underwater exploson problem. To valdate the statement on flud structure nteracton problems, we perform a smulaton of a sloshng problem. In the smulaton, the partcle spacng of SPH method needs to be at least two tmes fner than ALE mesh. A contact algorthm s performed at the flud structure nterface for both SPH and ALE formulatons. In Secton 2, the governng equatons of ALE formulaton are descrbed. In ths secton, we dscuss mesh moton as well as advecton algorthms used to solve mass, momentum and energy conservaton n ALE formulaton. Secton 3 descrbes the SPH formulaton, unlke FEM formulaton whch based of the Galerkn approach, SPH s a collocaton method. The last secton s devoted to numercal smulaton of flud sloshng nsde a movng rgd structure tank, usng both FEM and SPH methods. To get comparable results between FEM and SPH, the partcle spacng of SPH method needs to be at least two tmes fner than FEM mesh. 2. ALE FORMULATION A bref descrpton of the FEM formulaton used n ths paper s presented, addtonal detals can be provded n Benson [4]. To solve flud structure nteracton problems, a Lagrangan formulaton s performed for the structure and an ALE formulaton for the flud materal. In general ALE descrpton, an arbtrary referental coordnate s ntroduced n addton to the Lagrangan and Euleran coordnates. The materal dervatve wth respect to the reference coordnate can be descrbed n equaton (2.1). Thus substtutng the relatonshp between materal tme dervatve and the reference confguraton tme dervatve leads to the ALE equatons, f( X, t) = f( x, t) + w f ( x, t ) t t x (2.1)

3 Int. Jnl. of Multphyscs Volume 9 Number where X s the Lagrangan coordnate, x the Euleran coordnate, w s the relatve velocty. Let denote by v the velocty of the materal and by u the velocty of the mesh. In order to smplfy the equatons we ntroduce the relatve velocty w = v u. Thus the governng equatons for the ALE formulaton are gven by the followng conservaton equatons: () Mass equaton. ρ ρ = v t x ρ w x (2.2) () Momentum equaton. v ρ σ ρ ρ = + b w v t x, (2.3) σ s the stress tensor defned by σ = p + τ, where τ s the shear stress from the consttutve model, and p the pressure. The volumetrc compressve stress p s computed though an equaton of state, and the shear stress from materal consttutve law. () Energy equaton. E ρ σ ρ ρ = + v bv w E t x, (2.4) Note that the Euleran equatons commonly used n flud mechancs by the CFD communty, are derved by assumng that the velocty of the reference confguraton s zero, u = 0, and that the relatve velocty between the materal and the reference confguraton s therefore the materal velocty, w = v. The term n the relatve velocty n (2.3) and (2.4) s usually referred to as the advectve term, and accounts for the transport of the materal past the mesh. It s the addtonal term n the equatons that makes solvng the ALE equatons much more dffcult numercally than the Lagrangan equatons, where the relatve velocty s zero. There are two ways to mplement the ALE equatons, and they correspond to the two approaches taken n mplementng the Euleran vewpont n flud mechancs. The frst way solves the fully coupled equatons for computatonal flud mechancs; ths approach used by dfferent authors can handle only a sngle materal n an element as descrbed for example n Benson [4]. The alternatve approach s referred to as an operator splt n the lterature, where the calculaton, for each tme step s dvded nto two phases. Frst a Lagrangan phase s performed, n whch the mesh moves wth the materal, n ths phase the changes n velocty and nternal energy due to the nternal and external forces are calculated. The equlbrum equatons are: v ρ σ ρ t = + b,, E ρ σ ρ = v t + bv., (2.5) (2.6) In the Lagrangan phase, mass s automatcally conserved, snce no materal flows across element boundares.

4 212 SPH and ALE formulatons for sloshng tank analyss In the second phase, the advecton phase, transport of mass, energy and momentum across element boundares are computed; ths may be thought of as remappng the dsplaced mesh at the Lagrangan phase back to ts orgnal for Euleran formulaton or arbtrary locaton for ALE formulaton usng smoothng algorthms. From a dscretzaton pont of vew of (2.5) and (2.6), one pont ntegraton s used for effcency and to elmnate lockng, Belytschko et al. [3], zero energy modes are controlled wth an hourglass vscosty. A shock vscosty, wth lnear and quadratc terms developed by Von-Neumann and Rchtmeyer n earler fftes s used to resolve the shock wave. The resoluton s advanced n tme wth the central dfference method, whch provdes a second order accuracy for tme ntegraton. For each node, the velocty and dsplacement are updated as follows: n+ 1/2 n 1/2 1 u = u +Δ t. M.( F + F ) n+ 1 n 1 n+ 1/2 x = x +Δtu (2.7) where F nt s the nternal vector force and F ext the external vector force assocated wth body forces, couplng forces, and pressure boundary condtons, M s a dagonal lumped mass matrx. For each element of the mesh, the nternal force s computed as follows: B s the gradent matrx and Nelem the number of elements. The tme step sze, Δt, s lmted by the Courant stablty condton [4], whch may be expressed as: (2.8) where l s the characterstc length of the element, and c the sound speed of the element materal. For a sold materal, the speed of sound s defned as: exl t F = B. σ. dv nt Nelem k= 1 k Δt l c nt c = K ρ (2.9) where ρ s the materal densty, K s the module of compressblty MOVING MESH ALGORITHM The ALE algorthm used n the paper allows the flud nodes to move n order to mantan the ntegrty of the mesh. As the flud mpacts the plate, the flud mesh moves wth a mesh velocty that s dfferent from flud partcle velocty. The choce of the mesh velocty consttutes one of the maor problems wth the ALE descrpton. Dfferent technques have been developed for updatng the flud mesh doman. For problems defned n smple domans, the mesh velocty can be deduced through a unform or non unform dstrbuton of the nodes along straght lnes endng at the movng boundares. Ths technque has been used for dfferent applcatons ncludng water wave problems. For general computatonal domans, the mesh velocty s computed through partal dfferental equatons, wth approprate boundary condtons. For sloshng problem where the tank s movng wth a appled velocty that s tme dependent, the flud mesh moves as a rgd mesh followng the tank. Ths new ALE feature allows the mesh to stay regular, and the tme step, whch can be

5 Int. Jnl. of Multphyscs Volume 9 Number affected by mesh dstorton, to be stable. In other words, there s only mesh moton and no mesh dstorton due to the ALE formulaton. Ths method s very useful for movng or rotatng tanks, where the flud mesh wll move and rotate wth the tank wthout undergong any mesh deformaton. The ALE algorthm used n the paper allows the flud mesh to follow the movement of the structure. The ntegrty of mesh structure s mantaned. As the structure mpacts the rgd plate and then moves and rotates, the flud mesh moves as a rgd mesh n the coordnate system attached to the structure. Ths ALE algorthm can be appled to several problems n movng structure that are rgd or undergo small deformatons ADVECTION PHASE In the second phase, the transport of mass, momentum and nternal energy across the element boundares s computed. Ths phase may be consdered as a re-mappng phase. The dsplaced mesh from the Lagrangan phase s remapped nto the ntal mesh for an Euleran formulaton. To llustrate the advecton phase, we consder n Fgure 1.1, a smple problem wth 2 dfferent materals, one wth hgh pressure and the second a lower pressure. Durng the Lagrangan phase, materal wth hgh pressure expands, and the mesh moves wth the materal. Snce we are usng Euleran formulaton, the mesh s mapped to ts ntal confguraton, n the advecton phase, materal volume called flux s movng from element to element, but we keep separate materals n the same element, usng nterface trackng between materals nsde an element. Conservaton propertes are performed durng the Lagrangan phase; stress computaton, boundary condtons, contact forces are computed. The advecton phase can be seen as a remappng phase from a deformable mesh to ntal mesh for an Euleran formulaton, or to an arbtrary mesh for general ALE formulaton. In the advecton phase, volume flux of materal through element boundary needs to be computed. Once the flux on element faces of the mesh s computed, all state varables are updated accordng to the followng algorthm, usng a fnte volume algorthm (2.10), + + faces. = 1 V M V M Flux M = + (2.10) where the superscrpts and + refer to the soluton values before and after the transport. Values that are subscrpted by refer to the boundares faces of the element, through whch the materal flows, and the Flux are the volume fluxes transported through adacent elements. The flux s postve f the element receves materal and negatve f the element looses materal. The ALE mult-materal method s attractve for solvng a broad range of non-lnear problems n flud and sold mechancs, because t allows arbtrary large deformatons and enables free surfaces to evolve. The advecton phase of the method can be easly mplemented n an explct Lagrangan fnte element code. 3. SPH FORMULATION The SPH method developed orgnally for solvng astrophyscs problem has been extended to sold mechancs by Lbersky et al. [8] to model problems nvolvng large deformaton ncludng hgh mpact velocty. SPH method provdes many advantages n modelng severe deformaton as compared to classcal FEM formulaton whch suffers from hgh mesh dstorton. The method was frst ntroduced by Lucy [9] and Monaghan et al. [11], for gas dynamc problems and for problems where the man concern s a set of dscrete physcal

6 214 SPH and ALE formulatons for sloshng tank analyss partcles than the contnuum meda. The method was extended to solve hgh mpact velocty n sold mechancs, CFD applcatons governed by Naver-Stokes equatons, and flud structure nteracton problems. It s well known from prevous papers, Vla [17] that SPH method suffers from lack of consstency that can lead to poor accuracy of moton approxmaton. A detaled overvew of the SPH method s developed by Lu et al. [10], where the two steps for representng a contnuous functon, usng ntegral nterpolaton and kernel approxmaton are gven by (3.1) and (3.2), where the Drac functon satsfes: ux ( ) = uy ( ). δ( x ydy ) δ(x y) = 1 x = y δ(x y) = 0 x y (3.1) The approxmaton of the ntegral functon (3.1) s based on the kernel approxmaton W, that approxmates the Drac functon based on the smoothng length h, Fgure 3.1, 1 d W(, d h) = θ α. h h that represents support of the kernel functon. SPH nterpolaton s gven by the followng equaton (3.2): ux ( ) = ω. u. Wx ( x, h) where ω = m s the volume of the partcle. The sum n equaton (3.2) s over partcles n the ρ support doman of the kernel as descrbed n Fgure 3.1. Unlke FEM, where weak form Galerkn method wth ntegraton over mesh volume, s common practce to obtan dscrete set of equatons, n SPH snce there s no mesh, θ ( d / h) Fgure 3.1: Partcle spacng and Kernel functon 0 2 d/h

7 Int. Jnl. of Multphyscs Volume 9 Number collocaton based method s used. In collocaton method the dscrete equatons are obtaned by enforcng equlbrum equatons, mass, lnear momentum and energy, at each partcle. In SPH method, the followng equatons are solved for each partcle (3.3), d dt ρ m = ρ v ( v ) A ρ d σ σ = dt v m ρ A ρ A 2 2 (3.3) d dt e P = m ( v v ) A 2 ρ where ρ, v, e are densty, velocty and nternal energy of partcle, σ, m are Cauchy stress and partcle mass. A s the gradent of the kernel functon defned by (3.4) A = x W( x x, h) (3.4) It had be shown that convergence and stablty of the SPH methods depends on the dstrbuton of partcles n the doman. In order to treat problem nvolvng dscontnutes n the flow varables such as shock waves an addtonal dsspatve term s added as an addtonal pressure term. Ths artfcal vscosty should be actng n the shock layer and neglected outsde the shock layer. In ths smulaton a pseudo-artfcal pressure term μ derved by Monoghan et al. [12] s used. Ths term s based on the classcal Von Neumann artfcal vscosty and s readapted to the SPH formulaton as follow, π = β 2 α c, f v r < 0 (In the shock layer) (3.5) π = 0 (Outsde the shock layer) ( ) ( ) v. r ρ + ρ c + c Where μ =, ρ =,andc = are respectvely the average densty 2 2 r + εh 2 2 and speed of sound, s small perturbaton that s added to momentum and energy equatons to avod sngulartes, fnally α and β n equaton (5.1) are respectvely the lnear and quadratc coeffcents. 4. CONSTITUTIVE MATERIAL MODELS FOR WATER In ths paper a Newtonan flud consttutve materal law s used water, see Table 1. For pressure response, Me Grunesen equaton of state s used wth parameters defned n Table 2, whch defnes materal s volumetrc strength and pressure to densty rato. Pressure n Me Grunesen equaton of state s defned by equaton (4.1) 2 μμ ( 1) P = ρ. c. 0 ( μ s( μ 1)) 2 (4.1)

8 216 SPH and ALE formulatons for sloshng tank analyss ρ Where C s the bulk speed of sound, μ = 1, where ρ 0 and ρ are the ntal and current ρ0 denstes. The coeffcent s s the lnear Hugonot slope coeffcent of the shock velocty partcle velocty (U s U p ) curve, equaton (4.2), U s = c + s U p (4.2) U s s the shock wave velocty, U p s the partcle velocty. Ths equaton of state requres flud specfc coeffcent S, whch s obtaned through shock experment by curve fttng of the U s U p relatonshp. Shock experments on fluds and solds provde a relaton between the shock speed U s and the partcle velocty behnd the shock, U p along the locus of shocked states. An mportant phenomenon that arses durng hydrodynamc mpact s the formaton of shock, mathematcally equatons (3.1) (3.3) develop a shock, whch lead to non contnuous soluton and the problem s well posed only f the shock condtons are satsfed. These condtons called Rankne Hugonot condtons descrbe the relatonshp between the states on both sdes of the shock for conservaton of mass, momentum and energy across the shock, and are derved by enforcng the conservaton laws n ntegral form over a control volume that ncludes the shock. In the absence of numercal vscosty, hgh non physcal oscllatons are generated n the mmedate vcnty of the shock. 5. NUMERICAL SIMULATIONS 5.1. FLUID-STRUCTURE CONTACT ALGORITHM For SPH and ALE smulatons, a penalty type contact s used to model the nteracton between the flud and the plate. In computatonal mechancs, contact algorthms have been extensvely studed by several authors. Detals on contact algorthms can be found n Belyshko et al. [3]. Classcal mplct and explct couplng are descrbed n detal n Longatte et al. [12], where hydrodynamc forces from the flud solver are passed to the structure solver for stress and dsplacement computaton. In ths paper, a couplng method based on penalty contact algorthm s used. In penalty based contact, a contact force s computed proportonal to the penetraton depth, the amount the constrant s volated, and a numercal stffness value. In an explct FEM method, contact algorthms compute nterface forces due to mpact of the structure on the flud, these forces are appled to the flud and structure nodes n contact n order to prevent a node from passng through contact nterface. In contact one surface s desgnated as a slave surface, and the second as a master surface. The nodes lyng on both surfaces are also called slave and master nodes respectvely. The penalty method mposes a resstng force to the slave node, proportonal to ts penetraton through the master Table 1: Materal data for water Materal Densty (kg m 3 ) Dynamc Vscosty Water Table 2: Me-Grunesen Equaton of state Materal S Speed of sound C (m/s) water

9 Int. Jnl. of Multphyscs Volume 9 Number Fgure 5.1: SPH FEM contact descrpton segment, as shown n Fgure 4.1 descrbng the contact process. Ths force s appled to both slave and nodes of the master segment n opposte drectons to satsfy equlbrum. Penalty couplng behaves lke a sprng system and penalty forces are calculated proportonally to the penetraton depth and sprng stffness. The head of the sprng s attached to the structure or slave node and the tal of the sprng s attached to the master node wthn a flud element that s ntercepted by the structure, as llustrated n Fgure 4.1. Smlarly to penalty contact algorthm, the couplng force s descrbed by (5.1): F = k d (5.1) where k represents the sprng stffness, and d the penetraton. The force F n Fgure 4.1 s appled to both master and slave nodes n opposte drectons to satsfy force equlbrum at the nterface couplng, and thus the couplng s consstent wth the flud-structure nterface condtons namely the acton-reacton prncple. The man dffculty n the contact algorthms comes from the evaluaton of the stffness coeffcent k n Eq. (5.2). The stffness value s problem dependent, a good value for the stffness should reduce energy nterface n order to satsfy total energy conservaton, and prevent flud leakage through the structure. The value of the stffness k s stll a research topc for explct contact-mpact algorthms n structural mechancs. In ths paper, the stffness value s smlar to the one used n Lagrangan explct contact algorthms. The value of k s gven n term of the bulk modulus K of the flud element n the couplng contanng the slave structure node, the volume V of the flud element that contans the master flud node, and the average area A of the structure element connected to the structure node ALE MESH SENSITIVITY ANALYSIS FOR SPH METHOD A detaled fnte element model was developed to represent the sloshng problem. Before conductng the smulaton, mesh senstvty tests were performed to compute sloshng frequency of the fnte element model for whch analytcal soluton s avalable n the lterature. Three dfferent mesh denstes were used for mesh senstvty tests from to hexahedra elements for the flud mesh. Smulaton of the three fnte element meshes gves same results wth good correlaton expermental test results provded by Shao et al. [20]. The optmal model of elements, shown n Fgure 5.2, was taken as reference soluton for the ALE fnte element smulaton. In the smulaton, the tank s modeled as rgd materal shearng common nodes wth the flud mesh at the flud structure nterface. Dmensons of the rgd tank are presented n Fgure 5.2, wth a thckness of 1 mm. For the SPH smulatons, three dfferent models have been used, the frst model has a number of partcles smlar to the number of nodes n the ALE model, whch conssts of

10 218 SPH and ALE formulatons for sloshng tank analyss Fgure 5.2: Problem descrpton and ALE mesh approxmately partcles, the second model conssts of partcles and the thrd model wth partcles wth unform mass, both SPH models for and partcles are presented n Fgure 5.3 and 5.4. The tank s submtted to a horzontal velocty descrbed n the paper by Sho et al. [20] where expermental tme hstory for the heght of the waver for 10 seconds s provdes. The tank velocty n the horzontal drecton s gven by: v(t) = cos(2 π t/t) where T = 1.5 sec s the perod of the horzontal velocty moton of the tank. Fgure 5.3: Model wth SPH partcles

11 Int. Jnl. of Multphyscs Volume 9 Number Fgure 5.4: Model wth SPH partcles The ampltude of the water heght at the tank wall are computed for ALE and SPH smulatons and compared to expermental data. Good correlaton between ALE and expermental results usng dentcal parameters for the water and tank as shown n Table 3. The frst SPH smulaton usng partcles smlar to ALE model, dd not correlate well wth ether ALE nor expermental results n term of the ampltude of the wave at the tank nterface. To mprove SPH results and obtan good correlaton wth ALE model, fner partcle spacng needs to be performed for SPH smulaton. SPH refnement can be performed by decreasng partcle pacng by a factor from two to four, whch can be acheved by ncreasng the number of SPH partcles from to and partcles, where both SPH dscretzatons of two SPH models are shown n Fgures 5.3 and 5.4. By refnng the SPH model we acheved good correlatons between SPH and ALE models n term of the heght of the water wave, Fgure 5.11 shows tme hstory of heght of the water wave at the structure. The prce that need to be pad for effcency of SPH method, s that the SPH method may need larger number of partcles to acheve an accuracy comparable wth that of a mesh based method. Table 3: ALE and expermental data of peak wave ampltudes Hgh peaks ALE analyss Expermental results frst 0.21 m m second 0.29 m 0.30 m thrd 0.32 m 0.35 m

12 220 SPH and ALE formulatons for sloshng tank analyss Fgure 5.4: Water wave wth ALE smulaton at tme t = 9.6 sec Fgure 5.5: Water wave wth SPH at tme t = 9.6 s It s well known from desgn engneers and FEM analysts that expermental tests are costly and take long tme to perform. To reduce the number of expermental tests, numercal smulatons need to valdated and then performed on dfferent desgn product before settng up a prototype. In order to valdate the SPH technque descrbed n the paper, the ALE formulaton can be used for valdaton, snce ALE soluton s accurate for tmes where the mesh s deformed but not hghly dstorted, and has been valdated for dfferent applcatons. The bggest advantage the SPH method has over ALE methods s that t avods the heavy tasks of re-meshng. For some complex flud structure nteracton smulatons where

13 Int. Jnl. of Multphyscs Volume 9 Number Fgure 5.6: Water wave wth SPH at tme t = 6.8 s Fgure 5.7: Water wave heght for ALE and SPH smulatons SPH1 = SPH2 = SPH3 = partcles elements need to be eroded due to falure, the ALE remeshng method may fal, snce a new element connectvty needs to be regenerated. SPH method allows falure partcles by deactvatng faled partcles for the partcle loop processng. Ths s a maor advantage that SPH method has over classcal ALE and classcal FEM methods. To further mprove the accuracy of the SPH method for the smulaton of free surface and mpact problems, effcency and usefulness of the two methods, often used n numercal smulatons, are compared.

14 222 SPH and ALE formulatons for sloshng tank analyss CONCLUSION For structure ntegrty, several efforts have been made n automotve ndustry, for modelng sloshng tank analyss and ther effect on structures. In automotve and aerospace ndustres, engneers and FEM analysts move ther smulatons from mesh based method to SPH method to smulate water mpact on deformable structure. We also observed n defense ndustry where SPH method s recently used for undermne exploson problem and ther mpact on the surroundng structure. The bggest advantage the SPH method has over mesh based mesh methods s that t avods the heavy tasks of re-meshng for hydrodynamc problems or structural problems wth large deformaton. The prce to be pad for effcency s that the SPH method may need fner resoluton to acheve accuracy comparable wth that of a mesh based. As a result, SPH smulaton can be utlsed by usng fner partcle spacng for applcatons where mesh based method cannot be used because of remeshng problems due to hgh mesh dstortons. Snce the ultmate obectve s the desgn of a safer structure, numercal smulatons can be ncluded n shape desgn optmzaton wth shape optmal desgn technques (see Barras et al. [19]), and materal optmzaton (see Gabrys et al. [18]). Once smulatons are valdated by test results, t can be used as desgn tool for the mprovement of the system structure nvolved. REFERENCES [1] Aquelet, N., Soul, M. and Olovson, L. (2005): Euler Lagrange couplng wth dampng effects: Applcaton to slammng problems. Computer Methods n Appled Mechancs and Engneerng, Vol. 195, pp [2] Han Z. D and Atlur S. N (2014): On the Meshless Local Petrov-Galerkn MLPG-Eshelby Methods n Computatonal Fnte Deformaton Sold Mechancs Part II CMES: Computer Modellng n Engneerng & Scences, Vol. 97, No. 3, pp , [3] Belytschko, T., Lu, W. K. and B. Moran. (2000): Nonlnear Fnte Elements for Contnua and Structure. Wley & sons (2000). [4] Benson, D. J. (1992): Computatonal Methods n Lagrangan and Euleran Hydrocodes., Computer Methods n Appled Mechancs and Engneerng Vol. 99, pp [5] Colagross, A. and Landrn, M. (2003): Numercal smulaton of nterfacal flows by smoothed partcle hydrodynamcs. Journal of Computatonal Physcs, 191:448. [6] Gngold, R. A. and Monaghan, J. J. (1977): Smoothed partcle hydrodynamcs: theory and applcatons to non-sphercal stars. Monthly Notces of the Royal Astronomcal Socety Vol. 181, pp [7] Hallqust, J. O. (1998): LS-DYNA THEORY MANUEL. Lvermore Software Technology Corporaton, Calforna USA. [8] Lbersky, L. D., Petschek, A. G., Carney, T. C., Hpp, J. R. and Allahdad, F. A. (1993): Hgh Stran Lagrangan Hydrodynamcs: A Three-Dmensonal SPH CODE for Dynamc Materal Response. Journal of Computatonal Physcs, Vol. 109, pp [9] Lucy, L. B. (1977): A numercal approach to the testng of fsson hypothess. The Astronomcal Journal 82, pp [10] Lu M. B. and Lu G. R. (2010): Smoothed Partcle Hydrodynamcs (SPH): an Overvew and Recent Developments. Archves of Computatonal Methods n Engneerng, Vol. 17, pp

15 Int. Jnl. of Multphyscs Volume 9 Number [11] Monaghan J. J. and Gngold R. A. (1983): Shock Smulaton by partcle method SPH. Journal of Computatonal Physcs, Vol. 52, pp [12] Longatte L., Bendeddou Z. and Soul M., (2003): Applcaton of Arbtrary Lagrange Euler Formulatons to Flow-Inuced Vbraton problems. Journal of Pressure Vessel and Technology, Vol. 125, pp , [13] Ozdemr, Z., Soul, M. and Fahan Y. M. (2010): Applcaton of nonlnear flud structure nteracton methods to sesmc analyss of anchored and unanchored tanks. Engneerng Structures, Vol. 32, pp [14] Randles, P. W. and Lbersky, L. D. (1996): Smoothed Partcle Hydrodynamcs: Some recent mprovements and applcatons. Computer Methods n Appled Mechancs and Engneerng, Vol. 139, pp [15] Messahel R. and Soul M. (2013): SPH and ALE Formulatons for Flud Structure Couplng CMES, Computer Modelng n Engneerng & Scences Vol. 96(6), [16] Vgnevc, R, Reveles, J. and Campbell, J. (2006): SPH n a Total Lagrangan Formalsm. CMES Computer Modellng n Engneerng and Scence, Vol. 14, pp [17] Vla, J. (1999): On partcle weghted method and smoothed partcle hydrodynamcs, Mathematcal Models and Method n Appled Scence; 9; pp [18] Soul, M. and Gabrys, J. (2012): Flud Structure Interacton for Brd Impact Problem: Expermental and Numercal Investgaton. CMES, Computer Modelng n Engneerng & Scences Vol. 2137, No. 1, pp [19] Barras, G., Soul, M., Aquelet, N. and Couty, N. (2012): Numercal smulaton of underwater explosons usng ALE method. The pulsatng bubble phenomena. Ocean Engneerng, Vol. 41, pp [20] J. R Shao, H. Q L, G. R Lu and M. B Le: (2012): An mproved SPH model for modelng lqud sloshnd dynamcs. Computers and Structures pp (2012).

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