Three dimensional CFD simulations with large displacement of the geometries using a connectivity change moving mesh approach

Transcription

1 Engneerng wth Computers (2019) 35: ORIGINAL ARTICLE Three dmensonal CFD smulatons wth large dsplacement of the geometres usng a connectvty change movng mesh approach Ncolas Barral 1 Frédérc Alauzet 2 Receved: 8 January 2018 / Accepted: 26 March 2018 / Publshed onlne: 18 Aprl 2018 The Author(s) 2018 Abstract Ths paper deals wth three-dmensonal (3D) numercal smulatons nvolvng 3D movng geometres wth large dsplacements on unstructured meshes. Such smulatons are of great value to ndustry, but reman very tme-consumng. A robust movng mesh algorthm couplng an elastcty-lke mesh deformaton soluton and mesh optmzatons was proposed n prevous works, whch removes the need for global remeshng when performng large dsplacements. The optmzatons, and n partcular generalzed edge/face swappng, preserve the ntal qualty of the mesh throughout the smulaton. We propose to ntegrate an Arbtrary Lagrangan Euleran compressble flow solver nto ths process to demonstrate ts capabltes n a full CFD computaton context. Ths solver reles on a local enforcement of the dscrete geometrc conservaton law to preserve the order of accuracy of the tme ntegraton. The dsplacement of the geometres s ether mposed, or drven by flud structure nteracton (FSI). In the latter case, the sx degrees of freedom approach for rgd bodes s consdered. Fnally, several 3D mposed-moton and FSI examples are gven to valdate the proposed approach, both n academc and ndustral confguratons. Keywords Movng mesh Dynamc mesh Connectvty change Compressble flows ALE: Arbtrary Lagrangan Euleran Dscrete geometrc conservaton law Flud structure nteracton 6-DOF 1 Introducton Flud structure nteracton (FSI) smulatons are requred for a wde varety of subjects, from the smulaton of jellyfsh [23] to the releasng of a mssle [47]. The recent development of computng capactes has made t possble to run ncreasngly complex smulatons where movng bodes nteract wth an ambent flud n an unsteady way. However, engneers are stll far from performng such smulatons on a daly bass, largely due to the dffculty * Ncolas Barral n.barral@mperal.ac.uk Frédérc Alauzet Frederc.Alauzet@nra.fr 1 2 Department of Earth Scence and Engneerng, Imperal College London, London SW7 2AZ, UK Gamma3 Team, Inra Saclay Ile-de-France, Palaseau, France of handlng the movng meshes nduced by the movng geometres. When the dsplacement of the geometry s small enough, slghtly deformng the orgnal mesh [9, 19] can generally be acceptable. But when large deformaton of the boundares s consdered, the mesh quckly becomes dstorted and the numercal error due to ths dstorton quckly becomes too great, untl the elements of the mesh fnally become nvald, and the smulaton has to be stopped. Specfc strateges need to be developed to deal wth large dsplacement movng boundary problems. In the case of FSI problems, another dffculty arses from the fact that the dsplacement of the boundares s by defnton an unknown, and the deformaton of the mesh cannot be mposed a pror. Addressng the ssue of the mesh movement cannot be separated from addressng the ssue of the solver that wll compute on these movng meshes. Dependng on the strategy employed to deal wth the movement, specfc numercal methods must be desgned to take nto account the dsplacement of the mesh. The queston ths paper focuses on s: how can we effcently move the mesh for Vol.:( )

2 398 Engneerng wth Computers (2019) 35: large dsplacement 3D FSI smulatons and what numercal schemes need to be assocated to such a strategy? Three man approaches to address the mesh movement problem can be found n the lterature. The frst approach conssts n havng a sngle body-ftted mesh [11, 29], and movng t along wth the movng boundares. The mesh may thus undergo large deformaton. The second approach s the Chmera (or overset) method [12], n whch each movng body has ts own body-ftted sub-mesh, and the sub-meshes move rgdly together wth ther body and can overlap one another. Fnally the embedded boundary approach [14, 39] uses meshes that are not body-ftted at all: the bodes are embedded n a fxed grd, and technques such as level-sets are used to recover ther movng boundares. All three approaches have ther own strengths and weaknesses. Ths paper s algned wth our prevous works on ansotropc mesh adaptaton, wth the ultmate goal to use movng geometres n our adaptaton framework [8]. For ths reason, we focus on body-ftted approaches wth one sngle mesh. The frst strategy to handle body-ftted movng meshes s smply to move the mesh for as long as possble, and remesh (.e., generate a whole new mesh or a part of t) when the mesh qualty becomes crtcal [11, 29]. For each remeshng, the smulaton has to be stopped, a new mesh must be generated, and the soluton must be transferred to the new mesh. Ths approach can be effcent, especally when small dsplacements are consdered and very few remeshngs are necessary, because the solver and the meshng aspects are decoupled, and between two remeshngs the smulaton s fully ALE and free from nterpolaton errors. However, for larger dsplacements, the number of remeshngs ncreases to prevent nvald elements from appearng, and ths can both be costly and result n poor accuracy due to the soluton transfer step. Hence a second strategy has been developed [16, 20], based on the use of local remeshng operatons, such as vertex nserton, vertex collapse, connectvty changes and vertex dsplacements, to preserve a good mesh qualty throughout the smulaton. The advantage of ths method s that t mantans an acceptable mesh qualty wthout needng to stop, remesh and resume the smulaton. However, t requres fully dynamc mesh data structures that are permanently updated, whch can lead to a loss of CPU effcency, and the numerous mesh modfcatons can lead to a loss of accuracy. Our approach tres to overcome to these drawbacks and s descrbed n detal n [1]. It ams at movng meshes wth large dsplacements of the geometry wthout ever havng to remesh. [By remeshng, we mean stoppng the smulaton, generatng a new mesh (the entre mesh or part of t) and nterpolatng the soluton on the new mesh.] A lmted set of mesh modfcatons are used to preserve the mesh qualty throughout the smulaton: only connectvty changes (edge swaps) and vertex dsplacement are performed. Ths s for several reasons. Notably, performng local mesh modfcatons wthn the solver s far smpler than remeshng globally, and connectvty changes can be relatvely smply nterpreted n terms of evanescent cells for purposes of Arbtrary Lagrangan Euleran (ALE) numercal schemes. In many body-ftted movng mesh strateges, a lot of CPU tme s dedcated to computng the dsplacement of the mesh, n order to make t follow the movng boundares. Thanks to frequent mesh optmzatons, the cost of ths step s reduced by computng the mesh deformaton for a large number of solver tme steps (.e., we do t only a few tmes durng the smulaton). It s mportant to note that ths approach works best f vertex-centered solvers are consdered, because the connectvty changes preserve the number of degrees of freedom. Some studes try to mpose a mesh moton that s drectly adapted to the physcal phenomena n queston, usng for nstance ether so-called Movng Mesh PDEs [32] or a Monge Ampère equaton [15]. However, nterestng these approaches may be, they stll seem to be tme-consumng, especally n 3D, due to the soluton of a non-lnear equaton, and t s unsure whether they can handle complex 3D geometres. Therefore we prefer to prescrbe arbtrary movements to the mesh, and use our mesh adaptaton framework [42] when necessary. Regardng numercal solvers, we consder a classc framework for movng meshes: the Arbtrary Lagrangan Euleran (ALE) framework, whch s based on a formulaton of the equatons that takes nto account an arbtrary movement of the vertces. Ths technque was ntroduced n the 1970s n [21, 31, 33]. Snce then, so many developments have been made n that feld that a complete lst of them would not ft n ths paper. However, one may n partcular refer to [11, 24, 25, 29, 30, 46, 49], whch manly focus on mprovng temporal schemes for ALE smulatons. To our knowledge, very few examples of ALE solvers coupled wth connectvty-change movng mesh technques can be found n lterature. In [36] a conservatve nterpolaton s proposed to handle the swaps. In [28, 51] an ALE formulaton of the swap operator s bult. However, these studes are lmted to 2D. Drven by the requrements of ndustry, we are nterested n desgnng a method that works n 3D. In ths paper, a lnear nterpolaton s carred out after each swap nstead of usng a specfc ALE formulaton, and we wll evaluate the numercal error due to these swaps. The goal of the present paper s to demonstrate that three-dmensonal FSI smulatons can be run effcently by couplng an ALE solver to our connectvty-change movng mesh strategy. The frst part of ths paper focuses on recallng mportant aspects of the movng mesh algorthm. The second, thrd and fourth parts descrbe n detal the solver used. In the ffth part, some valdaton test cases are

3 Engneerng wth Computers (2019) 35: presented, and fnally some examples of complex 3D ALE smulatons are gven and analyzed. In ths paper, we only focus on rgd movements that are nvolved n rgd-body FSI. 2 Mesh connectvty change movng mesh strategy To handle movng boundares, we adopt a body-ftted approach, wth a sngle mesh: the nner vertces of the mesh are moved followng the movng boundares to preserve the valdty of mesh (.e., to prevent the mesh from gettng tangled). Our strategy nvolves two man parts: Computng the mesh deformaton Inner vertces are assgned a trajectory dependng on the dsplacement of the boundares, and thus a poston for future tme steps. Optmzng the mesh The trajectores computed n the mesh deformaton phase are corrected, and the connectvty of the mesh s modfed to preserve the qualty of the mesh. Ths strategy, detaled below, has proven to be very powerful n 3D [1], snce large dsplacement of complex geometres can be performed whle preservng a good mesh qualty wthout any global remeshng (.e., wthout ever generatng a whole new mesh). 2.1 Lnear elastcty mesh deformaton method Durng the mesh deformaton step, a dsplacement feld s computed for the whole computatonal doman, gven the dsplacement of ts boundares. Trajectores can thus be assgned to nner vertces, or n other words, postons at a future solver tme step. Several technques can be found to compute ths dsplacement feld: mplct or drect nterpolaton [13, 44], or solvng PDEs the most common of whch beng Laplacan smoothng [40], a sprng analogy [19] and a lnear elastcty analogy [6]. It s ths last method that we selected, due to ts robustness n 3D [58]. The computatonal doman s assmlated to a soft elastc materal, whch s deformed by the dsplacement of ts boundares. The nner vertces movement s obtaned by solvng an elastcty-lke equaton wth a P 1 fnte element method (FEM): dv(σ( )) = 0, wth = d + T d, 2 (1) 399 where σ and are, respectvely, the Cauchy stress and stran tensors, and d s the Lagrangan dsplacement of the vertces. The Cauchy stress tensor follows Hooke s law for an sotropc homogeneous medum. Drchlet boundary condtons are used and the dsplacement of vertces located on the doman boundary s strongly enforced n the lnear system. The lnear system s solved by a conjugate gradent algorthm coupled wth an LU-SGS pre-condtoner. An advantage of elastcty-lke methods s the opportunty they offer to adapt the local materal propertes of the mesh, especally ts stffness, accordng to the dstorton and efforts borne by each element. In partcular, the stffness of the elements s ncreased for small elements, n order to lmt ther dstorton. More detals can be found n [1]. 2.2 Improvng mesh deformaton algorthm effcency The computaton of the mesh deformaton here the soluton of a lnear elastcty problem s known to be an expensve part of dynamc mesh smulatons, and the fact that t s usually performed at every solver tme step makes t all the more so. We propose to combne several technques to mprove the tme effcency of ths step. Some regons are rgdfed, more specfcally a few layers around tny complex detals of the movng bodes, wth very small elements. They are moved wth exactly the same rgd dsplacement as the correspondng body, thus avodng very stff elements n the elastcty matrx. On the other hand, the elastcty can be solved only on a reduced regon, f the doman s bg compared to the dsplacement. A coarse mesh can also be used to solve the elastcty problem, the dsplacement of the vertces then beng nterpolated on the computatonal mesh. The major mprovement we proposed s to reduce the number of mesh deformaton computatons: the elastcty problem s solved for a large tme frame of length Δt nstead of dong t at each solver tme step δt. Whle there s a rsk of a less effectve mesh dsplacement soluton, t s a worthwhle strategy f our methodology s able to handle large dsplacements whle preservng the mesh qualty. Solvng the prevously descrbed mesh deformaton problem once for large tme frame could be problematc n the case of: (1) curved trajectores of the boundary vertces and (2) acceleratng bodes. To enhance the mesh deformaton prescrpton, accelerated-velocty curved,.e., hgh-order, vertex trajectores are computed. The paths of nner vertces can be mproved f a constant acceleraton a s provded to each vertex n addton to ts speed, whch results n an accelerated and curved trajectory.

4 400 Engneerng wth Computers (2019) 35: Durng tme frame [t, t +Δt], the poston and the velocty of a vertex are updated as follows: 3 2 x(t + δt) =x(t)+δt v(t)+ δt2 2 a v(t + δt) =v(t)+ δt a. e e 3 2 (face swappng) Prescrbng a velocty vector and an acceleraton vector to each vertex requres solvng two elastcty systems. For both systems, the same matrx, thus the same pre-condtoner, s consdered. Only boundary condtons change. If nner vertex dsplacement s sought for tme frame [t, t +Δt], boundary condtons are mposed by the locaton of the body at tme t +Δt 2 and t +Δt. These locatons are computed usng body velocty and acceleraton. Note that solvng the second lnear system s cheaper than solvng the frst one as a good predcton of the expected soluton can be obtaned from the soluton of the frst lnear system. Now, to defne the trajectory of each vertex, the velocty and acceleraton are deduced from evaluated mddle and fnal postons: Δt v(t)= 3 x(t)+4x(t +Δt 2) x(t +Δt) Δt 2 a =2x(t) 4x(t +Δt 2)+2x(t +Δt). 2 In ths context, t s mandatory to make sure that the mesh remans vald for the whole tme frame [t, t +Δt], whch s done by computng the sgn of the volume of the elements all along ther path [1]. 2.3 Local mesh optmzaton In order to preserve the mesh qualty between two mesh deformaton computatons, t has been proposed [1] to couple mesh deformaton wth local mesh optmzaton usng smoothng and generalzed swappng to effcently acheve large dsplacement n movng mesh applcatons. Connectvty changes are really effectve n handlng shear and removng hghly skewed elements. Here, we brefly recall the mesh optmzaton procedure. For 3D meshes, the qualty of an element s measured n terms of the element shape by the qualty functon: Q(K) = ( ) =1 l2 (e ) K [1, + ], where l(e) and K are edge length and element volume. Q(K) =1 corresponds to a perfectly regular element and Q(K) < 2 corresponds to excellent qualty elements, whle a hgh value of Q(K) ndcates a nearly degenerated element. (2) 5 possble trangulatons The frst mesh optmzaton tool s vertex smoothng whch conssts n relocatng each vertex nsde ts ball of elements,.e., the set of elements havng P as ther vertex. For each tetrahedron K j of the ball of P, a new optmal poston P opt for j P can be proposed to form a regular tetrahedron: P opt 2 = G j j + 3 n j l(n j ), 5 6 Fg. 1 Top left, the swap operaton n two dmensons. Top rght, edge swap of type 3 2 and face swap 2 3. Bottom left, the fve possble trangulatons of the pseudo-polygon for a shell havng fve elements. Bottom rght, an example of 5 6 edge swap. For all these fgures, shells are n black, old edges are n red, new edges n green and the pseudo-polygon s n blue. (Color fgure onlne) where F j s the face of K j opposte vertex P, G j s the center of gravty of F j, n j s the nward normal to F j and l(n j ) the length of n j. The fnal optmal poston P opt s computed as a weghted average of all these optmal postons {P opt } j Kj P, the weght coeffcents beng the qualty of K j. Ths way, an element of the ball s all the more domnant f ts qualty n the orgnal mesh s bad. Fnally, the new poston s analyzed: f t mproves the worst qualty of the ball, the vertex s drectly moved to ts new poston. The second mesh optmzaton tool to mprove mesh qualty s generalzed swappng/local-reconnecton (Fg. 1). Let α and β be the two tetrahedra vertces opposte the common face P 1 P 2 P 3. Face swappng conssts of suppressng ths face and creatng the edge e = αβ. In ths case, the two orgnal tetrahedra are deleted and three new tetrahedra are created. Ths swap s called 2 3. The reverse operator can also be defned by deletng three tetrahedra sharng such a common edge αβ and creatng two new tetrahedra sharng face P 1 P 2 P 3. Ths swap s called 3 2. A generalzaton of ths operaton exsts and acts on shells of tetrahedra [1, 26]. For an nternal edge e = αβ, the shell of e s the set of tetrahedra havng e as common edge. The dfferent edge swaps are generally denoted n m, where n e

5 Engneerng wth Computers (2019) 35: s the sze of the shell and m s the number of new tetrahedra. In ths work, edge swaps 3 2, 4 4, 5 6, 6 8 and 7 10 have been mplemented. In our algorthm, swaps are only performed f they mprove the qualty of the mesh. These operatons are well-known n the feld of mesh generaton [26, 56], but are not necessarly effcent n the context of ths work. Notably, performng too many of them results n slow codes, whereas the use of bad qualty functons results n poor qualty meshes. The nterest of the method used n ths paper les how and when optmzatons are performed. The mesh optmzatons are performed element by element, and only when they are needed. Smoothng s performed for every vertex, provded t ncreases the qualty of the correspondng ball. Swaps are only performed,.e., when the qualty of an element decreases and passes a certan threshold. Tetrahedra are treated n qualty order from the worst to the best one. The operaton s performed only f t verfes qualty crtera on the current poston of the mesh and on the fnal poston gven by the mesh deformaton. A key to performng effcent swaps n the movng mesh context s to allow a slght qualty degradaton n the future. Detals on ths optmzaton step can be found n [1]. CFL CFL 2.4 Handlng of boundares The mesh of the boundares s moved rgdly, and the vertces are not usually moved on the surface (no dsplacement n the tangental drectons). However, n some cases, such as when a body s movng very close to the boundng box of the doman, t can be useful to move the vertces of the boundng box as well. In ths case, we can allow tangental dsplacement on the boundary. The rsk of deformng a curved surface beng too great, we only do ths for planes algned wth the Cartesan frame. To do so, the dsplacements along the tangental axes are smply consdered as new degrees of freedom. For nstance, for a plane (x, y), the dsplacements along the x-axs and the y-axs are consdered as degrees of freedom and are added to the elastcty system. The dsplacement along the z-axs s stll set to 0, and thus s not added to the system. 2.5 Movng mesh algorthm The overall connectvty-change movng mesh algorthm s descrbed n Algorthm 1, where the dfferent phases descrbed above are put together. When coupled wth a flow solver (see Sect. 3), the flow solver s called after the optmzaton phase. In ths algorthm, stands for meshes, for solutons, Q for qualty (see Relaton (2)), d Ωh for the dsplacement on the boundary, and v and a for speed and acceleraton. Δt and δt are tme steps whose meanng s detaled below. In Algorthm 1, three tme steps appear: a large one Δt for the mesh deformaton computaton, a smaller one δt opt correspondng to the steps where the mesh s optmzed, and the solver tme step δt solver. Δt, s currently set manually at the begnnng of the computaton. After each mesh deformaton soluton, the qualty of the mesh n the future s analyzed: f the qualty s too low, the mesh deformaton s problem s solved agan wth a smaller Δt (Algorthm 1 step 2(d)). Moreover, f the mesh qualty degrades, a new mesh deformaton soluton s computed (Algorthm 1 step 3(g)). δt opt s computed automatcally, usng the CFL geom parameter as descrbed below. Determnng δt solver wll be dscussed n Sect. 3. If the solver tme step δt solver s greater than the optmzaton tme step, then the solver tme step s truncated to follow the optmzatons. If δt solver s smaller than the optmzaton tme step whch s almost always the case several teratons

6 402 Engneerng wth Computers (2019) 35: of the flow solver are performed between two optmzaton steps. 2.6 Movng mesh tme steps A good restrcton to be mposed on the mesh movement to lmt the apparton of flat or nverted elements s that vertces cannot cross too many elements on a sngle move between two mesh optmzatons. Therefore, a geometrc parameter CFL geom s ntroduced to control the number of stages used to perform the mesh dsplacement between t and t +Δt. If CFL geom s greater than one, the mesh s authorzed to cross more than one element n a sngle move. In practce, CFL geom s usually set between 1 and 8. The movng geometrc tme step s gven by: δt opt = CFL geom max P h(x ) v(x ), where h(x ) s the smallest heght of all the elements n the ball of vertex P. In practce, when coupled wth a flow solver, the actual tme step s the mnmum between the flow solver tme step and the geometrc one. 3 Arbtrary Lagrangan Euleran flow solver An Arbtrary Lagrangan Euleran (ALE) flow solver has been coupled to the movng mesh process descrbed n Algorthm 1. In ths secton, we dscuss n detal the mplemented solver, and all the choces that were made from the numerous possbltes avalable n the lterature. 3.1 Euler equatons n the ALE framework We consder the compressble Euler equatons for a Newtonan flud n ther ALE formulaton. The ALE formulaton allows the equatons to take arbtrary moton of the mesh nto account. Assumng that the gas s perfect, nvscd and that there s no thermal dffuson, the ALE formulaton of the equatons s wrtten, for any arbtrary closed volume C(t) of boundary C(t) moved wth mesh velocty w: d dt ( ) Wdx + ( (W) W w) n ds C(t) C(t) = F ext dx C(t) d ( ) Wdx + (F(W) W(w n)) ds dt C(t) C(t) = C(t) F ext dx, (3) (4) where W = (ρ, ρu, ρe) T s the conservatve varables vector (W) = ( ρu, ρu x u + pe x, ρu y u + pe y, ρu z u + pe z, ρuh ) s the flux tensor F(W) = (W) n = ( ρη, ρu x η + pn x, ρu y η + pn y, ρu z η + pn z, ρeη + pη ) T F ext = ( ) T 0, ρ f ext, ρu f ext s the contrbuton of the external forces, and we have noted ρ the densty of the flud, p the pressure, u =(u x, u y, u z ) ts Euleran velocty, η = u n, q = u, ε the nternal energy per unt mass, e = 1 2 q 2 + ε the total energy per unt mass, h = e + p ρ the enthalpy per unt mass of the flow, f ext the resultant of the volumc external forces appled on the partcle and n the outward normal to nterface C(t) of C(t). 3.2 Spatal dscretzaton As regards spatal dscretzaton of the solver, we use an edge-based fnte-volume approach, wth an HLLC Remann approxmate solver and second-order MUSCL gradent reconstructon. The man dfference when translatng these schemes from the standard formulaton to the ALE formulaton s the addton of the mesh veloctes n the wave speeds of the Remann problem Edge based fnte volume solver The doman Ω s dscretzed by a tetrahedral unstructured mesh. The vertex-centered fnte volume formulaton conssts n assocatng a control volume denoted C (t) wth each vertex P of the mesh and at each tme t. The dual fnte volume cell mesh s bult by the rule of medans. The common boundary C j (t) = C (t) C j (t) between two neghborng cells C (t) and C j (t) s decomposed nto several trangular nterface facets. The normal flux F j (t) along each cell nterface s taken to be constant (not n tme but n space), just lke the soluton W j on the nterface. Rewrtng System (4) for C(t) =C (t), we get the followng sem-dscretzaton at P : d ( C (t) W dt (t) ) + ( (5) C j (t) Φ j W (t), W j (t), n j (t), σ j (t) ) = 0, P j V W (t) s the mean value of state W n cell C at tme t V s the set of all neghborng vertces of P,.e., the mesh vertces connected connected to P by an edge

7 Engneerng wth Computers (2019) 35: n j s the outward normalzed normal (wth respect to cell C ) of cell nterface C j F j (t) = (W j (t)) n j (t) s an approxmaton of the physcal flux through C j (t). 1 σ j (t) = w C j (t) j (t) n j (t)ds s the normal veloc- Cj (t) ty of cell nterface C j (t) Φ j ( W (t), W j (t), n j (t), σ j (t) ) F j (t) W j (t)σ j (t) s the numercal flux functon used to approxmate the flux at cell nterface C j (t). The computaton of the convectve fluxes s performed monodmensonally n the drecton normal to each fnte volume cell nterface. Consequently, the numercal evaluaton of the flux functon Φ j at nterface C j can be acheved by solvng, at each tme step, a one-dmensonal Remann problem n drecton n j = n wth ntal values W L = W on the left of the nterface and W R = W j on the rght. The normal speed to the nterface s temporarly noted σ for clarty HLLC numercal flux The methodology provded by Batten [10] can be extended to the Euler equatons n ther ALE formulaton. The HLLC flux s then descrbed by three waves phase veloctes: S L = mn(η L c L, η c) and S R = max(η R + c R, η + c) S M = ρ R η R (S R η R ) ρ L η L (S L η L )+p L p R ρ R (S R η R ) ρ L (S L η L ) and two approxmate states: ρ L = ρ S L η L L S L S M p L W L = = ( )( ) p = ( ρ L ηl ) S L ηl S M + pl SL (ρu) L = η L ρul + ( ) p p L n, ( ) S L S M SL (ρe) L = η L ρel p L η L + p S M S L S M ρ R = ρ S R η R R S R S M p R W R = = ( )( ) p = ( ρ R ηr ) S R ηr S M + pr SR (ρu) R = η R ρur + ( ) p p R n, ( ) S R S M SR (ρe) R = η R ρer p R η R + p S M S R S M where η = u n s the nterface normal velocty and. are Roe average varables [52]. The HLLC flux through the nterface s fnally gven by: K j 403 F L σw L f S L σ>0 F Φ HLLC (W L, W R, n, σ) = L σw L f S L σ 0 < S M σ F R σw R f S M σ 0 S R σ. F R σw R f S R σ<0 The HLLC approxmate Remann solver has the followng propertes. It automatcally: (1) satsfes the entropy nequalty; (2) resolves solated contacts exactly; (3) resolves solated shocks exactly and (4) preserves postvty Hgh order scheme The prevous formulaton reaches at best a frst-order spatal accuracy. A MUSCL type reconstructon method has been desgned to ncrease the order of accuracy of the scheme. The dea s to use extrapolated values U j and U j of U at nterface C j to evaluate the flux, where U =(ρ, u, p) s the vector of physcal varables. The followng approxmaton s performed: Φ j = Φ(U j, U j, n j, σ j ) wth U j and U j lnearly nterpolated state values on each sde of the nterface: U j = U ( U) jp P j, and U j = U j ( U) jp P j. (6) In contrast to the orgnal MUSCL approach, the approxmate slopes ( U) j and ( U) j are defned for each edge usng a combnaton of centered, upwnd and nodal gradents. The centered gradent related to edge P P j, s defned mplctly along edge P P j va relaton: ( U) C j P P j = U j U. M P P j K j Fg. 2 Downstream K j and upstream K j tetrahedra assocated wth edge P P j Upwnd and downwnd gradents, whch are also related to edge P P j, are computed usng the upstream and downstream tetrahedra assocated wth ths edge. These tetrahedra are, respectvely, denoted K j and K j. K j (resp. K j ) s the unque tetrahedron of the ball of P (resp. P j ) whose opposte face s crossed by the straght lne prolongatng edge P P j ; see Fg. 2. Upwnd and downwnd gradents of edge P P j are then defned as: M j (7)

8 404 Engneerng wth Computers (2019) 35: ( U) U j = ( U) Kj and ( U) D j = ( U) Kj, where U K s the P 1 -Galerkn gradent on element K and φ P s the bass functon assocated wth P. Parametrzed nodal gradents are bult by ntroducng the β-scheme: where β [0, 1] s a parameter controllng the amount of upwndng. For nstance, the scheme s centered for β = 0 and fully upwnd for β = 1. The most accurate β-scheme s obtaned for β = 1 3, also called the V4-scheme. Ths scheme s thrd-order for the two-dmensonal lnear advecton problem on structured trangular meshes. In our case, for the non-lnear Euler equatons on unstructured meshes, a second-order scheme wth a fourth-order numercal dsspaton s obtaned [18]. The hgh-order gradents are gven by: Lmter The prevous MUSCL schemes are nether monotone nor postve. Therefore, lmtng functons must be coupled to the prevous hgh-order gradent evaluatons to preserve the monotoncty and postvty of the scheme. To ths end, the gradent of Relaton (6) s replaced by a lmted gradent denoted ( U lm ) j. Here, the three-entry lmter ntroduced by Derveux [17], whch s a generalzaton of the SuperBee lmter, wll be used : ( U lm ) ( U P D j P j = L( ) P j P j, ( U C) P j P j, ( U V4) ) P j P j wth = P K ( φp U P ) U j P P j =(1 β)( U) C j P P j + β ( U)U j P P j, U j P P j =(1 β)( U) C j P P j + β ( U)D j P P j, ( U) V4 j P P j = 2 3 ( U)C j P P j + ( U)U j P P j, ( U) V4 j P P j = 2 3 ( U)C j P P j + ( U)D j P P j. L(a, b, c) = { 0 f ab 0 sgn(a) mn(2 a,2 b, c ) otherwse. The operator L defned above s appled component by component. (8) ths condton has to take nto account the dsplacement of the body. Consequently, we mpose weakly 1 u n = σ, (9) where n s the DGCL compatble untary boundary face normal and σ s the boundary face velocty. The standard ALE slppng boundary flux of vertex P reduces to: ( ) 0 Φ slp W, n, σ = p p σ where C Γ = 1 K j P K 3 j s the nterface area, see K Sect , p s the vertex pressure, n = j P K j n Kj s the K j P K j mean outward normal of the boundary nterface and. However, when hgh-order numercal schemes are consdered, such a boundary condton creates oscllatons n the densty and the pressure when shock waves mpact normally the boundary. We thus prefer consderng a mrror state and apply an approxmate Remann state to dmnsh these oscllatons. We thus have to evaluate the flux between the state on the boundary W and the ALE mrror state W: ρ W = ρ u ρ E as the mrror state verfes To evaluate the boundary flux, we consder the HLLC numercal flux between the state and the mrror state: ( ) ( Φ slp W, n, σ = ΦHLLC W, W, n, σ ). (11) Note that by defnton we have n n C Γ, (10) and W = ρu 2 ρ (u n σ ) n ρe 2 ρ σ (u n σ ) p =p, ε = ε, c = c, u n = 2 σ u n and H =H 2 σ (u n σ ). Φ HLLC (W, W, n, σ )=F(W ) σ W. Thus, f Condton (9) s satsfed, then W = W and the flux Φ slp ( W, n, σ ) smplfes to the form n Relaton (10). In general, ths condton s not satsfed, so we use Relaton (11). ρ Boundary condtons Boundary condtons are computed vertexwse. Several condtons are used, but only one, the slppng condton, s appled to movng bodes. In the context of ALE smulaton, 1 We do not enforce the numercal soluton to verfy u.n = σ

9 Engneerng wth Computers (2019) 35: Tme dscretzaton Temporal dscretzaton s a more complex matter. In ths work, we chose an explct tme dscretzaton, whch s smpler than mplct dscretzatons. Our tme dscretzaton s complant wth the dscrete geometrc conservaton law, whch can be used to rgorously determne when the geometrc parameters that appear n the fluxes should be computed The geometrc conservaton law We need to make sure that the movement of the mesh s not responsble for any artfcal alteraton of the physcal phenomena nvolved, or at least, to make our best from a numercal pont of vew for the mesh movement to ntroduce an error of the same order as the one ntroduced by the numercal scheme. If System (4) s wrtten for a constant state, assumng F ext = 0, we get, for any arbtrary closed volume C = C(t): d( C(t) ) dt C(t) (w n) ds = 0. As the constant state s a soluton of the Euler equatons, f boundares transmt the flux towards the outsde as t comes, we fnd a purely geometrcal relaton nherent to the contnuous problem. For any arbtrary closed volume C = C(t) of boundary C(t), Relaton (12) s ntegrated nto: C(t + δt) C(t) = t+δt whch s usually known as the geometrc conservaton law (GCL). From a geometrc pont of vew, ths relaton states that the algebrac varaton of the volume of C between two nstants equals the algebrac volume swept by ts boundary. The role of the GCL n ALE smulatons has been analyzed n [22]. It has been shown that the GCL s nether a necessary nor a suffcent condton to preserve tme accuracy; however, volatng t can lead to numercal oscllaton [46]. In [24] the authors show that complance wth the GCL guarantees an accuracy of at least the frst order n some condtons. Therefore, most would agree that the GCL should be enforced at the dscrete level for a large majorty of cases Dscrete GCL enforcement t C(t) (w n) dsdt, wth t and t + δt [0, T], (12) (13) A new approach to enforcng the dscrete GCL was proposed n [46, 57, 58], n whch the authors proposed a framework to buld ALE hgh-order temporal schemes that reach 405 approxmately the desgn order of accuracy. The orgnalty of ths approach conssts n precsely defnng whch ALE parameters are true degrees of freedom and whch are not. In contrast to other approaches [35, 38, 48], they consder that the tmes and confguratons at whch the fluxes are evaluated do not consttute a new degree of freedom to be set thanks to the ALE scheme. To mantan the desgn accuracy of the fxed-mesh temporal ntegraton, the moment at whch the geometrc parameters, such as the cells nterfaces normals or the upwnd/downwnd tetrahedra must be computed, s entrely determned by the ntermedate confguratons nvolved n the chosen temporal scheme. The only degree of freedom to be set by enforcng the GCL at the dscrete level s σ. Incdentally, t s mplctly stated that w s never nvolved alone but only hdden n the term σ n whch represents the nstantaneous algebrac volume swept. In practcal terms, the nterfaces normal speeds are found by smply rewrtng the scheme for a constant dscrete soluton, whch leads to a small lnear system that s easly nvertble by hand. Ths procedure s detaled n the next secton for one Runge Kutta scheme. Any fxed-mesh explct RK scheme can be extended to the case of movng meshes thanks to ths methodology, and the resultng RK scheme s naturally DGCL. Even f ths has not been proven theoretcally, the expected temporal order of convergence has also been observed numercally for several schemes desgned usng ths method [57] RK schemes Runge Kutta (RK) methods are famous mult-stage methods to ntegrate ODEs. In the numercal soluton of hyperbolc PDEs, notably the Euler equatons, the favorte schemes among the huge famly of Runge Kutta schemes are those satsfyng the strong stablty preservng (SSP) property [53, 55]. In what follows, we denote by SSPRK(S,P) the S-stage RK scheme of order P. We adopt the followng notatons: f s = wth n η s j n Φ(W s, Ws j, ηs j, σs j ), j=1 the number of edges n the ball of vertex P the outward non normalzed normal to the porton of the nterface of cell C s around edge e j σ s j normal speed of the nterface around edge e j of cell C s. (14) Superscrpt notaton X s ndcates that the quantty consdered s the X obtaned at stage s of the Runge Kutta process.

10 406 Engneerng wth Computers (2019) 35: Table 1 Butcher and Shu Osher representatons of the thrd-order 4-step Runge Kutta scheme (SSPRK(4,3)) Butcher representaton Shu Osher representaton t s = t n + c s τ Y 0 n = Y Y 0 n = Y c 0 = 0, t 0 = t n Y 1 = Y 0 + τ 2 f 0 Y 1 = Y 0 + τ 2 f 0 c 1 = 1 2, t1 = t n τ Y 2 = Y 0 + τ ( f 0 + f 1 ) Y 2 2 = Y 1 + τ 2 f 1 c 2 = 1, t 2 = t n+1 Y 3 = Y 0 + τ ( f 0 + f 1 + f 2 ) 6 Y 3 = 2 3 Y 0 + Y 2 + τ 6 f 2 c 3 = 1 2, t3 = t n τ Y 4 = Y 0 + τ ( f 0 + f ) 3 f 2 + f 3 Y 4 = Y 3 + τ 2 f 3 c 4 = 1, t 4 = t n+1 For nstance, C s s the cell assocated wth vertex P when the mesh has been moved to ts sth Runge Kutta confguraton. Coeffcents ( c s ndcate the relatve poston n )0 s S tme of the current Runge Kutta confguraton: t s = t n + c s τ wth τ = t n+1 t n. Fnally, we denote by A s j the volume swept by the nterface around edge e j of cell C between the ntal Runge Kutta confguraton and the sth confguraton Applcaton to the SSPRK(4,3) scheme A 1 j A 2 j A 3 j A 4 j = τ η 0 j σ0 j η 1 j σ1 j η 2 j σ2 j η 3 j σ3 j η 0 j σ0 j η 1 j σ1 j η 2 j σ2 j η 3 j σ3 j = τ A 1 j A 2 j A 3 j A 4 j. (15) Ths approach was used, for example, to buld the thrdorder 4-step Runge Kutta scheme [51], whose Butcher and Shu Osher representatons are gven n Table 1. For ths scheme to be DGCL, t must preserve a constant soluton W = W 0, as stated n Sect In ths specfc case, our conservatve varable s Y = C W 0 and the purely physcal fluxes vansh, leadng to f s n = W 0 j=1 ηs j σs j. Therefore, the scheme reads: C 0 = Cn n C 1 C0 = A 1 j = τ n 2 j=1 n C 2 C0 = A 2 j = τ 2 j=1 j=1 n C 3 C0 = A 3 j = τ n 6 j=1 η 0 j σ0 j j=1 n j=1 ( ) η 0 j σ0 j + η1 j σ1 j ( ) η 0 j σ0 j + η1 j σ1 j + η2 j σ2 j n C 4 C0 = A 4 j = τ n ( η0 j σ0 j + η1 j σ1 j + η2 j σ2 j j=1 j=1 ) + η 3 j σ3, j where A s s the volume swept by the nterface around edge j e j of cell C between the ntal and the s th Runge Kutta confguraton. A natural necessary condton for the above relatons to be satsfed s to have, for each nterface around edge e j of each fnte volume cell C : Therefore, the normal speed of the nterface around edge e j of cell C must be updated n the Runge Kutta process as follows : σ 0 j = 2A1 j τ η 0 j σ 3 j = 2A3 j + 2A4 j, τ η 3 j, σ 1 j = 2A1 j + 2A2 j τ η 1 j and the η s j and As are computed on the mesh once t has been j moved to the sth Runge Kutta confguraton., σ 2 j = 2A2 j + 6A3 j, and τ η 2 j Practcal computaton of the volumes swept (16) The nterface of a fnte volume cell s made up of several trangles, connectng the mddle of an edge to the center of gravty of a face and the center of gravty of that tetrahedron; see Fg. 3. The two trangles of the nterface sharng one edge wthn a tetrahedron are coplanar (.e., the mddle of an edge, the center of gravty of the two faces neghborng ths edge and the center of gravty of tetrahedron are coplanar). The unon of these two trangles s called the facet assocated to the edge and the tetrahedron. At confguraton t s = t 0 + c s τ, (wth t 0 = t n ), the outward non-normalzed normal η s j and the volume swept As are comj

11 Engneerng wth Computers (2019) 35: P 0 The volume swept by the facet s: P 1 I 03 I 01 M 2 M 1 G I 02 M 3 A s j,k = c s τ (w G ) s j,k ηs j,k, wth the mean velocty wrtten as: (w G ) s j,k = 1 36 c s τ wth ( 13w s + 13ws j + 5ws m + 5ws l ) (18) I 0 I 2 M 1 P 2 G 21 G 20 T 20 puted as descrbed n [49]. As the cell nterface s made up of several facets, the total swept volume s the sum of the volumes swept by each facet. Let us assume that the facet consdered s assocated wth edge e j = P P j and belongs to tetrahedron K =(P 0, P 1, P 2, P 3 ). In what follows, j l m [[0, 3]], G denotes the center of gravty of tetrahedron K, M m denotes the gravty center of face F m =(P, P j, P l ) of tetrahedron K and M l the center of gravty of face F l =(P, P j, P m ). The outward non-normalzed normal of the facet s gven by: η s j,k =1 4 G0 M 0 m Gs M s l Gs M s m G0 M 0 l G0 M 0 m G0 M 0 l Gs M s m Gs M s l, where G 0 M 0 m = 1 ( ) P 0 + P 0 3P 0 12 j m + P0, l G s M s m = 1 ( ) P s 12 + Ps j 3Ps m + Ps, l G 0 M 0 = 1 ( ) P 0 + P 0 + P 0 l 12 j m 3P0 and l G s M s = 1 ( ) P s l 12 + Ps j + Ps m 3Ps l. G boundary trangle K T 21 M 0 G G T 02 T 12 G 01 T 01 G 10 T 10 P 2 P 0 M 2 P 1 Fg. 3 Interface of a fnte volume cell made up of facets (top) and boundary nterface (bottom) I 1 (17) w s ξ = Ps ξ P0 ξ. Fnally, the total volume swept by the nterface around edge e j of cell C s obtaned by summng over the shell of the edge,.e., all the tetrahedra sharng the edge: A s j = A s j,k. (19) K Shell(,j) It s mportant to understand that normals η s are pseudo- j,k normals whch are used only to compute the volumes swept by the facets. They must not be mstaken for normals to facets taken at t s, η s, whch are used for the computaton j,k of ALE fluxes Volumes swept by boundary nterfaces The pseudo-normals and swept volumes of boundary faces are computed n a smlar way. Let K =(P 0, P 1, P 2 ) be a boundary trangle, as n Fg. 3. Let M 0, M 1 and M 2 be the mddles of the edges and G the center of gravty of K. The trangle s made up of three quadrangular fnte volume nterfaces: (P 0, M 2, G, M 1 ) assocated wth cell C 0, (P 1, M 0, G, M 2 ) wth C 1 and (P 2, M 1, G, M 0 ) wth C 2. Each boundary nterface I s made of two sub-trangles, noted T j and T k wth j, k. The volume swept by nterface I between the ntal and current Runge Kutta confguratons s the sum of the volumes swept by ts two sub-trangles: A s,k = As T j + A s T k = c s τ w s G j η s j + c s τ w s G k η s k, (20) where w s G j s the velocty of the center of gravty G j of trangle T j and η s j s the pseudo-normal assocated wth T j, computed between the ntal and current Runge Kutta confguratons. The sx trangles T j are coplanar, so ther pseudo-normals have the same drecton. Moreover, as medan cells are used, ther pseudo-normals also have the same norm, whch s equal to one sxth of the norm of the pseudo-normal to trangle K. Ths common pseudo-normal s therefore equal to:

12 408 Engneerng wth Computers (2019) 35: η s = 1 6 ηs K = 1 6 thus A s,k where w s G j = [ 1 4 P0 0 P0 1 Ps 0 Ps Ps 0 Ps 1 P0 0 P P0 0 P0 1 P0 0 P Ps 0 Ps 1 Ps 0 Ps 2 ] = c sτ [w sgj + w sgk η s, Fnally, the total volume swept by the boundary nterface s: A s = A s,k. (23) MUSCL approach and RK schemes Regardng spatal accuracy, we have seen that the order of accuracy can be enhanced usng the MUSCL-type reconstructon wth upwndng. However, n the ALE context, one must determne how and when upwnd/downwnd elements should be evaluated to compute upwnd/downwnd gradents whch are necessary for the β-schemes. Ths queston s nether answered n the lterature and generally approxmatons are carred out. For nstance, some papers propose to use upwnd and downwnd elements at t n for the whole Runge Kutta process. However, ths choce should be consstent wth the consdered tme ntegraton scheme. Followng the framework of [57], t s clear that preservng the expected order of accuracy n tme mposes that the upwnd/ downwnd elements and the gradents are computed on the current Runge Kutta confguraton,.e., on the mesh at t s. Therefore, smlarly to geometrc parameters, the upwnd/ downwnd elements and the gradents should be re-evaluated at each step of the Runge Kutta stage Computaton of the tme step (21) The maxmal allowable tme step for the numercal scheme s: τ(p )= ], ( ) 1 11w s 18 c s τ + 5ws j + 2ws k K Ball() h(p ) c + u w, (22) wth w s ξ = Ps ξ P0 ξ. (24) where h(p ) s the smallest heght n the ball of vertex P, c s the speed of sound, u s the Euleran speed (the speed of the flud, computed by the solver) and w s the speed of the mesh vertex. The global tme step s then gven by τ = CFL mn P (τ(p )) Handlng the swaps An ALE formulaton of the swap operator, satsfyng the DGCL, was proposed n 2D n [51], but ts extenson to 3D s very delcate because t requres to handle 4D geometry, and t has not been carred out yet. Instead of consderng an ALE scheme for the swap operator, our choce n ths work s to perform the swaps between two solver teratons,.e., at a fxed tme t n. Ths consequently means that durng the swap phase, the mesh vertces do not move, and thus the swaps do not mpact the ALE parameters η and w, unlke [51] where the swaps are performed durng the solver teraton,.e., between t n and t n+1. After each swap, the soluton should be updated on the new confguraton. Two nterpolaton methods are consdered here. The frst, and smplest, one s to perform a lnear nterpolaton to recover the soluton. As only the connectvty changes and not the vertces postons, the soluton at the vertces does not change,.e., nothng has to be done. Ths nterpolaton s DGCL complant, snce the constant state s preserved (n fact, any lnear state s preserved), but t does not conserve the mass (.e., t does not conserve the ntegral of the conservatve varable) whch s problematc for conservatve equatons when dscontnutes are nvolved n the flow. The second method s the P 1 -exact conservatve nterpolaton followng [2, 5]. It s a smplfed verson of the latter because the cavty of the swap confguraton s fxed. The mass conservaton property of the nterpolaton operator s acheved by element element ntersectons. The dea s to fnd, for each element of the new confguraton, ts geometrc ntersecton wth all the elements of the prevous confguraton t overlaps and to mesh ths geometrc ntersecton wth smplces. We are then able to use a Gauss quadrature formula to exactly compute the mass whch has been locally transferred. Hgh-order accuracy s obtaned through the reconstructon of the gradent of the soluton from the dscrete data and the use of some Taylor formulae. Unfortunately, ths hgh-order nterpolaton can lead to a loss of monotoncty. The maxmum prncple s recovered by correctng the nterpolated soluton n a conservatve manner, usng a lmter strategy very smlar to the one used for fnte volume solvers. Fnally, the soluton values at vertces are reconstructed from ths pecewse lnear by element dscontnuous representaton of the soluton. The algorthm s summarzed n Algorthm 2, where

13 Engneerng wth Computers (2019) 35: m K stands for the ntegral of any conservatve quanttes (densty, momentum and energy) on the consdered element. Ths method s also complant wth the DGCL. 409 F ext denotes the resultant vector of the external forces appled on B, M G ( Fext ) the knetc moment of the external forces appled on B computed at G and g the gravty vector. We assume that the bodes are only submtted to forces of gravty and flud pressure. The equatons for sold dynamcs n an nertal frame then read: m d2 x G dt 2 = F ext = B p(s)n(s)ds + mg G d 2 θ dt Dscretzaton = M G ( Fext ) = B [( s xg ) p(s)n(s) ] ds. (25) Moreover, after each swap, the data of the fnte volume cells (volume and nterface normals) are updated, together wth the topology of the mesh (edges and tetrahedra). Ths requres to have a flow solver wth dynamc data. In Sects. 6.1 and 6.2, we wll quantfy the error due to the use of swaps n numercal smulatons. 4 FSI couplng The movng boundares can have an mposed moton, or be drven by flud structure nteracton. A smple sold mechancs solver s coupled to the flow solver descrbed prevously. The chosen approach s the 6-DOF (6 Degrees of Freedom) approach for rgd bodes. 4.1 Movement of the geometres In ths work, the bodes are assumed to be rgd, of constant mass and homogeneous,.e., ther mass s unformly dstrbuted n ther volume. The bodes we consder wll never break nto dfferent parts. Each rgd body B s fully descrbed by: Physcal quanttes Its boundary B and ts assocated nward normal n, ts mass m assumed to be constant, ts d d matrx of nerta G computed at G whch s symmetrc and depends only on the shape and physcal nature of the sold object. 2 Knematc quanttes The poston of ts center of gravty x G =(x(t), y(t), z(t)), ts angular dsplacement vector θ = θ(t) and ts angular speed vector dθ dt. 2 The moment of nerta relatve to an axs of drecton a passng through G where a s an arbtrary unt vector s gven by: J (Ga) = a T G a. The equaton governng the poston of the center of gravty of the body s easy to solve snce t s lnear. Its dscretzaton s straghtforward. However, the dscretzaton of the second equaton, whch controls the orentaton of the body, s more delcate. Snce the matrx of nerta G depends on θ, t s a non-lnear second-order ODE. The chosen dscretzaton s extensvely detaled n [50] and s based on rewrtng of the equatons n the frame of the movng body. 4.3 Explct couplng As the geometry must be moved n accordance wth the flud computaton, the same tme ntegraton scheme has been taken to ntegrate the flud and the sold equatons. Therefore, tme-advancng of the rgd bodes ODE System s performed usng the same RKSSP scheme as the one used to advance the flud numercal soluton. The couplng s loose and explct as the external forces and moments actng on rgd objects are computed on the current confguraton. 5 Implementaton and parallelzaton Most parts of the code are parallelzed wth p-threads usng an automatc p-thread parallelzaton lbrary [45] coupled wth a space fllng curve renumberng strategy [4]. The Hlbert space fllng curve based renumberng strategy s used to map mesh geometrc enttes, such as vertces, edges, trangles and tetrahedra, nto a one-dmensonal nterval. In numercal applcatons, t can be vewed as a mappng from the computatonal doman onto the memory of a computer. The local property of the Hlbert space fllng curve mples that enttes whch are neghbors on the memory 1D nterval are also neghbors n the computatonal doman. Therefore, such a renumberng strategy has a sgnfcant mpact on the effcency of a code. We can state the followng benefts: t reduces the number of cache msses durng ndrect addressng loops, and t reduces

14 410 Engneerng wth Computers (2019) 35: cache-lne overwrtes durng ndrect addressng loops, whch s fundamental for shared memory parallelsm. The automatc parallelzaton lbrary cuts the data nto small chunks that are compact n terms of memory (because of the renumberng), then uses a dynamc scheduler to allocate the chunks to the threads to lmt concurrent memory accesses. In the case of a loop performng the same operaton on each entry of a table, the loop s splt nto many sub-loops. Each sub-loop wll perform the same operaton (symmetrc parallelsm) on equallyszed portons of the man table and wll be concurrently executed. It s the scheduler s job to make sure that two threads do not wrte on the same memory locaton smultaneously. When ndrect memory accesses occur n loops, memory wrte conflcts can stll arse. To deal wth ths ssue, an asynchronous parallelzaton s consdered rather than of a classc gather/scatter technque,.e., each thread wrtes n ts own workng array and then the data are merged. Our movng mesh strategy performs frequent mesh optmzatons that mpact the ALE speeds of the vertces, so t s much more effcent to have the whole movng mesh part ncluded n the solver code. To handle the movng mesh, sem-dynamc data structures are necessary. Snce we do not do vertex nserton or deleton, only some of the data structures have to be dynamc (edge and tetrahedra lsts), whereas some reman statc (the lst of vertces, as well as all the data assocated to vertces). In the case of dynamc data, the approprate organzaton of the memory whch reduces concurrent accesses can be lost, and thus mposes re-sortng the data accordng to the Hlbert numberng from tme to tme n order to mantan a good speedup. Some parts of the code have not yet been parallelzed, n partcular the elastcty soluton and the mesh optmzaton step. The optmzatons are dffcult to parallelze because they are greatly dependent on the order n whch they are performed. Although changng ths order leads to the same overall result, t s not exactly the same, and thus, the dynamc handlng of the lbrary makes each run non reproducble. To have a parallel and reproducble mesh optmzaton step, the operators should be rewrtten wth dfferent algorthms to make them ndependent of the process order. Such algorthms are generally less effcent n terms of CPU tme. 6 Verfcaton of the solver In the followng secton, we present academc test cases n order to assess the behavor of the dfferent parts of the solver: the flud ALE solver, the rgd body dynamc solver and the FSI couplng. 6.1 Statc vortex n a rotatng mesh The frst test case s used to valdate the ALE solver, and study the mpact of the movng mesh on the accuracy of the soluton. We study the conservaton of a statc vortex (smlar to [34]) on a rotatng mesh. Ths case s an extenson of a 2D case: we consder an extruded cylnder, and the ntal soluton s constant along the z axs. The doman s a cylnder of radus 5 and of heght 1. The ntal soluton s defned as follows. Let r 0 = 1 be a characterstc radus and the reference state be: ρ = 1, u x = 0, u y = 0, u z = 0, p = 1 and γ = 1.4. (26) Let us defne the followng quanttes: D = 1 1 ρ, = 1 B 2 (2π) 2 p 8π 2 = 2r 2 0 γ 1 D γ, E =(2r 2 0 )2 B 2. For a pont of cylndrcal coordnates (r, θ, z), the enthalpy and speed of the vortex are: H vor = γ γ 1 p And fnally the ntal soluton s: Ths ntal confguraton results n a cylndrcal vortex rotatng around the z axs, that remans n a constant state over tme (densty, pressure and speed are preserved). To avod errors due to boundary condtons, the exact soluton s enforced for every vertex farther than 90% of the cylnder radus (typcally just a few layers of elements). The mesh s rotated rgdly around the z axs. Several rotaton speeds were consdered, all of whch led to the same conclusons. Here, we only present the results for an angular speed of 0.34 per tme unt, whch corresponds approxmately to the speed of the vortex at a radus r = 5. We address both space and tme convergence Space convergence ρ (u2 x + u2 y + u2 z ), v vor = 1 2π u x0 = u x v vor sn θ = v vor sn θ u y0 = u y + v vor cos θ = v vor cos θ u z0 = 0 ( ) ) 2D 2r p 0 = p exp E atan( 2 +B E π 2 γ ρ 0 = p γ 1 0. H vor 1 v 2 vor r r (27) For the space convergence study, we consder a set of unform meshes of szes varyng from 10,000 vertces to 1.8 mllon vertces; see Table 2. The smulaton s run untl

15 Engneerng wth Computers (2019) 35: Table 2 Test case of the statc vortex wthout swaps. Szes of the meshes used and number of solver tme steps performed Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 5 Mesh 6 Mesh 7 Mesh 8 Mesh 9 # vertces ( 10 3 ) # tetrahedra ( 10 3 ) ,218 # tme steps ,608 14,321 14,321 17,275 29,996 32,245 44,680 tme t = 540, whch corresponds to half a revoluton of the cylnder. Snce an exact soluton s known, t s easy to compute an error wth respect to ths exact soluton on each mesh. Two cases are compared: n one case the mesh s fxed, and n the other case the mesh s rotated as descrbed prevously. Snce no mesh optmzatons are performed, t allows us to make sure that the extra ALE terms are well computed. The results are shown n Fg. 4. On the left graph, we plot the error varyng wth tme for the dfferent meshes, fxed (red) and rotatng (blue). We can see that, as expected, the error dmnshes as the sze of the mesh grows, but more sgnfcantly the curves of the error for the movng and statc meshes are almost supermposed. Ths confrms the accuracy of the ALE scheme. To study the spatal order of convergence, the error s plotted wth respect to the mesh sze at dfferent tmes on the rght-hand graph. We can see that the order of convergence reached s slghtly greater than 2, whch s coherent wth the space scheme used Tme convergence Ths study was carred out n 2D for reasons of effcency. Our goal was to study the mpact of the Runge Kutta schemes used and the CFL parameter. We consder a dsc of radus 5, wth the same ntal soluton as prevously (Eq. 27) and rotatng wth the same speed. Three tme dscretzatons are used: a standard frst-order explct scheme (SSPRK(1,1)), a fvestep second-order Runge Kutta scheme (SSPRK(5,2)) and a four-step thrd-order Runge Kutta scheme (SSPRK(4,3)) descrbed n Sect Four CFL numbers are set for each case: CFL max, 3 4 CFL max, 1 2 CFL max and 1 10 CFL max. CFL max s, respectvely, 1, 4 and 2 for the schemes SSPRK(1,1), SSPRK(5,2) and SSPRK(4,3). The results are gathered n Fg. 5. On the frst graph, the error over tme s plotted for the dfferent temporal schemes and CFL numbers, whle on the second, we plot the error varyng wth the CFL number at dfferent tme steps. On both graphs, we can see that the error decreases wth the order of the Runge Kutta scheme and wth the CFL number. Whereas the standard explct scheme s very senstve to the tme step, the error does not vary much for the SSPRK(4,3) and SSPRK(5,2). What s nterestng to notce s that we have to use a CFL number of 0.1 CFL max for a standard explct scheme to reach the same level of accuracy as the SSPRK(4,3) scheme wth a CFL number at ts maxmum or the the SSPRK(5,2) scheme wth a CFL number of 0.75 tmes ts maxmum. To reach a non-dmensonal tme equal to 3 wth the standard explct scheme wth a CFL of 0.1, 30 tme steps are requred, whereas only 5 are requred for the SSPRK(5,2) scheme wth a CFL of 3, and 6 for the SSPRK(4,3) wth a CFL of 2. Thus we can go 6 tmes faster wth the hgh-order Runge Kutta schemes and stll obtan the same soluton accuracy, whch s obvously nterestng. Influence of swaps It s mportant to study the mpact of edge/face swaps on the soluton accuracy n 3D. However, t s dffcult to set up a meanngful test case wth swaps. The number of swaps should be constant (n proporton to the sze of the mesh), L 2 error L 2 error mesh 11k vert. ALE mesh 11k vert. statc mesh 45k vert. ALE mesh 45k vert. statc mesh 140k vert. ALE mesh 140k vert. statc mesh 190k vert. ALE mesh 190k vert. statc mesh 255k vert. ALE mesh 255k vert. statc mesh 370k vert. ALE mesh 370k vert. statc mesh 585k vert. ALE mesh 585k vert. statc mesh 965k vert. ALE mesh 965k vert. statc mesh 1.8M vert. ALE mesh 1.8M vert. statc tme (s) 10 3 tme 1s ALE tme 1s statc tme 8s ALE tme 8s statc tme 50s ALE 10 4 tme 50s statc tme 100s ALE tme 100s statc tme 200s ALE tme 200s statc tme 540s ALE 10 5 tme 540s statc Error order 1 Error order mesh sze (k-vertces) Fg. 4 Test case of the statc vortex (3D). Error curves comparng the smulaton on a fxed mesh (red) and on a movng mesh (blue) for several mesh szes. Top, the evoluton of the error over tme, bottom convergence curves wth respect to the sze of the mesh. (Color fgure onlne)

16 412 Engneerng wth Computers (2019) 35: L 2 error L 2 error RK 1 1, CFL=1.00 RK 1 1, CFL=0.75 RK 1 1, CFL=0.10 RK 3 4, CFL=2.00 RK 3 4, CFL=0.20 RK 2 5, CFL=4.00 RK 2 5, CFL=3.00 RK 2 5, CFL= tme ( s) 10 1 tme 1s RK 1 1 tme 1s RK 2 5 tme 1s RK tme 540s RK 1 1 tme 540s RK 2 5 tme 540s RK CFL/CFLmax Fg. 5 Test case of the statc vortex (2D). Error curves comparng several temporal schemes: standard explct Euler scheme (blue), SSPRK(4,3) (green), SSPRK(5,2) (red). Top, evoluton of the error over tme. Bottom, error at a fxed tme varyng wth the CFL number. (Color fgure onlne) and they should also be unformly dstrbuted n both tme and space. On all the smulatons run wth the connectvty-change movng mesh strategy, we notced that, on average, less than one swap per 10,000 tetrahedra and per tme step s performed. To ft ths observaton, n ths example, we swap all the bad elements to preserve the mesh qualty every 15 solver tme steps. Then, f not enough swaps were performed, we randomly swap elements to reach a number of 1 swap per 666 tetrahedra. Not all random swaps are actually performed, because some of them affect the qualty of the mesh too drastcally, so the number of swaps s not perfectly controlled but ths method allows us to have an average number of swaps close to one per 10,000 tetrahedra and per tme step for all the meshes, evenly dstrbuted n the doman, as stated n Table 3. We agan consder 3D meshes, rotatng wth the same speed of 0.34 per tme unt, and the smulatons are run up to 270s. In the absence of dscontnuty n the soluton, a lnear nterpolaton s performed after the swaps. The results are gathered n Fg. 6. The error s slghtly hgher wth swaps (n green) than wthout (n blue and red), however the error curves reman very close to those wthout swaps. The dscrepancy can manly be explaned by the non conservatve characterstc of the lnear nterpolaton. Even wthout a conservatve formulaton, the error ntroduced n ths example s not so large, and the same second-order convergence rate s observed. In the next example, we analyze the mpact of a conservatve formulaton n the case of dscontnutes n the soluton. To sum up, ths statc vortex test case allowed us to valdate our ALE solver. We asymptotcally reach an order of convergence of 2 for the spatal error, and we made sure there was no addtonal error ntroduced by the ALE terms. As regards temporal convergence, the mportance of a hghorder Runge Kutta scheme was establshed. Fnally, we showed that edge/face swappng artfcally created n ths case, but mandatory to preserve the qualty of the mesh (and thus the accuracy of the soluton) when complex geometrc dsplacements are nvolved only creates a small error, even f used wthout a specfc conservatve treatment. 6.2 Sod s shock tube wth a rotatng crcle To show the mpact of the choce of the nterpolaton when swaps are performed n the presence of dscontnutes, we consder the well-known Sod s shock tube problem [54]. The tube doman sze s [0, 1] [0, 0.2], and ntally, the tube s flled wth two fluds at rest (left part s x 0.5 and rght part s x > 0.5 ) verfyng: ρ left = 1, u left = 0, p left = 1 and ρ rght = 0.125, u rght = 0, p rght = 0.1. Table 3 Test case of the statc vortex wth swaps. Number of swaps for each mesh. On the last lne, the number of swaps s close to one swap per tme step and per 10,000 tetrahedra for all the meshes Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 5 Mesh 6 Mesh 7 Mesh 8 Mesh 9 # tetrahedra ( 10 3 ) ,218 # tme steps ,040 16,085 21,449 # swaps ( 10 3 ) ,838 26,719 # swaps/tme step/tet ( 10 4 )

17 Engneerng wth Computers (2019) 35: L 2 error L 2 error tme (s) mesh 11k vert. ALE mesh 11k vert. statc mesh 11k vert. swap mesh 45k vert. ALE mesh 45k vert. statc mesh 45k vert. swap mesh 140k vert. ALE mesh 140k vert. statc mesh 140k vert. swap mesh 255k vert. ALE mesh 255k vert. statc mesh 255k vert. swap mesh 370k vert. ALE mesh 370k vert. statc mesh 370k vert. swap mesh 585k vert. ALE mesh 585k vert. statc mesh 585k vert. swap mesh 965k vert. ALE mesh 965k vert. statc mesh 965k vert. swap mesh 1.8M vert. ALE mesh 1.8M vert. statc mesh 1.8M vert. swap tme 1s ALE tme 1s statc tme 1s swap tme 8s ALE tme 8s statc tme 8s swap tme 50s ALE tme 50s statc tme 50s swap tme 100s ALE tme 100s statc tme 100s swap tme 200s ALE tme 200s statc tme 200s swap tme 540s ALE tme 540s statc tme 540s swap Error order 1 Error order mesh sze (k-vertces) Fg. 6 Test case of the statc vortex (3D). Error curves comparng the smulaton on a fxed mesh (red), on a movng mesh wthout swaps (blue) and on a movng mesh wth swaps (green) for several mesh szes. Top, the evoluton of the error over tme, bottom, convergence curves wth respect to the sze of the mesh. (Color fgure onlne) The soluton s computed untl non-dmensoned tme t = 0.25, and a shock wave and a contact dscontnuty propagate on the rght-hand sde of the tube whle a rarefacton wave propagates on the left-hand sde. To analyze the ALE scheme and the mpact of the swaps, we defne a crcular regon of center (0.75, 0.1) and radus 0.05 whch performs two full rotatons wthn the smulaton tme frame; see Fg. 7. As the crcular regon s rotatng, the mesh s sheared and swaps are performed around the crcular regon 413 to mantan a good mesh qualty. As the shock wave and the contact dscontnuty move through the rotatng regon durng the smulaton tme frame, we can analyze the mpact of the nterpolaton stage when swaps are performed. For comparson, we also run the smulaton on the same fxed mesh. Fgure 8 shows the results on the fxed mesh (top), and on the movng mesh usng the lnear nterpolaton when swaps are performed (mddle) and usng the P 1 -exact conservatve nterpolaton (bottom). Extracton of the densty profle along the lne y = s gven n Fg. 9. We can observe that some defects n the soluton appear n the contact dscontnuty when the lnear nterpolaton s consdered due to the rotaton of a part of the doman. These defects dsappear when the conservatve nterpolaton s used. The defects are ponted out on the densty profle when we zoom on the contact dscontnuty; see Fg. 9 (rght). But, they reman mnor when we observe the overall profle. 6.3 Pston The thrd valdaton test case s an FSI pston case from [37]. It s orgnally a 1D problem, whch s extended to 3D. As shown n Fg. 10, we consder a gas contaned n a 3D cylndrcal chamber, closed at one end by a wall, and at the other end by a movng pston of mass m p and of cross-secton A p. Besdes the forces of pressure, the pston s submtted to a restorng force modeled by a sprng of rgdty k p. A natural frequency of the pston can be defned by f p = 1 2π k p m p. Let L 0 be the length of the chamber at rest, and u(t) be the dsplacement of the pston wth respect to the poston at rest. The gas s ntally at rest, and the pston s held n poston u 0. We consder that all the varables are unform on each cross-secton of the cylnder (1D assumpton). It s shown n [37] that the pston s submtted to a resultant force appled to ts center of gravty: F p = k p u(t)+a p (p(t) p(0)), (28) wth p(t) beng the pressure on the pston. No effects of gravty are taken nto account. In practce, the doman at t = 0 s a rectangular box of dmensons [ 1, 0.2] [ 0.5, 0.5] [ 0.5, 0.5]. The Fg. 7 Sod s shock tube problem wth a rotatng regon. Doman geometry where the green regon s fxed and the crcular red regon s rotatng (performng two full rotatons durng the smulaton). (Color fgure onlne)

18 414 Engneerng wth Computers (2019) 35: Fg. 8 Sod s shock tube problem wth a rotatng regon. From top to bottom, densty so-values and so-lnes at non-dmensoned tme t = 0.25 for the fxed mesh, the movng mesh wth lnear nterpolaton, and the movng mesh wth the P 1 -exact conservatve nterpolaton Fg. 9 Sod s shock tube problem wth a rotatng regon. Extracton of the densty profle along the lne y = at non-dmensoned tme t = Bottom, zoom on the contact dscontnuty regon vertcal face at x = 0.2 s moble, and all the other faces are fxed, whch corresponds to l 0 = 1 and u 0 = 0.2. The pston rgdty s set to k p = 10 7 Nm 1, and a varety of pston masses (and thus frequences) are consdered: m p = 4 kg, 10 kg, 100 kg and 1000 kg, correspondng, respectvely, to frequences f p = 252 Hz, 159 Hz, 50 Hz and 16 Hz. The other parameters are: ρ 0 = kg m 3, p 0 = 10 5 Pa, u 0 = 0ms 1. 3 Slppng condtons are mposed on all the boundary faces (fxed and moble). From a movng mesh 3 For more detals on the descrpton of the test n [37], see the source code referred n the paper.

19 Engneerng wth Computers (2019) 35: Two F117 arcraft crossng flght paths L 0 Gas A p m p,k p 0 u(t) Fg. 10 Pston test case. 2D representaton of the problem Table 4 Pston test case. Natural frequences and perods for the dfferent masses used m p (kg) f p (Hz) T p (s) pont of vew, t s essental to allow tangental movement of the vertces on the movng faces of the box. The quanttes of nterest are the dsplacement and the pressure of the pston. For the pressure, we consder the pressure at ts center of gravty. These two quanttes are plotted n Fg. 11. Regardng the pston movement, t s perodc, as expected, wth a perod T p = 1 f p (see Table 4), the ampltude of the oscllatons beng damped due to FSI effects. These curves are smlar to the results of [37] for the cases m p = 10 kg, 100 kg and 1000 kg. To hghlght the FSI effect, we added a case wth a smaller mass m p = 4 kg, where the reducton of the ampltude of the oscllatons s even clearer. As concerns the pston pressure, the results also correspond to [37]. We ran the case wth several mesh szes (from 200,000 to one mllon vertces) and several temporal schemes, and all results are consstent. The pressure felds at a tme around 2/5 of the fnal tme are shown n Fg. 12 for the four masses consdered. We can see that the 1D structure of the soluton s well preserved. 7 Some applcaton examples Fnally, we analyze the behavor of our strategy on two more complex, ndustral-lke, examples of smulatons n three dmensons. These examples are challengng due both to the sze of the meshes consdered, and to the large dsplacement of the geometres. Note that a lnear nterpolaton was used after edge/face swaps as no shocks are present n the flow-feld, and t does not alter the conclusons of these examples. x The frst example models two notonal F117 arcraft wth crossng flght paths. Ths problem llustrates well the effcency of the connectvty-change movng mesh algorthm n handlng large dsplacements of complex geometres wthout global remeshng. When both arcraft cross each other, the mesh deformaton encounters a large shearng due to the opposte flght drectons. The connectvty-change mesh deformaton algorthm easly handles ths complex dsplacement thanks to the mesh local reconnecton. Therefore, the mesh qualty remans very good durng the whole dsplacement, wthout any remeshng step. As concerns the flud smulaton, the arcraft are moved wth an mposed moton of translaton and rotaton at a speed of Mach 0.4, n an ntally nert unform flud: at t = 0 the speed of the ar s null everywhere. Transmttng boundary condtons are used on the sdes of the surroundng box, whle slppng condtons are mposed on the two F117 bodes. After a short phase of ntalzaton, the flow s establshed when the two F117s pass each other, and the densty felds around the arcraft and n ther wake nteract. Acoustc waves are created n front of the F117s due to the nstantaneous settng n moton of the arcraft. Ths s not realstc, but our am was to valdate our movng mesh approach rather than run a physcally realstc smulaton. In Fg. 13, a zoom on the geometry of the two arcraft s shown. In Fg. 14, we show both the movng mesh aspect of the smulaton and the flow soluton at dfferent tme steps. The mesh s composed of 585,000 vertces and (ntally) 3.5 mllon tetrahedra. The whole smulaton requres 22 elastcty solutons and 1600 optmzaton steps for a total of 2,500,000 swaps. The fnal mesh average qualty of 1.4 s excellent and we notce that 99.8% of the elements have a qualty smaller than 2 and only 52 elements have a qualty hgher than 5. Ths smulaton was run n a reasonable tme: 18 h were necessary to do 39,000 tme steps on a machne wth 20 hyperthreaded 7 cores at 2.5 GHz. Very few elastcty solutons are requres compared to the number of solver tme steps, and the total tme requred for the soluton of the elastcty problems s only 25 mn, whch represents only 2.3% of the total tme. The good qualty of the mesh ensures an acceptable soluton accuracy throughout the smulaton. The optmzaton steps (swappng and smoothng) only take 25 mn. As for the mpact of the swaps, t s dffcult to evaluate, snce the smulaton cannot be run wthout them. One can notce that on average, only 66 swaps per solver tme step were performed,

20 416 Engneerng wth Computers (2019) 35: Fg. 11 Pston test case. Dsplacement (rght) and pston pressure (left)

21 Engneerng wth Computers (2019) 35: Fg. 12 Pston test case. Pressure felds at 2/5 of the fnal tme. m = 10 and t = (top), m = 100 and t = (center), m = 1000 and t = (bottom). The moble pston s on the rght Fg. 13 Test case of the two F117s. Zoom on the arcraft mesh (top) and soluton (bottom) just before they pass each other whch s less than % of the number of elements of the mesh, and so s lkely to have lttle nfluence on the soluton. 7.2 Ejected cabn door The last example s the FSI smulaton of the ejecton of the door of an over-pressurzed arcraft cabn. Ths test case has been proposed by arcraft desgners, and the am s to evaluate when the door hnge wll yeld under cabn 417 pressure. Generally, such experments are done n a hangar and numercal smulatons must be able to predct f the door wll mpact the observaton area. From a purely movng mesh pont of vew, the dffculty s that the geometry s ansotropc and rolls over the elements whle progressng nsde a unform mesh composed of 965,000 vertces. Another dffculty les at the begnnng of the movement when the door s ejected from ts frame. The gap between the door and the frame s very small, and the mesh s sheared n the nterstce; see Fg. 15. However, no remeshng s needed. The connectvty-change movng mesh strategy s able to get rd of these skewed elements qute rapdly to fnally acheve an excellent qualty throughout the door dsplacement. From the FSI pont of vew, we present a smplfed verson of the case and thus probably not very representatve of the actual physcs nvolved. However, we make sure that the soluton s physcally coherent wth the smplfed ntal condtons. At tme t = 0, the volume s dvded nto two parts: outsde and nsde the cabn; see Fg. 15. The nsde of the cabn and the outsde are not nsulated, so the pressurzed ar leaks out of the cabn. The small volume surroundng the door s consdered ntally as beng outsde the cabn. The outsde of the cabn s ntalzed to classc atmospherc values at 10,000 m ( ρ = 0.44 kg m 3, u =(0, 0, 0) ms 1, p = 20 kpa). At ths alttude, the ar of the cabn should be pressurzed to smulate an alttude of 2500 m ( ρ = 0.96 kg m 3, u =(0, 0, 0) ms 1, p = 75 kpa). To smulate a blast, those values are multpled by 10 nsde the cabn ( ρ = 9.6 kg m 3, u =(0, 0, 0) ms 1, p = 750 kpa). The mass of the door s set to 100 kg. Snapshots of the soluton at dfferent tme steps are shown n Fg. 16. We can see features smlar to mach damonds at the rear of the door, due to the hgh speed of the door. As concerns the door movement, the door s blown away from the cabn as expected. However, t acqures a slow rotaton movement. For the tme frame consdered ( T end = 0.2s ), gravty has a neglgble mpact on the trajectory. The mesh s composed of 965,000 vertces and 5.6 mllons of tetrahedra. Due to the dffcult part where the door s movng n ts frame, the whole smulaton requres 29 elastcty solutons and 540 optmzaton steps for a total of 875,000 swaps. The fnal mesh average qualty of 1.33 s excellent and we notce that 99.95% of the elements have a qualty smaller than 2 and only 1 element has a qualty hgher than 5. The total tme of the smulaton s 2 h on a machne wth 20 hyperthreaded 7 cores at 2.5 GHz, for 1122 solver tme steps, ncludng 43 mn of elastcty soluton. The number of elastcty solutons s stll small compared to the number of tme steps (29 elastcty solutons and 1122 solver tme steps), and elastcty solutons do not account

22 418 Engneerng wth Computers (2019) 35: Fg. 14 Test case of the two F117s. Snapshots of the movng geometres and the mesh (left) and densty (rght) for more than a thrd of the total tme. Note that the a pror unpredctable trajectory of FSI problems generally results n a slghtly hgher number of elastcty steps than wth analytc mposed moton. The optmzaton step only takes 3 mn. Once agan, only an average of % swaps per tetrahedra and solver tme step are performed, whch s not a lot. 8 Concluson A strategy to run complex three-dmensonal smulatons wth movng geometres has been presented n ths paper. Ths strategy les frst on a robust connectvty-change movng mesh algorthm, whch couples an elastctylke mesh deformaton method and mesh optmzatons.

23 Engneerng wth Computers (2019) 35: Fg. 15 Test case of the door ejecton. Surface mesh and cut plane just after the begnnng of the smulaton. Note that the gap between the door and ts frame s very small, and meshed wth only one layer of elements, whch ncreases the dffculty of the movng mesh problem In partcular, a reduced number of elastcty solutons mproves the effcency of the algorthm, whle edge/face swappng makes t possble to deal wth the strong shearng of the mesh that occurs when the geometres undergo large dsplacement. To run CFD smulatons on such movng meshes, a fnte volumes flow solver for the compressble Euler equatons has been extended to the ALE framework. As regards the temporal accuracy, the SSPRK schemes consdered are based on the strct applcaton of the dscrete geometrc conservaton law 419 (DGCL), whch s supposed to preserve the accuracy of the schemes. The dsplacement of the geometres can be ether mposed a pror, or governed by a 6-DOF flud structure couplng. The goal of the paper was to demonstrate that ths strategy allows us to run complex three-dmensonal smulatons. Challengng examples of such smulatons, were presented and analyzed, wth mposed moton or fully FSI. Despte the large dsplacement of the geometres, no global remeshng was necessary to perform the smulatons, whle preservng a good mesh qualty. The unsteady solutons of these examples are physcally coherent, and the total CPU tme of the smulatons s reasonable. The use of hgh-order Runge Kutta schemes was shown to help reduce the cost of the smulaton. Whereas the handlng of the movng mesh usually takes up a large proporton of the smulaton total CPU tme, our connectvty-change movng mesh algorthm only accounts for a small fracton of that CPU tme, most of the tme beng spent on actually solvng the equatons. Some ssues stll need to be addressed. A thorough analyss of the mpact of swaps on the soluton accuracy needs to be carred out. We have shown that a conservatve nterpolaton method mproves sgnfcantly the accuracy of the soluton n the presence of dscontnutes. A comparson of ths approach wth the one descrbed n [51] would probably be revealng. The elastcty tme step s currently constant and fxed a pror for the whole smulaton. The effcency of the movng mesh part would be mproved f ths tme step could be automatcally adapted to the movement of the geometres. Fndng such an optmal tme step s harder than t seems. The case of non rgd bodes also has to be addressed: the movng mesh algorthm can handle deformable bodes [1, 7], but the FSI couplng would be much more complex. Fnally, ths entre strategy was devsed to ft wthn our metrc-based mesh adaptaton framework [42]. An unsteady verson of the adaptaton algorthm already exsts [3]. It was extended to the case of movng meshes n 3D recently [8], and a goal-orented verson of the process s developed. 9 Software used The solver descrbed n ths paper and used to run all the smulatons s Wolf. The meshes were generated usng GHS3d [27] and Feflo.a [43]. The vsualzaton of the meshes and solutons was done wth Vzr [41].

24 420 Engneerng wth Computers (2019) 35: Fg. 16 Test case of the door ejecton. Snapshots of the movng geometry and the mesh (left) and pressure (rght) Acknowledgements Ths work was partally funded by the Arbus Group Foundaton. The authors also gratefully acknowledge the support of EPRSC Grant EP/R029423/1. Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 4.0 Internatonal Lcense ( veco mmons.org/lcen ses/by/4.0/), whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal author(s) and the source, provde a lnk to the Creatve Commons lcense, and ndcate f changes were made.