ONE very challenging goal in the field of Computational Aeroelasticity (CA) is to be able to study fluid-structure

Transcription

1 43rd AIAA Aerospace Scences Meetng and Exhbt AIAA January 2005, Reno, Nevada DEFORMATION OF UNSTRUCTURED VISCOUS GRIDS Dens B. Kholodar,* Scott A. Morton, and Russell M. Cummngs Department of Aeronautcs, Unted States Ar Force Academy USAF Academy, CO ABSTRACT A mesh deformaton algorthm for unstructured grds s presented. It s desgned for hgh Reynolds number flow problems. Such grds are employed n aerodynamc and aeroelastc studes of wngs or complete arcraft confguratons n flows where the vscous effects are mportant. Gven a surface deformaton, the method effcently recalculates new locatons of hgh aspect rato cells that make up the vscous layers of the grd and then deforms the nvscd part of the grd usng an establshed method based on a torsonal sprng analogy technque. Results are presented for montorng the deteroraton of the qualty of the grd durng subsequent deformaton steps for aeroelastc studes as well as to ensure the tme effcency of the method. Results for grd deformaton of a 1.4 mllon cell AGARD wng grd desgned for flow at hgh Reynolds numbers due to typcal deformatons are also presented. Fnally, a dscusson of the parallelzaton performance and comparson of the runnng tme of the mesh deformaton algorthm to that used by the flow solver s made. I. INTRODUCTION ONE very challengng goal n the feld of Computatonal Aeroelastcty (CA) s to be able to study flud-structure nteractons and predct response of flght vehcles n hgh Reynolds number flows. Numercal smulatons of flud flow n such flght regmes requre employment of Computatonal Flud Dynamcs (CFD) grds that contan very thn, stretched (.e. hgh aspect rato) cells resolvng the boundary layer around the body surfaces. Deformatons of the surfaces requre movng the CFD grd surroundng the body. Robust algorthms for movng an unstructured grd s an mportant and recent subject n the felds of CFD, CA, controls and others. Movng an unstructured grd that s desgned for a hgh Reynolds number flow presents an even more challengng task. The development of an algorthm for ths type of grd (referred vscous grds n the paper) s shown n the current work. The complexty of a mesh deformaton algorthm that a CA program uses depends on the complexty of the problem, grd sze, computer resources, type of flud and structural solvers that communcate to the grd, and varous other factors. In partcular, deformaton attrbutes (large or small, translaton, rotaton or bodes n relatve moton), topology of the grd (structured or unstructured) and fnally presence of hgh aspect rato cells sutable for vscous flows determnes the dffculty of the problem. The frst class of methods based on creaton of a network of lnear sprngs connectng all vertces n the grd was ntroduced n Ref. 1. Ths method worked well for many applcatons, but s lmted to small deformatons. Snce then many advancements have been made ncludng those that permt large deformatons usng the sprng method (see, for example, Refs. 2-5). Most of the methods developed n these studes have been valdated on nvscd grds. The presence of hghly stretched cells n vscous grds wth aspect ratos n the thousands makes the deformaton problems even more challengng. That s why the past year saw a number of papers tryng to address ths problem. The authors of Ref. 6 consdered the tetrahedral mesh as a deformable elastc sold, whle movng the near-wall prsm cells as a rgd body. They also desgned coarsenng and refnement technques to allevate the nevtable dstorton of the grd durng large degrees of moton. Ref. 7 proposes a selectve sold-extenson mesh movng technque governed by the equatons of elastcty that allowed for lmtng the dstorton n thn layers and reduce the frequency of remeshng. In Ref. 8 the torsonal sprng method of the three-dmensonal grd s derved from the twodmensonal torsonal sprng method dfferently than that n the Ref. 2 by takng nto account the dynamcs of the threedmensonal object by analyzng the equatons of volume and area for a tetrahedron and ts faces and edges. 8 * Research Engneer, Member AIAA. Professor of Aeronautcs, Assocate Fellow AIAA. Professor of Aeronautcs, Assocate Fellow AIAA. 1 Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States.

2 a) b) Fg 1. F-18 Tal buffet problem: a) NASA F-18 Hgh Angle of Attack Research Vehcle (HARV). Photo courtesy of NASA Dryden, and b) Computatonal Detached Eddy Smulaton turbulence model. 9 In many current CFD and CA studes vscous unstructured grds of large sze are employed. For example, to correctly analyze the unsteady flowfeld phenomena of Fg. 1a, vortex breakdown nduced tal buffet, a grd desgned for a Reynolds number of 13 mllon s requred. Ths grd may have from 5 to 15 mllon cells for the half plane wth up to half of them n the boundary layer. These boundary layer cells wll have stretched cells wth aspect ratos n the thousands to accurately capture the boundary layer velocty profles. In a recent paper by the current authors on the F/A-18 tal buffet 9 problem, such a grd was necessary n order to apply varous computatonal turbulence models. One of these models - the so called Detached Eddy Smulaton approach - reproduced the expermentally observed unsteadness of the tal loads at flght condtons. Ths was accomplshed on a rgd grd but the goal s to perform a CA smulaton wth deformng meshes on the tals. The aspect rato of near-wall cells n the grd used n Fg. 1b s close to Ths makes the grd deformaton problem very challengng. In ths paper a deformaton strategy s presented for such stretched vscous grds. The approach s a combnaton of two separate technques: surface vector readjustment for movng grd ponts n the boundary layer, and the torson sprngs technque 2 for movng the nvscd porton of the grd. A descrpton of the method s presented below. II. METHOD Consder a typcal three-dmensonal unstructured vscous grd n the vcnty of a surface as shown n Fg 2. The vscous near-wall part of such a grd may have 8-20 layers of stretched cells. The grds used were generated by the grd generator VGRIDns developed at the NASA Langley Research Center. 10 The vscous cells are generated by the advancng layers method whle the remanng volume has tetrahedra generated by the advancng front method. Desgnng a robust method to deform such a mesh for an arbtrary deformaton of the surface s the objectve of the present work. A deformaton strategy should be developed wth the partcular knd of meshes n mnd, and other research groups may have vscous unstructured meshes that are dfferent. Therefore, we wll try to emphasze the lmtng factors of the current method. The man concern s obvously to prohbt the nter-penetraton of the neghborng cells as the grd s moved. The next challenge s to make sure that the grd qualty does not deterorate sgnfcantly after a number of steps, thus avodng local regeneraton of the grd durng consecutve applcatons of the method. Among other challenges s the parallelzaton and tme performance of the method. The presented procedure conssts of two mechansms: one for the vscous boundary layer and the other for the nvscd part of the grd. Fg 2. Example of a hybrd vscous unstructured grd. 2

3 Fg 3. Inserton of trangles wth torsonal sprngs nto a tetrahedron (courtesy to the authors of Ref. 2) wng (see Fg. 5) as was demonstrated n the orgnal paper. 2 One must note however that the unstructured AGARD wng grd used n Ref. 2 s made of 128 thousand nvscd tetrahedra. For the lnear sprng analogy method to perform well the grd consdered n Ref. 2 s too fne and the deformatons are too bg whle the torsonal analogy method worked well ths was all shown n Ref. 2. The AGARD wng s consdered n the current paper as well. Snce the target problems (e.g. F-18 tal buffet) requre fne vscous grds, the AGARD wng consdered here s fner than the one consdered n Ref. 2. Another very mportant consderaton s the parallel performance of the method. Grds of nterest can be of many mllons of cells. Thus, t becomes very mportant to have an effcent parallelzaton strategy. The torsonal sprng method conssts of two parts - computaton of the grd stffness matrx and soluton of a sparse lnear system of equatons. Both these tasks can be performed locally on all processors wth lmted communcaton tme ths ssue s addressed further n the parallel performance dscusson. These were some of the reasons behnd choosng a torsonal sprng analogy method for the grds of nterest. The nvscd part of the grd s deformed usng a three-dmensonal torsonal sprng analogy method proposed n Ref. 2. At the core of ths method s the ntroducton of twelve trangles each havng three torsonal sprngs of varyng stffness nto each computatonal tetrahedron to prevent t from collapsng durng the deformaton (see Fg. 3). The stffness coeffcents of the sprngs n the corners of the trangles are nversely proportonal to the squared sne of the trangle angles, makng t mpossble for the trangle to have ether 0 or 180 degree angles. Ths prevents the nodes that share an edge wth each other from colldng (somethng that a lnear sprng method can also do), but also prevents the grd nodes from crossng any edge or faces. The method has been tested on many confguratons and s shown to work n cases of dramatc deformatons when tradtonal lnear sprng methods faled. For example, consder a smple nne node (18 tetrahedra) benchmark system shown n Fg. 4. The two-dmensonal verson of ths system s consdered n Ref. 11, whle the three-dmensonal system s used n Refs. 4 and 8. In the current paper the system n Fg. 4 s consdered to test the grd deformaton algorthm by brngng down the top node (number 1) by 99.5% to pont A, whch s just above the plane of the bottom nodes 2, 3, and 4 n Fg 4. Results follow n secton III of the paper. A very good performance of the method n such cases of very large dsplacement was one of the reasons we consdered usng ths method. Another reason was the performance of the method on the aeroelastc case of the AGARD Fg 4. Benchmark nne node system. Very often cells other than tetrahedra are generated n the volume of the grd, e.g. prsms or pyramds. In that case a dvson of such cells nto three and two tetrahedra respectvely proved to be a very smple geometrc task that needs to be performed only once at the nput of the grd nformaton nto the deformaton module. Some unstructured grd generaton codes keep the nformaton about the tetrahedra that formed the hybrd cells, so n that case no dvson s necessary. It s also mportant to note that a flow solver wll contnue to use the hybrd cells as t only needs updated values of node coordnates from the mesh deformaton module. Fg 5. AGARD wng. There are two reasons why ths (torsonal sprng analogy) method or smlar deformaton methods may fal to deform a grd desgned for hgh Reynolds number flows,.e. vscous grds. Frstly, the cells n the boundary vscous layers may be thnner than the deformaton of the structure durng a sngle deformaton step, n whch case the method loses ts robustness n these layers. Secondly, even f the deformaton does not exceed the sze of the vscous cells, t s stll too large. After a number of deformaton steps the qualty of the 3

4 Fg 6. Surface vectors (courtesy to the author of Ref. 3). Fg 7. Deformed structured grd example. 15 vscous cells deterorates sgnfcantly whch may result n ether a creaton of negatve cells on the next mesh deformaton step or flow solver problems. In the orgnal development of the method for the two dmensonal case 11 ths restrcton s noted. For example, ctng Ref. 11: Even though we are nterested n flow problems where some boundares may undergo a moton wth large ampltude, we assume n the followng dervaton that the ncrement of mesh moton between two tme-steps s suffcently small to justfy the assumpton of small dsplacements and small rotatons. For the above reason, another deformaton approach s used n the boundary layer regon of the grd. Ths part of the grd s moved by frst recomputng the surface vectors (see Fg. 6) of the deformed surface and then redstrbutng the grd ponts n the boundary layer n the drecton of these vectors. Ths dea has proved to be a smple and very effcent method. No generaton of new nodes s requred durng ths process snce the grd structure remans ntact. Also, no strct requrement on the sze of the surface deformaton s present n ths method. Ths approach can be vewed as very smlar to some n the structured grd communty 15 the deformaton of the surface s propagatng along the nodal lnes that orgnate on the surface (see Fg. 7). The nodal lnes reman normal to the surface n the boundary layer n these methods. In the type of grds that are consdered here, the cell edges near the surface boundary (Fg. 2) ndeed resemble the nodal lnes of a structured grd. The man dffculty then s to fnd the consecutve nodes of the nodal lne of the unstructured volume grd, whch becomes more dffcult the further from the deformng surface one goes. However, t can be done usng the partculartes of a grd, e.g. the fact that the prsms n the vscous layers are stacked on top of each other. If no prsms are created by the grd generator, then one can use the fact that the next node n the nodal lne s the closest of all nearby nodes. Once the nodal lnes are known, a popular structured grd deformaton New Deformed approach such as algebrac method (see, for example, Ref. 15) can be C Surface Trangle ABC appled to deform the vscous part of B the grd. Also note that because these nodal lnes are n layers where the aspect rato of the cells s very hgh, the dstance between the nodes along the nodal lne s smaller than the dstance between the nodal lnes themselves. Therefore, the method for movng or deformng ths part of the grd can have lghter restrctons. In ths partcular work the method to move and redstrbute the nodal lnes was done as follows. The vscous layers were orgnally produced by the process of advancng layers along normal projectons from the surface trangular faces. Snce each surface node has multple faces, the normal for the advancng layer s a weghted average. To deform these vscous D V AD V AC n A n AC n AB A V AB D A C V V AC AD n A n AC n AB k Inertal Reference j Frame Fg 8. Vscous layer deformaton strategy. Orgnal Surface Trangle ABC V AB B 4

5 layers, one of the faces usng the surface node of nterest s arbtrarly chosen as the startng pont for computng a set of bass vectors (see Fg. 8). Follow-on methods may modfy ths to nclude the effect the surroundng faces on the surface to smooth the normal vector. Wth the current method for a surface trangle ABC, two surface vectors are formed from ponts A to B (V AB ) and ponts A to C (V AC ). A normal vector s formed by normalzng the cross product of V AB and V AC. Ths normal vector n A along wth vectors formed by normalzng V AB and V AC (n AB and n AC ) form a set of bass vectors. The grd generaton process produced prsm cells above ths surface trangle wth one of the nodes of each prsm approxmately along ths normal vector n A. These nodes make nodal lnes n the current method. Vectors from pont A on the surface to all the nodes of the nodal lne above t (e.g. node D n Fg. 8) are expressed n terms of the new bass vectors through the use of dot products between the vector V AD and the bass set, n A, n AB and n AC. After surface deformaton occurs, the new surface trangle may have changed shape, been rotated, and translated to new coordnates A, B, and C as shown n Fg. 8. Usng the new prmed coordnates and followng the same procedure as above, we create a new set of bass vectors n A, n AB and n AC. The new coordnate D s produced by applyng the components of V AD expressed n n A, n AB and n AC drectons to the new bass vectors (n A, n AB and n AC ). Ths method s very fast, mantans layers sutable for vscous boundary layer calculatons, and wll mantan good grd qualty (.e. no nodal lnes crossng to produce negatve cells) as long as the aspect rato of the cells s farly hgh. The contrbuton of ths work s to combne these two methods (for the nvscd and vscous parts of the grd), devse an nterface strategy between them, and apply them to a realstc confguraton n a hgh Reynolds number flow problem. III. RESULTS A. Benchmark Problem Fg 9. Orgnal (blue) and after deformaton cycle meshes: 90% (red) and 99.5% (green). The followng results descrbe the performance of the torsonal sprng grd deformaton method over many steps. A good example of when ths s mportant s an aeroelastc problem of lmt cycle oscllatons where a wng surface deforms back and forth n many cycles over tme. Consder agan a nne node mesh shown n Fg. 4. Ths very smple system s used to test the performance of the method. The effect of deformaton range s studed frst. The top Node 1 s moved downward toward the pont A located n the plane of the fxed nodes 2, 3, and 4 (see Fg. 4) wth a step of z=0.5%. So f ntally the Node 1 vertcal coordnate was z=1, then n the frst case of 180 steps t became z=0.10 (a 90% case), and n the second case n addtonal 19 steps (99.5% case) t becomes z=0.05. In both cases, Node 1 s then moved back to ts orgnal poston n the same manner. A pcture of the systems after the cycle of deformaton (red and green color) along wth the orgnal system (blue) s shown n Fg. 9. Obvously f these deformatons caused no damage to a grd, ts pcture should concde wth the orgnal. In ths case the grd 0.2 8,2,3, % 90% % 90 % d 0.1 9,5,6,7 d ,9,5, steps, steps, a) b) Fg 10. Effect of the sze of deformaton range: a) Hstory of nscrbed sphere dameter; b) Hstory of dameter rato. 5

6 that was deformed 90% and back s very close to ts orgnal state. The addtonal 19 steps of deformaton that almost flattened the second grd caused some sgnfcant damage to the grd. A smple but effectve way to montor these effects s by observng the dameter of the sphere nscrbed nto a tetrahedron, specfcally a rato of the dameters before and after a step of deformaton. A calculaton of the dameter of a sphere nscrbed nto a tetrahedron s a very nexpensve addtonal calculaton n the case when a volume of a tetrahedron s calculated, whch s done by the fnte volume CFD scheme of the flow solver and s also used for the purpose of checkng f negatve volumes appeared n the deformaton process. Fg. 10a, shows a tme hstory of such dameters for three out of eghteen tetrahedra present n ths system (the four nodes of each tetrahedron are marked on Fg. 10a), whle Fg. 10b shows a rato of these dameters, d, before and after each step durng the nterval from step 180 to 220. Ths rato devates sharply from the unty durng the steps when Node 1 reaches the bottom, z=0.05. Fg 11. Orgnal (blue) and after deformaton Thus a hgh deformaton rate (e.g. larger cycle meshes: small step (red) and large step (green). than d d = 0.1) leads to sgnfcant deteroratng changes n the grd. Ths s a very mportant concluson, and ts consequences reappear when examnng the case of the AGARD wng. A relatve effect of deformaton step sze s consdered next (see Fg. 11). Ths tme Node 1 s brought to z=0.1 and back wth steps of two dfferent szes a step z=0.005 (green color) and z= (red). In ths case one can see only a margnal edge shft n the deformed tetrahedra wth the larger dfference present n the case of larger z step comparng to the orgnal one. Fg. 12a shows tme hstores of the dameters for three nscrbed spheres, whle Fg. 12b shows a rato of the dameters before and after each step durng the process. Note that ths rato, d, n ths case stays very close to 1 and, of course, the largest devaton from 1 s hgher for the case of the larger deformaton step sze, z. When the deformaton sze used on every step s large, the fnal dsplacement of the grd nodes when the system returns to the orgnal poston s large as well. Thus, t was establshed that d s an nexpensve deformaton measure that allows montorng the changes n the element qualty. To keep the orgnal grd qualty durng deformatons t s desrable to keep ths value as close to unty as possble for all cells on all tme-steps. The exact value of an acceptable rato depends on the problem, cell locaton, and many other factors and should be programmed by the user. A detaled descrpton of a smlar but more nvolved and strct deformaton measure s avalable n Ref z=0.25% 8,2,3,4 z=0.5% z=0.25% z=0.5% d 0.1 8,5,6,7 9,5,6,7 d steps, steps, a) b) Fg 12. Effect of the sze of deformaton step: a) Hstory of nscrbed sphere dameter; b) Hstory of dameter rato. 6

7 B. AGARD Wng Problem Next, let us return to the AGARD wng case (see Fg. 5). Ths confguraton s used by many researchers for valdaton of aeroelastc models because of the number of expermental and computatonal results avalable for the transonc flutter problem of ths wng. A number of grds n the range from 1.5 mllon to 4.7 mllon tetrahedra were created usng the software package VGRIDns. 10 For example, n the case of 1.5 mllon tetrahedra, half of the tetrahedra are n the nvscd porton, whle the remanng tetrahedra are n the vscous porton and can be combned nto layers of approxmately 250 thousand prsms. The whole grd has 250 thousand nodes, 15 thousand of them beng on the wng surface (that surface mesh s shown n Fg. 5). For the wng root chord of 1, the smallest nscrbed dameter n a tetrahedron near the wng surface s ; the aspect rato for ths tetrahedron s equal to Suppose the wng s undergong a deformaton along ts frst bendng mode of the ampltude of 1% compared to the sze of ts root chord. In ths case, keepng the cell deformaton smaller than a cell sze requres more than a thousand tme steps. However, the above descrbed deformaton rato, d d, on each tme-step n that case s on the order of unty for the boundary tetrahedra. As has been already dscussed, ths leads to the deteroraton of the grd qualty wthn a few steps. Thus, one needs to perform many more thousand steps to deform the grd wth a smaller deformaton ncrement sze. Meanwhle usng the vscous deformer (.e. redstrbutng the nodes along the nodal lnes n the vscous part of the grd durng deformatons) leaves only remanng nvscd tetrahedra to deform, and the smallest nscrbed dameter n that part s (an mprovement by a factor of almost 40). Shown n Fg 13 are fve neghborng nodal lnes that penetrate 15 vscous layers of ths grd n the most deformed place of the wng durng the frst bendng mode deformaton,.e. the corner of the tralng edge and tp. The ntal nodal lnes when the wng s not deformed are ndcated by the red color. The same nodal lnes at 10% deformaton (based on the root chord sze) are ndcated by blue. Fnally, the same nodal lnes the 20% deformaton are ndcated by yellow. The surface vectors were recomputed n both cases and the nodes redstrbuted along the nodal lnes wthout any ntersecton of nodal lnes. These calculatons requre an nsgnfcant amount of tme compared to the nvscd cell deformatons. The remanng nvscd part of the grd can now be deformed. For example, the deformaton along the frst bendng mode of the ampltude of the 1% of the root chord sze can be acheved n fve hundred steps wth the deformaton rato of a most deformed tetrahedron on each step never 2 exceedng d d = The average deformaton rato n the most deformed layer of the nvscd terahedra n ths case does not exceed on each step. Admttedly, ths s stll rather a large number of steps for a deformaton of ths magntude, yet the grd s deformed and the grd qualty s preserved allowng for consecutve cycles of the grd moton wthout a need to regenerate nvald areas of the grd. As dscussed further below, the tme performance of the combned vscousnvscd grd deformaton method s good. Thus, even though a large number of steps s requred to mantan the qualty of the grd durng deformaton, the method performance s acceptable. C. Parallel Performance As has been already mentoned, the two man tasks of the sprng methods are: the stffness matrx computaton and solvng the large lnear system of equatons. A few detals of the process are dscussed here. For the grds of nterest, the stffness matrx s very large (despte beng sparse, so only nonzero elements are computed and stored). Storng t as a whole on each processor s often not possble due to memory requrements. To compute the matrx, t s necessary to dvde the entre grd between the avalable processors. So each processor has a certan number of tetrahedra to work wth. On the other hand, solvng the Fg 13. AGARD Wng Vscous Nodal Lne Deformaton. 7

8 system requres a dvson of nodes between processors (or to be precse, degrees-of-freedom that each node has). The optmal dvson of the grd by terahedra does not usually contan an optmal dvson by nodes. Numerous factors contrbute to ths fact. The node and tetrahedron numberng n the grd s not done n a very unform fashon durng the generaton of the grd. A number of boundary nodes can vary between processors, thus affectng the dmenson of the part of the system that each processor solves. For a gven node to have ts stffness coeffcents computed correctly, all tetrahedra wth that node must be consdered. For many nodes, ths requres ether passng some data between processors that have the partcpatng tetrahedra or computaton of overlappng terahedra regons, whch s faster. Yet dependng on the dvson and numberng of nodes and tetrahedra these overlappng regons can become very large especally as the number of processors grows. Once the vscous nodes were elmnated from the sprng method regon, the METIS 12 doman decomposton lbrary was used to dvde the grd nto nearly equal tetrahedron regons. METIS dvson can be performed wth an opton of keepng the boundares between the regons at mnmum, whch allows havng smaller overlappng regons. METIS C language routnes can be lnked to the code or used as a stand alone module to dvde the grd between the processors before the deformaton code s loaded. To solve the system of equatons n parallel, the AZTEC 13 lbrary was used. AZTEC s an teratve parallel solver that can use a number of methods and precondtoners. AZTEC C language lbrary was lnked to the torsonal sprng Fortran 90 code. The Message Passng Interface (MPI) lbrary s used to pass nformaton between processors n the code (ncludng AZTEC). Shown n Fg. 14a are the results of the torsonal sprng deformaton code parallel performance for three dfferent sze grds on a Pentum IV, 2.8 GHz cluster. The number of cells n Fg. 14a s the number of the tetrahedra that were deformed usng the torsonal sprng analogy method. Usng the above descrbed vscous layer deformer sgnfcantly reduces (at least by half) the number of cells that have to be deformed usng the torsonal sprng analogy method. For the consdered range of number of processors, each curve devates slghtly from the ntal parallel performance as the number of processors s ncreased. Ths s because the overall share of the overlappng regons for matrx computatons and the amount of data passed between processors for solvng a lnear system of equatons ncreases as the number of processors s ncreased. Ths declne n performance s clearly seen n Fg. 14b, where on one of the platforms the number of processors used vares from 8 to 128. a) b) Fg 14. Parallel Performance: a) Grds of dfferent sze on a cluster; b) Same grd on two clusters. D. Computaton Performance Comparng to Flow Solver Next let us compare the tme that a torsonal sprng analogy mesh deformaton code takes compared to the flow solver. In Ref. 2, the three-dmensonal torsonal sprng analogy method took 52% of the total tme of the AGARD wng aeroelastc smulatons on a 128 thousand tetrahedra grd. Thus, the grd deformaton tme was at least equal to the flow solver tme. In the current work on 32 processors (see Fg. 14a) t takes roughly 30 sec to handle the mesh deformaton of the 4.7 mllon AGARD wng tetrahedra grd wth the torsonal sprng analogy method (wthout usng a boundary layer deformer). The flow solver that s used n our numercal smulatons, such as n the tal buffet calculatons of Ref. 9, s the commercal verson of Cobalt developed by Cobalt Solutons. 14 It s a parallel, mplct, unstructured Euler/Naver-Stokes flow solver. In a typcal run a tme step wth three subteratons on the same 32 processors takes the Cobalt flow solver about 60 sec. Note that combnng the tetrahedra n the vscous layers nto prsms the flow solver operates wth only 3.6 mllon cells. Whether there s a need to update the grd on subteratons s not yet determned. In the best case scenaro, when a grd doesn t have to be deformed durng the subteratons, one obtans 1/2 rato of the grd deformaton module tme to the flow solver tme. However, after passng the 15 vscous layers present n ths grd to the vscous deformer, the torsonal sprng method needs to deform only the remanng 1.6 mllon tetrahedra (whch s about one-thrd of the ntal amount of tetrahedra). Thus the rato of 8

9 the consumed tme of the current grd deformaton module to the flow solver becomes 1/6 n the best case scenaro or 4/6 n the worst case scenaro, when a mesh update s necessary on each subteraton of the flow solver. Also, t must be noted that of the two operatons n the torsonal sprng analogy deformaton method, the constructon of the stffness matrx consumes sgnfcantly more tme compared to the tme t takes for the lnear system of equatons to converge. The exact soluton of these equatons s not at all necessary they are equatons of an magnary sprng system that reacts to gven surface deformatons to move the mesh. Thus, for example, wthn fve to ffteen Gauss-Sedel teratons a lnear system of equatons solver converges to a suffcent precson for the value of updated mesh node coordnates. That tme s about 20% of the tme that t takes to buld the stffness matrx. However, the constructon of the matrx s done wth the prevous coordnates of the grd nodes. Thus, one can be computng the stffness matrx on mesh deformng processors at the same tme the flow and structural solvers update the new locatons of aerodynamc surfaces on the other processors. Therefore, t s very plausble that separate processors are runnng a mesh deformaton module concurrently wth the flow and structural solver. The tme that the aeroelastc calculatons are slowed by the mesh update reduces dramatcally n ths parallel envronment.. IV. CONCLUSIONS In ths paper we presented a mesh deformaton method for unstructured computatonal flud dynamc grds that are employed n aerodynamc and aeroelastc studes of wngs or complete arcraft confguratons submerged n a hgh Reynolds number flow. Gven a surface deformaton, the method effcently recalculates new locatons of hgh aspect rato cells that make the vscous grd layer and deforms the nvscd part of the grd usng an establshed method of nserton of torson sprngs n the cells. A means for montorng of the mesh deformaton wth respect to detectng the changes n qualty of the grd durng deformaton s also consdered. Grd deformaton results are dscussed for a benchmark system as well as for a larger real applcaton type system (a vscous AGARD wng grd) under the deformatons exhbted n experments. Dscusson of the parallel performance and comparson of the runnng tme of the mesh deformaton code to that used by the flow solver s made. The method showed overall promsng results. The next step s to valdate the method on other confguratons. Fnally, the ultmate performance assessment of ths strategy wll be revealed when the method s coupled wth the flow and structural solvers n the aeroelastc smulatons. ACKNOWLEDGEMENTS The authors would lke to thank Dr. Chrstoph Degand 2 for sharng hs experence wth the torsonal sprng analogy method and Dr. Ray Tumnaro of Sanda Natonal Laboratores for help wth the AZTEC package. Fnally, the authors would lke to express ther grattude for the support of ths project by Dr. John Schmsseur of AFOSR. REFERENCES 1 Batna, J. T., Unsteady Euler Sprng Method for Three-Dmensonal Unstructured Dynamc Meshes, AIAA Paper , Jan Degand, C., and Farhat, C., A Three-Dmensonal Torsonal Sprng Analogy Method for Unstructured Dynamc Meshes, Computers and Structures, Vol. 80, pp , Przadeh, S. Z., An Adaptve Unstructured Grd Method by Grd Subdvson, Local Remeshng, and Grd Movement, AIAA Paper , Jun.-July Murayama, M., Nakahash, K., and Matsushma, K., Unstructured Dynamc Mesh for Large Movement and Deformaton, AIAA Paper , Jan Burg, C. O. E., Sreenvas, K, Hyams, D. G., and Mtchell, B., Unstructured Nonlnear Free Surface Flow Solutons: Valdaton and Verfcaton, AIAA Paper , June Cavallo, P. A., Snha, N., and Feldman, G. M., Parallel Unstructured Mesh Adaptaton for Transent Movng Body and Aeropropulsve Applcatons, AIAA Paper , 42 nd AIAA Aerospace Scences Meetng, Reno, Jan Sten, K., Tezduyar, T. E., and Benney, R., Automatc mesh update wth the sold-extenson mesh movng technque, Comput. Methods Appl. Mech. Engrg. 139, pp , Burg, C. O. E., A Robust Unstructured Grd Movement Strategy Usng Three-Dmensonal Torsonal Sprngs, AIAA Paper , 34 th AIAA Flud Dynamcs Conference, Portland, June-July Morton, S. A., Cummngs, R. M., and Kholodar, D. B., Hgh Resoluton Turbulence Treatment of F/A-18 Tal Buffet, AIAA Paper , Apr

10 10 Przadeh, S. Z., Unstructured Grd Generaton for Complex 3D Hgh-Lft Confguratons, AIAA Paper , Oct Farhat, C., Degand, C., Koobus, B., and Lesonne, M., Torsonal Sprngs for Two-Dmensonal Unstructured Flud Meshes, Comput. Methods Appl. Mech. Engrg., 163, pp , Karyps, G., and Kumar, V., A Fast and Hgh Qualty Multlevel Scheme for Parttonng Irregular Graphs, SIAM Journal of Scentfc Computng, Vol. 20, No. 1, pp , Tumnaro, R. S., Heroux, M., Hutchnson, S. A., and Shadd, J. N., Offcal Aztec User s Gude, Dec Strang, W.Z., Tomaro, R.F., and Grsmer, M.J., The Defnng Methods of Cobalt: A Parallel, Implct, Unstructured Euler/Naver- Stokes Flow Solver, AIAA Paper , Jan Melvlle, R., and Morton, S., Fully Implct Aeroelastcty on Overset Grd Systems, AIAA Paper , 36 th AIAA Aerospace Scences Meetng, Reno, Jan