THREE-DIMENSIONAL UNSTRUCTURED GRID GENERATION FOR FINITE-VOLUME SOLUTION OF EULER EQUATIONS

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1 ICAS 2000 CONGRESS THREE-DIMENSIONAL UNSTRUCTURED GRID GENERATION FOR FINITE-VOLUME SOLUTION OF EULER EQUATIONS Karm Mazaher, Shahram Bodaghabad Sharf Unversty of Technology Keywords: Unstructured grd generaton, Delaunay trangulaton, Euler equatons Abstract To solve the Euler equatons on a threedmensonal unstructured grd, an effcent procedure for generaton of relatvely hgh qualty elements s presented. The basc steps of ths method, whch s essentally based on Delaunay s trangulaton, s conssted of generatng surface ponts, ntal tessellaton, pont dstrbuton functon nterpolaton, nternal node generaton and update trangulaton. Specal treatments for surmountng degenerate cases are consdered wthn the algorthm. Valdaton and relablty of algorthm are checked by dfferent examples. An upwnd scheme s used to solve Euler equatons on these unstructured grds for steady-state problems. Spatal dscrtzaton s accomplshed by a cell-centered fnte-volume formulaton usng flux-dfference splttng method of Roe. Soluton s advanced n tme by a smple explct scheme. Local tme steppng s used to accelerate convergence to the steady state. Some applcaton cases are treated to test the method. Introducton Structured grds have been wdely used for a long tme n CFD applcatons []. Dfferent algorthms of generatng structured grds, solvng basc equatons on them and pre or postprocessng operaton on such grds are well developed. They use less memory per unt element compared wth unstructured one. Also, methods of generatng structured meshes are smpler. However, generatng structured grds about complex confguratons, especally automaton of t, s stll a dffcult task. Unstructured grds are stronger n coverng complex felds, automaton of generaton process s smple, varaton of element sze n grd s applable and adaptaton process s smpler. In versus of such advantages, data structure of unstructured grds s more complex than structured one and optmum process of grd generaton has a complex logc and need extra work. There are a varety of unstructured grd generaton methods. Many of these methods are essentally based on two famous schemes.e. Delaunay Trangulaton and Advancng Front Method algorthms. Delaunay trangulaton methods, wth applcatons n CFD, are presented by Lawson, Green, Sbson [2,3], Bowyer [4] and Watson [5]. In recent years varous modfcatons to Delaunay s method have been presented to optmze and elmnate ts practcal dffcultes. Frey [6] proposed an algorthm for 2D pont creaton based on crcumcrcle and nscrbed crcle centers. Ths algorthm was lnked wth Watson algorthm for trangulaton. Schroder and Shephard [7] have also used Watson algorthm. To keep the boundary conformng, they used a topologcal classfcaton for grd enttes, and generated a consstent grd for complex and non-manfold gven geometres. Muller and hs coworkers [8] presented a strong method for two dmensonal grd generaton by combnng Delaunay s trangulaton and advancng front method by pont creaton. Some efforts n automatc grd generaton for three-dmensonal applcatons are also made. Researches of Weatherll and Hassan [9], Golas and Tsbouks [0], Maccrum and Weatherll [] and heng 234.

2 Karm Mazaher, Shahram Bodaghabad et al [2] le n ths category. In present work a combned Delaunay/Advancng front method n 3D has been presented. Dfferent methods of outcomng degenerate cases n Delaunay trangulaton are consdered. Relatvely hgh qualty elements are created to be used n CFD applcatons. Euler equatons have been solved n some test problems, whch are meshed by current grd generaton algorthm. Followng sectons descrbe steps of grd generaton algorthm and the numercal soluton of the Euler equatons. Fnally results on some test problems are ncluded. 2 Grd generaton method Methods usng advancng front type pont placement and Delaunay connectvty have been developed by Delaunay [], Watson, Lawson, Sbson, Muller et al and George et al [3]. Ths dea s also used n three-dmensonal grd generaton by Rebey [4], Przadeh [5] and Khosravy. Current work s essentally based on work of Khosravy. Specal modfcatons are appled to mprove grd qualty, surmountng degeneraces and ncreasng robustness and relablty of algorthm. To better descrpton of the algorthm some basc defntons are presented. 2. Delaunay trangulaton Delaunay trangulaton s the geometrc dual of the Drchlet tessellaton [4]. Gven a set P of M ponts n an n-dmensonal space, there exsts a regon V such that V { x x P x P for all j} = ; () j The collecton V = = V s defned as the Drchlet tessellaton. Gven a Drchlet tessellaton V, one may form the Delaunay trangulaton by connectng any two ponts P and P j whose correspondng tles V and V j are neghbour. In the 3D case, Delaunay trangulaton gves a decomposton of the convex hull of P nto 3D smplexes (tetrahedra). An mportant property of Delaunay trangulaton M s that a crcumsphere of tetrahedron contans no ponts other than ts vertces. 2.2 The grd generaton process The algorthm n step-by-step manner can be expressed as follows:. Defne a set of ponts, whch form a convex hull contanng all other ponts. 2. Add the boundary nodes to the ntal grd, usng Watson algorthm. Obtan a vald trangulaton of them and recover all boundary surfaces (f they are lost). 3. Assgn a pont dstrbuton functon to each ntal boundary pont. 4. Calculate a crteron for qualty of elements. If qualty of an element s under a threshold value or element doesn t satsfy the pont dstrbuton functon, t s marked as a bad element. 5. Create a new pont for each bad element. 6. For each new pont, search for the element contanng t. 7. Interpolate the pont dstrbuton functon for the new pont from the contanng element. 8. Merge two new ponts that are too close to each other. 9. Reject new ponts that are too close to an exstng pont. 0. Insert each accepted new pont n the trangulaton usng the Watson algorthm.. Contnue from step 4; repeat untl no acceptable pont s created. 2. Smooth the grd dstrbuton. Watson algorthm begns by fndng tetrahedron, whch contans the new pont. Ths tetrahedron and all other tetrahedrons whose crcumsphere contan that pont are then deleted. Ths form a polyhedron bounded by trangular faces. Each face of ths polyhedron and the new pont form a new tetrahedron. Wth constructng all such tetrahedra, a new trangulaton obtaned

3 THREE-DIMENSIONAL GRID GENERATION FOR FINITE-VOLUME SOLUTION OF EULER EQUATIONS 2.3 Boundary pont creaton Generally there are four basc topologcal enttes n three dmensons. These are vertex, edge, surface and regon. Each entty s surrounded by one lower dmenson entty. Regons are surrounded by surfaces, boundary of each surface s an edge and each edge has two end vertces. Boundary nodes creaton s a procedure to create proper ponts n boundary edges and surfaces. Edges dvson s done due to local spacng requrements. Intally ponts may be dstrbuted unformly and then relocated usng proper methods [9]. Pont creaton on surfaces s a dffcult task n grd generaton. In the case of planar surfaces, algebrac methods can be used. For general curved surfaces, usng two-dmensonal grd generaton n surface parametrc space s a common method []. 2.4 Convex hull Convex hull s a smple large enough regon, whch contans all nodes n the mesh. A smple tetrahedron can be used, but experence showed that f the number of nodes on the convex hull were low, probablty of occurrng degenerate cases would be ncreased. For all examples presented here, a polyhedron wth one node n ts center s used. 2.5 Nodes nserton Node nserton algorthm s based on Watson algorthm. Ths algorthm deletes all neghbors of the contanng element whose crcumspheres contan the new node. New elements are created usng the new node and each face of the hollowed regon. The dffcult step of ths algorthm s the nsphere check. Ths nvolves makng a decson based upon floatng pont arthmetc. An ncorrect deleton of a tetrahedron can result n a dsjont, non-convex empty polyhedron whch, when new tetrahedra are created, ntersect each other. It has been found that such ncorrect decsons can also dsconnect and solate a prevously nserted pont such that t no longer appears n the data structure of the formng ponts. Such problems may even arse wth smple geometres, such as a set of ponts, whch defne a grd on the surface of a unt cube. Foundng degeneraces and surmountng them utlzes a lot of efforts n the three-dmensonal grd generaton development [7,9,0]. 2.6 Resolvng degenerate cases When more than four ponts of the pont set le on the crcumsphere of an element (wthn the resoluton of the numercal calculaton), the degenerate case occurs. Resolvng degeneraces requre a consstent decson for a partcular pont for beng nsde or outsde. In practce, the judgement whether a pont P s nsde a partcular sphere C of radus r and crcumcenter x c (Fgure ) can be made based on a tolerance factor ε: P C when P xc r ε Fgure Approxmate ncrcle test (2) Choosng ε>0 s equvalent to classfyng P outsde of all degenerate C. Generally ε s taken postve for convenence. The parameter ε n fact defnes an approxmate n/out sphere C of dfferent radus from the real crcumsphere C. By usng C nstead of C, the result s that the trangulaton may not satsfy the crcumsphere condton. But even f ε=0, the effect of numercal calculaton errors s to consder C rather than the real C. So, to have a vald mesh, the method should account for the 234.3

4 Karm Mazaher, Shahram Bodaghabad perturbed crcumsphere. Three problems can arse when computng wth C. Frst, the resultng trangulaton may not be a Delaunay s one wthn tolerance ε. In terms of generatng a computatonal mesh, as long as the trangulaton s a vald one and the elements qualty s acceptable, ths stuaton s not of concern. Secondly a pont P very close to pont P j, where P j s ncorporated nto the trangulaton pror to P, may not be nsde any C, and therefore wll not be partcpated n the trangulaton (Fgure 2). In ths case, t may be necessary to reduce the sze of ε. Thrdly and of most concern, s that nconsstences are created because the mesh doesn t globally satsfy the Delaunay property of the crcumsphere condton. Watson algorthm uses ths property to generate an nserton polyhedron to get a vald mesh. The nserton polyhedron must satsfy the condton of pont convexty [6], as descrbed below: A polyhedron wth faces F, for =,,n F s strctly pont convex to the pont P when the nward pontng normal to each polyhedron face, n, satsfes the condton of pont convexty (Fgure 3): ( P x ) 0 n. > (3) where x s an arbtrary pont on face F. Fgure 2 A pont not nsde any sphere When the nserton polyhedron s strctly pont convex, creatng tetrahedra by connectng P wth each face F produces a structurally consstent mesh. Pont convexty checkng s readly added to the Watson algorthm to prevent structural nconsstences. As each pont s nserted nto the trangulaton to create the nserton polyhedron, the condton for pont convexty s enforced. In partcular when n.( P x ) < δ where δ>0, the pont convexty condton s not satsfed. When a face F does not satsfy the pont convexty, the tetrahedron t s connected to, s deleted, formng a new nserton polyhedron. The added faces are agan evaluated for pont convexty and the procedure s repeated. Fgure 3 Pont convexty rule 2.7 Node creaton Dfferent crtera for checkng bad elements are presented [2]. Two of them are the rato of the smallest edge length to the largest one and the rato of the nscrbed sphere radus to the crcumsphere radus. After taggng bad elements, those that le adjacent to a good one or the boundary, generates the front and new node s created adjacent to each front face. New node s located such that local spacng (based on spacng functon) s satsfed and a regular tetrahedron s expected to be created. 2.8 Mesh smoothng Mesh smoothng (node relaxaton) s a technque of ncreasng the overall qualty by sutable repostonng of nodes. Node relaxaton s appled to nternal and surface nodes. Surface nodes can be moved only on ther geometrc surfaces. One of the most usual methods s Laplacan smoothng [6], whch s used n the 234.4

5 THREE-DIMENSIONAL GRID GENERATION FOR FINITE-VOLUME SOLUTION OF EULER EQUATIONS current work, too. Dfferent qualty crtera can be used [2]. In present work the rato of volume to cubc power of the largest edge length s used. In order to have a pcture of the mesh overall qualty, the mean (Q m ) and jont (Q j ) qualtes are ntroduced []: Q Q m j = N = N N = N = Q Q (4) where N s the number of tetrahedra and Q s ts qualty. The jont qualty factor has a value that s very senstve to elements wth low qualtes. In smoothng procedure, for each node repostonng, the mean and jont qualtes values are calculated n new poston. If these values ncrease respect to ther old values, the node wll be moved. 2.9 Data structure In three-dmensonal grd generaton the followng nformaton are stored: cell vertces, cell neghbors, cell goodness/badness flag, node coordnates and node spacng functon. So, the requred storage space s 9N nteger and 4M floatng words, where M s the number of nodes. 3 Soluton scheme of the Euler equatons The Euler equatons can be wrtten as: U F G H (5) = 0 t x y z The state vector U ( = [ ρ ρu ρv ρw ρe] T ) and the flux vectors F, G, H are expressed n terms of the conservatve varables. ρ s densty, (u, v, w) are velocty components and E s the total energy. E can be related to the other varables by the perfect gas state equaton. A cell-centered method, n whch grd elements are used as doman control volumes, s used. Each cell has four faces and fluxes are F computed on these faces. Flux quanttes are computed usng Roe s flux-dfference-splttng [7]. The flux across each cell face s computed usng [ ] ( U, U ) F( U ) + F( U ) A ( U U ) L R = 2 L Here subscrpts L and R denote to left and rght of the cell face. The Jacoban matrx A s evaluated wth Roe-averaged quanttes [8]. Soluton procedure starts by defnton of the ntal condtons, and then advances n tme by usng a smple explct scheme,.e. U = U t V n+ n 4 R = F n S R L (7) (6) where F n s the normal flux across face, S s the face area, V s the cell volume and t s the soluton tme step. Superscrpts n and n+ denote the current and the next tme step. Steady state soluton s acheved when resdual term of ths equaton s lower than a threshold value. Further dscusson of ths method can be found n many references lke [8] and [9]. Intal condton, boundary condtons and stablty In external flows, t s common that free stream condton s appled as ntal condton. In nternal flows, lke flow n supersonc ducts and cascade flows, unform flow based on nlet condton s usually used. For sold boundares (ncludng symmetry planes), the flow tangency condton (slp condton) s mposed by settng the velocty on the boundary ghost cells as mrror mage of the boundary cell velocty. Densty and pressure n the ghost cells are smply set equal to the boundary cell values. For far-feld boundary, characterstc boundary condtons are appled usng the fxed and extrapolated Remann nvarants correspondng to the ncomng and outgong waves. An approxmate method s used by proper selecton of the conservatve varables to be extrapolated from outsde the doman or to be 234.5

6 Karm Mazaher, Shahram Bodaghabad nterpolated from the nsde due to the local Mach number. Stablty condton s appled by Courant- Fredrcks-Lewy (CFL) condton. Local tme steppng accelerates convergence to the steady state by advancng the soluton at each cell n tme at a CFL number near the local stablty lmt. The expresson for the local tme step can be represented as [3]: V t = β u = + 4 ( c ) S Here β s a safety factor, u, c and (8) S are the normal velocty, sound speed and area of the face. Bar quanttes show Roe-averaged values. 4 Results In ths secton sample generated meshes are presented to show valdty of the proposed algorthm. Also the supersonc flow around a wedge and a cone are smulated usng the Eluer equatons. 4. Sample meshes A smple rectangular doman around a wedge s the frst example. Fgure 4 shows a cross secton of ths grd. The qualty dstrbuton for ths mesh s shown n Fgure 5. The second example s a grd around a cone. Because of the symmetry, only one-half of the cone s consdered. The far-feld boundary s a cylndrcal surface. A cross secton of ths grd s shown n Fgure 6. Thrd sample mesh s a more complex doman,.e. grd around ONERA M6 wng. A sphercal surface s used as the farfeld boundary. Ths mesh s shown n Fgure 7. It s appear from Fgure 5 shows that the qualty factor has a dstrbuton smlar to the normal dstrbuton. Average qualty for ths mesh s about 0.4 (ths s true for other examples, too). Of course, the dstrbuton s preferred to be as close as possble to unty. Smoothng could move the peak of the dstrbuton curve to one, but n all examples, the number of cells wth qualty greater than 0.9 s decreased, as t can be observed n Table. In other words, smoothng dstrbutes the badness of cells among all cells. Snce the overall qualty of the grd s mportant (and not qualty of a specal cell), t can be deduced that smoothng procedure s advantageous. An mportant factor, whch s also observed n Table, s the very low value of the mnmum qualty n the grd. It means that some bad cells always present n grd. These cells may affect the soluton accuracy and n the worst case deterorate the stablty. 4.2 Applcatons Supersonc flow around a wedge s a good test problem, whch s a really two-dmensonal problem, so comparson of the threedmensonal soluton wth the 2D one s possble. Snce the exact soluton exsts for ths problem. A supersonc flow wth Mach number M =2.5 around a wedge wth half-angle of 5 degree s consdered. Mach contours n a 2D secton of the doman are shown n Fgure 8. Mach number and pressure rato across the shock s extracted from the soluton and s compared wth the two-dmensonal and the exact solutons n Fgure 9. 2D soluton s obtaned from an adaptve solver [20]. Adaptve 2D grd s at least two levels fner than the 3D grd and has a hgher qualty. Snce the wedge flow s essentally a 2D flow, the dfference between 2D and 3D solutons s acceptable. Steady state soluton of the supersonc flow wth freestream Mach number of 2 wth 0 degrees angle of attack around a cone wth 0 degrees half-angle s the second test problem. Ths s a three-dmensonal problem. Mach contours on the cone surface are shown n Fgure 0. Pressure coeffcent on the cone surface s compared wth Shankar [2] soluton n Fgure. As expected, the pressure reaches to ts mnmum on the leeward sde of the cone and then ncreases. In the current soluton ths mnmum s detected almost accurately. Weak shock waves are not well developed on the leeward sde, and strong wndward shocks domnate the physcs of ths problem. Indeed the 234.6

7 THREE-DIMENSIONAL GRID GENERATION FOR FINITE-VOLUME SOLUTION OF EULER EQUATIONS hgher order schemes and adaptaton can lead to better results n the soluton. References [] Thompson J F, Son B K, Weatherll N P. Handbook of grd generaton. st edton, CRC Press, 999. [2] Barth T J. On unstructured grds and solvers. Von Karman Seres n Computatonal Flud Dynamcs, pp -65, 990. [3] Khosrav R. Soluton of compressble flow equatons on 3D unstructured grds. MS Thess of Mechancal Engneerng, Department of Mechancal Engneerng, Sharf Unversty of Technology, 997. [4] Bowyer A. Computng Drchlet tessellatons. The Computer Journal, Vol. 24, No. 2, pp 62-66, 98. [5] Watson D F. Computng n-dmensonal Delaunay tessellaton wth applcaton to Voron polytopes. The Computer Journal, Vol. 24, No. 2, pp 67-72, 98. [6] Frey W. Selectve refnement: a new strategy for automatc node placement n graded trangular meshes. Internatonal Journal for Numercal Methods n Engneerng, Vol. 24, pp , 987. [7] Schroeder W J, Shephard M S. Geometry-based fully automatc mesh generaton and Delaunay trangulaton. Internatonal Journal for Numercal Methods n Engneerng, Vol. 26, pp , 988. [8] Muller J D, Roe P L, Deconnck H. A frontal approach for nternal node generaton n Delaunay trangulatons. Internatonal Journal for Numercal Methods n Fluds, 992. [9] Weatherll N P, Hassan O. Effcent threedmensonal Delaunay trangulaton wth automatc pont creaton and mposed boundary constrants. Internatonal Journal for Numercal Methods n Engneerng, Vol. 37, pp , 994. [0] Golas N A, Tsbouks T D. An approach to refnng three-dmensonal tetrahedral based on Delaunay transformatons. Internatonal Journal for Numercal Methods n Engneerng, Vol. 37, pp , 994. [] Macrum D L, Weatherll N P. Unstructured grd generaton usng teratve pont nserton and local reconnecton. AIAA Journal, Vol. 33, No. 9, pp , 995. [2] heng Y, Lews R W, Gethn D T. Three-dmensonal unstructured mesh generaton part -3. Computer Methods n Appled Mechancs and Engneerng, Vol. 34, pp , 996. [3] George P L, Hermelne F. Delaunay s mesh of a convex polyhedron n dmenson d. Applcaton to arbtrary polyhedra. Internatonal Journal for Numercal Methods n Engneerng, Vol. 33, pp , 997. [4] Rebay S. Effcent unstructured mesh generaton by means of Delaunay trangulaton and Bowyer-Watson algorthm. Journal of Computatonal Physcs, Vol. 06, pp 25-38, 993. [5] Przadeh Sh. Three-dmensonal unstructured vscous grds by the advancng-layers method. AIAA Journal, Vol. 34, No., pp 43-49, 996. [6] Schroeder W J, Shephard M S. A combned octree/delaunay method for fully automatc 3-D mesh generaton. Internatonal Journal for Numercal Methods n Engneerng, Vol. 29, pp 37-55, 990. [7] Roe P L. Approxmate Remann solvers, parameter vectors and dfference schemes. Journal of Computatonal Physcs, Vol. 83, pp 25-38, 98. [8] Hrsch C. Numercal computaton of nternal and external flows. 2 nd edton, John Wley & Sons, 988. [9] Frnk N T. Upwnd scheme for solvng the Euler equatons on unstructured tetrahedral meshes. AIAA Journal, Vol. 30, No., pp 70-77, 992. [20] Lessan B. Unsteady soluton of two-dmensonal compressble flow on unstructured grds. MS Thess of Mechancal Engneerng, Department of Mechancal Engneerng, Sharf Unversty of Technology, 997. [2] Shankar V. Treatment of concal and nonconcal supersonc flows by an mplct marchng scheme appled to the full potental equaton. Proc ASME/AIAA conference on Computers n Flow Predctons and Flud Dynamcs Experments, Washngton D.C., pp 63-70,

8 Karm Mazaher, Shahram Bodaghabad Y 2500 Qualty Dstrbuton for Wedge Grd Before Smoothng After Smoothng.8.6 z.4.2 Number of Tetrahedron y Fgure 4 Wedge mesh cross secton x Qualty Fgure 5 Qualty dstrbutons for wedge grd Y 2 Y Fgure 6 Cross secton of mesh over cone x 0 Fgure 7 Mesh around ONERA M6 wng Y= Const. Secton n Wedge Flow M = 2.5 Θ =5deg Fgure 8 Mach contours n a secton of wedge Level Mach Mach D 3D Exact Fgure 9 Comparson of 3D, 2D and exact solutons P 234.8

9 THREE-DIMENSIONAL GRID GENERATION FOR FINITE-VOLUME SOLUTION OF EULER EQUATIONS Y M =2 α =0 Current Soluton Shankar Soluton 0.2 C P 0.5 Cross Vew α =0deg M =2 Mach φ φ Fgure 0 Mach contours on cone surface Fgure Crcumferental pressure coeffcent around cone Table Qualty comparson of meshes before and after smoothng Mesh Q m Q j Q mn Q max Before After Before After Before After Before After Wedge Cone M6 wng

R s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes

R s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 1 of 1 Notes Physcs Tool box Concave Mrror If the reflectng surface takes place on the nner surface of the sphercal shape so that the centre of the mrror bulges