, July 6-8, 2011, London, U.K. Enhanced Advancng Front Technque wth Extenson Cases for Intal Trangular Mesh Generaton Z. Abal Abas and S. Salleh Abstract Durng the element creaton procedure of ntal mesh generaton usng the standard advancng front technque (SAFT), there s a possblty where a conflct of undecded element occurs whch do not follow the normal cases. We proposed a drect approach to tackle ths knd of problem by extendng the two normal cases n SAFT to the total number of fve cases of consderaton durng the element creaton procedure n ntal mesh generaton. The approach s called as Enhanced Advancng Front Technque-1 (EAFT-1). The am of the work s to mprove the SAFT n order to tackle drectly such problem of sudden nvald trangular element wthout havng to re-order back the Front lst and wthout erasng the exstng element n the ntal mesh generaton. A smulaton of ths approach has been done and the qualty of the mesh s examned usng a number of defned measurements n Gambt software. The resulted mesh s appled for modelng the radaton heat transfer problem usng Dscrete Ordnate method. Index Terms Intal mesh, Enhanced Advancng Front Technque, extended cases and radaton. S I. INTRODUCTION TRUCTURED or unstructured trangular mesh formaton s an mportant step for approxmatng solutons to boundary value problems. Recently, unstructured grd has been predomnant because of ts ablty of modelng complex geometres, besdes ts natural envronment for adaptvty [1]-[3]. Trangular element s chosen n ths work snce t s reported that these type of element are the most flexble for automatc mesh generaton partcularly when mesh gradng s requred [1], [2]. Among the famous method of generatng unstructured mesh s Delaunay trangulaton and Advancng front technque (AFT). The AFT s guaranteed to preserves boundary ntegrty as well as t has the capacty to create trangular elements wth hgh aspect rato n boundary-layer regon [2]. In the present paper, we extend the work done n [2] and the ssue rased n [4] by llustratng two dfferent condtons Manuscrpt receved Jan 13, 2011; revsed Feb 5, 2011. Ths work was supported by Mnstry of Hgher Educaton, Malaysa under SLAB scholarshp. Z. Abal Abas s now wth Department of Mathematcs, Faculty of Scence, Unverst Teknolog Malaysa, 81310 Skuda, Johor, Malaysa, currently on study leave from Unverst Teknkal Malaysa Melaka. (e-mal: {zuradaa@utem.edu.my). S. Salleh s wth the Department of Mathematcs, Faculty of Scence, Unverst Teknolog Malaysa, 81310 Skuda, Johor, Malaysa. (emal: ss@utm.my). of undecded element creaton durng the teraton, regarded as conflct durng element creaton procedure usng the Standard Advancng Front Technque (SAFT) whch wll be descrbed n detal n secton II. A bref concept of SAFT and numercal method s descrbed n secton III. We present our drect approach of encounterng ths specfc problem by ncorporatng the fve extended cases of consderaton durng the element creaton procedure. The approach s called Enhanced Advancng Front Technques-1 (EAFT-1) where each of the cases wll be explaned n Secton IV. At the end of the paper, we apply the unstructured mesh for modelng the radatve heat transfer problem usng Dscrete Ordnate method for a smple two dmensonal doman. II. PROBLEM STATEMENT In general, the followng cases are possble durng the procedure of element creaton: (1) A new node (deal pont) s created and been selected as a vertex to form the new trangle element where ths new node satsfes certan condton lsted n the SAFT. (2) The already exstng front node and actve edges are selected to form the new trangle element where ths exstng node satsfes certan condton lsted n the SAFT. However, there are also condtons whch do not belong to any of the above cases. Ths ssue has been rased n [4] where an effcent mesh generaton algorthm should be able to tackle such problem when nether the normal case lsted above occurs durng the element creaton. Before llustratng the problem n detal, a few terms need to be defned. We denote an edge as (a,b) whch means that the edge s formed by node a and node b. The current selected edges beng consdered s called as base edge. Front s a lst or a set of actve edges whch s currently avalable for element trangulaton. It must be noted that all the edges belong to the current Front wll be consdered as actve edges whle edges that had been drop from the current Front lst wll be consdered as passve edges. At the same tme, any nodes whch compose the actve edges n the current poston of the Front lst are also known as actve node. In the creaton of the trangular element, once base edge has been dentfed, the poston of deal pont (IP) on the perpendcular bsector of the base edge s computed n such a way that an equlateral trangle s formed wth IP as vertex.
, July 6-8, 2011, London, U.K. III. STANDARD ADVANCING FRONT TECHNIQUE AND THE NUMERICAL METHOD A. The general basc step of standard Advancng Front Technque wth background grd Fg. 1. Edge (7,9) as the base edge. The IP s out of the boundary and no pont le wthn the crcle Fgure 1 llustrates the frst condton that does not belong to nether of the normal case. The boundary curve has been dscretsed nto a set of boundary edges where edge (7,8) s a part of the boundary. The procedure of the trangular element creaton s conducted smoothly as outlned n the SAFT for the frst sx elements but when t comes to create the next trangular element, wth (7,9) as the base edge, the problem occur when the IP constructed as n Fgure 1 s out of the boundary curve and at the same tme there s no actve node le wthn the crcle whch s approprate to replace the IP. Approprate here means the node needs to be nsde the boundary as well as the node do not les wthn other exstng trangular element. Another condton that mght occur n whch we classfed as the second problem s llustrated n Fgure 2 where the base edge beng consdered s (a,b). There s also a possblty that the IP created s stll nsde the boundary, but t s too close to the exstng element, and one cannot connect the base edge drectly to the IP because the sde of the new element wll ntersect the exstng sde of the trangle element. Furthermore, no actve nodes whch can replace the IP approprately le wthn the crcle. These second condton also leads to conflcts or undecded element creaton procedure durng the element creaton. Fg 2. The second problem where the base edge s (a,b) and t cannot connect to the IP drectly. Accordng to the algorthm of SAFT n [2], f these happen, one needs to re-order the Front and take the second shortest actve edge and repeat the trangulaton process agan. At some later stage, the same condton may occur and agan, one needs to re-order the Front agan and select the next actve edge. Imagne f we have thousand of edges n a bgger computatonal doman, ths procedure mght be complcated and may take longer tme to complete the task. Other author n [4] suggests that one mght consder to erase a few exstng elements f ths problem occurs. Therefore, ths scenaro shows that the algorthm s not suffcent to cover all condtons, hence an mprovement need to be done to tackle the problem drectly. Our work focus on how to mprove the algorthm of SAFT n order to tackle such problem descrbe above drectly wthout havng to re-order back the Front lst and wthout erasng the exstng element n the ntal mesh generaton. The approach nvolves studyng all types of enttes or objects that fall nto the crcle for further consderaton so that the trangular element can be generated drectly. The followng basc steps of SAFT n ths secton are based on [2]. The boundary curves are dscretzed by dvdng the boundary curves nto straght lne segments and hence, nodes are placed on each of the segment. The man approach of controllng the grd n terms of sze and shape of the trangular element nvolves the defnton of the requred grd-cell characterstc through background grd generaton [1]-[7]. The grd-cell characterstcs or known as grd cell parameters are the sze parameter, the stretchng s and fnally the orentaton of the cells. These parameters are requred to be defned at each node formng the background grd. Grd cell parameters for all nodes n the Front are nterpolated usng the value defned at the background grd. The edge wth shortest length l n the Front s selected as a departure zone, n order to create the trangular element. In order to obtan for the next step of calculaton, the nterpolated values of correspondng to the two nodes of the edges s averagng. The poston of IP on the perpendcular bsector of the sde s computed and the emprcal formula s used to obtan the radus, r. A crcle wth centre at IP s constructed wth the followng emprcal formula for the radus, r 08. *, where 0. 55* l; 0. 55* l ; 0. 55* l 2. 0* l 2. 0* l; 2. 0* l Actve nodes that le wthn the crcle are searched (f any) and ther dstances from IP are lsted. The best canddate for the thrd vertex of the trangular element would be the closest one to the IP. The equlateral trangle wll be formed f there are no such actve nodes les wthn the crcle. However, the valdty of the IP becomng the vertex must be check n whch t need to satsfy the followng condtons: (1) The coordnate of IP do not le nsde another exstng trangular element. (2) There s no ntersecton between the sde of the new trangular element and any exstng sdes of the actve front. Generally, n a marchng process, the Front moved nto the nteror of the doman, n whch new nodes and edges are created and at the same tme old related edges are deleted from the Front lst n order to produce trangular element. The new trangular element s constructed by two nodes of a segment of a Front and another node ether a newly created or already been exsted n the Front. The Front wll be updated every tme an element s constructed. If the above step of creatng the trangular element faled, then one needs to re-order the Front and then select the second shortest actve edges as the departure zone for element creaton procedure. Ths process wll be contnued untl there are no (1)
, July 6-8, 2011, London, U.K. longer actve edges n the Front or the whole computatonal doman has been trangulated completely. The general scheme for SAFT s llustrated n Fgure 3. B. The concept of radaton and Dscrete Ordnate method Radaton wll be emtted more at hgher temperature materals. Emssve power E s a rate of heat flow per unt surface area emtted by a radatng surface. For a black body emssve power, E s gven by Eb 4 b T (2) 8 Where 5. 6710 s Stefan-Boltzmann s constant and T s absolute temperature. The ntensty I s the rate of heat flow receved per unt area perpendcular to the rays and per unt sold angle. The formula for ntensty of the black body, I s b No No Fg. 3. The general scheme of SAFT 4 Ib E b / T / (3) Equaton (3) s used to compute the emtted radaton ntensty from the surfaces and fluds. When combuston occurs, there exst large amount of carbon doxde and water vapor where these combuston products are both strong absorbers/emtters n the nfrared part of the spectrum and ths flud s termed as partcpatng medum [8], [9]. The nteracton between partcpatng flud medum and radaton can be measured n terms of absorpton coeffcent and ts scatterng coeffcent where the sum of these two propertes s called s Defne grd cell parameter at background grd and dscretsed boundary curve. Select shortest edge n the Front Construct element based on 2 normal cases Element created successfully Yes Update the Front lst Empty front Yes END extncton coeffcent s [9]. The general relatonshp that governs the changes n radaton ntensty at a pont along a radaton ray due to emsson, absorpton and non-scatterng n a flud medum s as follows [10] di r,s ds I b I r r,s (4) I r,s s the ntensty of radaton at locaton ndcated by poston vector r, n drecton s. The frst and second terms n the rght-hand sde equaton are the emtted ntensty and the absorbed ntensty respectvely. Equaton (4) s ntegrated over each trangular element n the mesh obtaned. It s also subject to the boundary condton as follows [10] 1 I r w,s Ib rw I dω 2 r w,s n.s (5) rw s a poston vector for a pont on a dffusvely emttng and reflectng opaque surface and s surface emssvty. I s ncdent ntensty and n s outward surface normal. Usng Dscrete Ordnate method, equaton (4) s solved for a set of n varous drectons s, 12,,...n. A numercal quadrature replaces the ntegral over drecton, or [10] 4 Where n 2 f ( s )d w f ( s ) (6) w s the quadrature weght assocated wth drecton s. Therefore, the approxmaton of equaton (4) s done by a set of n equatons [10], di r,s Ib r Ir,s (7) ds Wth 12,,...n subject to the boundary condtons [10] 1 r n w w w b w j w j j j1 I r,s r I r w I r,s n.s IV. THE EAFT-1 WITH THE EXTENSION CASES Our contrbuton n ths paper s to ntroduce fve extended cases whch wll be used as a gudance to decde the element creaton drectly. The frst two cases are the normal cases as n SAFT whle the thrd, fourth and ffth cases are newly created one. These drect approach nvolved n the consderaton of trangular element creaton procedure depends on the condton of objects le wthn the crcle or at the crcum crcle. The objects here mean any actve nodes or actve edges. Ths s where the constructon of the trangular element wll be based on. Case 1: The frst case (Case 1) s the smplest one. Let say the current base edge beng consdered s (a,b), as shown n Fgure 4a. A crcle at centre IP wth radus followng the emprcal rule s constructed. If there s no object lyng wthn the crcle, the IP wll turn out to be the thrd vertex for the trangular element constructon. The base edge wll be the sde of the trangle. The new second and thrd sdes wll be (a,c) and (a,b). Snce all three sdes have equal length, the trangular element formed wll be an equlateral trangle element as llustrated n Fgure 4b. (8)
, July 6-8, 2011, London, U.K. Fg. 4a. No object le wthn the crcle Fg. 4b. An equlateral trangle s constructed Case 2: Let say the base edge beng consdered s (a,b) as shown n Fgure 5a to descrbe the constructon of a trangular element based on Case 2. A crcle at center IP wth radus followng the emprcal rule s constructed agan. There are two types of objects that le wthn the crcle. Frst s node d and the second type of object s two actve edges ntersected wth the crcle whch are edge (c,d) and edge (d,e). Among nodes and edges, ths category of extenson cases gves prorty to nodes when t comes to the selecton f both objects occur smultaneously wthn the crcle. If there are multple actve nodes n the crcle, calculate all the dstance of every actve node to the IP. The node whch s closest one to IP s selected as the thrd vertex for trangular element. As llustrated n Fgure 5b, node d s selected as the vertex, therefore, a trangular element s constructed wth edge (a,b), edge (a,d) and edge (d,b) as the three sdes. Fg. 5a. Two types of object le wthn the crcle whch are actve nodes and edges Fg. 5b. Node d s selected as the thrd vertex for trangular element Case 3: At the same tme, there s another trangular element beng constructed automatcally snce the three edges are connected to themselves. Ths condton belongs to Case 3. As llustrated n Fgure 5b, a trangular element s constructed automatcally when edge (b,d) s formed. The three edges (b,c), (c,d) and (b,d) are connected to themselves, and they become the sdes of the second trangular element. Case 4: Fgures 6a and 6b show the constructon of trangulaton element followng the fourth case (Case 4). Let say the base edge beng consdered s (a,b) where a crcle wth radus followng the emprcal rule and at centre IP had been constructed. Based on the fgure, t can be seen that the only object nvolved s an actve edge whch s ntersected wth the crcle. The actve edge ntersected wth the crcle s (b,c) as llustrated n Fgure 6a. The ntersecton pont of the crcum crcle wth the actve edge (b,c) has been marked as ponts p1 and p2. Based on the algorthm of EAFT, we need to determne the length between the ntersecton ponts. In other words, f referrng to Fgure 6a, we need to determne the length between pont p1 and pont p2. If the length s less than the radus of the crcle, therefore, the ntersected actve edge s dsregarded. Instead, the IP wll become the thrd vertex of the trangular element. Based on Fgure 6b, the IP has become node d n whch a trangle element s formed wth edges (a,b), (a,d) and (b,d) as the three sdes of the trangular element. Fg. 6a. An actve edge ntersected wth the crcle Fg. 6b. The IP become the new actve node for trangular element Case 5: Slghtly dfferent from Case 4, f the length between the ntersecton pont p1 and p2 s more than the length of the radus, the actve edge ntersected wth the crcle wll be consdered n whch the node of the actve edge wll be selected as the thrd vertex. In other words, referrng to Fgure 7a, snce the length of pont p1 and pont p2 s more than the length of the radus, therefore, node c, whch s the node assocated wth actve edge (b,c) wll be the vertex to form trangle element as llustrated n Fgure 7b. Ths condton belongs to Case 5. However, f there are multple edges ntersected wth the crcle and they are all satsfyng the condton of Case 5, therefore, the assocated node whch s the closest to IP wll be selected as the thrd vertex. Fg. 7a. The ntersected actve edge s (b,c). The length of ntersecton pont p1 and p2 wth the crcle s more than the length of radus Fg. 7b. The trangular element s constructed by connectng the node assocated wth the ntersected actve edge wth the base edge
, July 6-8, 2011, London, U.K. The general scheme of EAFT-1s demonstrated as n Fgure 8. It must be noted that most of the basc steps n EAFT-1 s stll followng the SAFT except that the constructon of the element s now based on fve extended cases. No Defne grd cell parameter at background grd and dscretsed boundary curve. Select shortest edge n the Front Construct element based on 5 cases Update the Front lst Empty front V. RESULTS Yes Fg. 8. The general scheme of Enhanced Advancng Front Technque -1. END Our approach s appled at a rectangular computatonal doman as llustrated n Fgure 9 wth ts boundary segment dscretzed nto a set of boundary edges. We use the default value for the sze parameter whch s fve percent from the length of the dagonal of the background grd. The grd generaton parameters on the background grd are specfed to be 0. 56 and s 1 at each pont of the background grd. The result of ntal unstructured mesh wth EAFT-1 s llustrated n Fgure 10. No re-order Front or erasng a few exstng element are needed durng the element creaton procedure snce our extenson cases offer a drect approach of decdng element creaton when the problem descrbe n secton II occur. The total number of element generated usng our approach s 72 and each of the trangular elements has been numbered from 1 to 72. In order to examne the qualty of the mesh, the EquSze Skew, Q whch s the measure of skewness s used where t s defned as [11] Q S eq S S eq In the above equaton, S s the area of the mesh elements, S s the maxmum area of equlateral trangle the eq crcumscrbng radus of whch s dentcal to that of the mesh element. The value of the examned qualty based on ths equaton provded by [11] s n the range of 0 1 where Q 0 f the element s equlateral Q EAS trangle and Q 1f the trangular element s degenerate (poorly shape). It must be noted that generally, hgh-qualty meshes poses average value of Q 01. for two dmensonal doman [11]. Fgure 11a-11c llustrate the correspondng trangular element n the mesh wth the range of the qualty examned. It can be seen that 83.33% of the elements are n the range of 0 Q 0. 1 whch can be consdered as a hgh qualty meshes. Therefore, the trangular elements generated usng our approach of extendng the normal cases to fve cases n the standard AFT for element generaton procedure prove to work well when there s not normal condtons occur as descrbed n secton II. In order to prove that the trangular mesh work well and s approprate for computatonal analyss, the governng equaton of radatve heat transfer has been ncorporated wth the mesh. Usng FLUENT verson 6.0, smulaton had been conducted to solve the radatve heat transfer by usng the Dscrete Ordnate Methods. Snce ths s a conceptual model, we mposed the value of the wall temperature (K) so that t can be used as boundary values n the calculaton. The smulaton s conducted untl convergence s reached. Fgure 12 shows the flue gas temperature dstrbuton contours between the walls respectvely. As shown n the smulaton result, the temperature dstrbuton of the flue gas s nonunform, whch wll further nfluence the heat transfer between the walls. (9) Fg. 9. Rectangular computatonal doman wth a set of boundary edges Fg. 11a. Element wth qualty 0 Q 0. 1 Fg. 10. The resulted unstructured trangular mesh.
, July 6-8, 2011, London, U.K. ACKNOWLEDGMENT The authors thanks the Mnstry of Hgher Educaton, Malaysa and Unverst Teknkal Malaysa Melaka (UTeM) for sponsorng the research work REFERENCES Fg. 11b. Element wth qualty 0. 2 Q 0. 3 Fg. 11c. Element wth qualty 0. 4 Q 0. 5 [1] P. R. M. Lyra and D. K. Carvalho, "A computatonal methodology for automatc two-dmensonal ansotropc mesh generaton and adaptaton," Journal of the Brazlan Socety of Mechancal Scences and Engneerng, vol. 28, 2006, pp. 399-412. [2] M. Farrashkhalvat and J. P. Mles, Basc structured grd generaton wth an ntroducton to unstructured grd generaton: Butterworth Henemann, 2003, ch.8. [3] R. Lohner, "Progress n Grd Generaton va the Advancng Front Technque," Engneerng wth Computers, 1996, pp. 186-210. [4] N. Kovac, S. Gotovac, and D. Poljak, "A New Front Updatng Soluton Appled to Some Engneerng Problems," Archves of Computatonal Methods n Engneerng, vol. 9, 2002, pp. 43-75. [5] J. Zhu, T. Blacker, and R. Smth, "Background overlay grd sze functons," n Proceedngs of the 11th nternatonal meshng roundtable, 2002, pp. 65 7. [6] E. Seveno, "Towards an adaptve advancng front method," n 6th Internatonal Meshng Roundtable, 1997, pp. 349 362. [7] J. Perare, J. Pero, and K. Morgan, "Advancng front grd generaton," n Handbook of Grd Generaton, N. P. Weatherll, B. K. Son, and J. F. Thompson, Eds.: CRC Press LLC, 1999. [8] G. Stefands, B. Merc, G. Heynderckx, and G. Marn, "Gray/nongray gas radaton modelng n steam cracker CFD calculatons," AIChE Journal, vol. 53, 2007, pp. 1658 1669. [9] H. K. Versteeg and W. Malalasekera, An Introducton to Computatonal Flud Dynamcs. England: Pearson Educaton, 2007, ch. 13. [10] M. F. Modest, Radatve Heat Transfer, 2 ed.: Academc Press, 2003. [11] "Gambt 2.3 Documentaton - Gambt User Gude." vol. 2010: FLUENT Incorporated. Fg. 12. The contour of flue gas temperature dstrbuton between walls. VI. CONCLUSION In the present work of ntal unstructured trangular mesh generaton usng EAFT-1, the algorthm stll uses SAFT method of specfyng the grd cell parameter at the background mesh n order to control the mesh sze as well as stll usng the emprcal rule to construct the crcle for element consderaton. The only dfferent s that we proposed the normal case n the SAFT to be extended to fve cases n order to encounter the problem of conflct or undecded element creaton durng the mesh generaton procedure. Our EAFT-1 approach of ncorporatng the extended cases of element creaton s able to tackle the problem of element creaton drectly by consderng the enttes that le wthn the crcle constructed. The examned results of ntal mesh proved that the approach s able to produce hgh qualty meshes. The smulaton results obtaned from the FLUENT also support our fndngs n EAFT-1 for effectvely approxmatng the radaton ntensty and temperature values n the two dmensonal doman. Zurada Abal Abas was born n Perak, Malaysa n 1981. She receved the BSc n Industral Mathematcs wth frst class honors from Unverst Teknolog Malaysa, Malaysa n 2002 and MSc n Operatonal Research from London School of Economcs, UK n 2003. Her current research nterests nclude mesh generaton, numercal methods and operatonal research technques. Shaharuddn Salleh receved the BA n Mathematcs from Calforna State Unversty, MSc. n Mathematcs from Portland State Unversty, Oregon and Ph.D n Computatonal Mathematcs from UTM. He s currently a Professor at Faculty of Scence, Unverst Teknolog Malaysa. Hs current research nterests nclude numercal modelngs and smulatons, graph theoretcal applcatons, moble computng and parallel algorthms. Hs has wrtten four books whch nclude three at the nternatonal level. There are also 65 techncal papers n varous journals and conferences at the natonal and nternatonal levels.