In this paper we discuss the automatic construction of. are Delaunay triangulations whose smallest angles are bounded and, in

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1 uomaic consrucion of qualiy nonobuse boundary Delaunay riangulaions Nancy Hischfeld and ara-ecilia Rivara Deparmen of ompuer Science, Universiy of hile, casilla 2777, Saniago, HILE bsrac. In his paper we discuss he auomaic consrucion of qualiy nonobuse boundary Delaunay riangulaions of polygons such as needed for conrol volume or nie elemen mehod applicaions. These are Delaunay riangulaions whose smalles angles are bounded and, in addiion, whose boundary riangles do no have obuse angles opposie o any boundary or inerface edge. The mehod we propose in his paper consiss on: (1) The consrucion of a consrained (good qualiy) Delaunay riangulaion of he polygon by using a Lepp-Delaunay algorihm (based on he longes-edge propagaion pah of arge riangles); (2) posprocess sep which eliminaes obuse angles by Delaunay inserion of a nie number of adequae poins on boundary or inerface edges. keywords. Nonobuse riangulaion, Delaunay meshes, Voronoi diagram, conrol volume discreizaion mehod, box mehod. 1 Inroducion The numerical soluion of parial dierenial equaions (pdes) is invaluable in design and opimizaion in many elds of engineering. The spaial discreizaion (mesh) of he srucure o be simulaed, i.e. is subdivision in cells, is key o he accuracy of he compued soluion. n appropriae mesh should fulll several requiremens. Firs, i mus provide a reasonable approximaion of he geomery o be modeled, in paricular of is boundary and inernal maerial inerfaces. Second, i is exremely imporan o accuraely approximae all inernal quaniies relevan o he soluion of he pdes. Third, each cell mus fulll cerain geomeric consrains imposed by he numerical inegraion Submied o a journal, Juni Technical Repor TR/D

2 mehod: if he pdes are solved wih he nie elemen mehod, no angle mus be smaller han some bound supplied a priori. However, if he equaions are solved using a conrol volume discreizaion mehod(cvm)[1], he cener of he smalles circumcircle ha surrounds each boundary elemen mus be inside he region of he elemen [2]. For wo dimensional geomeries (2-D) his means ha he angle opposie o a boundary edge mus be a nonobuse angle. The cvm is very popular in he numerical simulaion of semiconducor devices [1, 3, 4, 5]. In 2-D, boh riangulaions and mixed elemen meshes have been used. review of previous work on his area can be found in [6, 7]. more recenly approach is he one presened in [8] based on he sphere packing echnique [9]. ll hese approaches generae meshes wihou obuse angles. This paper presens a new algorihm o generae good qualiy 2-D meshes for boh conrol volume discreizaion and nie elemen mehod which exends he Lepp-Delaunay algorihm inroduced by Rivara in [10]. This kind of meshes are also very useful when problems are solved combining boh mehods. For example, urgler [3] uses he cvm mehod (voronoi diagram) o obain he numerical soluion of he Poisson equaion and he nie elemen mesh for grid adapion and error esimaion. This requires he combinaion of good qualiy meshes and well shaped Voronoi boxes. In paricular, he minimum angle should be bounded and boundary riangles should no have obuse angles opposie o any boundary edge or inerface edge. The mehod we propose in his paper, based on he use of longes-edge bisecion echniques [11], consiss on wo seps: (1) The consrucion of a good qualiy (consrained) Delaunay riangulaion (DT) of he polygon having inerior angles comprised beween 30 and 120 [10]; (2) posprocess sep which eliminaes boundary obuse riangles by combining longes-edge inserion poins, he Delaunay algorihm and a new reamen for a cerain ype of boundary obuse riangles. The consrucion of he good qualiy (consrained) Delaunay riangulaion consiss of: (a) The generaion of an iniial consrained Delaunay riangulaion (which essenially uses he polygon verices), and b) he use of an Lepp-Delaunay algorihm which improves he qualiy of he mesh so ha he minimum angle is greaer han or equal o 30. The basic Lepp-Delaunay improvemen sraegy uses he Longes-Edge Propagaion Pah of he arge riangles (o be eiher rened and/or improved in he mesh) in order o decide which is he bes poin o be insered, o produce a good-qualiy disribuion of poins. This sraegy is repeaedly used unil he arge riangle is desroyed. The sep designed o eliminae boundary obuse riangles of polygonal regions considers hree cases: (a) riangles wih only one boundary edge which is opposie o he obuse angle, (b) riangles wih wo boundary edges and one of hem opposie o he obuse angle, and (c) riangles wih hree boundary 2

3 edges. The case (a) is solved by insering he midpoin a he boundary edge. Since he obuse angle is smaller han or equal o 120, he inserion of only one poin is required. Some diagonal swapping migh be necessary. For he case (b), a boundary isosceles riangle of he larges edges equal o half he smalles boundary edge of he arge riangle is consruced. This consrucion can produce an obuse riangle wih one boundary edge, which is in urn eliminaed by he Delaunay inserion of he midpoin of is longes edge. This poin inserion can again produce a new boundary obuse riangle, wih larges angle smaller han he previous one and so on. The boundary obuse riangles are eliminaed afer he inserion of a nie number of poins. riangle wih hree boundary edges (case (c)) is a paricular case. Depending on he angles of he riangle, one or wo isosceles riangles wih wo boundary edges are generaed and a nie number of poins insered. For obuse riangles wih one inerface edge he same sraegy as for case (a) is applied. For riangles wih wo or more inerface edges adjacen o oher obuse riangles wih wo boundary edges, isosceles riangles for he new riangles which keep wo inerface edges are consruced. Since he boundary obuse riangles are adjacen, he number of poins insered on shared edges is bounded by he edge ha requires he highes number of poin inserions. The nal mesh (having no boundary obuse angles and wihou inerface obuse angles) is a Delaunay riangulaion even for he riangles lying a he boundary and inerface. 2 asic conceps and deniions Deniion 1 boundary riangle is any riangle ha has a leas one edge on he geomeric boundary or on a maerial inerface (boundary edge). Deniion 2 boundary obuse riangle is any riangle ha has a boundary edge opposie o is obuse angle. Deniion 3 n inerface obuse riangle is any riangle ha has an inerface edge opposie o is obuse angle. Deniion 4 n 1-edge boundary riangle is any riangle ha has exacly one boundary or inerface edge Deniion 5 n 2-edge boundary riangle is any riangle ha has exacly wo boundary or inerface edges. Deniion 6 boundary consrained angle is an angle ha is dened by wo boundary edges. This angle can no be modied. 3

4 Deniion 7 Le P be a polygon wih maerial inerfaces. essellaion T of P is appropriae for he V [1, 2] (well-shaped) if (i) T is a Delaunay essellaion, (ii) The cener of he circumcircle (Voronoi Poin) ha surrounds each boundary riangle lies in he same polygon as he boundary riangle. p j v p j e ij p i p i p k b j (a) (b) Figure 1: 2-D Delaunay riangulaions and heir Voronoi diagrams. Figure (a) shows an accepable riangulaion for he cvm, and Figure (b) shows an unaccepable riangulaion (he Voronoi poin v is ouside he boundary obuse riangle dened by p j ; p k ; p i, where p j p k is a boundary edge) Theorem 1 (Thales) Le be a riangle dened by he verices,, and. If he riangle lies on a circumcircle so ha one of is edges is equal o he diameer of he circumcircle, hen he riangle is a righ riangle (Figure 2(a)). The Thales heorem can be also inerpreed in he following way: If he circle wih cener in he midpoin of and diameer includes he poin on is boundary, he angle is a righ angle and he cener of he circumcircle of he riangle ( 1 ) coincides wih (Figure 2(a)); if he poin is ouside his circle, is a nonobuse angle and he cener of he circumcircle of he riangle ( 1 ) is inside he riangle (Figure 2(b)) and if he poin is inside he circle, is an obuse angle and he cener of he circumcircle of he riangle ( 1 ) is ouside he riangle (Figure 2(c)). This analysis will be used laer o show if a paricular Voronoi poin 1 is inside or ouside a riangle. 3 Preliminary conceps: he Lepp() and geomerical properies This secion reviews he Lepp concep [10] and summarizes some geomerical properies inroduced in [12]: 4

5 α α α, 1 (a) (b) (c) Figure 2: (a) Thales heorem: = 90, (b) ener ( 1 ) of he circumcircle ha surrounds is inside he riangle (c) ener ( 1 ) of he circumcircle ha surrounds is ouside he riangle ( > 90 ) Deniion 8 For any riangle 0 of any conforming riangulaion T, he Longes- Edge Propagaion Pah of 0 will be he ordered lis of all he riangles 0, 1, 2,... n 1, n, such ha i is he neighbor riangle of i 1 by he longes edge of i 1, for i = 1,2,.., n. In addiion we shall denoe i as he Lepp( 0 ). Proposiion 1 For any riangle 0 of any conforming riangulaion of any bounded 2-dimensional geomery, he following properies hold: (a) for any, he Lepp() is always nie; (b) The riangles 0, 1,..., n 1, have sricly increasing longes edge (if n > 1); (c) For he riangle n of he Longes-Edge Propagaion Pah of any riangle 0, i holds ha eiher: (i) n has is longes edge along he boundary, and his is greaer han he longes edge of n 1, or (ii) n and n 1 share he same common longes edge. Deniion 9 Two adjacen riangles (, *) will be called a pair of erminal riangles if hey share heir respecive (common) longes edge. In addiion will be a erminal boundary riangle if is longes edge lies along a boundary side. Noe ha he Longes-Edge Propagaion Pah of any riangle corresponds o an associaed polygon, which in cerain sense measures he local qualiy of he curren poin disribuion induced by. To illusrae hese ideas, see Figure 7(a), where he Longes-Edge Propagaion Pah of 0 corresponds o he ordered lis of riangles ( 0, 1, 2, 3 ). oreover he pair ( 2, 3 ) is a pair of erminal riangles. The deniion 8 should be slighly modied o consider he case where he longes edge is no unique. In such a case, he longes edge ha produces he shores pah should be seleced. 5

6 Deniion 10 For any inpu PLSG (planar sraighline graph) ha denes a general polygon o be riangulaed, he geodesic disance beween wo poins of he polygon is dened as he shores pah ha says wihin he inerior of he polygon. In addiion poins P and Q will be called geodesic inerior poins if he geodesic disance beween boh poins is equal o he shores Euclidean disance beween poins P and Q (see Figure 3). Oherwise hey will be geodesic exerior poins. P Q R Figure 3: Poins P and Q (P and R) are geodesic inerior (exerior) poins. Deniion 11 For any inpu PLSG which denes a general polygon o be riangulaed, hree poins ; ; conribue o a valid (Delaunay consrained) riangle in a DT if (a) The verices,,, are geodesic inerior poins beween hem; and (b) The circumcircle hrough he poins,, conains no oher geodesic inerior poin (wih respec o he poins,,,) in is inerior (see Figure 4). P Figure 4: Triangle is a valid Delaunay riangle 6

7 Proposiion 2 Le be any riangle = (; ; ) of longes-edge. Then for any neighbor riangle ha shares side wih, he pair (; ) forms a pair of Delaunay erminal riangles if and only if he hird verex of belongs o he region R = \, where ; ; y are circles of radius equal o he lengh of and respecive ceners ; and ; and is he circumcircle of riangle. Proof. I follows from he fac ha he pair (,*) is a pair of erminal riangles over a Delaunay riangulaion and he condiions ha such a pair of riangles hold.2 Figure 5 illusraes hree dieren cases of regions R. deailed proof can be found in [12]. r T Q P R S r P R S Q T r/2 R (a) (b) (c) r Figure 5: Geomerical place for he 4h verex (denoed by region R) in a pair of erminal riangles The following Theorem [12] saes he geomerical condiions which assure ha (; ) is a pair of Delaunay erminal riangles. In his case he hird verex of mus belong o R 6=. Theorem 2 Le be any DT. Then for any pair of Delaunay erminal riangles (; ) in, he following propery holds: is an obuse riangle if and only if he disance d beween he circumcener P of and he longes-edge of saises ha 0 < d r, where r is he circumradius of. 2 Proof. The resul follows by nding he limi case where R reduces o one poin ( inersecs only in one poin), which holds for d=r/2 (See Figure 5(c)).2 orollary 3 For any pair of Delaunay erminal riangles (; ), is an obuse riangle if and only if is larges angle holds ha 120 ; and is an acueangled riangle. 7

8 4 Lepp-Delaunay improvemen riangulaion algorihm and properies In his secion we use an improved version of he Lepp-Delaunay algorihm (inroduced in [10]) which allows he qualiy improvemen of any riangulaion in he sense ha a minimum angle of 30 is obained for any angle ha is a non-boundary consrained riangle. The basic ackward-le-delaunay improvemen procedure uses he Longes- Edge Propagaion Pah of he arge riangles (o be eiher rened and/or improved in he mesh) in order o decide which is he bes poin o be insered, in order o produce a good-qualiy disribuion of poins. This procedure is repeaedly used unil he riangle is desroyed. Noe ha his basic algorihm does no consider he fac ha could be a boundary riangle. asic ackward-le-delaunay-improvemen (, T)f while remains wihou being modified do Find he Longes-Edge Propagaion Pah of Perform a Delaunay inserion of he poin p (midpoin of he longes edge of he las riangle in he Lepp()) end while g Figure 6: ackward-le-delaunay improvemen procedure We have used he word improvemen insead of bisecion or renemen. This is o explici he fac ha one sep of he procedure does no necessarily produce a smaller riangle. ore imporan however, is he fac ha he procedure improves he riangle in he sense of Theorem 4. The proof of his heorem can be found in [10]. Theorem 4 For any Delaunay riangulaion T, he repeiive use of he ackward- LE-Delaunay-Improvemen echnique over he wors riangles of he mesh wih smalles angle < 30 produces a qualiy riangulaion of smalles angles greaer han or equal o 30. orollary 5 The use of Lepp-polygon riangulaion algorihm wih " = 30 produces a Delaunay riangulaion such ha obuse riangles have angles smaller han or equal o 120. For an illusraion of his idea see Figure 7 where he riangulaion (a) is his iniial Delaunay riangulaion wih Lepp( 0 ) = 0 ; 1 ; 2 ; 3, and he riangulaion (b), (c) and (d) illusrae he complee sequence of poin inserions 8

9 needed o improve 0. In his example, he improvemen (modicaion) of 0 implies he auomaic Delaunay inserion of hree addiional Seiner poins. Each one of hese poins is he midpoin of he las riangle of he curren Lepp( 0 ). I should be poined ou here ha each Delaunay poin inserion essenially improves he local poin disribuion in he curren Lepp( 0 ) a) b) c) d) Figure 7: ackward Longes-Edge Delaunay improvemen of riangle 0 (a) (b) (c) Figure 8: oundary reamen echnique y combining he basic Lepp-procedure and adequae boundary consideraions, a simple 2-dimensional qualiy-riangulaion algorihm is obained. The special boundary reamen echnique is o avoid he inserion of undesirable inerior poins. To illusrae his idea consider he simple example of Figure 8(a). In his case he naive use of he Lepp poin inserion algorihm would produce undesirable inerior poins (as shown in Figure 8(b)). The Lepp-improvemen algorihm including he special boundary reamen can be formulaed as shown in Figure 9. 9

10 g Qualiy-Polygon-Triangulaion ( P, ) f Inpu: general polygon P (defined by a se of verices and edges); and a olerance parameer ( < 30^) onsruc T, a consrained (boundary) Delaunay riangulaion of P. Find S, he se of he wors riangles of T (of smalles for angle < ) each in S do ackward-le-delaunay-improvemen (T, ) Updae he se S (by adding he new small-angled riangles and eliminaing hose desroyed hroughou he process) end for g ackward-le-delaunay-improvemen (T, ) f while remains wihou being modified do if (* has a boundary edge l, and l is no he smalles edge of,) selec p, he midpoin of l else Find he Lepp(), and * he las riangle in he Lepp() selec p midpoin of he longes edge of * end if Perform he Delaunay inserion of p end while Figure 9: Lepp-procedure wih boundary consideraions Noe ha: (1) is a hreshold parameer less han or equal o 30 ha can be easily adjused; (2) in pracice we have worked wih a consrained Delaunay riangulaion of he 2-dimensional geomery (hew, 1989); (3) he qualiy-riangulaion algorihm mainains and processes he riangles of he se S in any order. 10

11 5 Nonobuse boundary Delaunay riangulaions In his secion we shall show ha by using a posprocess sep over he qualiy mesh generaed wih he Lepp-polygon riangulaion algorihm described in he previous secion (wih an " = 30 ), he unaccepable riangles having obuse angles opposie o a boundary or inerface edge are eliminaed. Furhermore, we shall show ha he resuling riangulaion is a Delaunay riangulaion and no a consrained Delaunay riangulaion. 5.1 Non-obuse boundary riangles for polygons wihou inerfaces oundary obuse riangles wih one, wo or hree boundary edges require differen sraegies o eliminae he obuse angle Triangles wih one boundary edge Theorem 6 Le be any improved Delaunay riangulaion of any PSLG geomery (wih smalles angle greaer han or equal o 30 ). Le be a boundary obuse riangle of and e he unique boundary edge of. Then (a) he obuse riangle is eliminaed by insering he midpoin of e and (b) he new generaed boundary riangles are nonobuse riangles. Proof. Le be any boundary obuse riangle of of verices,,p where is he unique boundary edge of. In order o prove par (a) of he heorem consider he exreme case of he isosceles obuse riangle of longes edge equal o, larges angle equal o 120 and smalles angles equal o 30 ha resric he geomery of (See Figure 10(a)). In eec he verex P of mus belong o he region limied by he prolongaion of he edges and and he circle of diameer. We shall show ha for he exreme riangle, (1) he inserion of he midpoin of generaes wo nonobuse boundary riangles and and (2) he Delaunay poin inserion sep (diagonal swapping) does no inroduce obuse angles of verex (riangles and are boundary nonobuse riangles). Le suppose ha riangle has an obuse angle of verex. In such a case he poin of his riangle should be inside of he circle wih cener 1 and diameer. However, his is no possible because according o heorem 2, for he specic riangle, he disance beween and is r=2 (where r is he radius of he circumcircle ha surrounds ) and he radius of he circles wih ceners 1 and 2 is less han r. Thus, region DE does no inersec hese circles, which implies ha he circles do no conain he verex P of, for any valid verex P. 11

12 To proceed wih he second par of he proof consider he region in Figure 10(b) which idenies he locaion of a verex D so ha a diagonal swapping is required afer he inserion of he poin. The diagonal swapping generaes wo new riangles where one of hose, he riangle D, is a boundary riangle. Since he smalles angle of riangle D is 30, he new boundary riangle is nonobuse because he circle wih diameer does no include D. The shores disance beween D and he edge is r and he circle wih cener 2 has a radius less han r.2 D E Ω D r/2 r r (a) (b) Figure 10: (a)the shadow region shows he possible locaion of he P verex so ha P is an obuse riangle on P, (b) Diagonal inerchange ( o D) does no produce a new boundary obuse riangle orollary 7 For any improved Delaunay riangulaion of any PLSG geomery (wihou inerfaces) he number of poin inserions (N 1b ) required o eliminae N 1-edge boundary obuse riangles is equal o N Triangles wih wo boundary edges The eliminaion of 2-edge boundary obuse riangles can be divided ino wo cases: 1. The smalles edge and he longes edge of he riangle are boundary edges (edges and in Figure 11). Noe ha in his case, he boundary consrained angle is wih 30 (The angle of verex is smaller han or equal o he angle of verex ). The sraegy of he previous secion also applies o his case because he inserion of he midpoin 12

13 does no creae a new boundary obuse riangle. In addiion, noice ha mus be greaer han or equal o 30 because if is less han 30, would be less han 30 oo and hen, he Lepp improvemen procedure would no have nished ye. D E β γ α β >= α α >= 30 r/2 r Figure 11: Region of valid poins for 2. The smalles edge is an inerior edge. In his case, he boundary consrained angle mus be less han. We can no apply he same sraegy as for he 1-edge boundary obuse riangles because afer wo applicaions of he sraegy, a new riangle similar o he original one will be obained, as shown in Figure 12 ( o is similar o 4 ). One addiional problem is ha because of he boundary resricions he minimum angle of his riangle can be less han 30 and consequenly, he obuse angle can be greaer han 120. o Figure 12: o is similar o 4 The essenial ideas of he algorihm o handle case 2 are he followings: n 2-edge boundary isosceles riangle of he larges edges equal o half he smalles boundary edge of he arge riangle is consruced (Figure 13(b)). This consrucion can produce an 1-edge boundary obuse riangle 1, which 13

14 is in urn eliminaed by he Delaunay inserion of he midpoin of he longes edge of 1 (Figure 13(c)). This can again produce a new boundary obuse riangle 1, wih larges angle smaller han he previous one and so on. The boundary obuse riangles are eliminaed afer he inserion of a nie number of poins. Noe however ha, since he boundary consrained angle can be less 1 (a) (b) 1 (c) (d) Figure 13: Eliminaion of 2-edge boundary obuse riangles han 30, some 1-edge boundary obuse riangles wih obuse angle greaer han 120 can be produced. To illusrae see Figure 13. The algorihm o handle 2-edge boundary obuse riangles where he smalles edge is an inerior edge can be schemaically described as shown in Figure 14. Theorem 8 Le be a 2-edge boundary obuse riangle wih inerior smalles edge. (1) If he angle of verex is greaer han 0 (where 0 is a consan o be deermined laer), he obuse angle is eliminaed by inserion of exacly wo poins by creaing a 2-edge boundary isosceles riangle (2) If he angle of verex is less han 0 as shown in Figure 16, an isosceles riangle is creaed as in poin (1) and if 1-edge boundary obuse riangles are generaed, hey are eliminaed by insering a number of poins bounded by i N. Proof. In order o eliminae 2-edge boundary obuse riangles wih inerior smalles edge ( is he smalles angle of he riangle), we build a 2-edge boundary isosceles riangle as shown in Figure 15. The consrucion of an isosceles riangle avoids he propagaion of obuse angles opposie o a boundary edge. The inserion of only wo poins is required o eliminae he obuse angle if he angle is greaer han or equal o 0 because in his case 1 90 (see Figure 15). The value of 0 ha produces 1 = 90 can be found by using he isosceles properies of riangle N and he cosines heorem. Thus, he following hree equaions ha relae and are obained. They allow o compue giving values o. 14

15 Inpu: is a 2-edge boundary obuse riangle wih smalles inerior angle and T is he curren riangulaion ompue he midpoin of he smalles boundary edge of ( See Figure 15) ompue he poin N so ha he lengh of segmen is equal o he lengh of segmen N Perform he Delaunay inserion of N and (This reduces o join poins N and j; and poins N and ) S = if riangle 1 of verices N is a 1-edge boundary obuse angle S = f 1 g end if while S is no empy Ge one of he riangles of S Perform he Delaunay inserion of he longes edge midpoin of if a new riangle 1 is an 1-edge boundary obuse riangle S = S U f 1g end if end while Figure 14: lgorihm o eliminae 2-edge boundary obuse riangles m 2 = 2d 2 2d 2 cos() x 2 = m 2 + d 2 2md cos( ) m 2 = d 2 + x 2 2dx cos() In addiion, since mus be greaer han, he condiion = 0 is imposed o compue he maximum value of 1 for any riangle ha saises he condiions of his heorem and has a boundary consrained angle 0. The following relaion is hus obained: 15

16 1 = o 90 Numerically, we obain ha if o is greaer han 32:54, is greaer or equal o 24:93, and hen 1 is less han 90. D E d o d m β0 d N δ γ x d 1 α α > β 0 β 0 > δ > γ 1 < 90 Figure 15: inimum value of so ha i is no necessary o inser addiional poins In case 0 as shown in Figure 16(a), he new 1-edge boundary riangle N migh be obuse. If his riangle is a boundary obuse riangle, a boundary edge midpoin is insered and he Delaunay crieria is applied. fer his inserion, new 1-edge boundary obuse riangles migh appear. n upper bound of he number of poin inserions can be obained if we consider ha he poin inserions nish when he boundary edge is he smalles edge of he 1- edge boundary riangles. Since he smalles edge of he quadriaeral N 0 is N 0, he number of poin inserions on each boundary edge is bounded by d 0 e.2 0 N orollary 9 The number of poins insered (V 2b ) o eliminae N 2-edge boundary obuse riangles wih smalles inerior edge is: V 2b ( j ) 2N + 2 NX j=1 j 0j d e 0j N j where j is he 2-edge boundary obuse riangle j. 5.2 Nonobuse boundary for PLSG geomeries oundary obuse angles opposie o a polygon inerface as shown in Figure 17 can be handled in he same way as 1-edge boundary obuse riangles. nalogously o his previous case, he inserion of he midpoin of he inerface 16

17 D E D E D E r/2 30 o 120 γ β α βο 30 β α βο N N r/2 r/2 o 120 γ N i βi r r r (a) (b) (c) Figure 16: oundary obuse riangle wih a consrained angle less han 0 (common) edge desroys boh obuse angles and does no generae obuse angles opposie o he inerface edges. orollary 10 The number of verices (V 1i ) insered o eliminae N 1-edge inerface obuse riangles is bounded as follows: N V 2 1i N D 1 2 Figure 17: Obuse angles opposie o a maerial inerface Figure 18 illusraes he more complex case arising when inerfaces whih several inerface edges converge o a common verex. In his case we eliminae he obuse angles by insering he midpoin of he smalles inerface edge and a poin N j on each edge j so ha he disance beween N j and is 17

18 equal o he disance beween and. Thus, isosceles riangles are generaed around. We hen eliminae he 1-edge boundary obuse riangles using he same eliminaion sraegy applied o 1-edge boundary obuse riangles shown in Figure 14. Since he boundary obuse riangles are adjacen, he number of poins insered on shared edges is dened by he riangle ha requires he highes number of poin inserions. The previous sraegy is also applied if some of he riangles of he group of adjacen 2-edge boundary riangles are nonobuse riangles. Oherwise, he inserion of poins o desroy only he 2-edge boundary obuse angles of he group can produce new 2-edge boundary obuse riangles in he adjacen riangles ha were 2-edge boundary nonobuse riangles. orollary 11 The number of verices (V 2a ) insered o eliminae N convergen boundary obuse riangles is: NV (j) = min( jn j ; j+1 N j+1 ) d e; 1 j N N j+1 N j V 2a () <= N (N + 1) max (NV (j); NV (j + 1)) 1j<N N 4 N N N 1 1 Figure 18: djacen boundary obuse riangles Proposiion 3 The number of poin inserions o eliminae N boundary obuse angles is O(N). Proof. Le be N 1b he number of 1-edge boundary obuse riangles, N 1i he number of -edge inerface obuse riangles, N 2b he number of isolaed 2-edge boundary obuse riangles and N 2a he number of nodes ha concenraes 18

19 adjacen 2-edge boundary obuse riangles. The oal number of insered poins V is: V <= N 1b + N 1i + XN 2b j=1 V 2b ( j ) + NX 2a j=1 V 2a ( j ) Le be N k he number of riangles associaed o he node k and o be he 2-edge boundary obuse riangle ha requires he highes number of poin inserions o eliminae is obuse angle. In order o idenify o, we consider each 2-edge boundary obuse riangle independenly. Then, he previous expression can be bound as follows: V <= N 1b + N 1i + (N 2b + NX 2a j=1 N k )V 2b ( o ) = O(N) orollary 12 Nonobuse boundary and inerfaces riangles => Delaunay riangulaions. 6 Examples This secion discusses he resuls obained by applying he algorihm presened in his paper o several es examples wih dieren geomerical complexiy. To illusrae he pracical behavior of he algorihm, four es problems of dieren geomerical complexiy have been considered: he righ angled spiral of Figure 19(a); he srip geomery wih "inerior" inerface edge of Figure 20(a), he wo circle polygon wih addiional inerior inerface edges of Figure 21(a) and he polygon wih several consrained angles of Figure 22(a). Tables 1, 2, 3 and 4 summarize he geomerical informaion of he meshes generaed hroughou he auomaic improvemen process. Each able conains informaion abou he number of verices (verices), he number of riangles (riangles), he minimum angle (min. angle), he average value of he minimum angles, he maximum angle (max. angle), he average value of he maximum angles and he number of boundary obuse riangles (b-obuse riangles) ha sill remains afer applying a Delaunay algorihm (Delaunay), afer applying he Lepp-Delaunay sraegy (Lepp-Delaunay) and afer applying he sraegy o eliminae boundary obuse angles (Final mesh). In paricular, when he mesh has 2-edge boundary obuse riangles wih smalles inerior edge (riangles whose qualiy can be only parially improved by he Lepp-improvemen procedure because of heir boundary consrained angles), he rows ha give angle informaion conain wo values: he lef one corresponds o he se of Lepp-improvable riangles and he righ one considers he se of riangles wih boundary consrained angles. 19

20 (a) (b) (c) (d) Figure 19: Example 1 Some riangles wih angles less han 30 can be inroduced while eliminaing 2-edge boundary obuse riangles wih smalles inerior edge. They are locaed in he neighborhood of he original boundary obuse riangles. The number of riangles wih minimum angle less han 30 is shown able 3 and 4 close o he number of riangles of he nal mesh. For example, in example 3, he number of riangles wih boundary consrained angle less han 30 are 4 and he number of riangles wih angle less han 30 generaed while eliminaing he 2-edge boundary obuse riangles are 16. The number of involved riangles depends on he number of poin inserions and on he number of diagonal swapping made o eliminae he boundary obuse angle. 20

21 Example 1 Delaunay Lepp-Del Final mesh verices riangles min. angle aver. min. angle max. angle aver. max. angle b-obuse riangles Table 1: Saisical informaion for he example 1 (Figure 19) (a) (b) (c) (d) Figure 20: Example 2 Example 2 Delaunay Lepp-Del. Final mesh verices riangles min. angle aver. min. angle max. angle aver. max. angle b-obuse riangles Table 2: Saisical informaion for he example 2 (Figure 20) 21

22 (a) (b) (c) (d) Figure 21: Example 3 Example 3 Delaunay Lepp-Del Final mesh verices riangles (16,4) min. angle aver. min. angle max. angle aver. max. angle b-obuse riangles Table 3: Saisical informaion for he example 3 (Figure 21) 22

23 (a) (b) (c) (d) Figure 22: Example 4 Example 4 Delaunay Lepp-Del. Final mesh verices riangles (5,4) min. angle aver. min. angle max. angle aver. max. angle b-obuse riangles Table 4: Saisical informaion for he example 4 (Figure 22) 23

24 Tabla 5 compares he heoreically expeced number of poin inserions of he posprocess algorihm o eliminae boundary obuse angles wih he number of poin inserions obained in pracice. The able shows ha he implemened algorihm conrm he expeced heoreical resuls. Number of poin inserions during he eliminaion of boundary obuse riangles N 1b N 1i N 2b expeced insered Example (or. 7) 28 Example N 21 (or. 10) 17 Example (Prop. 3) 19 Example (Prop. 3) 12 Table 5: Number of poin inserions while eliminaing boundary obuse angles 7 onclusions In his paper we presen a new auomaic algorihm o generae good qualiy meshes for he conrol volume discreizaion and he nie elemen mehods. The resuling riangulaions are qualiy Delaunay riangulaions, whose boundary riangles do no have obuse angles opposie o boundary or inerface edges. The algorihm consiss of wo seps: (1) The generaion of good qualiy consrained Delaunay riangulaion. The qualiy of any mesh is improved using he Lepp-Delaunay sraegy: he angles are bounded by 30 and 120. In pracice, he 2-dimensional riangulaions obained is size-opimal [13]. The use of his improvemen echnique simplies very much he nex sep. In addiion, he qualiy mesh has very few boundary obuse angles. (2) The eliminaion of boundary obuse riangles. The posprocess o eliminae boundary obuse riangles inroduces a linear number of poins wih respec o he number of boundary obuse riangles. For meshes whose domain geomery does no have boundary consrained angles less han 32:54, he number of insered poins is bounded by he number of boundary obuse riangles. Oherwise, he posprocess insers a nie number of poins ha is proporional o he number of boundary obuse riangles. The posprocess ha eliminaes boundary obuse angles guaranees ha: (1) if afer he Lepp improvemen algorihm, he mesh has only 1-edge boundary or inerface obuse riangles, he angles of he riangulaion are bounded by 30 and 120. (2) If he mesh has 2-edge boundary obuse riangle wih boundary consrained angles greaer han 0 = 32:54, he angles of he riangulaion are also bounded by 30 and 120 excep in a number of riangles 24

25 equal o he number of 2-edge boundary obuse riangles. (3) For meshes wih any ype of boundary obuse riangles, mos of he angles are bounded by 30 and 120. The few riangles ha are no bounded are in he neighborhood of he 2-edge boundary obuse riangles. 8 cknowledgmen The rs auhor hanks o Norber Srecker for a valuable ineracion abou he subjec. The programming of he algorihm was done by auricio Palma. This work was suppored by Fondecy projec No and Fondap N-1 projec. References [1]. R. Pino. omprehensive Semiconducor Device Simulaion for Silicon ULSI. PhD hesis, Sanford Universiy, [2] N. Hischfeld, P. oni, and W. Fichner. ixed Elemens Trees: Generalizaion of odied Ocrees for he Generaion of eshes for he Simulaion of omplex 3-D Semiconducor Devices. IEEE Trans. on D/IS, 12:1714{1725, November [3] J. F. urgler. Discreizaion and Grid dapaion in Semiconducor Device odeling. PhD hesis, ETH Zurich, published by Harung-Gorre Verlag, Konsanz, Germany. [4] G. Heiser. Design and Implemenaion of a Three Dimensional, General Purpose Semiconducor Device Simulaor. PhD hesis, ETH Zurich, published by Harung-Gorre Verlag, Konsanz, Germany. [5] Sephan uller. n objec-oriened approach o mulidimensional semiconducor device simulaion. PhD hesis, Swiss Federal Insiue of Technology, [6] N. Hischfeld. Grid Generaion for Three-dimensional Non-Recangular Semiconducor Devices. PhD hesis, ETH Zurich, Series in icroelecronics, Vol. 21, PhD hesis published by Harung-Gorre Verlag, Konsanz, Germany. [7] G. Garreon, L. Villablanca, N. Srecker, and W. Fichner. new approach for 2-d mesh generaion for complex device srucures. In NUPD V - Technical Diges, Honolulu, US, June

26 [8] Gary L. iller, Dafna Talmor, Shang-Hua Teng, Noel Walkingon, and Han Wang. onrol volume meshes using sphere packing: generaion, renemen and coarsening. In Proceedings of he 5h Inernaional meshing Roundable, pages 47{61, Pisburgh, Pennsylvania, [9] Jim Rupper arshall ern, Sco ichell. Linear nonobuse riangulaion of polygons. In Proc. 10h annu. sympos. compuaional geomery, pages 231{241, S.Louis, [10].. Rivara. New longes-edge algorihms for he renemen and/or improvemen of unsrucured riangulaions. Inernaional journal for numerical mehods in Engineering, 40:3313{3324, [11].. Rivara and G. Iribarren. The 4-riangles longes-side Pariion of Triangles and linear Renemen lgorihms. ahemahics of ompuaion, 65(216):1485{1501, ocober [12].. Rivara and N. Hischfeld. Geomerical properies of he leppdelaunay algorihms for he qualiy riangulaion problem. Deparmen of ompuer Science, U. de hile, [13] J Rupper. delaunay renemen algorihm for qualiy 2-d mesh generaion. Journal of algorihms, 18:548{585,