Subdivision of Boundary Cells in 3DChristoph P aumInstitut f�ur Angewandte Mathematik und Statistik,Universit�at W�urzburge-mail: pflaum�mathematik.uni-wuerzburg.de

Abstra tUsing a suitable grid generator, it is possible to approximate generaldomains in 3D { with the ex eption of slit domains { by a pure tensorprodu t grid in the interior of the domain and 12 types of boundary ells.In this paper, we des ribe how to subdivide these boundary ells by tetra-hedral su h that no large interior angles appear and su h that it is possibleto onstru t a ontinuous �nite element spa e on su h a grid.1 Introdu tionNumeri al algorithms for the dis retization of partial di�erential equations an beimplemented more eÆ ient on tensor produ t grids than on unstru tured grids.There are several reasons for this. One is the ompli ated data stru ture in aseof unstru tured grids. Another reason is that a �nite element dis retization ona tensor produ t grid an be des ribed by simple sten ils as the �nite di�eren edis retization (see [2℄). On the other hand, it is not possible to dis retize generaldomains by a pure tensor produ t grid. Therefore, optimal dis retization gridsare omposite grids, whi h mainly onsist of a tensor produ t grid, but whi huse an unstru tured grid near the boundary or between two pat hes of a tensorprodu t grid (see [3℄). More referen es on automati grid generation an be foundin [3℄.In this paper, we use a tensor produ t grid in the interior of the domain. Toobtain a minimal unstru tured grid near the boundary, the unstru tured grid is ompletely ontained in boundary ells. These boundary ells must be dividedby several tetrahedral su h that� the �nite element spa e onsists of ontinuous fun tions,� the maximal interior angle 'edge between two edges of a tetrahedral isbounded by 180Æ, and 1

� the maximal interior angle 'fa e between two fa es of a tetrahedral is boundedby 180Æ.See [1℄ for more details on �nite element spa es with no large interior angles. The on ept of marking ells by interior and exterior ells an be found in [4℄.To redu e the number of types of boundary ells, we use a spe ial grid gen-erator. In a subsequent paper, we dis uss in detail the approximation propertiesof �nite element spa es obtained by this grid generator. In this paper, we onlyexplain the subdivision of all types of boundary ells. Due to the grid generatorthere exist only 12 types of boundary ells. The grid generator an be applied togeneral domains in 3D with the ex eption of slit domains. The maximal interiorangels an be estimated by (see Theorem 2):'edge � 144Æ and 'fa e � 163Æ:The grid generator and the subdivision of boundary ells is implemented in thelibrary EXPDE:http://ifamus.mathematik.uni-wuerzburg.de/�expde/index.html .

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2 Boundary ells in 2DBefore we an study the 3D ase, we have to analyze boundary ells in 2D.Figure 1 shows all types of boundary ells whi h appear in 2D. Boundary ellsas in Figure 2, are so alled boundary ells of \double type".type a) type b) type ) type d)Figure 1: Di�erent types of boundary ells in 2D

Figure 2: Boundary ell with double type.Using a suitable grid generator, it is possible to avoid boundary ells of typed). All other boundary ells an be divided by triangles in the following way,where is the dis retization domain:Type a) ells: Let be a type a) boundary ell. Then, the domain \ an be approxi-mated by one triangle �h. The interior anglesof this triangle are smaller or equal 90Æ.Maximal interior angle: 'edge � 90Æ �h

�3

Type b) ells: Let be a type b) boundary ell. Then, the domain \ an be approx-imated by two triangles �h;1 and �h;2. Obvi-ously, the interior angles of triangle �h;1 aresmaller or equal 90Æ. The interior angles oftriangle �h;2 are smaller or equal 135Æ. Letus prove this. Obviously, it is 45Æ � �3 � 90Æand �1 � 60Æ. This implies�2 � 180Æ � �1 � �3 � 135Æ.Maximal interior angle: 'edge � 135Æ �h;1 �h;2�1 �2�3 �Type ) ells: Let be a type ) ell. Then,the domain \ an be approximated bythree triangles �h;1, �h;2 and �h;3. The inte-rior angles of the triangles �h;1 and �h;3 aresmaller or equal 90Æ. The interior angles ofthe triangle �h;2 are smaller or equal 135Æ.Let us prove this. Obviously, it is �3 � 90Æ.Furthermore, observe that �1 � 45Æ. Thisimplies �1 � 180Æ � �1 � 135Æ. Analogously,we obtain �2 � 135Æ.Maximal interior angle: 'edge � 135Æ �h;1 �h;2�h;3�1 �2�3�1 �2�

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3 Subdivision of boundary ells in 3D3.1 Classi� ation of boundary ellsThe fa es of a boundary ell in 3D are 2D ells of type a), b), ), d) as in se tion2 or a square. By introdu ing additional edges, it is possible to avoid boundary ells with fa es of type d). Su h a grid generator will be explained in more detailin a subsequent paper.Theorem 1 (Classi� ation of boundary ells)There exist exa tly 12 boundary ells with fa es of type a), b), ) or a square.These boundary ells an be lassi�ed in the following way:1. There are 5 boundary ells, su h that at least one fa e is a square.2. There are 5 boundary ells, su h that no fa e is a square and at least onefa e is of type ).3. There is 1 boundary ell, su h that no fa e is a square or type ) and atleast one fa e is of type a) and at least one of type b).4. There is 1 boundary ell, su h that all fa es are of type a).Idea of the proof: For proving this theorem, one has to study whi h ver-ti al edges of the boundary ell are ontained in the domain, where the fa e ofmaximal type is the bottom or top fa e (see se tion 3.5). Here a square is a fa eof maximal type and the type a) fa e has the minimal type.A ording to the above Theorem we all the boundary ells� type 1.1, type 1.2, type 1.3, type 1.4, type 1.5,� type 2.1, type 2.2, type 2.3, type 2.4, type 2.5,� type 3.1, and� type 4.1.For example, observe that the type 4.1 boundary ell is just one tetrahedral.5

3.2 Continuity of the �nite element spa eTo guarantee ontinuity of the �nite element spa e, the fa es of the interior ellsand boundary ells must be divided by triangles a ording a unique on ept. Letus explain this on ept:Continuity on ept of fa es:� fa e of type a): unique subdivision by one triangle (see se tion 2).� fa e of type ): unique subdivision by three triangles (see se tion 2).� fa e of type b): introdu e an edge su h that the fa e an be subdivided bytwo triangles a ording se tion 2. There are two possibilities to introdu esu h an edge. To obtain ontinuity of the �nite element spa e, we intro-du e the edge su h that the slope of the edge is negative with respe t tothe 2 dimensional oordinate system obtained by omitting the oordinatedire tion orthogonal to the fa e.� square fa e: introdu e an edge su h that the fa e an be subdivided by twotriangles. To obtain ontinuity of the �nite element spa e, we introdu e theedge su h that the slope of the edge is negative with respe t to the 2 di-mensional oordinate system obtained by omitting the oordinate dire tionorthogonal to the fa e.3.3 Subdivision of elementary bodiesBefore, we explain the subdivision of the boundary ells, we have to onstru tsuitable subdivisions for elementary bodies. These elementary bodies are a tetra-hedral (Tet), a pyramid (Pyr), a prism (Prism), and a ube for interior ells.3.3.1 Tetrahedral01 2 3Subdivision:Tet: 0, 1, 2, 36

3.4 Pyramid012 3 4Case 1. Edge between P1 and P3. Subdivision:Tet: 0, 1, 3, 4Tet: 0, 1, 2, 3Case 2. Edge between P2 and P4. Subdivision:Tet: 0, 1, 2, 4Tet: 0, 2, 3, 43.4.1 Prism 53 4

102y fa e

x fa eLet x fa e be the fa e P2; P5; P3; P0.Let y fa e be the fa e P1; P4; P3; P0.We distinguish two di�erent fa e types for the fa es x fa e and y fa e. Thesetypes depend on the situation at the point P0:Let us write:y fa e type = spitz, if there is an edge between P0 and P5.y fa e type = an, if there is an edge between P2 and P3.Let us write:x fa e type = spitz, if there is an edge between P0 and P4.x fa e type = an, if there is an edge between P1 and P2.7

Prism ase 1. x fa e type = an and y fa e type = an.Prism version 1.1:Subdivision:Tet: 0, 2, 3, 1Tet: 2, 3, 1, 4Tet: 5, 3, 2, 4

5 410

2 3

Prism version 1.2:Subdivision:Tet: 0, 2, 3, 1Tet: 2, 3, 1, 5Tet: 5, 3, 1, 45 4

102 3

Prism ase 2. x fa e type = spitz and y fa e type = an.Subdivision:Tet: 2, 0, 1, 4Tet: 3, 0, 2, 4Tet: 5, 3, 2, 4

5 410

2 3

8

Prism ase 3. x fa e type = an and y fa e type = spitz.Subdivision:Tet: 5, 3, 0, 1Tet: 2, 5, 0, 1Tet: 4, 5, 1, 3

5 410

2 3Prism ase 4. x fa e type = spitz and y fa e type = spitz.Prism version 4.1:Subdivision:Tet: 5, 2, 1, 0Tet: 5, 3, 0, 4Tet: 1, 5, 0, 4

5 410

2 3

Prism version 4.2:Subdivision:Tet: 5, 2, 4, 0Tet: 5, 3, 0, 4Tet: 1, 2, 0, 45 4

102 3

9

3.4.2 CubeSubdivision of the interior ells ( ube):

WSD ESDENDENTEST

WNTWST WNDTet: WNT, WND, WST, ESTTet: EST, WND, WST, ESDTet: WND, WSD, WST, ESDTet: EST, WND, ESD, ENDTet: ENT, WNT, EST, ENDTet: WNT, WND, EST, ENDThis shows that the ube is subdivided by two prisms:

WSDWNDWST WNT

ESDEST WND

WNT ENTEND

ESDEST

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3.5 Subdivision of boundary ellsUsing the subdivisions in se tion 3.3, we are able to divide the 12 boundary ellsby tetrahedral su h that no large interior angels appear.Theorem 2 (Maximal interior angles in boundary ells)Every boundary ell with fa es of type a), b), ) or a square an be divided intetrahedral su h that� the ontinuity on ept in se tion 3.2 is satis�ed,� the maximal edge angle satis�es'edge � 144Æ;and� the maximal fa e angle satis�es'fa e � 163Æ:For proving this theorem, we have to study every type of the boundary ells.To this end, let us normalize every boundary ell su h that the ube of theboundary ell is [0; 1℄3.3.5.1 At least one fa e is a square.Type 1.1: No verti al edge in the domain.

12

345 6

78

Subdivision:Prism: 5, 7, 6, 1, 3, 2Prism: 6, 7, 8, 2, 3, 4The prisms are subdivided a ording se tion 3.4.1.Maximal interior angles: 'edge � 135Æ and 'fa e � 135Æ.11

Type 1.2: One verti al edge in the domain.

1 2 345 6 78 910Case 1. Edge between P1 and P3. Subdivision:Tet: 1, 5, 6, 7Tet: 1, 6, 9, 7Tet: 1, 7, 9, 10Tet: 1, 6, 8, 9Pyr: 1, 10, 9, 3, 4Pyr: 1, 9, 8, 2, 3Case 2. Edge between P2 and P4. Subdivision:Tet: 1, 5, 6, 7Prism: 9, 8, 10, 3, 2, 4Pyr: 1, 8, 2, 4, 10Case 2.1 P7P5 < P6P5: Hide P7. Additional tetrahedral:Tet: 1, 7, 6, 10Tet: 1, 6, 8, 10Case 2.2 P7P5 � P6P5: Hide P6. Additional tetrahedral:Tet: 1, 7, 6, 8Tet: 1, 7, 8, 10Maximal interior angles: 'edge � 135Æ and 'fa e � 145Æ.

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Type 1.3: Two verti al adja ent edges in the domain.

15 8 6 7

103492Let smax = maxnP9P4; P10P3; P7P6; P8P5o :Case 1. smax > 0:5.We introdu e an additional point at:P11 := (0:4; 0:5; 0:4):Subdivision:// east surfa ePyr: 11, 10, 9, 3, 4// west surfa ePyr: 11, 5, 1, 2, 6// top surfa ePyr: 11, 8, 5, 6, 7// bottom surfa ePyr: 11, 2, 1, 4, 3// south surfa eTet: 11, 8, 1, 5Tet: 11, 8, 9, 1Tet: 11, 9, 4, 1// north surfa eTet: 11, 7, 6, 2Tet: 11, 7, 2, 10Tet: 11, 10, 2, 3

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Case 1.1. smax = P9P4 or smax = P7P6.// rest of surfa eTet: 11, 9, 8, 7Tet: 11, 9, 7, 10Case 1.2. otherwise:// rest of surfa eTet: 11, 9, 8, 10Tet: 11, 8, 7, 10Case 2. otherwise:Prism: 4, 9, 1, 3, 10, 2Prism: 5, 1, 8, 6, 2, 7Prism: 1, 9, 8, 2, 10, 7These prisms are subdivided a ording se tion 3.4.1. One has to hoose the rightversion (see subdivision of type 3.1).Maximal interior angles: 'edge � 135Æ and 'fa e � 145Æ.

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Type 1.4: Two verti al opposite edges in the domain.

1 432

10 8 97 65 1112We introdu e an additional point at:P11 := 14(P2 + P4 + P5 + P8):Subdivision:Tet: 5, 6, 7, 4Tet: 8, 10, 9, 2Tet: 13, 10, 2, 9Tet: 13, 7, 6, 4Tet: 13, 7, 4, 12Tet: 13, 6, 11, 4Tet: 13, 11, 9, 2Tet: 13, 10, 12, 2Tet: 13, 12, 4, 1Tet: 13, 12, 1, 2Tet: 13, 11, 2, 3Tet: 13, 11, 3, 4Pyr: 13, 2, 1, 4, 3Tet: 13, 7, 12, 10Tet: 13, 6, 9, 11

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Additional tetrahedral are:Tet: 13, 7, 10, 9Tet: 13, 7, 9, 6orTet: 13, 7, 10, 6Tet: 13, 6, 10, 9One has to hoose one of these two versions, su h that the largest interior angleof the re tangle Re (10; 9; 6; 7) is divided by an edge.Maximal interior angles: 'edge � 135Æ and 'fa e � 135Æ.

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Type 1.5: Three verti al edges in the domain.

412 3

765 9 810We introdu e an additional point at:P11 := (0:5; 0:5; 0:5):Subdivision:Pyr: 11, 5, 1, 2, 6Pyr: 11, 4, 3, 2, 1Pyr: 11, 6, 2, 3, 7Tet: 11, 1, 5, 9Tet: 11, 1, 10, 4Tet: 11, 1, 9, 10Tet: 11, 6, 9, 5Tet: 11, 6, 8, 9Tet: 11, 6, 7, 8Tet: 11, 3, 8, 7Tet: 11, 3, 10, 8Tet: 11, 3, 4, 10Tet: 11, 9, 8, 10Maximal interior angles: 'edge � 135Æ and 'fa e � 145Æ.

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3.5.2 No square is a fa e and at least one type ) fa e.Type 2.1: No verti al edge in the domain. (Looks like two legs!)1 2 36784 5

Let A fa e be the fa e P4; P1; P2; P5.Let B fa e be the fa e P5; P2; P3; P6.We distinguish two di�erent fa e types for the fa es A fa e and B fa e. Thesetypes depend on the situation at the point P2:Let us writeA fa e type = spitz, if there is an edge between P2 and P4.A fa e type = an, if there is an edge between P1 and P5.Let us writeB fa e type = spitz, if there is an edge between P2 and P6.B fa e type = an, if there is an edge between P5 and P3.Case 1. P1P4 < P4P8 or A fa e type == spitz :Pyr: 8, 4, 1, 2, 5Case 1.1 P7P6 < P3P6 and B fa e type == an :Tet: 7, 8, 5, 3Tet: 6, 7, 5, 3Tet: 5, 3, 8, 2Case 1.2 P7P6 � P3P6 or B fa e type == spitz :Tet: 5, 7, 8, 2Pyr: 7, 6, 5, 2, 318

Case 2. P1P4 � P4P8 and A fa e type == an :Case 2.1. P7P6 � P3P6 or B fa e type == spitz :Pyr: 7, 5, 2, 3, 6Tet: 5, 1, 2, 7Tet: 5, 7, 8, 1Tet: 5, 4, 1, 8Case 2.2. P7P6 < P3P6 and B fa e type == an :Tet: 5, 1, 2, 3Tet: 5, 6, 7, 3Tet: 5, 7, 8, 3Tet: 5, 8, 1, 3Tet: 5, 4, 1, 8Maximal interior angles: 'edge � 135Æ and 'fa e � 135Æ.

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Type 2.2: One opposite verti al edge in the domain. (Looks like three legs!)

9 1 210 345678

Subdivision:Tet: 10, 2, 9, 1Tet: 10, 6, 7, 8Tet: 10, 4, 5, 3Let us introdu e three labels:Let us writeLA = l if P9P1 < P2P1.LA = r if P9P1 � P2P1.Let us writeLB = l if P3P4 < P5P4.LB = r if P3P4 � P5P4.Let us writeLC = l if P7P6 < P7P8.LC = r if P7P6 � P7P8.Case 1. LA = LB = LC = l: Subdivision:Tet: 10, 8, 9, 2Tet: 10, 2, 3, 5Tet: 10, 5, 6, 8Tet: 10, 8, 2, 520

Case 2. LA = LB = LC = r: Subdivision:Tet: 10, 3, 9, 2Tet: 10, 5, 6, 3Tet: 10, 6, 8, 9Tet: 10, 9, 3, 6Case 3. All other ases.Without loss of generality assume LB = r and LC = l:Case 3.1. LA = r. Subdivision:Tet: 10, 9, 2, 3Tet: 10, 3, 8, 9Tet: 10, 5, 6, 3Tet: 10, 6, 8, 3Case 3.2. LA = l. Subdivision:Tet: 10, 8, 9, 2Tet: 10, 8, 2, 3Tet: 10, 5, 6, 3Tet: 10, 6, 8, 3Maximal interior angles: 'edge � 135Æ and 'fa e � 135Æ.

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Type 2.3: One verti al edge in the domain at one side.To be pre ise there are two symmetri ases.

65 4 3211 107 98

65 4 321097 8

11Let us study only the left of these ases. We introdu e an additional point at:P12 := (0:75; 0:5; 0:25):Subdivision:// bottom surfa eTet: 12, 5, 6, 3Tet: 12, 2, 5, 3Tet: 12, 6, 4, 3// east surfa eTet: 12, 10, 4, 9Tet: 12, 10, 3, 4Tet: 12, 8, 9, 4// west surfa eTet: 12, 11, 5, 2// top surfa eTet: 12, 8, 7, 9// north surfa ePyr: 12, 10, 11, 2, 3// south surfa ePyr: 12, 7, 8, 4, 6// rest of the surfa eTet: 12, 7, 11, 9Tet: 12, 9, 11, 10Tet: 12, 7, 5, 11Tet: 12, 7, 6, 5Maximal interior angles: 'edge � 144Æ and 'fa e � 163Æ.22

Type 2.4: Two opposite verti al edges in the domain.

65 4 32107 9813 12 11

We introdu e an additional point at:P14 := 15(P2 + P3 + P4 + P8 + P12):Subdivision:// bottom surfa eTet: 14, 5, 6, 3Tet: 14, 2, 5, 3Tet: 14, 6, 4, 3// east surfa eTet: 14, 10, 4, 9Tet: 14, 10, 3, 4Tet: 14, 8, 9, 4// north surfa eTet: 14, 10, 11, 2Tet: 14, 10, 2, 3Tet: 14, 11, 12, 2// top surfa eTet: 14, 8, 7, 9Tet: 14, 13, 12, 11// south surfa ePyr: 14, 8, 7, 6, 4// west surfa ePyr: 12, 13, 5, 2, 12// rest of the surfa eTet: 14, 9, 11, 10Tet: 14, 7, 5, 13Tet: 14, 7, 6, 5 23

Additional tetrahedral are:Tet: 14, 7, 13, 11Tet: 14, 7, 11, 9orTet: 14, 7, 13, 9Tet: 14, 9, 13, 11One has to hoose one of these two versions, su h that the largest interior angleof the re tangle Re (11; 13; 7; 9) is divided by an edge.Maximal interior angles: 'edge � 135Æ and 'fa e � 147Æ.

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Type 2.5: Three verti al edges in the domain.10

13 12 91134675 28

We introdu e an additional point at: P14 := 14(P2 + P4 + P8 + P13).Subdivision:// bottom surfa eTet: 14, 5, 6, 3;Tet: 14, 2, 5, 3;Tet: 14, 6, 4, 3// east surfa eTet: 14, 11, 9, 4;Tet: 14, 9, 3, 4;Tet: 14, 8, 11, 4// top surfa eTet: 14, 12, 10, 13;Tet: 14, 12, 11, 10;Tet: 14, 11, 8, 10// north surfa eTet: 14, 12, 2, 9;Tet: 14, 12, 13, 2;Tet: 14, 9, 2, 3// south surfa eTet: 14, 8, 6, 7;Tet: 14, 8, 4, 6;Tet: 14, 8, 7, 10// west surfa eTet: 14, 13, 7, 5;Tet: 14, 13, 5, 2;Tet: 14, 13, 10, 7 25

// rest of the surfa eTet: 14, 7, 6, 5;Tet: 14, 11, 12, 9Maximal interior angles: 'edge � 135Æ and 'fa e � 145Æ.

An isomorphi ase : Two adja ent verti al edges in the domain.10 13 12 11 8

3467 592 10 8

3467 5 911 1213 2

These boundary ells are isomorphi to type 2.4.

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3.5.3 Only type b) and type a) fa es and at least one of them.Type 3.1: One prism.1 234

56Subdivision:Prism: 1, 2, 3, 4, 5, 6The prism is subdivided a ording se tion 3.4.1.� In ase of Prism ase 1: (notation as in 3.4.1){ Prism version 1.1 has to be hosen, if P0P2 > P0P1.{ Prism version 1.2 has to be hosen, if P0P2 � P0P1.� In ase of Prism ase 4: (notation as in 3.4.1){ Prism version 4.1 has to be hosen, if P3P5 > P3P4.{ Prism version 4.2 has to be hosen, if P3P5 � P3P4.Maximal interior angles: 'edge � 135Æ and 'fa e � 145Æ.

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3.5.4 Only type a) fa es.Type 4.1: One tetrahedral.1 2

34Subdivision:Tet: 1, 2, 3, 4Maximal interior angles: 'edge � 90Æ and 'fa e � 90Æ.Referen es[1℄ T. Apel. Anisotropi �nite elements: Lo al estimates and appli ations. SFB-Report 393/03/99 A, Te hnis he Universitt Chemnitz, 1999. Habilitationss- hrift.[2℄ W. Ha kbus h. Theorie und Numerik elliptis her Di�erentialglei hungen.Teubner, Stuttgart, 1986.[3℄ W.D. Henshaw. Automati grid generation. A ta Numeri a, pages 121{148,1996.[4℄ C. W. Hirt and J. P. Shannon. Free-surfa e stress onditions forin ompressible- ow al ulations. Journal of Computational Physi s, 2, 1968.

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