HOUSTON JOURNAL OF MATHEMATICS

Volume 21, No. 3, 1995

SUCCESSIVE SUBDIVISIONS OF TETRAHEDRA AND

MULTIGRID METHODS ON TETRAHEDRAL MESHES

SHANGYOU ZHANG

Communicated by Ridgway Scott.

ABSTRACT. Unlike the case of refining triangles in the multigrid method, even an equilateral tetrahedron cannot be subdivided into eight similar subtetrahe- dra. Special techniques are needed in successive refinement of tetrahedra in 3D multigrid methods. In this paper, several methods are proposed and analyzed for refining tetrahedra. It is shown that by a special method the measure of quasiuniformity for the nested refined grids remains bounded where the bound is given explicitly. In fact, there are at most six different types of tetrahedra in the sequence of successively subdivided tetrahedra for any given tetrahedron by the method. Thorough study is given to the tetrahedra which can be re- fined into 8 congruent or equivalent subtetrahedra. Numerical implementation is described.

1. Introduction.

Because of the optimal order (the number of arithmetic operations for solving a linear system is proportional to the number of unknowns in the system) or nearly optimal orders of computation, the multigrid method, the hierarchical basis method, the domain decomposition method and their ap- plications have been studied extensively recently (cf. [3], [4], [8], [9] and references in [17]). However, little work on these methods has been done in the 3-D case. A key issue is to subdivide a tetrahedron into a nested, nondegenerate sequence of subtetrahedra. For example, the general nested multigrid method theory (cf. [3]) remains valid in 3D if a nested sequence of

AMS (MOS) subject classifications. 65N30, 65F10

541

542 SHANGYOU ZHANG

quasi-uniform grids exists. In this paper, we solve this problem, and conse- quently we show the optimal computational order of the multigrid method in 3D.

By linking the middle points of the three edges of a triangle, the triangle is subdivided into four congruent subtriangles, all of which are similar to the original one. Therefore, a nested family of quasiuniform triangulations is defined automatically if an initial one is given. Unfortunately, there is no way to subdivide a regular (equilateral) tetrahedron into eight identical tetrahedra. If the further subdivisions are not done properly, we may get a degenerate sequence of subtetrahedra (volume=o(h 3) where h is the length of the longest edge of a tetrahedron) and/or a sequence of non-quasiuniform subtetrahedra (hmax/hmin, the ratio of the maximal size over the minimal size of all tetrahedra in one level, goes to oo). The goal of the paper is to show some implementable methods for the multigrid refinement of tetrahedra. We discuss also the method of applying Voronoi-tessellation with Delaunay- triangulation ([16], [25] and [14])to tetrahedron subdivision, and the relation between this method and one of our subdivision methods.

The successive refinement of tetrahedral meshes is also important in the finite element methods where the elements are based on tetrahedral meshes.

The subdivisions of tetrahedra proposed in this paper are needed for generat- ing tetrahedral meshes and for adaptive tetrahedral refinement. The nested refinement of tetrahedra are necessary for hierarchical-basis methods and domain-decomposition methods ([32], [26-28], [4-5], [7], [9], [11], [31], [S], [15] and [24] ). The subdivision of tetrahedra has been studied by others, for example, in [18-19, 20]. Alternatively, one may consider the use of nonnested grids in 3-D multigrid methods (cf. [30] and [22]), where the multilevel grids are not naturally nested.

2. Successive Subdivision of a Regular Tetrahedron.

In multigrid finite elment methods, triangular grids are refined succes- sively by connecting the middle points. All the subtriangles are similar to the starting one. If the same method is done on a tetrahedron (cut it by planes through the midpoints on its edges), the subtetrahedra may not look like the starting tetrahedron any more. The subdivision of tetrahedra we are going to discuss is depicted in Figure 1.

By a unit, regular tetrahedron, we mean the tetrahedron which has all

MULTIGRID METHODS ON TETRAHEDRAL MESHES 543

6 edges of length 1. When refining it into half-sized tetrahedra, the 6 middle points on the edges are new vertices (see Figure 1) in addition to the 4 original vertices. The four corner subtetrahedra are still regular. In general, there are three ways to cut the inner regular octahedron. But by symmetry, all three ways of cutting this octahedron are the same here. After the cutting, a new edge is generated. All the rest of the edges of the eight subtetrahedra have length 1/2, and are on the faces of the big tetrahedron.

The length of this new, internal edge can be computed by the next lemma, which applies to general tetrahedra. We need to introduce a few notations. Ta•a2a3a4 denotes a tetrahedron T with four vertices a•, a2, a3 and a4 as depicted in Figure 1. We use l•, la, 13, 14, 15 and 16 to denote the lengths of the six edges. {hi, i = 1, 2,... , 6} denote the midpoints of the six edgesß

I• I, . I• C'ul off d comer lelrahcdra

,.•

( ...... -alcd by 2) •.• b I •.•

• ,•'% ,,

Figure 1. Subdivide a tetrahedron into 8 subtetrahedra.

Lemma 2.1. Let tetrahedron Taxa2a3a4 be as in Figure 1. The distance be- tween the two middle points of a pair skew edges of Taxa2a3a4 is given by

1

lo = Iblb2l = •x/la 2 + 14 • + l• + l• 2 -l• -12 2.

Proof. For any triangle ABC, we have the following edge relation:

(2.2) CDI 2 •(2 •-• 2 + 2 BC 2 •-• 2) -• • -- ß

544 SHANGYOU ZHANG

where ICD is the distance of the vertex C to the mid-point, D, of the edge AB. One can prove (2.2) by embedding /XABC into a parallelogram. Now (2.1) follows by using (2.2) three times:

4102 = 4 b•b2 2 = 2 a-•22 2 + 2 a2b2l 2 -la-•--•l 2

= «[(21a-122 a-7l 2 2) + (2 a'-•-'• 2 + 2 a'-•-ff 2 -la'•--• 2) _ 2 a'•--d'• 21 =t] +t4 +t] []

By Lemma 2.1, when only vertices and midpoints are used, a unit regular tetrahedron can only be subdivided into 4 regular tetrahedra and 4 subtetra- hedra which have five edges of length 1/2 and one edge of length x/-•/2. When we do the next level subdivision, there is no unique subdivision of the inner 4 subtetrahedra. We define two types of tetrahedra:

• - {T•2• 4 ß 11: l•, i = 2,.-. , 6},

•r2 --- {Tala2a3a 4 'll : •/•li, i: 2,''', 6).

We have shown above that subtetrahedra of a T• tetrahedron are either T• or T2. In general, there are 3 ways to subdivide a tetrahedron into 8 subtetrahe- dra. That is, one of the vertices of the internal new edge (b•b2 in Figure 1) can be the middle point of edge ala2, edge ataa or edge a•a4. We use Ta•a21asa4 to denote an oriented tetrahedron so that the subdivision is unique: the new, interior edge connects the midpoints of edge a•a2 and edge a3a4.

Definition 2.2 (subdivision of tetrahedra). For a tetrahedron Ta•a2asa4 (see Figure 1), there are three subdivisions: Ta•a2iaaa4, Talaaia2a 4 and Ta•41•2• a. For example (see Figure 1),

Ta•a21aaa4 -- •{Ta•b•bab•, Tbxa2b4bs, Tbab4aab2, Tbsb•b2a4, Tb• b2bsb4 , Tb• b2b4b5 , Tb• b2bsb• • Tbl b2b•b3 }.

Let us consider the further subdivisions of the 4 new, inner subtetrahe- dra obtained from subdividing a regular tetrahedron. As shown in Figure 1,

MULTIGRID METHODS ON TETRAHEDRAL MESHES 545

these 4 subtetrahedra are identical (type T2): 5 edges of length 1/2 and one edge (the inner edge) of length x/•/2. Let us consider one of them, Tblb2b3b4. There are two different further subdivisions on it: Tblb2[b4b 3 and Tb•b4[b3b • (the third one, Tblb3]bab4, is the same as the latter due to the geometry of T2 tetrahedron). By Lemma 2.1, we can see that the subdivision Tb•&al&4&3 produces 6 T2 subsubtetrahedra and 2 T1 subsubtetrahedra (from the inner octahedron). Therefore, we obtain

Theorem 2.3. If we let the long edge ofT2 tetrahedra be the first edge in the successive subdivision, then the subtetrahedra of T1 and T2 are either in T1 or in T2.

By theorem 2.3, we can generate a sequence of quasiuniform triangula- tions in the multigrid method if the initial grid consists of T1 and T2 tetrahedra only. Let us consider the other subdivision of a T2 tetrahedron: Tbib4[b3b 2. We can view the new subdivision in Figure I as well. We map Tblb4[b•b 2 back to Ta•a•]a•a 4 in Figure 1:

bl '-• al, b2 --• a4, b3 • a3, b4 • a2.

Now 13 -- X/-•c and the rest of li = c, where c = 1/2. This time, 10 = x/•c 2 by Lemma 2.1 and the lengths of edges of one (Tv2641v•v•) of the sub- subtetrahedra are c 2, c •, c 2, c 2, x/•c •, x/r•c 2. A new type of tetrahedron is gen- erated. To repeat this fashion of subdivision, we do another map

bl --• a2, b2 --• a4, b3 --• al, b4 -• a3,

and another subdivision. The lengths of edges of (Tbxb2b3b4) now are c 3, v/•c 3, c a, c a, x/•c a and x/•c a. Repeating more, we get subtetrahedra of which the lengths of 6 edges are

{ 1, vf•, 1, 1, V•, v/•}c 4 , { 1, V•, 1, 1, v/•, x/T6}c s, { 1, •, 1,1, cx/-5•-•_1 , v/•}c k,

where c2j = j2 + j + 1, c2j+l -- j2 + 2j + 2, j = 0, 1, 2,.-- . Therefore the ratio of the longest edge over the short edge for some tetrahedra after k subdivisions is greater than k. We can see that the new tetrahedra

546 SHANGYOU ZHANG

tend to be very sharp and long. As k goes to infinity, we get degenerate tetrahedra. The measure of degeneracy (see section 3) tends to oc if this sub- division is used. Therefore, some refinement methods can lead to a sequence of degenerate tetrahedral grids in the multigrid method. Caution is needed.

3. Nested Refinement of a General Tetrahedral Grid.

In section 2, we have seen that Tx and • tetrahedra could be refined into Tx and • only if the subdivision is done properly. We will generalize the method. The method above labels the new, inner edge to be the first edge for the further subdivision. That is, the interior edge of a tetrahedron must have the midpoint of the first edge as one of its vertices.

Definition 3.1 (labeled-edge subdivision). The subdivision of a tetra- hedron Taxa21asa4 (cf. Definition 2.2 and Figure 1) is

(3.1) Tala2lasa 4 -- •J(Tax bx lbsbe , Tbx a•lb4b• , Tbsb4[asb• , Tb•belb•a4 ,

This method has been implemented in [21], from which we get

subroutine subtet( new, listtet, nodetet ) integer new, listtet(1), nodetet(1), nodeord(32) data nodeord/1,5,6,7, 5,2,8,10, 6,8,3,9, 7,10,9,4,

5,9,8,6, 5,9,6,7, 5,9,7,10, 5,9,10,8/ doi=l, 8

doj -- 1,4 listtet(new•Ci, j) - nodetet( nodeord((i - 1).4 •c j) )

enddo

end

Here, riodeter contains the 10 (4 vertices plus 6 midpoints) node iden- tities of the tetrahedron to be subdivided. The 10 nodes are

{al, a3, a2, a4, b3, bl, b6, b4, bs, b2• in Figure 1. The subroutine returns 8 sub- tetrahedra by listing their node identities in listtet. A standard multi- grid method requires nested, quasiuniform grids. We need to show that the labeled-edge subdivision produces such a sequence of grids from an initial grid.

MULTIGRID METHODS ON TETRAHEDRAL MESHES 547

Definition 3.2. A measure of degeneracy of a tetrahe&on T is defined by

(3.2) 0.T : hT /PT ,

where h T and PT are the longest edge and the diameter of the biggest ball inscribed in T respectively. A measure of quasiuniformity of a tetrahedral grid {7 is defined by

maxT•6 hT (3.3) q• =

minTeg; PT

Clearly 0.T is greater than the ratio of the longest edge over the shortest edge of a tetrahedron T. However, the latter ratio does not give a measure of degeneracy. For example, the ratio is x/• for tetrahedra {T: ll = Y/•li,i = 2,-.. , 6). But these tetrahedra have zero volume as they are in fact planar parallelograms with two diagonal lines. In general, it is not simple to find the measure of degeneracy for a given tetrahedron. Here we use the following fact.

Lemma 3.3. Let V T and A T be the volume and the surface area of a tetra- hedron T. Then

(3.4) PT =6VT/AT.

Proof. Connecting the four vertices of the tetrahedron T with the center of the greatest ball inscribed in T, we will get 4 subtetrahedra. They have the same height, the radius of the ball. (3.4) follows by the volume formula of tetrahe&on. []

Lemma 3.4. Let 0.1 and 0.2 denote the measure of degeneracy for tetrahedra in T1 and T2 respectively. We have

(3.5) 0.1 ---- V• and 0' 2 -- 2

Proof. For a unit T1 tetrahedron, the volume is -v/-•/12 and the surface area is Vf•. A simple computation also shows that for a unit T• tetrahedron, the volume is 1/24 and the surface area is (2 + V/-•)/4. (3.5) is proven by (3.4) and(3.2). []

Now we are ready to show the main theorem of the paper.

548 SHANGYOU ZHANG

Theorem 3.5 (quasiuniform refinement). Let •o be a given initial grid, and 6i (i > O) the grids generated by the successive refinement (3.1) on 6o. Let

si = maxaw and qi = %, for i = O, l, . . . Then

2 (3.6) si _• a•a2so -- 9.143o and qi •_ a2q0 • 13.9q0, for i _• 1,

where O' 1 and • are defined in (3.5).

Proof. For any tetrahedron T• 6o, there is an affine mapping:

B(x) =Ax+y ß R 3 •R 3, such that B(T1) =T, where T• is the unit regular tetrahedron. B maps the subtetrahedra of T1 bijectively to the subtetrahedra of T if the labeled-edge subdivision method (3.1) is applied to T1 and T. Suppose the longest edge in T is 1 and the diameter of the greatest ball in T is p, then it follows (cf. Ciarlet [10])

I 1 IIA • •, and IIA-111•-.

Pw• P

Let T(i) • •i be a subtetrahedron of T. T(i) is the image of either a • or a • subtetrahedron of Tx under the mapping B. The two cases are considered below, where T2 denotes a • tetrahedron whose long edge is of length

h• IIAII/2 (3.7) •, p•, - p•llA-111-x/2• - ppT•2 ß

h•, h•llAII/2 • -- •

Here [[-[[ denotes the matrix norm associated with the Euclidean norm in •3. Therefore, we have shown the first estimate in (3.6). Similarly, we can show the second inequality in (3.6).

The best constants in (3.6) are unknown. They will be improved a little in the next section. We can find some lower bounds as well. For example, they are no less than •/• = (2 + •)/• • 1.5236. By Theorem 3.5, one ca• a..•y a sta•aara mu•tigria theory ([3] a•a [•0]) to sho• that mu•tigria method can provide optimal order algorithm for elliptic problems in 3-D. We do not repeat the theory here. Readers may find it in the thesis of the author [•].

MULTIGRID METHODS ON TETRAHEDRAL MESHES 549

4. Tetrahedra which can be subdivided into same shape tetrahe- dra. In section 2, we have seen that a regular tetrahedron can be subdivided successively into only type T• and type T• tetrahedra if the subdivision is done properly. But a regular tetrahedron cannot be subdivided into 8 regular subtetrahedra. In section 3, it is shown that by the labeled-edge subdivision, any tetrahedron can be refined into a sequence of quasiuniform subtetrahe- dra. A natural question would be to ask how many types of tetrahedra are produced when subdividing a tetrahedron successively by (3.1).

Definition 4.1. Two tetrahedra T• and T• are said to be identical, or to be congruent, denoted by T• •T•, if the two tetrahedra can be matched by a rigid motion. T• and T• are said to be equivalent, denoted by T• _•T•, if the two tetrahedra can be matched by a rigid motion plus a mirror reflection (if needed). T• and T• are said to be of the same type if T• is identical to a tetrahedron which can be scaled to T•.

Theorem 4.2. At most six different types of tetrahedra can be generated when successively subdividing any given tetrahedron by the labeled-edge subdivision defined in (3.1). Further, all the subtetrahedra are equivalent to at most 3 different types of tetrahedra.

Proof. We need to do three levels of subdivision on a given tetrahedron and to identify the subtetrahedra there with one of the tetrahedra on a level above. Readers can find the list of the subtetrahedra in author's thesis [29]. We do not repeat the proof here as it is only a long, but straightforward listing. []

When the labeled-edge subdivision is used, the face triangles of subte- trahedra of all levels are parallel to one of the six planes (2 cutting planes plus 4 face planes of the given tetrahedron) of the initial subdivision. Now if one combines a given tetrahedron with 5 others to make a parallelepiped (in many ways), one can subdivide then this parallelepiped into 8 identical sub- parallelepipeds, and recursively subdivide them further. If one decompose all of the last level subparallelepipeds into 6 tetrahedra in the same way that the big parallelepiped is formed, the given tetrahedron has been subdivided auto- matically. This inherited subdivision is exactly the same as the labeled-edge subdivision. The successive subdivision for the tetrahedra cut from a cube

has been studied by Ong and Adams [20]. This embedding-subdivision can subdivide a tetrahedron into k 3 subtetrahedra at each step too. IT is simple

55O SHANGYOU ZHANG

to extend the labeled-edge subdivision to subdivide a tetrahedron into 33. Also we need to remark that in the homotopy theory we have the Freuden- thal triangulation. The triangulation subdivides R n or C n into n-cubes and then into n-simplexes [13]. This is the same as the discussion of embedding a tetrahedron into parallelepiped above.

Theorem 4.3. There is a unique type of tetrahedron which can be subdivided into 8 identical subtetrahedra of the same type. This type of tetrahedron is called type T•:

•3 '-- {T'/1 --- 12 z 21i/vf•, i = 3,4,5,6).

Proof. By reading Figure 1, we can see that 10 = 11 - 12 and 14 - 15 = 16 if we want the inner 4 subtetrahedra to be the same type as the big tetrahedron. The reasons are as follows. In the inner 4 subtetrahedra, either 11 or 12 is replaced by 10. All/4, 15 and 16 are skew edges of an edge of length 13 in some of the 8 subtetrahedra. Next, by the formula (2.2) and relations above on edges, we can show that 311 • = l• 4- 31• and that ll •/: 14. Comparing the East and the West subtetrahedra in the subdivision of the inner octahedron (see Figure 1), we get 13 = 14. (4.1) then follows by the formula (2.2). []

Lemma 4.4. The measure of degeneracy for T• tetrahedra is

(4.2) o'3 = x/•.

Proof. The volume and the surface area of the unit T3 tetrahedron are 1/12 and v• respectively. (4.2)is obtained by applying (3.2) and (3.4). []

Using the type T• tetrahedra and (4.2), we can improve Theorem 3.5 where the constants ala2, a• and a• in (3.8) can now be replaced by a•2 = 8.

Theorem 4.5 (quasiuniform refinement). Let •o be a given initial grid, and 6i (i > O) the grids generated by the refinement (3.1). Let

si = maxa• and qi = q• for i = 0,1, . . . Then, Teg,

(4.3) si _< 8so and qi _< 8qo, for i _> 1.

MULTIGRID METHODS ON TETRAHEDRAL MESHES 551

Next, we look for tetrahedra whose 8 subtetrahedra are all equivalent to a tetrahedron of its own type. This can be easily done when we read the proof for Theorem 4.3, where only 13 can vary independently. Therefore we get the following theorem, which has been established by Moore and Warren earlier and independently.

Theorem 4.6 (Theorem 7 in [18]). The set of tetrahedra, which can be sub- divided into 8 subtetrahedra all equivalent to a tetrahedron of its own type, can be characterized with a parameter c• as follows:

(4.4) T, = {T : x,/-•l• = x,/-•12 = 2/4 = 215 = 2/6, 13 = o•}.

Clearly T3 C •. Further, via MAPLE (by the University of Waterloo), the author has shown that the smallest measure of degeneracy for tetrahedra in T, is that for T3 tetrahedra. We omit the boring calculation here. A typical • tetrahedron might be a one cut from a cube (cut into 6 equivalent tetrahedra), whose edges' lengths are: x/-•, x/-•, x/•, 1, 1 and 1. If we vary the third edge we get all types of tetrahedra in T,. For example, if we change the length of that length x/• edge to a one of length 1, then we get a T3 tetrahedron. One can see that T3 tetrahedra are "fattest" among •.

5. Other Possible Subdivisions for Tetrahedra.

We note that in the labeled edge subdivision of T2 and T3 tetrahedra, the longest edge (or one of the longest edges) is labeled as the first edge. This can be stated in another way, i.e., the subdivision will make the new interior edge be the shortest (of the 3 possible edges). From Figure 1, one can see such a subdivision can make the new subtetrahedra less degenerate. We would like to extend this idea to a new method.

Definition 5.1 (short-edge subdivision). In the successive short-edge sub- division, a tetrahedron is labeled as T•a2la3a4 (see Figure 1) such that

l• 2 + 122 >_ max{l] + ls 2, 142 + 162}.

In case the labeling is not unique, the default labeling (3.1) will be applied.

Clearly in the short-edge subdivision we have a small total surface area for the 4 subtetrahedra when cutting the inner octahedron (see Figure 1). All

552 SHANGYOU ZHANG

three ways of cutting this inner octahedron have the same volume (1/8 of the volume of the original tetrahedron) for the 4 subtetrahedra. Therefore, the short-edge subdivision tends to minimize the measure of degeneracy for the new subtetrahedra (cf. Lemma 3.3). However, by the short-edge subdivision, more and more new types of tetrahedra might be generated in the successive subdivision. Infinitely many new types may be generated if the length of one of the six edges is not a rational number, nor a square root of a rational number. Generally speaking, the new type tetrahedra are smaller in size than their siblings (especially those from corner of the initial tetrahedron). For example, for a tetrahedron with 6 edges

(0.96595955500379, 0.43182715424336, 1.04559100535698, 0.00609160051681, 0.33822372788563, 0.51489443890750),

the short-edge subdivision generates 683 different types of tetrahedra (after 108 levels). At some level k, if we scaled the tetrahedra by 2 k, we have the above (5.2) tetrahedron as well as a 1000 times smaller (in the size of the longest edge) tetrahedron

(0.00177626263350, 0.00087364766048, 0.00100123129087, 0.00052954611601, 0.00085554497782, 0.00032450391445).

However, in multigrid application, this may not be a problem as there are only a few grid levels in practice. We would like to remark that the volumes of all subtetrahedra at a given level k have a same volume, 1/8 k-• of the volume of the given tetrahedra. In particular, the volumes of the two tetrahedra in (5.2) and (5.3) are the same.

Lemma 5.2. If a tetrahedron T can be initially labeled so that (see Figure 1)

(5.4) min{/• + 12 2, l(• + l•, l(• + 12 2} > max{/• + l], 3 <_ i < j _< 6},

then the successive subdisions of labeled-edge and short-edge are the same.

Proof. Condition (5.4) guarantees the labeling in the labeled-edge subdivision to satisfy (5.1). This can been seen easily by examining Figure 1. []

MULTIGRID METHODS ON TETRAHEDRAL MESHES 553

Lemma 5.3. If a tetrahedron T can be labeled such that (5.1) holds but (5.4) fails to hold, then there is at least one obtuse face triangle on T.

Proof. Without loss of generality, to contradict (5.4), we can assume that l0 2 + 11 • < l] + l• 2. By the formula (2.2) for 10, it follows l• 2 + l• 2 < l•. Therefore, the face triangle /ka2a3a4 of T in Figure 1 is an obtuse one by the cosine law. []

Combining above two lemmas, we get

Theorem 5.4. For any tetrahedron T which has no obtuse face-triangle, if T is labeled such that (5.1) holds initially, then the successive labeled-edge and short-edge subdisions are the same. []

In 2D, we have a theory which shows that any polygonal domain can be triangulated by acute and right-angle triangles (cf [2]). It is possible to extend this theory to show that any polyhedral domain can be triangulated by tetrahedra which have no obtuse face-angles, and consequently no obtuse face-triangles. On such grids, the above two subdivision methods are the same in the multigrid refinement.

We now study the Voronoi tessellation and the Delaunay triangulation (cf. [16] and [25]). For references of their applications in numerical analysis, we refer to [12], [1], [6] and [23]. Given n vertices {ai}L 1 in the plane, there is a convex-polygonal subdivision of the plane such that each convex polygon contains exactly one vertex of {ai}*i•l in it. In this subdivision, the convex polygon containing ai is the intersection of all n- i half-planes. Here, the n - i half-planes are the half-planes containing ai obtained by the bisector perpendicular to some line segments aiaj, j • i. Linking all neighboring vertices, we get a triangulation (not necessarily unique) on the convex hull of the n vertices {ai}i•l. The polygonal subdivision is called a Voronoi tessellation and the triangulation is called a Delaunay triangulation. This process is dimension-independent, i.e., works for any N-dimensional convex polygon. The application of the Delaunay triangulation in the 3D multigrid method would be similar as that of 2D. We consider 2D case first.

554 SHANGYOU ZHANG

Figure 2.

A. Applying the standard multigrid subdivision; B. Applying the Delaunay triangulation to the same vertices; C. Applying the Delaunay triangulation within each triangle.

In the standard multigrid refinement, we add the midpoints of all edges to the next-level triangulation as vertices (see Figure 2A). If we apply the Delaunay triangulation to the set of new vertices, the two triangulations may not be nested. For example, the fine-level triangle a•blb4 is not a subtriangle of any coarse level triangle in Figure 2B. Then one may try a natural modi- fication of the Delaunay triangulation that the Voronoi-Delaunay method is applied within each coarse-level triangle (see Figure 2C). By this new method, clearly we have a nested multigrid refinement in 2D which is different from the standard one, and in which better triangles can be obtained (comparing triangles in Figure 2A and 2C). Surprisingly, this method can be applied to 3D grids as well except a special care should be taken when the Voronoi- Delaunay is not unique (need to match interface triangles). If we apply the Voronoi-Delaunay method within the inner octahedron only in subdividing a tetrahedron, then the method is precisely the short-edge subdivision. Acknowledgements Special thanks to the author's advisor, Professor Ridg- way Scott, for introducing the problem to the author and for encouraging the author to publish this work.

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MULTIGRID METHODS ON TETRAHEDRAL MESHES 555

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DEPARTMENT OF MATHEMATICAL SCIENCES, UNIVERSITY OF DELAWARE, NEWARK, DA 19716

E-mail address: szhang@math.udel.edu