filling space with tetrahedra

13
*Correspondence to: D. J. Naylor, Department of Civil Engineering, University of Wales Swansea, Swansea SA2 8PP, U.K. E-mail: d.j.naylor@swansea.ac.uk CCC 00295981/99/101383 13$17.50 Received 18 August 1997 Copyright ( 1999 John Wiley & Sons, Ltd. Revised 4 August 1998 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng. 44, 1383 1395 (1999) FILLING SPACE WITH TETRAHEDRA D. J. NAYLOR* Department of Civil Engineering, ºniversity of ¼ ales Swansea, Swansea SA28PP, º.K. SUMMARY In the context of 3D finite element meshes various options for filling an indefinite space (such as would be approached within a fine mesh) with tetrahedra are considered. This problem is not trivial as it is in 2-D since, unlike equilateral triangles, regular tetrahedra cannot be fitted together to fill space. Various groupings, or assemblies, which can be repeated indefinitely to fill space are considered. By altering the shape of the tetrahedra in one of these to minimize a suitable function a unique shape of tetrahedron is obtained which optimizes the conditioning. The mesh thus produced is shown to be better conditioned than alternatives based on assemblies of different shaped tetrahedra. A number of conditioning measures are used to confirm this. Finally, actual meshes which fit boundaries are briefly considered. Copyright ( 1999 John Wiley & Sons, Ltd. KEY WORDS: finite elements; tetrahedra; three dimensions 1. INTRODUCTION Tetrahedra are increasingly used in finite element meshes. In computational fluid dynamics four-noded linear tetrahedra are used and applications involving in the order of one million such elements have been reported [1, 2]. Ten-noded quadratic (linear strain) tetrahedra are good for general use in 3-D structural or geomechanical applications [3]. When used with four integrating points, these elements provide a nice balance between the constraints imposed by the integration and the nodal degrees of freedom. 3-D mesh generation has been the subject of much research, particularly in the aerospace industry over the last two decades [1, 2, 4 10]. The generation procedures can be classified under two headings: structured and unstructured. In the former, the mesh is generated by a procedure in which the configuration is established a priori. Schemes include interpolation, conformal map- ping and the use of differential equations [10]. An example of the last is the use of the Laplace equation to generate an orthogonal grid which defines approximate squares in two dimensions and cubes in three. The cubes may then be subdivided into tetrahedra. In unstructured mesh generation the configuration is determined during the generation process. In 3-D it typically involves first the generation of a mesh of triangles over the boundary followed by the filling of the interior with tetrahedra by means of the Delaunay triangulation technique [1, 2].

Upload: d-j-naylor

Post on 06-Jun-2016

218 views

Category:

Documents


2 download

TRANSCRIPT

Page 1: Filling space with tetrahedra

*Correspondence to: D. J. Naylor, Department of Civil Engineering, University of Wales Swansea, Swansea SA2 8PP,U.K. E-mail: [email protected]

CCC 0029—5981/99/101383—13$17.50 Received 18 August 1997Copyright ( 1999 John Wiley & Sons, Ltd. Revised 4 August 1998

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)

FILLING SPACE WITH TETRAHEDRA

D. J. NAYLOR*

Department of Civil Engineering, ºniversity of ¼ales Swansea, Swansea SA2 8PP, º.K.

SUMMARY

In the context of 3D finite element meshes various options for filling an indefinite space (such as would beapproached within a fine mesh) with tetrahedra are considered. This problem is not trivial as it is in 2-Dsince, unlike equilateral triangles, regular tetrahedra cannot be fitted together to fill space. Variousgroupings, or assemblies, which can be repeated indefinitely to fill space are considered. By altering theshape of the tetrahedra in one of these to minimize a suitable function a unique shape of tetrahedron isobtained which optimizes the conditioning. The mesh thus produced is shown to be better conditioned thanalternatives based on assemblies of different shaped tetrahedra. A number of conditioning measures are usedto confirm this. Finally, actual meshes which fit boundaries are briefly considered. Copyright ( 1999 JohnWiley & Sons, Ltd.

KEY WORDS: finite elements; tetrahedra; three dimensions

1. INTRODUCTION

Tetrahedra are increasingly used in finite element meshes. In computational fluid dynamicsfour-noded linear tetrahedra are used and applications involving in the order of one million suchelements have been reported [1, 2]. Ten-noded quadratic (linear strain) tetrahedra are good forgeneral use in 3-D structural or geomechanical applications [3]. When used with four integratingpoints, these elements provide a nice balance between the constraints imposed by the integrationand the nodal degrees of freedom.

3-D mesh generation has been the subject of much research, particularly in the aerospaceindustry over the last two decades [1, 2, 4—10]. The generation procedures can be classified undertwo headings: structured and unstructured. In the former, the mesh is generated by a procedure inwhich the configuration is established a priori. Schemes include interpolation, conformal map-ping and the use of differential equations [10]. An example of the last is the use of the Laplaceequation to generate an orthogonal grid which defines approximate squares in two dimensionsand cubes in three. The cubes may then be subdivided into tetrahedra. In unstructured meshgeneration the configuration is determined during the generation process. In 3-D it typicallyinvolves first the generation of a mesh of triangles over the boundary followed by the filling of theinterior with tetrahedra by means of the Delaunay triangulation technique [1, 2].

Page 2: Filling space with tetrahedra

While it is difficult to generate a 3-D mesh an added challenge is to ensure that it is of goodquality. That is where this study comes in. It does not deal at all with how a mesh may begenerated, nor does it consider the important matter of the constraints imposed by the bound-aries. Instead it considers different patterns of tetrahedra that can be fitted together to fill a spaceof unlimited extent, such as might be approached in regions of a fine mesh remote from theboundary. Unlike the situation in two dimensions this cannot be achieved using regular tet-rahedra (i.e. having faces that are equilateral triangles). Either a mix of different shapes or possiblyan arrangement of identical but irregular tetrahedra is inevitable.

A fascinating history of the search for an answer to the question of which polyhedra fill space isgiven by Marjorie Senechal [11]. The five regular solids of Plato—the tetrahedron, cube,octahedron, dodecahedron and icosahedron—were first considered. Of these only the cube canfill space but apparently Aristole thought that the regular tetrahedron could as well. This was notsorted out until the 17th century! An important more recent figure is that of Sommerville(1879—1934). Noting that a triangular prism is space filling and can be divided into threetetrahedra, he considered four classes of prism [12]. The constituent tetrahedra in each classwere geometrically identical, i.e. congruent, and as triangular prisms can be fitted together to fillspace, so also could these tetrahedra. Field and Smith [6] make use of this in their application ofthe ‘octree’ approach to the generation of tetrahedral meshes. A Type II Sommerville tetrahed-ron—the ‘parent’ from which the others are derived—forms the starting point for their meshgeneration. As will be seen, this tetrahedron has special significance in the present work, althoughhere the focus is on structured mesh generation rather than on the Delaunay triangulationapproach used by Field and Smith.

The most obvious way of filling space with tetrahedra is to first fill it with cubes and thensubdivide these into tetrahedra. The minimum number of tetrahedra into which a cube can bedivided is five, the next in number being six. These two, designated the cub5 and cub6 assemblies,respectively, are considered first. Two others based on the octahedral crystal structure ofdiamond are then investigated. In this arrangement, carbon atoms are located at the corners ofregular tetrahedra which are stacked in such a way that the gaps between them are regularoctahedra. It was thought that an atomic layout which resulted in such a stable material mightalso be a good arrangement for finite element nodes. Here the octahedra are divided into eitherfour or eight tetrahedra, referred to as the oct4 and oct8 assemblies, respectively.

The blocks for these four assembles could also be fitted together if they were distorted, in thecase of the cubes into parallelepipeds. The undistorted blocks are first considered and then theeffect of distortion is investigated.

Field [13] has proposed an interesting way of filling space with tetrahedra. He groups theminto sets of 20 to form icosahedra. These are then assembled into a lattice with the gaps betweenthem filled with tetrahedra having four different shapes. The arrangement produces a well-conditioned mesh because the 20 icosahedra-filling tetrahedra are nearly regular and account fornearly two-thirds of the volume. This arrangement will be considered in detail later in the paper.At the end of the paper, Field writes: ‘Another interesting and useful question is how to producea space filling lattice which maximises the average normalised shape ratio’. This is precisely thequestion addressed here.

The quality of the mesh is assessed by three criteria. The first, believed to be the mostimportant, is the conditioning of the element. A number of parameters have been used in theliterature to measure this, and some of these are referenced in the next section. Here the measure ischosen to lie in the range 0—1, increasing as the conditioning improves, and becoming unity when

1384 D. J. NAYLOR

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)

Page 3: Filling space with tetrahedra

the tetrahedron is regular. The second criterion is that there should not be too much variationin the volume of the tetrahedra within each repeating block. This criterion is, however, qualifiedby the recognition that a better conditioned tetrahedron should perhaps be larger than a lesswell-conditioned neighbour. The third criterion is the ergonomics of mesh generation, i.e. the easewith which the blocks can be generated to fill a domain. This will depend on whether the mesh isstructured or unstructured in the sense used above. It is the only part of this paper which toucheson the boundary conditions.

2. ANALYSIS

2.1. Conditioning parameters

Five measures are used in this paper. The first, identified by g without a suffix and referred to asthe ‘conditioning efficiency’, is used for the main investigation. Comparisons are then madeagainst the other measures. As the efficiency of the tetrahedron is being measured, values areexpressed as percentages. The numerical values are therefore 100 times the definitions givenbelow. The measures are defined as follows. Throughout, » is the volume of the tetrahedron, A itstotal surface area and e

ithe length of edge i.

2.1.1. Conditioning efficiency g. g is defined as the volume of the actual tetrahedron divided bythe volume of the regular tetrahedron which has the same average edge length (e

2). Noting that

the volume of a regular tetrahedron is e32/(6J2),

g"6J2»

e32

with e2"

1

6

6+i/1

ei

(1)

This is similar to the parameter a used by Weatherill et al. [2]. a is defined as e32/» so that

g"6J2/a. The only reason for using g in preference to a is that it lies in the range 0—1 rather

than infinity to 6J2.

2.1.2. Area conditioning ratio gA. This is defined as the surface area of a regular tetrahedron

having the same volume as the actual tetrahedron divided by the surface area of the actualtetrahedron:

gA"

31@66»2@3

A(2)

2.1.3. ¸iu and Joe’ conditioning measure gt(14). This measure was proposed as an alternative to

the ratio of the inscribing to circumscribed spheres (gSbelow) and the minimum corner angle (g)

below). In addition to lying in the range 0—1 its authors promote it on the grounds of simplicity. Itis given by

gL"

12(3»)2@3

+6i/1

e2i

(3)

FILLING SPACE WITH TETRAHEDRA 1385

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)

Page 4: Filling space with tetrahedra

Figure 1. Assemblies of tetrahedra into a cube

2.1.4. Ratio of inscribed to circumscribed spheres gS. This ratio is 3 for a regular tetrahedron so,

for the measure to lie in the range 0—1:

gS"

3r

R(4)

in which r is the radius of the inscribed sphere and R the radius of the circumscribed sphere. Thismeasure is used by Field and Smith [6], who call it the ‘normalized shape ratio’.

2.1.5. Solid angle measure g) . The solid angle ) has the value )3%'

"0)551285 for the fourcorners of a regular tetrahedron. Since the minimum solid angle )

.*/cannot exceed )

3%'the

measure is defined as

g)")

.*/)

3%'

(5)

2.2. Cubic patterns

The alternative cub5 and cub6 patterns are illustrated in Figure 1. Adjacent cubes in the cub5must be rotated 90° relative to each other (about any of the three axes normal to the cubes faces)to prevent crossing of the diagonal edges. This does not apply to the cub6 assembly. The cub6also has the advantage that it can be subdivided into two wedges (right triangular prisms) withthree tetrahedra in each. The plane ACGE in Figure 1(b) divides the two wedges. They are mirrorimages of each other so that, as with cub5 assemblies, the diagonals on the shared plane do notcross.

g values calculated using equation (1) are given in Table I. In calculating the average g theindividual values are weighted according to the volumes occupied by the tetrahedra. Thus for thecub5 assembly in which the central tetrahedron has twice the volume of a corner tetrahedron,g for the former has twice the weight of the latter. The cub6 tetrahedra all have the same volumeand therefore have equal weights.

1386 D. J. NAYLOR

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)

Page 5: Filling space with tetrahedra

Figure 2. Octahedral arrangement

Table I. g for cube assemblies

cub5 cub6

Tetrahedron n w g per cent Tetrahedron n w g per cent

Corner 4 2/3 80)4 a 2 1/3 80)4Central 1 1/3 100)0 b 2 1/3 70)7— — — — c 2 1/3 60)2

Average g 86)9 Average g 70)4

n is the number of tetrahedra and w is the proportion of the total volume they occupy. The corners in the cub5 assemblyare at A, C, F and H in Figure 1(a). For cub6, a identifies DEGH and AFCB in Figure 1(b), b: AGDC and AFGE and c:AEGD and GCAF

2.3. Octahedral patterns

Figure 2 shows one block of the octahedral assembly. CDEFAG is the octahedron. Its divisioninto the eight non-regular tetrahedra of the oct8 assembly is indicated by the short broken lines.These are combined in pairs to form four in the oct4 assembly. DEGH and ACFB are the tworegular tetrahedra. Alternative assemblies which could be fitted together to fill an indefinite spacecould be defined by attaching the two regular tetrahedra to different faces of the octahedron.There is a particular reason for choosing the arrangement shown in Figure 2, as will becomeapparent later.

The two regular tetrahedra occupy one-third the total volume of the assembly. Consequently,in the oct4 arrangement, all the tetrahedra have the same volume, whereas in the oct8 eachtetrahedron in the octahedron is half the size of one of the regular tetrahedra. The oct4 and oct8g values are given in Table II.

The oct8 arrangement is actually the same as the cub5, as can be seen by supposing that one ofthe eight tetrahedra in the octahedron, e.g. AFCO in Figure 2, is a corner tetrahedron in the cub5assembly and that the adjacent regular tetrahedron, AFCB, is the central tetrahedron. OB wouldbe a diagonal of this cube.

FILLING SPACE WITH TETRAHEDRA 1387

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)

Page 6: Filling space with tetrahedra

Table II. g for octahedral assemblies

oct4 oct8

Tetrahedron n w g per cent n w n per cent

In octahedron 4 2/3 81)9 8 2/3 80)4Regular 2 1/3 100)0 2 1/3 100)0Average g 87)9 86)9

n and w as in Table I

2.4. The effect of distortion

The question is, would distortion of the regular assemblies in such a way that they can still befitted together affect the conditioning, as measured by the average g? The cube assemblies couldbe distorted into parallelepipeds and still be fitted together. So also could the octahedral. Theeffect of distorting the cubes will be considered first, then the octahedral. Since, however, the oct8assembly is the same as the cub5, the benefit or otherwise of distortion will therefore apply toboth. A similarity between the cub6 and oct4 assemblies (which is not obvious) will be revealedlater. This will answer the question about the effect of distorting the oct4.

To find out how the average g is affected by distortion, its variation with respect to parametersdefining the shape of the assembly will be investigated. Rather than seek the maximum ofg directly a function relating to its inverse and defined in terms of the squares of the edge lengths isused so that a least-squares minimization can be applied. Let this function be F* (later to bemodified and designated F) and be defined as

F*"n+k/1

wkR

k»k

(6)

where n is the number of tetrahedra in the assembly, i.e. five for the cub5 and six for the cub6, wkis

a weighting, »kthe volume of the tetrahedron and

Rk"

6+i/1

e2i

in which, as before eiis the edge length.

Let the parallelepiped into which the cube is to be distorted be defined by three vectors a, b, calong the initially orthogonal edges and let the orthogonal axes (x, y, z) be chosen so that afterdistortion a continues to lie along the x-axis and b to lie in the x—y plane. The volume of theparallelepiped, »

T, is then the scalar triple product of the vectors a, b and c, i.e.

»T"a · b]c

which, on expanding and substituting ay"a

z"b

z"0, gives

»T"a

xbycz

(7)

The volumes of the component tetrahedra can also be obtained in terms of this scalar tripleproduct and it can easily be verified that the volumes of the four corner tetrahedra in the cub5

1388 D. J. NAYLOR

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)

Page 7: Filling space with tetrahedra

assembly and all six tetrahedra in the cub6 are each »T/6 and that the cub5 central tetrahedron is

»T/3. The length of the edges are either the lengths of the vectors a, b or c or various sums or

differences of them. Using suffix c1 to denote the corner tetrahedron ABDE in the cub5 assembly(Figure 1(a)):

R#1"a2#b2#c2"Db!a D2#Dc!b D2#Da!c D2

in which a2"a · a, etc., and Db!a D2"(b!a ) · (b!a), etc. Expanding these leads to

R#1"3(a2#b2#c2)!2(a · b#b · c#c · a)

Proceeding similarly for the other three corner tetrahedra (CBDG, HDGE and FEBG):

R#2"3(a2#b2#c2)!2(a · b!b · c!c · a)

R#3"3(a2#b2#c2)#2(a · b!b · c#c · a)

R#4"3(a2#b2#c2)#2(a · b#b · c!c · a)

and, for the central tetrahedron (BEDG) (designated by suffix o).

R0"4(a2#b2#c2)

Since all the corner tetrahedra have the same volume they will be assigned the same weight w#,

and the central tetrahedron a weight w0. (w

0is now not necessarily 2w

#; see below.) Substituting in

equation (6), noting that »T/6 and »

T/3 are the respective volumes of corner and central

tetrahedra and using suffix c5 to identify the cub5 assembly.

F*#5"

6w#

»T

4+k/1

R#k#

3w0

»T

R#0

on summing R#k

(k"1, 4) the scalar products a · b, b · c and c · a cancel, so that

F*#5"

12

»T

(6w##w

0) (a2#b2#c3)

For the purpose of the minimization »T

will be kept constant. Also, w#and w

0are constants, the

values of which will not affect the outcome. Consequently, the function to be minimized can beredefined as

F#5"a2#b2#c2 (8)

Proceeding similarly for the cub6 assembly the expressions for R for the pairs of tetrahedradesignated a, b and c in Table I are

Ra"3 (a2#b2#c2)#2(a · b!b · c#c · a)

Rb"4a2#3(b2#c2)#2(2a · b#b · c#2c · a)

Rc"3a2#4(b2#c2#a · b#c · a)

FILLING SPACE WITH TETRAHEDRA 1389

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)

Page 8: Filling space with tetrahedra

Figure 3. The par6 assembly

All the tetrahedra in the cub6 assembly have the same volume (»T/6) so that the weights can be

taken as unity. Substituting in equation (6),

F*#6"

12

»T

(Ra#Rb#Rc)

F*#6"

120

»T

(a2#b2#c2#a · b#c · a)

Note that in contrast to the cub5 case only b · c of the scalar products cancel. Again, as »T

isa constant the function to be minimized can be defined as

F#6"a2#b2#c2#a · b#c · a (9)

Let l be the side length of the undeformed cube so that »T"l3. Then, minimizing F

#5and F

#6with respect to the variation of the six non-zero components of a, b and c and subject to theconstraint of equation (7) (see the Appendix), for the cub5 assembly,

a"[l, 0, 0]T, b"[0, l, 0]T, c"[0, 0, l]T (10)

and for the cub6 assembly,

a"[p, 0, 0]T, b"[!p/2, p/J2, 0]T, c"[!p/2, 0, p/J2 ]T (11)

in which p3"2l3. In both cases, the second partial derivatives of F with respect to the variablesare all positive, thus confirming that the stationary values are minima.

These results show that for the cub5 assembly the undeformed shape is optimum, i.e.the average conditioning efficiency g will be a maximum for this shape. This also applies tothe oct8 assembly since, as explained above, it is the same as the cub5. It does not applyto the oct4 case, as will become clear in the next section. The optimum shape for thecub6, however, is not a cube but the parallelepiped illustrated in Figure 3. It will be designatedthe ‘par6’ assembly. It has some interesting and important properties which will now beconsidered.

1390 D. J. NAYLOR

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)

Page 9: Filling space with tetrahedra

2.5. The par6 assembly

The distortion of the cube into a parallelepiped involves a 35)3° rotation of the edges about they- and z-axis. The original a, b and c tetrahedron become the same so that the assembly comprisessix identical tetrahedra. Furthermore, the four faces of these tetrahedra are all the same—an

isosceles triangle with one side of length p and the other two of length J3p/2. To reflect this thename ‘isotet’ is suggested for it. It is in fact Sommerville’s Type II tetrahedron referred to earlier.Its g value is 93)6 per cent which, since all the tetrahedra are the same, is also the average. Thatthere is a single shape so close to a regular tetrahedron which can fill space does not appear tohave been noted before.

The par6 assembly shown in Figure 3 is only one of a number of possible ways of fitting sets ofsix isotets together to fill space. In each no single tetrahedron will have the same orientation. Alsothere are six different face orientations.

Returning to the question of whether distortion of the oct4 assembly increases its efficiency,interestingly it does. A simple compression in the direction of the vertical axis AG in Figure 2 by

a factor of J2 relative to the horizontal axes DF and CE converts the oct4 rhombohedron intothe par6 assembly of Figure 3. The same lettering sequence has been used in Figures 2 and 3 sothat this likeness can be visualized.

So far it has been shown (in Tables I and II) that the par6 assembly is more efficient than thecube and octahedral assemblies, at least based on g. Two questions remain: is this affected ifdifferent measures are used and are there are other ways of filling space which might be moreeffective? In the next Section one other assembly will be considered followed by a comparison ofall of them using the five efficiency measures.

2.6. Field+s icosahedral assembly

The space-filling assembly proposed by Field [13] is interesting because it has such a highproportion of near-regular tetrahedra, albeit at the expense of complexity. First the assembly willbe described. This is done in a different way to that in Field’s paper to highlight what is therepeating pattern. Then the average conditioning will be compared with the other assembliesconsidered above.

The icosahedra are arranged in layers so that a pair of parallel opposite faces lie in ‘horizontal’planes. The icosahedra touch each other along edges. Viewed in plan these contiguous edges formregular hexagons. The touching corners alternate above and below a horizontal plane throughthe centre of the icosahedra. Each icosahedron can therefore be fitted into a right hexangularprism with one face flush with the top of the prism and one with the bottom. The contiguous edgeforms a zig-zag line (with six straights) round its girth (Figure 4). Clearly, such prisms can beassembled to fill the space. Field found that four types of infill tetrahedra, designated F

1, . . . , F

4,

were needed to fill the gaps between the icosahedra. These are indicated in Figure 4. The prismboundaries divide tetrahedra F

1and F

3into two and F

2into three. The tetrahedra F

4are fully

inside the prism. Noting that these filler tetrahedra occur at both ends it can be seen that for eachprism there are six tetrahedra of types F

1, F

3and F

4, two of type F

2and 20 of what shall be

referred to as type C, i.e. filling the icosahedron itself. As mentioned in the introduction the strongpoint about this assembly is that the type C tetrahedra are very nearly regular and account formost of the volume (63)7 per cent).

FILLING SPACE WITH TETRAHEDRA 1391

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)

Page 10: Filling space with tetrahedra

Figure 4. Field’s icosahedral assembly

Table III. g and gSfor icoshedral assembly

Tetrahedron w g per cent gSper cent

C 0)6366 99)7 99)6F1

0)1459 86)1 87)8F2

0)0301 82)5 82)7F3

0)0787 73)2 76)8F4

0)1087 80)4 77)0

w as in Table 1

sField gave gSto four decimal places. The present work, which was carried out using six significant figures, confirmed his

values for types C, F1

and F2

but there were small differences for F3and F

4. 0)7683 and 0)7700, respectively, were obtained

compared with Field’s values of F3"0)7697 and F

4"0)7681

In Table III the g and gS

values are compared for the five component tetrahedra of Field’sassembly. The proportions by volume (the weighting w) of each group of tetrahedra in theassembly are included in the table. g

Shas been chosen because Field used this measure in his

paper.s

2.7. Comparison of the five conditioning measures

In Table IV the weighted average conditioning measures for the four assemblies consideredhere plus the icosahedral assembly are compared using all five conditioning measures. It can beseen that the par6 assembly of isotets provides the best conditioning by all the measures exceptgA, followed closely by the icosahedral assembly. Apart from this one exception all the measures

put all the assemblies in the same order. The difference in any case is very small (97)5 per cent forthe icosahedral assembly as against 97)2 per cent for the par6).

1392 D. J. NAYLOR

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)

Page 11: Filling space with tetrahedra

Table IV. Comparison of conditioning measures for the various assemblies

Assembly g per cent gSper cent g

Aper cent g

Lper cent g) per cent

cub6 70)4 69)1 87)0 76)1 47)5cub5 ("oct8) 86)9 82)1 94)8 89)3 74)4oct4 87)9 86)9 95)2 90)4 74)4par6 93)6 94)9 97)2 95)2 95)0Icosahedral 93)0 93)4 97)5 94)2 81)7

Field and Smith tabulate gS

(their ‘normalized shape ratio’) for the four Sommerville tet-rahedra. They obtained the same value as that obtained here (94)9 per cent) for Type II, i.e. theisotet, and lower values for the others, the highest of which was 70)1 per cent for Type I. This isconsistent with the finding here that the isotet is of special significance, being both wellconditioned and capable of being fitted together to fill space.

3. CONCLUDING REMARKS

Three criteria have been suggested for assessing the quality of the different assemblies: theconditioning efficiency, the (lack of ) variation in volume and the ergonomics of mesh generation.The relevant comparisons are between the cub5 (which is the same as the oct8), the oct4, the cub6,the par6 and Field’s icosahedral assembly. They compare as follows.

First the conditioning: Four of the five measures put the par6 assembly with its identical isotetstop with an average of the five measures of 95 per cent . No particular significance is attached tothe marginally higher g

Aobtained by the icosahedral assembly. This assembly is slightly less

efficient with a corresponding average of 92 per cent. Next come the oct4 with an average of87 per cent, followed closely by the cub5 ("oct8) with 86 per cent. The cub6 comes last with70 per cent. It is paradoxical that this, the worst conditioned of the assemblies considered,becomes the best when the cube is distorted into the par6 parallelepiped.

Next the variation in volume: The icosahedral and cub5 assemblies are favoured because theirbest conditioned tetrahedra are bigger than the other ones. In the cub5 assembly the centralregular tetrahedron has twice the volume of a corner tetrahedron and in the icosahedral the 20tetrahedra comprising the icosahedron are roughly twice as large as the others. (The range is1)3—2)4 times.) All the tetrahedra are the same size in the other assemblies.

¹he final criterion—ease of mesh generation—is clearly of major importance to analysts. Thecube assemblies are the most straightforward to generate and have conventionally been used forstructured mesh generation. Difficulties arise, however, in accommodating non-orthogonalboundaries. No mention is made in Field’s paper of how an icosahedral mesh might be generatedbut one imagines that this would be quite difficult. The 40 elements illustrated in Figure 4 formonly part of the repeating block since alternate layers need to be rotated 180° (actually anymultiple of 60°) relative to each other to make the connecting equilateral triangular faces match.The par6 assembly is relatively easy to generate. The main problem is to match orthogonalboundaries. A strategy used successfully by the writer has been to subdivide some boundaryisotets to minimize the inevitable distortion from the ideal shape. These elements will then beinherently less well conditioned but this will be a local effect confined to boundaries.

FILLING SPACE WITH TETRAHEDRA 1393

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)

Page 12: Filling space with tetrahedra

It is interesting that octahedral assemblies (and therefore also the cub5) which resemble thearrangement of carbon atoms in diamond are less efficient than the par6 assembly. It must be thatnature applies different criteria, specifically that carbon atoms must lie at the corners of regulartetrahedra.

The structured generation of an actual mesh involves a mapping from an idealized assembly,such as have been considered above, to the actual mesh. The originating mesh may be viewed asa subset of the infinite space-filling assembly with boundaries which crudely fit the real bound-aries. The mapping then distorts this mesh to provide a better fit, the objective being as far aspossible to retain the ideal shape but sometimes also to grade the mesh. For example, in structuralapplications the elements should be smaller where stress is concentrated. This process willnormally result in average conditioning efficiencies well below at least the best four of theassemblies considered here. The important point is that the space-filling assemblies providea target which can only be approached by the average conditioning efficiency of an actual mesh.Thus, if a cub5 mesh generator were used this ceiling would be 87 per cent by the measure usedabove, 93 per cent were the mesh icosahedral and 94 per cent for an isotet mesh.

In addition to their role of measuring the average conditioning the measures can also be usedto pick out any particularly badly shaped elements, for example the near flat tetrahedra referredto as ‘slivers’ [2] which can be produced in unstructured mesh generation.

As, apart from the one exception noted above, all the efficiency measures put the differentassemblies in the same order there is little to choose between them. g and g

Lare relatively easy to

calculate and may be preferred for that reason. gA

is not particularly recommended. Apart from itbeing the source of the exception just referred to it gives values much higher than the others. Thewriter is not aware of a precedent for its use.

The main conclusion is that the space-filling assembly involving just one shape of tetrahed-ron—the isotet—is the best conditioned. The question posed by Field is answered.

APPENDIX

Minimization of F for cub5 and cub6 assemblies

Equations (8) and (9) are to be minimized. They are conveniently combined by the introductionof a parameter m with m"0 for equation (8) and m"1 for equation (9), i.e.

F"a2#b2#c2#m(a · b#c · a)

Expanding the scalar products, i.e. a2"a · a, etc., and noting that ay"a

z"b

z"0,

F"a2x#b2

x#b2

y#c2

x#c2

y#c2

z#ma

x(b

x#c

x) (12)

This is to be minimized subject to the constraint of equation (7) so that there are five independentvariables. Equation (7) gives c

z"»

T/(a

xby) in which »

Tis constant. The partial derivatives of c

zwith respect to a

xand b

yare first obtained. These are

Lcz

Lax

"!

cz

ax

andLc

zLb

y

"!

cz

by

1394 D. J. NAYLOR

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)

Page 13: Filling space with tetrahedra

Table V. First and second partial derivatives of F

Variable (l) LF/Ll L2F/Ll2

ax

2ax!2c2

z/a

x#m (b

x#c

x) 2#6c2

z/a2

xbx

2bx#ma

x2

by

2by!2c2

z/b

y2#6c2

z/b2

ycx

2cx#ma

x2

cy

2cy

2

The partial derivatives of F with respect to the five independent variables ax, . . . , c

yare now

obtained by differentiating equation (12) and noting that LF/Lax

and LF/Lby

contain the terms2c

z(Lc

z/La

x) and 2c

z(Lc

z/Lb

y), respectively. These and also the second partial derivatives of F are

given in Table V. The sets of values defining the minima are found by equating LF/Lv(v"a

x, . . . , c

y) to zero. That these are minima is confirmed by the positive values of L2F/Lv2 (for

all v). The values are as follows: cub5 assembly (m"0): ax"b

y"c

z"l, b

x"c

x"c

y"0; in which

l is the side length of the undistorted cube, i.e. l3"»T. cub6 assembly (m"1): a

x"p,

bx"c

x"!p/2, b

y"c

z"p/J2, c

y"0, in which p3"2l3 with l again the side length of the

undistorted cube. These values define the vectors a, b and c given as equations (10) and (11) in thetext.

REFERENCES

1. Weatherill NP, Hassan O. Efficient three-dimensional Delaunay triangulation with automatic point creation andimposed boundary constraints. International Journal for Numerical Methods in Engineering 1994;37:2005—2039.

2. Weatherill NP, Hassan O, Marcum DL. Compressible flowfield solutions with unstructured grids generated byDelaunay triangulation. AIAA Journal 1995;33(7):1196—1204.

3. Naylor DJ. Canales dam—a 3D collapse settlement study. Internal Report No. CR/902/95, Department of CivilEngineering, University of Wales Swansea., 1995.

4. Bowyer A. Computing Dirichlet tessellations. Computer Journal 1981;24(2):162—166.5. Cavendish JC, Field DA, Frey WH. An approach to automatic three-dimensional finite element mesh generation.

International Journal for Numerical Methods in Engineering 1985;21:329—347.6. Field DA, Smith WD. Graded tetrahedral finite element meshes. International Journal for Numerical Methods in

Engineering 1991;31:413—425.7. Lohner R, Parikh P. Three-dimensional grid generation by the advancing front method. International Journal for

Numerical Methods in Fluids 1988;8:1135—1149.8. Peraire J, Peiro J, Formaggia L, Morgan K, Zienkiewicz OC. Finite element Euler computations in three dimensions.

International Journal for Numerical Methods in Engineering 1988;26:2135—2159.9. Watson DF. Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. Computer

Journal 1981;24(2):167—172.10. Weatherill NP. Mesh generation for aerospace applications. SaJ dhanaJ 1991;16(1):1—45.11. Senechal M. Which tetrahedra fill space? Mathematics Magazine 1981;54:227—243.12. Sommerville DMY. Space-filling tetrahedra in Euclidean space. Proceedings of the Edinburgh Mathematical Society

1923;41:49—57.13. Field DA. Implementing Watson’s algorithm in three dimensions. Proceedings of the 2nd Annual ACM Symposium on

Computational Geometry 1986:246—259.14. Liu A, Joe B. On the shape of tetrahedra from bi-section. Mathematics of Computation 1994;63(207):141—154.

FILLING SPACE WITH TETRAHEDRA 1395

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1383—1395 (1999)