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A comparison of tetrahedron quality measures
Finite Elements in Analysis and Design 15 (1993) 255-261
255
Elsevier FINEL 348 Technical Note
A comparison of tetrahedron quality measures V.N. P a r t h a s a r a t h y General Electric Consulting Services, Schenectady, IVY 12301, USA
C.M. G r a i c h e n a n d A . F . H a t h a w a y Solid Mechanics Laboratory, General Electric Corporate Research and Development, Schenectady, N Y 12301, USA
Received May 1992 Revised June 1993
Abstract. One of the difficult requirements imposed on fully automatic tetrahedral mesh generators has been the ability to produce "good" quality graded elements. This note is concerned with the choice of a distortion quality measure used in post-meshing activities such as (i) quality evaluation, and (ii) quality improvement of a finite element mesh consisting of a disjoint collection of tetrahedra. In that light it provides (i) a comparison of the fidelity of various existing tetrahedron quality measures to a set of distortion sensitivity tests, and (ii) a comparison of the computational expense of such measures. The topics of mesh generation and mesh quality improvement are not discussed.
Introduction T h e versatility o f t e t r a h e d r a l m e s h e s to a p p r o x i m a t e very c o m p l e x g e o m e t r i e s is well a c k n o w l e d g e d in the c o m p u t a t i o n a l s t r u c t u r a l / f l u i d m e c h a n i c s , a n d e l e c t r o m a g n e t i c s communities. A l t h o u g h c o n s i d e r a b l e p r o g r e s s has b e e n r e p o r t e d , r e s e a r c h in a u t o m a t i c t r i a n g u l a tion a n d t e t r a h e d r i z a t i o n o f c o n v e x / n o n - c o n v e x 3D d o m a i n s c o n t i n u e s to b e a topic o f interest. R e s e a r c h e r s in c o m p u t a t i o n a l g e o m e t r y have also p r o p o s e d m a n y solutions to this p r o b l e m . N e v e r t h e l e s s , such a l g o r i t h m s a r e n o t always directly u s a b l e d u e to a s s u m p t i o n s o f d e g e n e r a c y , a c c e p t a b l e quality, etc. R e c e n t l y , i n c r e a s e d a t t e n t i o n is b e i n g p a i d to t h e quality of the resulting t e t r a h e d r i z a t i o n t h a t an a l g o r i t h m p r o d u c e s . This stems from t h e r e a l i z a t i o n t h a t a g o o d quality m e s h is a p r e - r e q u i s i t e for i n c r e a s e d c o n f i d e n c e in the results o f any s u b s e q u e n t analysis a n d / o r a d a p t i v e p r o c e d u r e s . F o r discussion p u r p o s e s of this note, the p a r t i c u l a r a l g o r i t h m a d o p t e d (i.e., O c t r e e , D e l a u n a y , A d v a n c i n g F r o n t or M e d i a l Axis T r a n s f o r m a t i o n ) to p r o d u c e t h e finite e l e m e n t m e s h is i n c o n s e q u e n t i a l .
Tetrahedron quality measures and their sensitivity F i g u r e 1 shows s o m e c o m m o n c o n f i g u r a t i o n s of " p o o r l y " s h a p e d t e t r a h e d r a . R e s e a r c h e r s in the p a s t have p r o p o s e d v a r i o u s m e a s u r e s to c h a r a c t e r i z e t h e t e t r a h e d r o n shape. T h e e n t i r e Correspondence to: V.N. Parthasarathy, Room K1 - 3A32, GE-CR&D, P.O. Box 8, One River Road, Schenectady, NY
12301, USA. 0168-874X/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0168-874X(93)E0071-8
256
V.N. Parthasarathy et al. / Comparison of tetrahedron quality measures
"Thin" element
"Flat" element
"Wedge-like" element
"Sliver" element - 4 co-planar points Fig. 1. Some "poorly" shaped tetrahedra.
mesh consisting of tetrahedrons is judged using the maximum value of the measure or the distribution of elements above a given threshold value or variations thereof. Cavendish et al. [1] characterize a tetrahedron by the ratio of the circumscribing sphere radius to the inscribed sphere radius. In an effort to categorize thin and flat tetrahedra, Baker [2] proposed the combined use of the ratios maximum edge length to inscribed sphere radius and maximum edge length to the minimum edge length. Cougny et al. [3] use tetrahedron volume and the four composing facet areas to define a normalized aspect ratio. Dannenlogue and Tanguy [4] found that a ratio involving the tetrahedron volume and the average edge length of the six composing edges suffices to characterize a tetrahedron. An extension of their measure is hereby proposed by replacing the average length with the root mean square of the edge lengths. The advantage of this new measure is the use of edge-length-squared vis-a-vis the edge length in aspect ratio computation. Table 1 gives the mathematical description of the measures mentioned above. Also in Table 1, let quantities with an asterisk (*) superscript denote values of such measures when applied to a equilateral tetrahedron whose edge lengths are all equal to unity. Normalized shape distortion value for any tetrahedron can then be obtained by dividing any measure by its corresponding asterisk value. What follows is a set of sensitivity tests which evaluate the fidelity of the quality measures sighted previously. The tests have mainly been drawn from observations of the nature of distortions that occur during the meshing and smoothing (i.e., element quality improvement) stages of a tetrahedral mesh generator [5]. Although the suite of tests may not be viewed as comprehensive, they can be considered representative enough to provide a good understanding of a given measure's distortion fidelity.
Test A : In this test, beginning with an equilateral tetrahedron the apex is moved in small increments along the vertical line joining the apex to the centroid of the opposite facet. For apex locations close to the base, this test simulates the shape of a flat element of Fig. 1 and for large apex heights, the shape of a thin element is simulated. Figure 2 shows the plot of location of the apex versus the normalized shape ratio. Test B: Here beginning with an equilateral tetrahedron, the apex is allowed to trace, in small increments, a quarter-arc defined using the height of the tetrahedron as the radius. At the final position, the apex lies on the plane defined by the opposite facet, thus simulating a sliver element. This test also captures some of the notorious shapes that a tetrahedron can assume. Figure 3 shows the element configuration and a plot of angle of inclination of the line joining the apex to the centroid and the bottom facet versus the normalized shape ratio.
V.N. Parthasarathy et al. / Comparison o f tetrahedron quality measures
257
Table 1 A list of t e t r a h e d r o n s h a p e m e a s u r e s u s e d in l i t e r a t u r e A s p e c t ratio m e a s u r e
V a l u e for a e q u i l a t e r a l t e t r a h e d r o n
U s e d in
CR /3 = - IR
/3 * = 3.0
[1]
tr * = 4.898979
[2]
to* = 0.612507
[2]
r * = 1.0
[2]
K* = 4.58457e - 04
[3]
o~* = 8.479670
[4]
3'* = 8.479670
[5]
Smax
tr = - IR CR to =
r =
Smax Smax Stain V4
K=
a = 7 =
i=4
]3
iE=lSA2J S3v~ V
S3ms V
Nomenclature: C R = r a d i u s of the c i r c u m s c r i b e d sphere, I R = r a d i u s of the inscribed sphere, S i = l e n g t h of any e d g e i, Smax = max(Si) (i = 1... 6), Smi n = m i n ( S i ) (i = 1 . . . 6), S A = surface a r e a of a t r i a n g u l a r facet, Savg = a v e r a g e ( S i) (i = 1 . . . 6), Srm s = root m e a n s q u a r e ( S i) (i = 1 . . . 6 ) , V = v o l u m e of the t e t r a h e d r o n , a n d
_1, i
Xl
YI
X2
Y2
113, Z2
li=6
4V IR =
S~ms = I / ~1 xL' $2i •
i = ~ '
10
• x / x * it
• ala**
•/V#* e~
--N--
0)
r/r* '~ ~'D' * 0)/(.0 *
L o c u s o f t h e a p e x in 3 - s p a c e
...,..-
Test A
E
Z
0
1
2
3
Distance of the apex from the base
Fig. 2. Sensitivity of s h a p e m e a s u r e s to test A.
V.N, Parthasarathy et al. / Comparison of tetrahedron quality measures
258
10
•
riJ
9
• a / a * * r/r * "2
I][]
'/'"
8
L o c u s o f t h e a p e x in 3 - s p a c e
l lH
7
ca, 6 5
E Z
4 3 2 1
0 10
0
2O
30
40
50
60
70
8O
90
Fig. 3. Sensitivity of shape measures to test 13.
Inclination of the apex from the vertical axis
Test C: A commonly occurring tetrahedron configuration is the wedge-like element (sometimes known as Sommerville Type II tetrahedron in Ref. [6]), where two edges of the same length are positioned perpendicular to each other, but are displaced by some finite distance in space. The other four edges are formed by completing the remaining possible connections and are longer than the initial two edges. In this test, one of the two shorter edges of a wedge-like tetrahedron is subject to length changes, while preserving its original
10 9 0 t~
t
•• axlx / a **÷ x o/o* r/r*
8
• ~/t~" • y/r*
7
r t~
- °~/'°----------L" -~-
N
6
L e n g t h ( L M ) = 0.1
5 4 o Z
3 2 N
1
L e n g t h ( L M ) = 1.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
1.1
[ N o d e s N , K r e m a i n fixed]
Length of element edge LM. Fig. 4. Sensitivity of shape measures to test C.
V.N. Parthasarathy et al. / Comparison of tetrahedron quality measures
q
10
,c/~* • a/a* •/V/~* -N"
~
8
I
259
x ./a*l * ~/~*l • ~'/y*[ o,/~,* I
I
6
J N
"~
rJ3
Initial Configuration
z 2
Final Configuration 10
20
30
40
50
60
70
80
90
[Nodes NK trace a quarter-circle, Angle between element edges LM and NK. while nodes LM remain fixed.] Fig. 5. Sensitivityof shape measures to test D. configuration. Figure 4 shows the element configuration and the plot of edge length against the normalized shape ratio. Test D: In this test one of the shorter edges of a wedge-like tetrahedron is made parallel to the other edge by allowing the edge's two nodes to simultaneously trace a quarter-arc on an x - y plane. This test simulates a practical situation that occurs during independent smoothing boundary surface triangulation. Typically, an edge in the surface triangulation belongs to this tetrahedron, whose other five edges lie in the model region interior. Therefore, nodes which lie on the boundary surface are mobile while the other two nodes are immobile as they are located in the interior. The behavior of various quality measures is shown in Fig. 5. From Figs. 2 - 5 the following observations can be made: the measures a , / 3 , o', 3' are able to characterize distortions of all four tests. The measure z is successful only in distortions where at least one of the edge length undergoes a significant change, i.e., tests A and C. The fidelity of the measure to is limited to distortions of test B. The ratio K is also significantly sensitive to all the tests except that, unlike the other measures, a lower value denotes higher distortion and vice-versa. This is due to the nature of its formulation, where the volume term appears in the numerator, as opposed to the denominator in other measures.
Computational effort Although it is not apparent at the outset, the particular choice of a quality measure does have a significant impact on the CPU time for the overall element quality evaluation of finite element meshes, especially when the mesh size is large (number of elements > 50,000). For a three-dimensional model, acceptability of a mesh depends on the acceptability of (i) the boundary surface mesh quality, and (ii) the interior mesh quality. Hence, an evaluation of mesh quality always implies independent evaluations of surface and interior meshes. Typically, a mesh generator evaluates the quality of a mesh at various stages - before smoothing, after smoothing, to name a few.
260
1AN. Parthasarathy et al. / Comparison of tetrahedron quafity measures
For the measure /3, although the circum-radius is computed as part of some mesh generation schemes, the post-meshing activity of quality improvement procedure re-positions the nodes, thus requiring its re-computation. Moreover, in some robust approaches of smoothing [7], the mesh quality is monitored by repeated evaluations, as a sub-part of a larger smoothing algorithm. The cumulative effect of such repeated quality evaluations on the overall smoothing operation time, and hence the mesh generation time can be noticeably high [8]. For the above reasons, it is desirable to use a computationally inexpensive quality measure which possesses the required fidelity. It is well recognized that data structures play a crucial role in mesh g e n e r a t i o n / improvement time efficiency. However, in this note, for fair comparison purposes, the assumption is made that the quality "evaluator" acts as a separate "black box". In such a context, the individual computational effort involved to determine a measure for a tetrahedron can be broken down as follows: a - the six edge lengths and element volume y - the six edge lengths' squared and element volume /3 - C R - the solution of a 3 × 3 algebraic equation system for determining circum-center and subsequent computation of circum-radius. IR computation of the four triangular facet areas and element volume ~r - maximum edge length and I R Among the quality measures mentioned in the previous Section, due to fidelity considerations the choice has been narrowed to - a , / 3 , 7, ~r. Also, since all measures are d i m e n s i o n l e s s , the dependence of the measures' value is invariant with respect to the element size. As is obvious, for a given element, the number of basic computations ( + , - , x , / , f ) is minimum for the measure 7, and maximum for/3. Also, unlike/3 or ~r, a and 3' do not involve any face area computation. Hence, further savings are achieved simply because of the fact that in any tetrahedra! mesh, the number of triangular faces is always considerably higher than the number of finite element edges. The inexpensiveness of a and 7 was also confirmed by some CPU timing experiments of our implementation of the above measures, on meshes of various sizes, which showed that evaluation of a and y were nearly 50% less expensive than evaluation of/3. However, depending on the particular implementation the amount of savings can differ. The measure 3' is now being used in Ref. 5. (It is also observed that a naive zero-volume check has to be performed to avoid a zero divide for certain measures.)
Conclusions Existing tetrahedron quality were reviewed. A new tetrahedron quality measure was proposed by extending an existing measure. A set of sensitivity tests which capture some important types of tetrahedron distortion was described. Of the measures found sensitive, a comparison of computational cost showed that the quality measure which uses the root mean square of the edge lengths and volume was the least expensive and can be used to reduce the overall computational effort of element quality a s s e s s m e n t / e n h a n c e m e n t operations.
Acknowledgements The authors would like to thank Mr. Peter M. Finnigan, Manager, Mechanical Design Methods Program, G E - C R D , for providing support to pursue the work described in this paper and for his thoughtful comments. The authors would also like to thank a referee on
V.N. Parthasarathy et al. / Comparison of tetrahedron quality measures
261
his/her detailed comments on Sommerville tetrahedron, normalization, and general clarity of the original version of the paper.
References [1] J.C. CAVENDISH,D.A. FIELD and W.H. FREY, "An approach to automatic three-dimensional finite element mesh generation", Int. J. Num. Meth. Eng. 21, 1985, pp. 329-347. [2] T.J. BAKER, "Element quality in tetrahedral meshes", Proc. 7th Int. Conf. on Finite Element Methods in Flow Problems, Huntsville, AL, April 3-7 1989. [3] H.L CoucNv, M.S. SHEPHARD and M.K. GEORGES, "Explicit Node Point Smoothing within Octree", Report No. 10-1990, SCOREC, RPI, Troy, NY, 1990. [4] H.H. DANNELONGUE and P.A. TANGUY, "Three-dimensional adaptive finite element computations and applications to non-Newtonian flows", Int. J. Num. Meth. Fluids 13, 1991, pp. 145-165. [5] C.M. GRAICHEN, A.F. HATHAWAY and V.N. Parthasarathy, "OCTREE Theoretical Manual", GE-CRD Report, September 1991. [6] M. SENECHAL,"Which tetrahedra fill space?", Mathematics Magazine 54(5) 1981, pp. 227-243. [7] V.N. PARTHASARATHY and S. KODIYALAM, "A constrained optimization approach to finite element mesh smoothing", Finite Elem. Anal. Des. 9, 1991, pp. 309-320. [8] L. FORMAGG1A,"An unstructured mesh generation algorithm for three-dimensional aeronautical configurations", Proc. Numerical Grid Generation in CFD and Related Fields, edited by A.S. ARCILLA, J. HAUSER, P.R. EXSEMAN and J.F. THOMPSON, North-Holland, 1991.
255
Elsevier FINEL 348 Technical Note
A comparison of tetrahedron quality measures V.N. P a r t h a s a r a t h y General Electric Consulting Services, Schenectady, IVY 12301, USA
C.M. G r a i c h e n a n d A . F . H a t h a w a y Solid Mechanics Laboratory, General Electric Corporate Research and Development, Schenectady, N Y 12301, USA
Received May 1992 Revised June 1993
Abstract. One of the difficult requirements imposed on fully automatic tetrahedral mesh generators has been the ability to produce "good" quality graded elements. This note is concerned with the choice of a distortion quality measure used in post-meshing activities such as (i) quality evaluation, and (ii) quality improvement of a finite element mesh consisting of a disjoint collection of tetrahedra. In that light it provides (i) a comparison of the fidelity of various existing tetrahedron quality measures to a set of distortion sensitivity tests, and (ii) a comparison of the computational expense of such measures. The topics of mesh generation and mesh quality improvement are not discussed.
Introduction T h e versatility o f t e t r a h e d r a l m e s h e s to a p p r o x i m a t e very c o m p l e x g e o m e t r i e s is well a c k n o w l e d g e d in the c o m p u t a t i o n a l s t r u c t u r a l / f l u i d m e c h a n i c s , a n d e l e c t r o m a g n e t i c s communities. A l t h o u g h c o n s i d e r a b l e p r o g r e s s has b e e n r e p o r t e d , r e s e a r c h in a u t o m a t i c t r i a n g u l a tion a n d t e t r a h e d r i z a t i o n o f c o n v e x / n o n - c o n v e x 3D d o m a i n s c o n t i n u e s to b e a topic o f interest. R e s e a r c h e r s in c o m p u t a t i o n a l g e o m e t r y have also p r o p o s e d m a n y solutions to this p r o b l e m . N e v e r t h e l e s s , such a l g o r i t h m s a r e n o t always directly u s a b l e d u e to a s s u m p t i o n s o f d e g e n e r a c y , a c c e p t a b l e quality, etc. R e c e n t l y , i n c r e a s e d a t t e n t i o n is b e i n g p a i d to t h e quality of the resulting t e t r a h e d r i z a t i o n t h a t an a l g o r i t h m p r o d u c e s . This stems from t h e r e a l i z a t i o n t h a t a g o o d quality m e s h is a p r e - r e q u i s i t e for i n c r e a s e d c o n f i d e n c e in the results o f any s u b s e q u e n t analysis a n d / o r a d a p t i v e p r o c e d u r e s . F o r discussion p u r p o s e s of this note, the p a r t i c u l a r a l g o r i t h m a d o p t e d (i.e., O c t r e e , D e l a u n a y , A d v a n c i n g F r o n t or M e d i a l Axis T r a n s f o r m a t i o n ) to p r o d u c e t h e finite e l e m e n t m e s h is i n c o n s e q u e n t i a l .
Tetrahedron quality measures and their sensitivity F i g u r e 1 shows s o m e c o m m o n c o n f i g u r a t i o n s of " p o o r l y " s h a p e d t e t r a h e d r a . R e s e a r c h e r s in the p a s t have p r o p o s e d v a r i o u s m e a s u r e s to c h a r a c t e r i z e t h e t e t r a h e d r o n shape. T h e e n t i r e Correspondence to: V.N. Parthasarathy, Room K1 - 3A32, GE-CR&D, P.O. Box 8, One River Road, Schenectady, NY
12301, USA. 0168-874X/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0168-874X(93)E0071-8
256
V.N. Parthasarathy et al. / Comparison of tetrahedron quality measures
"Thin" element
"Flat" element
"Wedge-like" element
"Sliver" element - 4 co-planar points Fig. 1. Some "poorly" shaped tetrahedra.
mesh consisting of tetrahedrons is judged using the maximum value of the measure or the distribution of elements above a given threshold value or variations thereof. Cavendish et al. [1] characterize a tetrahedron by the ratio of the circumscribing sphere radius to the inscribed sphere radius. In an effort to categorize thin and flat tetrahedra, Baker [2] proposed the combined use of the ratios maximum edge length to inscribed sphere radius and maximum edge length to the minimum edge length. Cougny et al. [3] use tetrahedron volume and the four composing facet areas to define a normalized aspect ratio. Dannenlogue and Tanguy [4] found that a ratio involving the tetrahedron volume and the average edge length of the six composing edges suffices to characterize a tetrahedron. An extension of their measure is hereby proposed by replacing the average length with the root mean square of the edge lengths. The advantage of this new measure is the use of edge-length-squared vis-a-vis the edge length in aspect ratio computation. Table 1 gives the mathematical description of the measures mentioned above. Also in Table 1, let quantities with an asterisk (*) superscript denote values of such measures when applied to a equilateral tetrahedron whose edge lengths are all equal to unity. Normalized shape distortion value for any tetrahedron can then be obtained by dividing any measure by its corresponding asterisk value. What follows is a set of sensitivity tests which evaluate the fidelity of the quality measures sighted previously. The tests have mainly been drawn from observations of the nature of distortions that occur during the meshing and smoothing (i.e., element quality improvement) stages of a tetrahedral mesh generator [5]. Although the suite of tests may not be viewed as comprehensive, they can be considered representative enough to provide a good understanding of a given measure's distortion fidelity.
Test A : In this test, beginning with an equilateral tetrahedron the apex is moved in small increments along the vertical line joining the apex to the centroid of the opposite facet. For apex locations close to the base, this test simulates the shape of a flat element of Fig. 1 and for large apex heights, the shape of a thin element is simulated. Figure 2 shows the plot of location of the apex versus the normalized shape ratio. Test B: Here beginning with an equilateral tetrahedron, the apex is allowed to trace, in small increments, a quarter-arc defined using the height of the tetrahedron as the radius. At the final position, the apex lies on the plane defined by the opposite facet, thus simulating a sliver element. This test also captures some of the notorious shapes that a tetrahedron can assume. Figure 3 shows the element configuration and a plot of angle of inclination of the line joining the apex to the centroid and the bottom facet versus the normalized shape ratio.
V.N. Parthasarathy et al. / Comparison o f tetrahedron quality measures
257
Table 1 A list of t e t r a h e d r o n s h a p e m e a s u r e s u s e d in l i t e r a t u r e A s p e c t ratio m e a s u r e
V a l u e for a e q u i l a t e r a l t e t r a h e d r o n
U s e d in
CR /3 = - IR
/3 * = 3.0
[1]
tr * = 4.898979
[2]
to* = 0.612507
[2]
r * = 1.0
[2]
K* = 4.58457e - 04
[3]
o~* = 8.479670
[4]
3'* = 8.479670
[5]
Smax
tr = - IR CR to =
r =
Smax Smax Stain V4
K=
a = 7 =
i=4
]3
iE=lSA2J S3v~ V
S3ms V
Nomenclature: C R = r a d i u s of the c i r c u m s c r i b e d sphere, I R = r a d i u s of the inscribed sphere, S i = l e n g t h of any e d g e i, Smax = max(Si) (i = 1... 6), Smi n = m i n ( S i ) (i = 1 . . . 6), S A = surface a r e a of a t r i a n g u l a r facet, Savg = a v e r a g e ( S i) (i = 1 . . . 6), Srm s = root m e a n s q u a r e ( S i) (i = 1 . . . 6 ) , V = v o l u m e of the t e t r a h e d r o n , a n d
_1, i
Xl
YI
X2
Y2
113, Z2
li=6
4V IR =
S~ms = I / ~1 xL' $2i •
i = ~ '
10
• x / x * it
• ala**
•/V#* e~
--N--
0)
r/r* '~ ~'D' * 0)/(.0 *
L o c u s o f t h e a p e x in 3 - s p a c e
...,..-
Test A
E
Z
0
1
2
3
Distance of the apex from the base
Fig. 2. Sensitivity of s h a p e m e a s u r e s to test A.
V.N, Parthasarathy et al. / Comparison of tetrahedron quality measures
258
10
•
riJ
9
• a / a * * r/r * "2
I][]
'/'"
8
L o c u s o f t h e a p e x in 3 - s p a c e
l lH
7
ca, 6 5
E Z
4 3 2 1
0 10
0
2O
30
40
50
60
70
8O
90
Fig. 3. Sensitivity of shape measures to test 13.
Inclination of the apex from the vertical axis
Test C: A commonly occurring tetrahedron configuration is the wedge-like element (sometimes known as Sommerville Type II tetrahedron in Ref. [6]), where two edges of the same length are positioned perpendicular to each other, but are displaced by some finite distance in space. The other four edges are formed by completing the remaining possible connections and are longer than the initial two edges. In this test, one of the two shorter edges of a wedge-like tetrahedron is subject to length changes, while preserving its original
10 9 0 t~
t
•• axlx / a **÷ x o/o* r/r*
8
• ~/t~" • y/r*
7
r t~
- °~/'°----------L" -~-
N
6
L e n g t h ( L M ) = 0.1
5 4 o Z
3 2 N
1
L e n g t h ( L M ) = 1.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
1.1
[ N o d e s N , K r e m a i n fixed]
Length of element edge LM. Fig. 4. Sensitivity of shape measures to test C.
V.N. Parthasarathy et al. / Comparison of tetrahedron quality measures
q
10
,c/~* • a/a* •/V/~* -N"
~
8
I
259
x ./a*l * ~/~*l • ~'/y*[ o,/~,* I
I
6
J N
"~
rJ3
Initial Configuration
z 2
Final Configuration 10
20
30
40
50
60
70
80
90
[Nodes NK trace a quarter-circle, Angle between element edges LM and NK. while nodes LM remain fixed.] Fig. 5. Sensitivityof shape measures to test D. configuration. Figure 4 shows the element configuration and the plot of edge length against the normalized shape ratio. Test D: In this test one of the shorter edges of a wedge-like tetrahedron is made parallel to the other edge by allowing the edge's two nodes to simultaneously trace a quarter-arc on an x - y plane. This test simulates a practical situation that occurs during independent smoothing boundary surface triangulation. Typically, an edge in the surface triangulation belongs to this tetrahedron, whose other five edges lie in the model region interior. Therefore, nodes which lie on the boundary surface are mobile while the other two nodes are immobile as they are located in the interior. The behavior of various quality measures is shown in Fig. 5. From Figs. 2 - 5 the following observations can be made: the measures a , / 3 , o', 3' are able to characterize distortions of all four tests. The measure z is successful only in distortions where at least one of the edge length undergoes a significant change, i.e., tests A and C. The fidelity of the measure to is limited to distortions of test B. The ratio K is also significantly sensitive to all the tests except that, unlike the other measures, a lower value denotes higher distortion and vice-versa. This is due to the nature of its formulation, where the volume term appears in the numerator, as opposed to the denominator in other measures.
Computational effort Although it is not apparent at the outset, the particular choice of a quality measure does have a significant impact on the CPU time for the overall element quality evaluation of finite element meshes, especially when the mesh size is large (number of elements > 50,000). For a three-dimensional model, acceptability of a mesh depends on the acceptability of (i) the boundary surface mesh quality, and (ii) the interior mesh quality. Hence, an evaluation of mesh quality always implies independent evaluations of surface and interior meshes. Typically, a mesh generator evaluates the quality of a mesh at various stages - before smoothing, after smoothing, to name a few.
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For the measure /3, although the circum-radius is computed as part of some mesh generation schemes, the post-meshing activity of quality improvement procedure re-positions the nodes, thus requiring its re-computation. Moreover, in some robust approaches of smoothing [7], the mesh quality is monitored by repeated evaluations, as a sub-part of a larger smoothing algorithm. The cumulative effect of such repeated quality evaluations on the overall smoothing operation time, and hence the mesh generation time can be noticeably high [8]. For the above reasons, it is desirable to use a computationally inexpensive quality measure which possesses the required fidelity. It is well recognized that data structures play a crucial role in mesh g e n e r a t i o n / improvement time efficiency. However, in this note, for fair comparison purposes, the assumption is made that the quality "evaluator" acts as a separate "black box". In such a context, the individual computational effort involved to determine a measure for a tetrahedron can be broken down as follows: a - the six edge lengths and element volume y - the six edge lengths' squared and element volume /3 - C R - the solution of a 3 × 3 algebraic equation system for determining circum-center and subsequent computation of circum-radius. IR computation of the four triangular facet areas and element volume ~r - maximum edge length and I R Among the quality measures mentioned in the previous Section, due to fidelity considerations the choice has been narrowed to - a , / 3 , 7, ~r. Also, since all measures are d i m e n s i o n l e s s , the dependence of the measures' value is invariant with respect to the element size. As is obvious, for a given element, the number of basic computations ( + , - , x , / , f ) is minimum for the measure 7, and maximum for/3. Also, unlike/3 or ~r, a and 3' do not involve any face area computation. Hence, further savings are achieved simply because of the fact that in any tetrahedra! mesh, the number of triangular faces is always considerably higher than the number of finite element edges. The inexpensiveness of a and 7 was also confirmed by some CPU timing experiments of our implementation of the above measures, on meshes of various sizes, which showed that evaluation of a and y were nearly 50% less expensive than evaluation of/3. However, depending on the particular implementation the amount of savings can differ. The measure 3' is now being used in Ref. 5. (It is also observed that a naive zero-volume check has to be performed to avoid a zero divide for certain measures.)
Conclusions Existing tetrahedron quality were reviewed. A new tetrahedron quality measure was proposed by extending an existing measure. A set of sensitivity tests which capture some important types of tetrahedron distortion was described. Of the measures found sensitive, a comparison of computational cost showed that the quality measure which uses the root mean square of the edge lengths and volume was the least expensive and can be used to reduce the overall computational effort of element quality a s s e s s m e n t / e n h a n c e m e n t operations.
Acknowledgements The authors would like to thank Mr. Peter M. Finnigan, Manager, Mechanical Design Methods Program, G E - C R D , for providing support to pursue the work described in this paper and for his thoughtful comments. The authors would also like to thank a referee on
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his/her detailed comments on Sommerville tetrahedron, normalization, and general clarity of the original version of the paper.
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