Civil Engineering Reference
In-Depth Information
Cap: zero short edge
Wedge: one short edge
Slat: two short edges
Needle: three short edges
Figure 6.1
Degenerate tetrahedra with zero, one, two and three short edges.
were plotted for easy visualisation (Knupp 2003a). A shape measure for tetrahedral elements
based on the condition number of the transformation matrix was also proposed, which can,
in turn, be used to define a shape measure for hexahedral elements (Knupp 2003b).
Degenerate tetrahedral elements.
A tetrahedron is said to be degenerated when it shrinks
to zero volume, where all its vertices are either on the same plan, on the same line or all in
one point. Some researchers would only take degenerate tetrahedral elements as tetrahe-
dra with zero volume but non-zero faces or edges; hence, the big crunch of a tetrahedron
shrinking to a point is not considered as a proper case of degeneracy (Knupp 2001). Liu
and Joe (1994a) listed a number of practical degenerate cases by considering the number
of short edges in a tetrahedron, as shown in Figure 6.1. By a progressive reduction of the
length of the short edges, the behaviour of some common shape measures was monitored
and compared. A systematic classification of ten degenerate cases for tetrahedral elements
degenerating on a plane and on a line can be found in Dompierre et al. (2005) in which the
number of short edges, how faces are collapsed and the limiting circumradius for degenerate
tetrahedra were studied.
6.2.1 Common simplex shape measures
Simplex shape measures aroused much interest of Liu and Joe (1994a,b) when they worked
on establishing a bound for the quality of tetrahedral elements subject to repeated subdivi-
sions. The solid angle in three dimensions, with natural geometrical relationship for object
visualisation, measures how large an object appears to an observer looking it from a point.
The solid angle and the ratio between the in-radius and the circumradius of a tetrahedron
are among the earliest shape measures used for simplices. The mean ratio defined by the geo-
metric mean and the arithmetic mean of the eigenvalues of the transformation matrix turns
out to have also a strong geometrical interpretation related to the volume and the edges of
a simplex. With the introduction of the distortion matrix between a simplex and its regular
counterpart, other shape measures, following more on algebraic derivations rather than
geometric considerations, can be defined in terms of some scalar functions of the associated
transformation matrix.
6.2.1.1 Minimum solid angle
θ
The solid angle θ
i
at vertex
x
i
of tetrahedron T(
x
1
,
x
2
,
x
3
,
x
4
) is given by the surface area
formed by projecting each point on the face opposite to
x
i
to the unit sphere centred at
x
i
, as
shown in Figure 6.2. Since the area of a unit sphere is 4π, we have
0 ≤ θ
i
≤ 2π
