Civil Engineering Reference
In-Depth Information
up with tetrahedral elements and other element types, it is necessary to assess the quality
of these elements so as to have an overall evaluation of the FE mesh used in the analysis.
Trilinear eight-node hexahedral elements are the most popular, and it is crucial that we
could develop a generic scheme to measure its shape properties.
For the four-node tetrahedral element, the deformation is homogeneous and the Jacobian
(transformation) matrix is a constant, which can be evaluated at any point within the ele-
ment. As for distorted hexahedral elements, the deformation is not homogeneous, and the
Jacobian matrix depends on the point where it is evaluated. This was the major difficulty
defying early attempts in defining logical and consistent shape measures for hexahedral
elements. For simplicity, shape measured can be defined based on the Jacobian matrix
evaluated at some interior points of the element such as the centroid or the Gaussian points.
However, this computation is more expensive, and worse still, the Jacobian at the centroid
may not be sensitive enough to detect degenerate cases on the boundary, as shown in Figure
6.7. By extending the shape measure for simplices, Knupp (2003a,b) proposed a genius
shape measure for quadrilaterals and hexahedra, which turns out to be a generic scheme
applicable to general polyhedron. For each vertex of a hexahedral element H, there is an
associated tetrahedron formed by the vertex and the three neighbouring vertices (neigh-
bouring vertices are on the same edge of the hexahedron). As shown in Figure 6.8, the asso-
ciated tetrahedron for vertex 1 is tetrahedron T
1
= T(1,2,4,5), and similarly, the associated
tetrahedron for vertex 8 is tetrahedron T
8
= T(8,7,5,4). As the reference tetrahedron for a
cube is given by the identity matrix, the transformation
F
1
from the reference tetrahedron
to tetrahedron T
1
is given by
F
1
= [
a
1
a
2
a
3
] = [
x
2
-
x
1
x
4
-
x
1
x
5
-
x
1
]
η
ξ
Figure 6.7
Degeneracy on the boundary cannot be detected by Jacobian at an interior point.
8
8
7
7
5
4
5
6
Reference
tetrahedron
a
3
a
2
6
F
1
1
3
4
.
3
a
1
1
2
2
Figure 6.8
Associated tetrahedron at vertices.
