Civil Engineering Reference
In-Depth Information
(a)
(b)
(d)
(c)
Figure 6.20
(a) A domain to be modeled. (b) Triangular elements.
(c) Rectangular elements. (d) Rectangular and quadrilateral
elements.
closer to the actual geometry. However, also note that the elements in the inner
“rows” become increasingly slender (i.e., the height to base ratio is large). In gen-
eral, the ratio of the largest characteristic dimension of an element to the smallest
characteristic dimension is known as the
aspect ratio
. Large aspect ratios increase
the inaccuracy of the finite element representation and have a detrimental effect
on convergence of finite element solutions [8]. An aspect ratio of 1 is ideal but
cannot always be maintained. (Commercial finite element software packages pro-
vide warnings when an element's aspect ratio exceeds some predetermined limit.)
In Figure 6.20b, to maintain a reasonable aspect ratio for the inner elements, it
would be necessary to reduce the height of each row of elements as the center of
the sector is approached. This observation is also in keeping with the convergence
requirements of the
h-
refinement method. Although the triangular element can be
used to closely approximate a curved boundary, other considerations dictate a
relatively large number of elements and associated computation time.
If we consider rectangular elements as in Figure 6.20c (an intentionally
crude mesh for illustrative purposes), the problems are apparent. Unless the
elements are very small, the area of the domain excluded from the model (the
