Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

108 Incompressible laminar ¯ow
This approximation is made to be a least square approximation to the actual
distribution of curvatures, i.e.
…
N T N @2
"
@xi@x j
#
h
@2
ÿ d
@xi@x j
ˆ 0 …4:29†
and integrating by parts to give
@2 @xi@xj ˆ M ÿ1
…
N T 2 h
@
@xi@x j
d ˆÿM ÿ1
… T
@N @N
d
@xi @xj ~ …4:30†
where M is the mass matrix given by
…
M ˆ N T N d …4:31†
which of course can be `lumped'.
4.5.2 Element elongation
Elongated elements are frequently introduced to deal with `one-dimensional'
phenomena such as shocks, boundary layers, etc. The ®rst paper dealing with such
elongation was presented as early as 1987 by Peraire et al. 22 and later by many authors
for ¯uid mechanics and other problems. 70ÿ73 But the possible elongation was limited
by practical considerations if a general mesh of triangles was to be used. An
alternative to this is to introduce a locally structured mesh in shocks and boundary
layers which connects to the completely unstructured triangles. This idea has been
extensively used by Hassan et al., 39;53;56 Zienkiewicz and Wu 50 and Marchant
et al. 65 in the compressible ¯ow context. In both procedures it is necessary to establish
the desired elongation of elements. Obviously in completely parallel ¯ow phenomena
no limit on elongation exists but in a general ®eld the elongation ratio de®ning the
maximum to minimum size of the element can be derived by considering curvatures.
Thus the local error is proportional to the curvature and making h 2 times the
curvature equal to a constant, we immediately derive the ratio h max=h min.
In Fig. 4.13, X 1 and X 2 are the directions of the minimum and maximum principal
values of the curvatures. Thus for an equal distribution of the interpolation error we
can write for each nodey
h 2 min
@ 2
@X 2 2
which gives us the stretching ratio s as
ˆ h 2 max
s ˆ hmax ˆ
hmin v
u
t
@ 2
@X 2 1
@ 2
@X 2 2
@ 2
@X 2 1
ˆ C …4:32†
…4:33†
y Principal curvatures and directions can be found in a manner analogous to that of the determination of
principal stresses and their directions. Procedures are described in standard engineering texts.