Civil Engineering Reference
In-Depth Information
Figure 3.4
Block decomposition of a multi-connected domain into super-elements.
A point
p
in the reference domain, which is a triangle of convenient regular shape, say, a
right-angled triangle, is mapped onto the physical real domain through the use of appropri-
ate interpolation functions, as shown in Figure 3.3. For straight-edge triangles, we have
x
= H
1
x
1
+ H
2
x
2
+ H
3
x
3
and for six-node curved triangles with quadratic edges, we have
x
= H
1
x
1
+ H
2
x
2
+ H
3
x
3
+ H
4
x
4
+ H
5
x
5
+ H
6
x
6
where
x
i
are the nodal points of the triangle under consideration. By means of the FE interpola-
tion, a mesh of the reference domain can be reproduced on the actual physical domain by map-
ping properly all the points on the reference domain point by point onto the physical domain.
Based on the block decomposition (Bykat 1976; Sezer and Zeid 1991; Srinivasan et al.
1992; Egidi and Maponi 2008), a complex or multi-connected domain can be divided into
a number of super-elements, each of which can be further subdivided into smaller elements
of the same type, as shown in Figure 3.4.
3.2.2 Transfinite mapping
In the event that the curved boundary cannot be satisfactorily represented by FE polynomial
interpolation functions, more general analytical functions have to be used. The transfinite
mapping provides a means to map a unit square into a region bounded by four arbitrary
analytical curves. Let
C
1
(t),
C
2
(t),
C
3
(t) and
C
4
(t), t ∈ [0, 1], be the parametric bounding
curves of a region, as shown in Figure 3.5. For instance, a straight line between two points
A and B with parameter t is given by
C(t) = (1 - t)A + tB t ∈ [0, 1]
and a circular arc from point A to point B is given by
C(t) = M + r(cos(ϕ), sin(ϕ))
where ϕ = (1 − t)θ
a
+ tθ
b
, mid-point of AB, M = (A + B)/2, r = ||AB||/2 and θ
a
and θ
b
are,
respectively, the angles of MA and MB making with the x-axis.
