Steady-state heat transfer in a nonuniform. plate in the (x,y) plane is governed by
where T is the temperature, f(x,y) is the internal heat generation per unit volume, and k is
the thermal conductivity. We consider the heat flow problem on the quadrilateral domain Ω of
Figure 1. The bottom and top boundaries are kept at temperatures T0 and 2T0, respectively,
while the other boundaries are insulated, meaning that the heat flux through these boundaries
is zero, i.e., ∂T/∂x = 0.
We shall assume a uniform. heat source and take units such that a = 1, k = 1 and f = 1.
a) Solve the FE equation for the nodal temperatures on the mesh depicted in Figure 2. Use
the local and global node numbering indicated.
b) Do the same for the refined mesh of Figure 3. Compare the two solutions.
c) For the solution in b) compute the heat flux through the boundary at global nodes 2 and
7.
Problem 2:
a) Show that the shape functions for the normal 6-node triangular element
are given by
1
N1(x,y) = (1−x−y)(1−2x−2y),
N2(x,y) = x(2x−1),
N3(x,y) = y(2y −1),
N4(x,y) = 4x(1−x−y),
N5(x,y) = 4xy,
N6(x,y) = 4y(1−x−y).
b) Use the element in a) as the parent element in an isoparametric transformation to compute
the stiffness matrix for the Laplacian ∂2∂x2 + ∂2∂y2 on the following 6-node curved element in
the shape of a quarter of a circular disc:
c) The equation governing small lateral deflections z of a uniform. membrane subjected to a
lateral (dimensionless) pressure p is given by
∂2z
∂x2 +
∂2z
∂y2 +p = 0.
Solve an appropriate FE equation to find the deflection at the centre of a circular mem-
brane of radius 1 that is fixed (i.e., z = 0) at the boundary. Assume uniform. pressure
p = 1.
Hint: use the result in b).
d) Discuss the solution. Compute and physically interpret the boundary vector.
(Due in date: 26 January 2018)