Problem 7.9#
The local linear shape functions are defined in 7.56 as
\[\begin{equation*}
N_a (\xi) = \frac{1}{2} \left[ 1 + (-1)^a \, \xi \right]
\end{equation*}\]
and the local diffusivity matrix is defined in 7.63 as
\[\begin{equation*}
k_{ab}^A = \int_{x_A}^{x_{A+1}} \left( \partial_x N_a \right) \left( \partial_x N_b \right) \: \mathrm{d}x \quad.
\end{equation*}\]
To get the derivatives of the shape functions, we need to use the chain rule
\[\begin{equation*}
\partial_x N_a = \frac{\mathrm{d}N_a}{\mathrm{d} \xi}\frac{\mathrm{d} \xi}{\mathrm{d} x}.
\end{equation*}\]
yielding
\[\begin{equation*}
k_{ab}^A = \int_{x_A}^{x_{A+1}} \frac{\mathrm{d}N_a}{\mathrm{d} \xi} \frac{\mathrm{d}N_b}{\mathrm{d} \xi} \left[ \frac{\mathrm{d} \xi}{\mathrm{d} x} \right]^2 \: \mathrm{d}x \quad.
\end{equation*}\]
Next we can change the variable of integration as
\[\begin{equation*}
\mathrm{d}x = \frac{\mathrm{d}x }{\mathrm{d}\xi}\: \mathrm{d}\xi
\end{equation*}\]
and the limits
\[\begin{align*}
x = x_A &\rightarrow \xi = -1 \\
x = x_{A+1} &\rightarrow \xi = +1
\end{align*}\]
to produce
\[\begin{equation*}
k_{ab}^A = \int_{-1}^{1} \frac{\mathrm{d}N_a}{\mathrm{d} \xi} \frac{\mathrm{d}N_b}{\mathrm{d} \xi} \left[ \frac{\mathrm{d} \xi}{\mathrm{d} x} \right]^2 \: \frac{\mathrm{d}x }{\mathrm{d}\xi}\: \mathrm{d}\xi = \int_{-1}^{1} \frac{\mathrm{d}N_a}{\mathrm{d} \xi} \frac{\mathrm{d}N_b}{\mathrm{d} \xi} \frac{\mathrm{d} \xi}{\mathrm{d} x} \: \: \mathrm{d}\xi
\end{equation*}\]
as in 7.64. Now let us evaluate this integral using
\[\begin{equation*}
\frac{\mathrm{d}N_a}{\mathrm{d} \xi} = \frac{1}{2} (-1)^a \quad,
\end{equation*}\]
\[\begin{equation*}
\frac{\mathrm{d}N_b}{\mathrm{d} \xi} = \frac{1}{2} (-1)^b \quad,
\end{equation*}\]
\[\begin{equation*}
\frac{\mathrm{d}\xi}{\mathrm{d} x} = \frac{2}{\Delta x_A} \quad,
\end{equation*}\]
which gives
\[\begin{align*}
k_{ab}^A &= \frac{1}{4} \int_{-1}^{1} (-1)^a (-1)^b \frac{2}{\Delta x_A} \: \: \mathrm{d}\xi \\
&= \frac{1}{4} \int_{-1}^{1} (-1)^{a+b} \frac{2}{\Delta x_A} \: \: \mathrm{d}\xi \: = \frac{(-1)^{a+b}}{\Delta x_A}
\end{align*}\]