Variational approach for one-dimensional heat conduction

Governing equation and functional

Steady 1D conduction with heat generation:

$$kA\frac{d^2T}{dx^2} + QA = 0$$

The equivalent functional (analogous to the total potential energy) is

$$\Pi = \int_0^L \left[\frac{kA}{2}\left(\frac{dT}{dx}\right)^2 - QA\,T\right]dx$$

The solution of the differential equation makes $\Pi$ stationary (this can be shown by taking $\delta\Pi = 0$ and integrating by parts, which gives back the governing equation).

Element approximation

For a linear element with nodal temperatures $T_1$, $T_2$:

$$T = N_1T_1 + N_2T_2,\quad N_1 = 1 - \frac{x}{L},\quad N_2 = \frac{x}{L}$$ $$\frac{dT}{dx} = \frac{1}{L}[-1\ \ 1]\begin{Bmatrix} T_1 \\ T_2 \end{Bmatrix} = [B]\{T\}$$

Minimising the functional

Substitute into $\Pi$:

$$\Pi = \frac{1}{2}\{T\}^T\left[\int_0^L kA\,[B]^T[B]\,dx\right]\{T\} - \{T\}^T\int_0^L QA\,[N]^T dx$$

Setting $\dfrac{\partial\Pi}{\partial T_i} = 0$ gives $[k]\{T\} = \{f\}$, where

$$[k] = \int_0^L kA\,[B]^T[B]\,dx = \frac{kA}{L}\begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$$ $$\{f\} = \int_0^L QA\begin{Bmatrix} N_1 \\ N_2 \end{Bmatrix}dx = \frac{QAL}{2}\begin{Bmatrix} 1 \\ 1 \end{Bmatrix}$$

Element equation

$$\frac{kA}{L}\begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}\begin{Bmatrix} T_1 \\ T_2 \end{Bmatrix} = \frac{QAL}{2}\begin{Bmatrix} 1 \\ 1 \end{Bmatrix} + \begin{Bmatrix} q_1 \\ q_2 \end{Bmatrix}$$

where $q_1$, $q_2$ are nodal heat flows at the ends (natural boundary conditions). The generated heat is shared equally between the two nodes. Convection at an end with coefficient $h$ and ambient temperature $T_\infty$ adds $hA$ to the stiffness term at that node and $hAT_\infty$ to the load.

The assembly, boundary conditions ($T$ specified at a node) and solution follow the same steps as for the bar problem.