flw2ts¶

Purpose:

Compute heat flux and temperature gradients in a triangular heat flow element.

Syntax:

[es, et] = flw2ts(ex, ey, D, ed)

Description:

flw2ts computes the heat flux vector es and the temperature gradient et (or corresponding quantities) in a triangular heat flow element.

The input variables ex, ey and the matrix D are defined in flw2te. The vector ed contains the nodal temperatures \(\mathbf{a}^e\) of the element and is obtained by the function extract as

\[\mathbf{ed} = (\mathbf{a}^e)^T = [\;T_1\;\; T_2\;\; T_3\;]\]

The output variables

\[\mathbf{es} = \mathbf{q}^T = \left[\; q_x \; q_y \;\right]\]

\[\mathbf{et} = (\nabla T)^T = \left[\begin{array}{l} \frac{\partial T}{\partial x}\;\;\frac{\partial T}{\partial y} \end{array} \right]\]

contain the components of the heat flux and the temperature gradient computed in the directions of the coordinate axis.

Theory:

The temperature gradient and the heat flux are computed according to

\[\nabla T = \bar{\mathbf{B}}\;\mathbf{C}^{-1}\;\mathbf{a}^e\]

\[\mathbf{q} = - \mathbf{D} \nabla T\]

where the matrices \(\mathbf{D}\), \(\bar{\mathbf{B}}\), and \(\mathbf{C}\) are described in flw2te. Note that both the temperature gradient and the heat flux are constant in the element.