flw2i4s¶

Purpose:

Compute heat flux and temperature gradients in a 4 node isoparametric heat flow element.

Syntax:

[es, et, eci] = flw2i4s(ex, ey, ep, D, ed)

Description:

flw2i4s computes the heat flux vector es and the temperature gradient et (or corresponding quantities) in a 4 node isoparametric heat flow element.

The input variables \(\mathbf{ex}\), \(\mathbf{ey}\), \(\mathbf{ep}\) and the matrix \(\mathbf{D}\) are defined in flw2i4e. The vector \(\mathbf{ed}\) contains the nodal temperatures \(\mathbf{a}^e\) of the element and is obtained by extract as

\[\mathbf{ed} = (\mathbf{a}^e)^T = [\;T_1\;\; T_2\;\; T_3\;\; T_4\;]\]

The output variables

\[\begin{split}\mathbf{es} = \bar{\mathbf{q}}^T = \left[ \begin{array}{cc} q^1_x & q^1_y \\ q^2_x & q^2_y \\ \vdots & \vdots \\ q^{n^2}_x & q^{n^2}_y \end{array} \right]\end{split}\]

\[ \begin{align}\begin{aligned}\begin{split}\mathbf{et} = (\bar {\nabla} T)^T = \left[ \begin{array}{cc} \frac{\partial T}{\partial x}^1 & \frac{\partial T}{\partial y}^1 \\ \frac{\partial T}{\partial x}^2 & \frac{\partial T}{\partial y}^2 \\ \vdots & \vdots \\ \frac{\partial T}{\partial x}^{n^2} & \frac{\partial T}{\partial y}^{n^2} \end{array} \right]\end{split}\\\begin{split}\qquad \mathbf{eci} = \left[ \begin{array}{cc} x_1 & y_1 \\ x_2 & y_2 \\ \vdots & \vdots \\ x_{n^2} & y_{n^2} \end{array} \right]\end{split}\end{aligned}\end{align} \]

contain the heat flux, the temperature gradient, and the coordinates of the integration points. The index \(n\) denotes the number of integration points used within the element, cf. flw2i4e.

Theory:

The temperature gradient and the heat flux are computed according to

\[\nabla T = \mathbf{B}^e\,\mathbf{a}^e\]

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

where the matrices \(\mathbf{D}\), \(\mathbf{B}^e\), and \(\mathbf{a}^e\) are described in flw2i4e, and where the integration points are chosen as evaluation points.