flw2qe¶

Purpose:

Compute element stiffness matrix for a quadrilateral heat flow element.

Syntax:

Ke = flw2qe(ex, ey, ep, D)
[Ke, fe] = flw2qe(ex, ey, ep, D, eq)

Description:

flw2qe provides the element stiffness (conductivity) matrix Ke and the element load vector fe for a quadrilateral heat flow element.

The element nodal coordinates \(x_1\), \(y_1\), \(x_2\) etc, are supplied to the function by ex and ey, the element thickness \(t\) is supplied by ep and the thermal conductivities (or corresponding quantities) \(k_{xx}\), \(k_{xy}\) etc are supplied by D.

\[\begin{split}\begin{array}{l} \mathbf{ex} = [\, x_1 \;\; x_2 \;\; x_3 \;\; x_4 \,] \\ \mathbf{ey} = [\, y_1 \;\; y_2 \;\; y_3 \;\; y_4 \,] \end{array} \qquad \mathbf{ep} = \left[\, t \,\right] \qquad \mathbf{D} = \left[ \begin{array}{cc} k_{xx} & k_{xy} \\ k_{yx} & k_{yy} \end{array} \right]\end{split}\]

If the scalar variable eq is given in the function, the element load vector \(\mathbf{fe}\) is computed, using

\[\mathbf{eq} = \left[\, Q \,\right]\]

where \(Q\) is the heat supply per unit volume.

Theory:

In computing the element matrices, a fifth degree of freedom is introduced. The location of this extra degree of freedom is defined by the mean value of the coordinates in the corner points. Four sets of element matrices are calculated using flw2te. These matrices are then assembled and the fifth degree of freedom is eliminated by static condensation.