Element Stiffness Matrix¶
The element stiffness matrix in its matrix formulation is given by
\[\boldsymbol{K}^{e}
=
\int \limits_{\Omega^{e}}
{\boldsymbol{B}^{e}}^{T} \boldsymbol{C}^{e} \boldsymbol{B}^{e}
\,\mathrm{d} \Omega.\]
After transforming the integral from real space to reference space the expression becomes
\[\boldsymbol{K}^{e}
=
\int \limits_{-1}^{1}
\int \limits_{-1}^{1}
{\boldsymbol{B}^{e}}^{T} \boldsymbol{C}^{e} \boldsymbol{B}^{e} \det\bigl(\boldsymbol{J}^{e}\bigr)
\,\mathrm{d} \xi
\,\mathrm{d} \eta.\]
In many cases this integral can not be solved analytically. There are, however, a handful of ways to approximate it numerically. The so-called Gauss quadrature is a widely employed method in finite element software for the approximation of this integral.
| Number of points, \(n\) | Points, \(x_{i}\) | Weights, \(w_{i}\) |
|---|---|---|
| \(1\) | \(0\) | \(2\) |
| \(2\) | \(\pm \sqrt{\dfrac{1}{3}}\) | \(1\) |
| \(3\) | \(0\) | \(\dfrac{8}{9}\) |
| \(\pm \sqrt{\dfrac{3}{5}}\) | \(\dfrac{5}{9}\) |
In our case we are going to use two Gauss points with their corresponding weights.
\[\boldsymbol{K}^{\mathrm{e}}
=
\sum \limits_{i = 1}^{2}
\sum \limits_{j = 1}^{2}
{\boldsymbol{B}^{\mathrm{e}}}^{T}(\xi_{i}, \eta_{j})
\boldsymbol{C}^{\mathrm{e}}
\boldsymbol{B}^{\mathrm{e}}(\xi_{i}, \eta_{j})
\det\bigl(\boldsymbol{J}^{\mathrm{e}}(\xi_{i}, \eta_{j})\bigr)
w_{j}
w_{i}\]
And now we can implement a routine to give us the element stiffness matrix.
-
Element.stiffness_matrix()¶ Return the stiffness matrix of the element.
Returns: The stiffness matrix of the element. Return type: numpy.ndarray