Polynomial shape functions

The basis function vector is generated with row-stacking of the individual lagrange polynomials. Each polynomial defined in the interval \([-1,1]\) is a function of the parameter \(r\). The curve parameters matrix \(\boldsymbol{A}\) is of symmetric shape due to the fact that for each evaluation point \(r_j\) exactly one basis function \(h_j(r)\) is needed.

\[\boldsymbol{h}(r) = \boldsymbol{A}^T \boldsymbol{r}(r)\]

Curve parameter matrix

The evaluation of the curve parameter matrix \(\boldsymbol{A}\) is carried out by boundary conditions. Each shape function \(h_i\) has to take the value of one at the associated nodal coordinate \(r_i\) and zero at all other nodal coordinates.

\[ \begin{align}\begin{aligned}\boldsymbol{A}^T \boldsymbol{R} &= \boldsymbol{I} \qquad \text{with} \qquad \boldsymbol{R} = \begin{bmatrix}\boldsymbol{r}(r_1) & \boldsymbol{r}(r_2) & \dots & \boldsymbol{r}(r_p)\end{bmatrix}\\\boldsymbol{A}^T &= \boldsymbol{R}^{-1}\end{aligned}\end{align} \]

Interpolation and partial derivatives

The approximation of nodal unknowns \(\hat{\boldsymbol{u}}\) as a function of the parameter \(r\) is evaluated as

\[\boldsymbol{u}(r) \approx \hat{\boldsymbol{u}}^T \boldsymbol{h}(r)\]

For the calculation of the partial derivative of the interpolation field w.r.t. the parameter \(r\) a simple shift of the entries of the parameter vector is enough. This shifted parameter vector is denoted as \(\boldsymbol{r}^-\). A minus superscript indices the negative shift of the vector entries by \(-1\).

\[ \begin{align}\begin{aligned}\frac{\partial \boldsymbol{u}(r)}{\partial r} &\approx \hat{\boldsymbol{u}}^T \frac{\partial \boldsymbol{h}(r)}{\partial r}\\\frac{\partial \boldsymbol{h}(r)}{\partial r} &= \boldsymbol{A}^T \boldsymbol{r}^-(r) \qquad \text{with} \qquad r_0^- = 0 \qquad \text{and} \qquad r_i^- = \frac{r^{(i-1)}}{(i-1)!} \qquad \text{for} \qquad i=(1 \dots p)\end{aligned}\end{align} \]

n-dimensional shape functions

Multi-dimensional shape function matrices \(\boldsymbol{H}_{2D}, \boldsymbol{H}_{3D}\) are simply evaluated as dyadic (outer) vector products of one-dimensional shape function vectors. The multi-dimensional shape function vector is a one-dimensional representation (flattened version) of the multi-dimensional shape function matrix.

\[ \begin{align}\begin{aligned}\boldsymbol{H}_{2D}(r,s) &= \boldsymbol{h}(r) \otimes \boldsymbol{h}(s)\\\boldsymbol{H}_{3D}(r,s,t) &= \boldsymbol{h}(r) \otimes \boldsymbol{h}(s) \otimes \boldsymbol{h}(t)\end{aligned}\end{align} \]