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.
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.
Interpolation and partial derivatives
The approximation of nodal unknowns \(\hat{\boldsymbol{u}}\) as a function of the parameter \(r\) is evaluated as
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 \(-1\).
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.
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