Notes on quadrature

In Kynema, we represent beams that can have highly variable properties along the length and we use a single high-order element for the whole beam. Material properties, including the sectional mass matrix \(\underline{\underline{M}}^*\), stiffness matrix \(\underline{\underline{C}}^*\), and twist \(\tau\), are defined by the user at stations located along the beam reference line. In typical finite-element beam implementations, \(P\)-point Gauss-Legendre quadrature is common. While that is often sufficient for uniform or linearly varying properties, it can be inadequate for highly variable material properties. Kynema provides users the option of Guass-Legendre or trapezoid-rule quadrature, with the former being suitable for constant-material-property beams and the latter for variable-property beams. Trapezoid-quadrature locations at each user-defined material point, and additional points are added between points in a user-defined manner that divides sections between user-defined propery points in an equidistant fashion.