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 commonly used for evaluating the underlying weak-form integrals. 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 Gauss-Legendre (GL) or Gauss-Lobatto-Legendre (GLL) quadrature. GL or GLL quadrature can be applied over the whole beam with a user-specified number of points, or in a composite fashion between user-defined material points. In the latter, a composite GLL quadrature with 2 points would be equivalent to using trapezoid rule integration over the full blade with an integration point at each user-defined material station.