Hello,

It is reccommended to use one member with high order shape functions in combination with a trapezoidal scheme. However, this sounds contradicting to me. I would expect that a trapezoidal scheme is used only in combination with low order functions on multiple members. Wouldn’t a Gaussian quadrature be more suitable for high order functions so that the inner product of the FE formulation is calculated more accurately?

I’m grateful for any thoughts on this matter.

Regards,

Jelmer Polman

Dear Jelmer,

Gauss quadrature in BeamDyn is implemented in reduced form to improve efficiency and avoid shear locking. There is one fewer Gauss point than finite-element nodes. So, with e.g. a typical 7th order finite element, there will be only 6 Gauss quadrature points and the specified distributed mass and stiffness will be interpolated to these points. For smooth mass and stiffness, this works well, but for large variations in mass and stiffness, much of the variation is lost by the interpolation to a few points.

Trapezoidal quadrature in BeamDyn is implemented based on the number of blade input stations and can be further refined using Refine > 1. So, all of the distributed mass and stiffness data will be utilized. Often there are many more blade input stations (often > 20) than finite-element nodes, so trapezoidal quadrature is typically much more refined and not dependent on interpolation of the specified mass and stiffness distribution.

I hope that helps.

Best regards,

Dear Jason,

I’ve also read the article by Wang et al, 2017*. As I understood from your answer and the article, with a quassian quadrature, the structural data is linearly interpolated to the FE nodes using the neirest input stations and these FE nodes are also used as quadrature points. Hence, if the number of FE nodes is (much) less than the number of input stations, much of structural data is unused. Furthermore, if the data varies greatly inbetween stations, this interpolation might be inaccurate.

With a Trapezoidal quadrature, all quadrature points are forced to coincide with the input stations, and even more points are created if Refine is more than one. I thought that the structural data is used only in the FE nodes to properly weigh the shape functions, and the quadrature points are used to integrate the shape functions for the FE weak formulation. This would mean that just because the station location is used as an integration point, it does not necessarily mean that the known structural data is utilized at these points? What is the difference between the quadrature points and the FE nodes, and how exactly is the data utilized in the quadrature points?

Best regards,

Jelmer Polman

[*] Q. Wang et al: BeamDyn: a high-fidelity wind turbine blade solver in the FAST modular framework, Wind Energy 2017, 20, 1439-1462.

Dear Jelmer,