A Legendre Spectral Finite Element Implementation of Geometrically Exact Beam Theory

Abstract

This paper considers Legendre spectral finite elements (LSFEs) for linear and nonlinear elastic deformation of composite beams. LSFEs are high-order Lagrangian-interpolant finite elements with nodes located at the Gauss-Lobatto-Legendre quadrature points. Geometrically exact beam theory (GEBT) is adopted as the theoretical framework, where coupling effects (which usually exist in composite structures) and geometric nonlinearity are taken into consideration. Preliminary results are shown for two example problems. In the first example, the planar linear deflection and natural frequencies of a tapered beam are calculated with LSFEs and with first-order finite elements found in a commercial code. In the second example, the planar nonlinear deflection of a beam subjected to a tip moment is examined with LSFEs and first-order finite elements. For both cases, the LSFEs exhibit exponential convergence rates and are dramatically more accurate than low-order finite elements for a given model size.

Legendre spectral finite elements for Reissner–Mindlin composite plates

“…Looking forward, we hope to extend this work in the creation of shell elements for linear and geometrically nonlinear deformation. We will build upon our success in applying the Legendre spectral element method to geometrically exact beam theory [24,25] for large elastic deformations.…”

Nonlinear Legendre Spectral Finite Elements for Wind Turbine Blade Dynamics

“…In this paper, we present a three-dimensional displacement-based implementation of the geometrically exact beam theory using LSFEs. This work builds on previous efforts that showed the implementation of 3D rotation parameters 9 and a demonstration example of two-dimensional nonlinear spectral beam elements 21 for static deformation. The code implemented in this work is in accordance with the new FAST modularization framework 26 , which allows coupled aero-hydro-servo-elastic simulation of both land-based and offshore wind turbine under realistic operating conditions.…”

“…LSFEs have seen successful use in the simulation of fluid dynamics [11][12][13] , two-dimensional elastic wave propagation in solid media in geophysics 14 , elastodynamics 15 , and acoustic wave propagation 16 . LSFEs have been applied to the linear-response analysis of beams [17][18][19][20][21] and plate elements [22][23][24] . Xiao and Zhong 25 reported a displacement-based implementation by LSFEs for two-dimensional static nonlinear beam deformation.…”

“…Xiao and Zhong 25 reported a displacement-based implementation by LSFEs for two-dimensional static nonlinear beam deformation. Their LSFEs were compared against a mixed-formulation low-order-FE GEBT code by Wang and Sprague 21 ; it was shown that the LSFEs provide exponential convergence rates, while the low-order FEs were limited to an algebraic convergence rate.…”

Partitioned nonlinear structural analysis of wind turbines using BeamDyn

“…Recently, researchers at the National Renewable Energy Laboratory (NREL) developed BeamDyn [1][2][3][4] , an opensource nonlinear structural dynamics module for composite wind turbine blade analysis, in the FAST modularization framework. BeamDyn, which is founded on geometrically exact beam theory (GEBT), was created to accurately simulate the large, nonlinear deflections of modern wind turbine blades that are designed with aero-elastic tailoring and complicated composite structures.…”