Space-Time Finite Element Method for Fully Intrinsic Equations of Geometrically Exact Beam

Abstract

In this paper, a space-time finite element method based on a Galerkin-weighted residual method is proposed to solve the nonlinear fully intrinsic equations of geometrically exact beam which are first-order partial differential equations about time and space. Therefore, it is natural to discretize it in time and space simultaneously. Considering the continuity and intrinsic boundary conditions in the spatial direction and the continuity and periodic boundary conditions in the time direction, the boundary value scheme of space-time finite element for solving the full intrinsic equations is derived. This method has been successfully applied to the static analysis and dynamic response solution of the fully intrinsic equations of nonlinear geometrically exact beam. The numerical results of several examples are compared with the analytical solution, existing algorithms, and literature to illustrate the applicability, accuracy and efficiency of this method.

Mass and Force Lumping: An Essential Enhancement to the Intrinsic Beam Finite Element Discretization

“…For high-altitude, long-endurance aircraft with large flexibility and a high aspect ratio, the geometrically exact beam model proposed by Hodges [2] is one of the most widely used approaches. A number of very flexible aircraft simulation frameworks based on the geometrically exact beam model have been created [3][4][5][6][7], often using nonlinear beam models to represent the structural response, combined with low-order aerodynamic descriptions to reduce coupled nonlinear aeroelastic problems to a manageable scale for time domain calculations. There are several versions of the geometrically exact beam model, such as the displacement-based formulation [8], strain-based formulation [9,10], and fully intrinsic formulation [11,12].…”

“…For the fully intrinsic formulation, there are many forms of spatial discretization, such as the central difference method [17][18][19], Galerkin approach [3,6,20], and generalized differential quadrature method [21,22], etc. Upon examining the spatial discretization outcomes of the fully intrinsic beam formulation in the aforementioned literature, we discovered that, with the exception of [3], none have delved into the analysis of the system's equation form post spatial discretization.…”