Variable-order finite elements for nonlinear, fully intrinsic beam equations

Abstract

Fully intrinsic equations and boundary conditions involve only force, moment, velocity, and angular velocity variables, but no displacement or rotation variables. This paper presents variable-order finite elements for the geometrically exact, nonlinear, fully intrinsic equations for both nonrotating and rotating beams. The finite element technique allows for hp-adaptivity. Results show that these finite elements lead to very accurate solutions for the static equilibrium state as well as for modes and frequencies for infinitesimal motions about that state. For the same number of variables, the accuracy of the finite elements increases with the order of the finite element. The results based on the Galerkin approximation (which is a special case of the present approach) are the most accurate but require evaluation of complex integrals. Cubic elements are shown to provide a near optimal combination of accuracy and complexity.

Nonlinear free vibrations of centrifugally stiffened uniform beams at high angular velocity

“…(29), (33) and (35), we note that, for example, the generalized displacement vector is composed of three parts: the initial displacement q 0 , the contribution of the trivial solution of the singular tangent operator, and, finally, the third part, which is defined by the general form P n À 1 j ¼ 1 a j :q j . This latter is orthogonal to the trivial solution and thus also to the second part.…”

“…This model has been used in a study of nonlinear forced vibrations of rotating anisotropic beams [17]. Different techniques of approximations are used in the literature for numerical computation, including the finite element method (FEM) in [31][32][33], the finite difference method (FDM) in [34] and the Galerkin method in [17,35]. In this present study, we apply a Galerkin approximation to this model, by choosing the weighting functions that represent the assumed modes themselves.…”

Chebyshev collocation method for the free vibration analysis of geometrically exact beams with fully intrinsic formulation

“…Patil and Althoff (2011) have proposed a Galerkin's method in which the Legendre polynomials are used as trial functions for the dynamic and free vibration analysis of fully intrinsic beam formulation. This approach has been extended further through the implementation of a variable order finite element by Patil and Hodges (2011).…”

Dynamic Instability of Beams Under Tip Follower Forces Using Geometrically Exact, Fully Intrinsic Equations

“…They are limited to simple discretization of the equations (Sotoudeh et al (2010) and Mardanpour et al (2014)), variable order finite element method (Patil and Hodges (2011)), Galerkin method (Patil and Althoff (2011)), and Chebyshev collocation method Ovesy (2014, 2015)). All these methods are successfully applied to the fully intrinsic beam equations for different problems.…”