Introducing the Logarithmic finite element method: a geometrically exact planar Bernoulli beam element

Abstract

We propose a novel finite element formulation that significantly reduces the number of degrees of freedom necessary to obtain reasonably accurate approximations of the low-frequency component of the deformation in boundary-value problems. In contrast to the standard Ritz-Galerkin approach, the shape functions are defined on a Lie algebra-the logarithmic space-of the deformation function. We construct a deformation function based on an interpolation of transformations at the nodes of the finite element. In the case of the geometrically exact planar Bernoulli beam element presented in this work, these transformation functions at the nodes are given as rotations. However, due to an intrinsic coupling between rotational and translational components of the deformation function, the formulation provides for a good approximation of the deflection of the beam, as well as of the resultant forces and moments. As both the translational and the rotational components of the deformation function are defined on the logarithmic space, we propose to refer to the novel approach as the "Logarithmic finite element method", or "LogFE" method.

Simulation of viscoelastic Cosserat rods based on the geometrically exact dynamics of special Euclidean strands

“…Other methods consider the fundamental role of the special Euclidean group and the associated algebra . Sander, 23 Chirikjian, 24 and Sonneville et al 25,26 discretize the degrees of freedom at the group level, applying suitable techniques for the dynamics on a Lie manifold, while Zupan and Saje, 27,28 Češarek et al, 29 Su and Cesnik, 30 and Schröppel and Wackerfuß 31 perform the discretization on elements of the Lie algebra that are precisely the generalized strains of rod theory.…”

Simulation of viscoelastic Cosserat rods based on the geometrically exact dynamics of special Euclidean strands

“…Other methods consider the fundamental role of the special Euclidean group SE (3) and the associated algebra se (3). Sanders [24], Chirikjian [10], and Sonneville, Cardona & Brüls [28,29] discretize the degrees of freedom at the group level, applying suitable techniques for the dynamics on a Lie manifold, while Zupan & Saje [33,34], Češarek, Saje & Zupan [9], Su & Cesnik [31], and Schröppel & Wackerfuß [25] perform the discretization on elements of the Lie algebra that are precisely the generalized strains of rod theory.…”

Co‐rotational extension of the Logarithmic finite element method

“…In the Logarithmic finite element (LogFE) method, a novel finite element approach proposed by the authors [1,2], shape functions are defined on the Lie algebra of the deformation function, i.e.…”