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

Christian Schröppel,Jens Wackerfuß

doi:10.1186/s40323-016-0074-8

Christian Schröppel, Jens Wackerfuß

Open Access

https://doi.org/10.1186/s40323-016-0074-8

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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.

Highlights

We propose a novel finite element formulation, the Logarithmic finite element, or “LogFE” method, that significantly reduces the number of degrees of freedom necessary to obtain accurate approximations of boundary-value problems
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
We aim to identify, in a function space generated by polynomial shape functions on the logarithmic space, i.e. the space of the Lie algebra, a deformation function that, in the case of the Bernoulli beam, transforms the neutral axis of the given initial configuration so as to obtain an equilibrium configuration

Summary

Introduction

We propose a novel finite element formulation, the Logarithmic finite element, or “LogFE” method, that significantly reduces the number of degrees of freedom necessary to obtain accurate approximations of boundary-value problems. In order to keep the exposition as simple as possible, we restrict the model presented in this paper to the case of a planar Bernoulli beam, i.e. a beam endowed with Bernoulli kinematics embedded in the Euclidean plane. While we restrict the numerical examples to the evaluation of a beam consisting of one single element only, we explicitly show that degrees of freedom related to adjacent finite elements can be linked together by linear maps, based on geometrically meaningful continuity. The construction of a global finite element system based on the beam elements presented in this work is possible

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