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

Schröppel and Wackerfuß Adv. Model. and Simul. in Eng. Sci. (2016) 3:27 DOI 10.1186/s40323-016-0074-8 RESEARCH ARTICLE Open Access Introducing the Logarithmic finite element method: a geometrically exact planar Bernoulli beam element Christian Schröppel and Jens Wackerfuß * * Correspondence: [email protected] Emmy Noether Research Group MISMO “Mechanical Instabilities in Self-similar Molecular Structures of Higher Order”, Institute of Structural Analysis, University of Kassel, Mönchebergstraße 7, 34125 Kassel, Germany 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. Keywords: Logarithmic finite element method, Geometrically exact beam, Finite rotations, Large deformations, Lie group theory, Bernoulli kinematics Background 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. The LogFE method focuses on the low-frequency part of a deformation and minimizes spurious high-frequency components in the solution. 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. In addition, we limit the degrees of freedom to coefficients related to rotations and dilatations at the nodes of the element. 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 © 2016 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 0123456789().,–: vol

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

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