Keywords: geometrically exact beam, finite volume method, finite rotations, large displacements, Newton-Raphson

This study focuses on Finite-Volume (FV) discretisation of geometrically-exact beam theory and its efficacy to solve established benchmark cases. Several authors have investigated a range of geometrically nonlinear problems subjected to large displacements and rotations mostly using finite element (FE) approximations [1, 2]. Recent developments of Tuković et al. [3] have focused on the implementation of the geometrically-exact beam theory using the FV method to model shear-deformable circular cross-sectional beams. In this work, we use a FV solver to simulate distinct benchmark cases and provide an analogy of the FV method to the widely accepted FE approximations. The FV method is mostly popular in the field of computational fluid dynamics and only in the past few decades, has been applied to the field of solid mechanics [4]. This paper uses a cell-centred FV approach to discretise the spatial domain of the beam model into a set of uniform control volumes (CVs) and the governing equations are balanced across all the internal faces enclosed by the CV. The nonlinear stress resultant equations of spatial forces and moments are linearised and iteratively solved for incremental displacements and incremental rotations using a Newton-Raphson procedure. For the benchmark cases, the accuracy of results obtained using the FV method is compared against the existing numerical results achieved using the FE approach. The results of the research show that the simple, yet powerful and naturally conservative FV approach can be used as an alternative to the FE approximations for such large displacement and rotation problems.

Acknowledgments

This publication has emanated from research [conducted with the financial support of/supported in part by a grant from] Science Foundation Ireland under Grant number RC2302_2. For the purpose of Open Access, the author has applied a CC BY public copyright license to any Author Accepted Manuscript version arising from this submission.
Financial support is gratefully acknowledged from the Irish Research Council through the Laureate programme, grant number IRCLA/2017/45. Additionally, the authors want to acknowledge project affiliates, Bekaert, through the Bekaert University Technology Centre (UTC) at University College Dublin (www.ucd.ie/bekaert), and I-Form, funded by Science Foundation Ireland (SFI) Grant Number 16/RC/3872, co-funded under European Regional Development Fund and by I-Form industry partners. Provision of computational facilities and support from the DJEI/DES/SFI/HEA Irish Centre for High-End Computing (ICHEC, www.ichec.ie) and ResearchIT Sonic cluster, funded by UCD IT Services and the Research Office is gratefully acknowledged.

References

1. Simo, J.C., 1985. A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Computer methods in applied mechanics and engineering, 49(1), pp.55-70.

2. Ibrahimbegovic, A., 1997. On the choice of finite rotation parameters. Computer Methods in Applied Mechanics and Engineering, 149(1-4), pp.49-71.

3. Tuković Ž., De Jaeger P., Cardiff P., Ivanković A., 2019. A finite volume solver for geometrically exact Simo–Reissner beams. In: ECCOMAS MSF Thematic conference. Sarajevo, Bosnia and Herzegovina.

4. Cardiff, P. and Demirdžić, I., 2021. Thirty years of the finite volume method for solid mechanics. Archives of Computational Methods in Engineering, pp.1-60.