A cell‐centered finite volume formulation of geometrically exact Simo–Reissner beams with arbitrary initial curvatures

This paper presents a novel total Lagrangian cell-centred finite volume formulation of geometrically exact beams with arbitrary initial curvature undergoing large displacements and finite rotations. The choice of rotation parametrisation, the mathematical formulation of the beam kinematics, conjugate strain measures and the linearisation of the strong form of governing equations is described. The finite volume based discretisation of the computational domain and the governing equations for each computational volume are presented. The discretised integral form of the equilibrium equations are solved using a block-coupled Newton-Raphson solution procedure. The efficacy of the proposed methodology is presented by comparing the simulated numerical results with classic benchmark test cases available in the literature. The objectivity of strain measures for the current formulation and mesh convergence studies for both initially straight and curved beam configurations are also discussed.

[1]  Ignacio Romero,et al.  The interpolation of rotations and its application to finite element models of geometrically exact rods , 2004 .

[2]  K. Bathe,et al.  Large displacement analysis of three‐dimensional beam structures , 1979 .

[3]  王东东,et al.  Computer Methods in Applied Mechanics and Engineering , 2004 .

[4]  Debasish Roy,et al.  A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization , 2009 .

[5]  M. Géradin,et al.  Kinematics and dynamics of rigid and flexible mechanisms using finite elements and quaternion algebra , 1988 .

[6]  J. Argyris.  An excursion into large rotations , 1982 .

[7]  J. Mäkinen.  Total Lagrangian Reissner's geometrically exact beam element without singularities , 2007 .

[8]  J. C. Simo,et al.  A finite strain beam formulation. The three-dimensional dynamic problem. Part I , 1985 .

[9]  M. Géradin,et al.  A beam finite element non‐linear theory with finite rotations , 1988 .

[10]  Miran Saje,et al.  The quaternion-based three-dimensional beam theory , 2009 .

[11]  A. Ibrahimbegovic,et al.  Computational aspects of vector-like parametrization of three-dimensional finite rotations , 1995 .

[12]  A. Ibrahimbegovic.  On the choice of finite rotation parameters , 1997 .

[13]  I. Demirdzic,et al.  Thirty Years of the Finite Volume Method for Solid Mechanics , 2018, Archives of Computational Methods in Engineering.

[14]  P. Ming,et al.  Static and free vibration analysis of functionally graded beams by combination Timoshenko theory and finite volume method , 2016 .

[15]  M. Todd,et al.  The mathematical theory of a higher-order geometrically-exact beam with a deforming cross-section , 2020, International Journal of Solids and Structures.

[16]  Robert Schmidt,et al.  Instability of Clamped-Hinged Circular Arches Subjected to a Point Load , 1975 .

[17]  J. C. Simo,et al.  A three-dimensional finite-strain rod model. Part II: Computational aspects , 1986 .

[18]  Miran Saje,et al.  Finite-element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures , 2003 .

[19]  A Finite Volume Method for Plate Buckling Analysis , 2006 .

[20]  Enzo Marino,et al.  Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams , 2016 .

[21]  Wolfgang A. Wall,et al.  An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods , 2014 .

[22]  Gordan Jelenić,et al.  A kinematically exact space finite strain beam model - finite element formulation by generalized virtual work principle , 1995 .

[23]  Hrvoje Jasak,et al.  A tensorial approach to computational continuum mechanics using object-oriented techniques , 1998 .

[24]  A. Ghanbari,et al.  A displacement finite volume formulation for the static and dynamic analysis of shear deformable circular curved beams , 2017 .

[25]  Alexander Popp,et al.  Geometrically Exact Finite Element Formulations for Slender Beams: Kirchhoff–Love Theory Versus Simo–Reissner Theory , 2016, 1609.00119.

[26]  Rakesh K. Kapania,et al.  A formulation and implementation of geometrically exact curved beam elements incorporating finite strains and finite rotations , 2003 .

[27]  Ahmed K. Noor,et al.  Mixed models and reduced/selective integration displacement models for nonlinear analysis of curved beams , 1981 .

[28]  M. Crisfield.  A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements , 1990 .

[29]  F. Delfanian,et al.  The orthogonal meshless finite volume method for solving Euler–Bernoulli beam and thin plate problems , 2011 .

[30]  N. Fallah,et al.  A cell vertex and cell centred finite volume method for plate bending analysis , 2004 .

[31]  M. Crisfield,et al.  Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[32]  Ignacio Romero,et al.  An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy–momentum conserving scheme in dynamics , 2002 .

[33]  M. Ebrahimnejad,et al.  Finite volume analysis of adaptive beams with piezoelectric sensors and actuators , 2014 .

[34]  M. Wheel,et al.  A finite volume method for analysing the bending deformation of thick and thin plates , 1997 .

[35]  Debasish Roy,et al.  Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam , 2008 .

[36]  Gordan Jelenić,et al.  Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics , 1999 .