Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Abstract We extend the isogeometric collocation method to the geometrically nonlinear beams. An exact kinematic formulation, able to represent three-dimensional displacements and rotations without any restriction in magnitude, is presented without the introduction of the moving frame concept. A displacement-based formulation is adopted. Full linearization of the strong form of the governing equations is derived consistently with the underlying geometric structure of the configuration manifold. Incremental rotations are parametrized through Eulerian rotation vectors and configuration updates are performed by means of the exponential map. Numerical tests demonstrate that the proposed combination of isogeometric collocation method with the chosen rotations parametrization results in an efficient computational scheme able to model complex problems with high accuracy.

[1]  T. Hughes,et al.  Isogeometric collocation for elastostatics and explicit dynamics , 2012 .

[2]  Thomas J. R. Hughes,et al.  Consistent linearization in mechanics of solids and structures , 1978 .

[3]  Jean-Louis Batoz,et al.  On the role of geometrically exact and second-order theories in buckling and post-buckling analysis of three-dimensional beam structures , 1996 .

[4]  K. C. Gupta,et al.  An historical note on finite rotations , 1989 .

[5]  Roland Wüchner,et al.  Nonlinear isogeometric spatial Bernoulli Beam , 2016 .

[6]  Thomas J. R. Hughes,et al.  Isogeometric collocation for large deformation elasticity and frictional contact problems , 2015 .

[7]  Alessandro Reali,et al.  Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods , 2012 .

[8]  Victor M. Calo,et al.  The role of continuity in residual-based variational multiscale modeling of turbulence , 2007 .

[9]  Simon R. Eugster,et al.  Geometric Continuum Mechanics and Induced Beam Theories , 2015 .

[10]  Alessandro Reali,et al.  An Introduction to Isogeometric Collocation Methods , 2015 .

[11]  Ignacio Romero,et al.  A comparison of finite elements for nonlinear beams: the absolute nodal coordinate and geometrically exact formulations , 2008 .

[12]  Yuri Bazilevs,et al.  Isogeometric fluid–structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines , 2012 .

[13]  P. Wriggers,et al.  Isogeometric large deformation frictionless contact using T-splines , 2014 .

[14]  Thomas J. R. Hughes,et al.  A large deformation, rotation-free, isogeometric shell , 2011 .

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

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

[17]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

[18]  Alessandro Reali,et al.  An isogeometric collocation approach for Bernoulli–Euler beams and Kirchhoff plates , 2015 .

[19]  I. Akkerman,et al.  Isogeometric analysis of free-surface flow , 2011, J. Comput. Phys..

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

[21]  Alessandro Reali,et al.  Isogeometric collocation methods for the Reissner–Mindlin plate problem , 2015 .

[22]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[23]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

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

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

[26]  G. S. Sekhon,et al.  Large Deformation -I , 2003 .

[27]  Peter Wriggers,et al.  Isogeometric contact: a review , 2014 .

[28]  Stuart S. Antman,et al.  Kirchhoff’s problem for nonlinearly elastic rods , 1974 .

[29]  Alessandro Reali,et al.  Locking-free isogeometric collocation methods for spatial Timoshenko rods , 2013 .

[30]  Leopoldo Greco,et al.  B-Spline interpolation of Kirchhoff-Love space rods , 2013 .

[31]  Jari Mäkinen,et al.  Rotation manifold SO(3) and its tangential vectors , 2008 .

[32]  Robert L. Taylor,et al.  On the role of frame-invariance in structural mechanics models at finite rotations , 2002 .

[33]  Dinar Camotim,et al.  On the differentiation of the Rodrigues formula and its significance for the vector‐like parameterization of Reissner–Simo beam theory , 2002 .

[34]  Leopoldo Greco,et al.  An isogeometric implicit G1 mixed finite element for Kirchhoff space rods , 2016 .

[35]  Alessandro Reali,et al.  Isogeometric collocation: Cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations , 2013 .

[36]  F. Auricchio,et al.  Single-variable formulations and isogeometric discretizations for shear deformable beams , 2015 .

[37]  Bernd Simeon,et al.  A finite volume method on NURBS geometries and its application in isogeometric fluid-structure interaction , 2012, Math. Comput. Simul..

[38]  T. Hughes,et al.  A Simple Algorithm for Obtaining Nearly Optimal Quadrature Rules for NURBS-based Isogeometric Analysis , 2012 .

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

[40]  M. Epstein,et al.  Differentiable manifolds and the principle of virtual work in continuum mechanics , 1980 .

[41]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[42]  Adnan Ibrahimbegovic,et al.  Quadratically convergent direct calculation of critical points for 3d structures undergoing finite rotations , 2000 .

[43]  Jari Mäkinen,et al.  Critical study of Newmark-scheme on manifold of finite rotations , 2001 .

[44]  H. Nguyen-Xuan,et al.  An extended isogeometric thin shell analysis based on Kirchhoff-Love theory , 2015 .

[45]  W. Rossmann.  Lie Groups: An Introduction through Linear Groups , 2002 .

[46]  Marion Kee,et al.  Analysis , 2004, Machine Translation.

[47]  P. M. Naghdi,et al.  Finite Deformation of Elastic Rods and Shells , 1981 .

[48]  Thomas J. R. Hughes,et al.  Isogeometric analysis of nearly incompressible large strain plasticity , 2014 .

[49]  T. Hughes,et al.  ISOGEOMETRIC COLLOCATION METHODS , 2010 .

[50]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

[51]  Alain Combescure,et al.  Locking free isogeometric formulations of curved thick beams , 2012 .

[52]  T. Hughes,et al.  Efficient quadrature for NURBS-based isogeometric analysis , 2010 .

[53]  Yuri Bazilevs,et al.  3D simulation of wind turbine rotors at full scale. Part I: Geometry modeling and aerodynamics , 2011 .

[54]  Roland Wüchner,et al.  Isogeometric shell analysis with Kirchhoff–Love elements , 2009 .

[55]  E. Reissner.  On one-dimensional finite-strain beam theory: The plane problem , 1972 .

[56]  J. Stuelpnagel.  On the Parametrization of the Three-Dimensional Rotation Group , 1964 .

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

[58]  Gerhard A. Holzapfel,et al.  Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science , 2000 .

[59]  Yuri Bazilevs,et al.  3D simulation of wind turbine rotors at full scale. Part II: Fluid–structure interaction modeling with composite blades , 2011 .

[60]  A. Ibrahimbegovic.  On finite element implementation of geometrically nonlinear Reissner's beam theory: three-dimensional curved beam elements , 1995 .

[61]  Alessandro Reali,et al.  Isogeometric Analysis of Structural Vibrations , 2006 .

[62]  J. C. Simo,et al.  On the dynamics of finite-strain rods undergoing large motions a geometrically exact approach , 1988 .

[63]  F. Auricchio,et al.  On the geometrically exact beam model: A consistent, effective and simple derivation from three-dimensional finite-elasticity , 2008 .

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

[65]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

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

[67]  Alessandro Reali,et al.  AN ISO GEOMETRIC ANALYSIS APPROACH FOR THE STUDY OF STRUCTURAL VIBRATIONS , 2006 .

[68]  John A. Evans,et al.  Isogeometric collocation: Neumann boundary conditions and contact , 2015 .

[69]  José Paulo Moitinho de Almeida,et al.  A hybrid-mixed finite element formulation for the geometrically exact analysis of three-dimensional framed structures , 2011 .

[70]  T. Hughes,et al.  Isogeometric fluid-structure interaction: theory, algorithms, and computations , 2008 .