Improved numerical integration for locking treatment in isogeometric structural elements. Part II: Plates and shells

Abstract B-spline reduced quadrature rules are proposed in the context of isogeometric analysis. When performing a full Gaussian integration, the high regularity provided by spline basis functions strengthens the locking phenomena and deteriorates the performance of Reissner–Mindlin elements. The uni-dimensional B-spline-based quadrature rules, given in a previous paper (part I), are extended to multi-dimensional problems such as plates and shells. The improved reduced integration schemes are constructed using a tensor product of the uni-dimensional schemes. A single numerical quadrature is performed for bending, transverse shear and membrane terms, without introducing Hourglass modes. The proposed isogeometric reduced elements are free from membrane and transverse shear locking. Convergence is first assessed in plate problems with several aspect ratios and then in the shell obstacle course problems. The resulting under-integrated elements exhibit better accuracy and computational efficiency.

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

[2]  Habibou Maitournam,et al.  Selective and reduced numerical integrations for NURBS-based isogeometric analysis , 2015 .

[3]  Thomas J. R. Hughes,et al.  A simple and efficient finite element for plate bending , 1977 .

[4]  G. Prathap.  The Finite Element Method in Structural Mechanics , 1993 .

[5]  T. Belytschko,et al.  Membrane Locking and Reduced Integration for Curved Elements , 1982 .

[6]  Thomas J. R. Hughes,et al.  Blended isogeometric shells , 2013 .

[7]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[8]  Roberto Frias,et al.  The deformation of cylindrical shells subjected to radial loads , 2010 .

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

[10]  Manfred Bischoff,et al.  The discrete strain gap method and membrane locking , 2005 .

[11]  Thomas J. R. Hughes,et al.  Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis , 2014 .

[12]  Manfred Bischoff,et al.  Numerical efficiency, locking and unlocking of NURBS finite elements , 2010 .

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

[14]  O. C. Zienkiewicz,et al.  Reduced integration technique in general analysis of plates and shells , 1971 .

[15]  Gangan Prathap,et al.  Mechanics of field-consistency in finite element analysis—a penalty function approach , 1987 .

[16]  Gangan Prathap,et al.  Reduced integration and the shear-flexible beam element , 1982 .

[17]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

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

[19]  E. Ramm,et al.  A unified approach for shear-locking-free triangular and rectangular shell finite elements , 2000 .

[20]  Ray W. Clough,et al.  Improved numerical integration of thick shell finite elements , 1971 .

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

[22]  R. L. Harder,et al.  A proposed standard set of problems to test finite element accuracy , 1985 .

[23]  J. Batoz,et al.  Modélisation des structures par éléments finis: poutres et plaques , 1990 .

[24]  Thomas J. R. Hughes,et al.  Isogeometric shell analysis: The Reissner-Mindlin shell , 2010 .

[25]  Miran Saje,et al.  On A proposed standard set of problems to test finite element accuracy: the twisted beam , 2004 .

[26]  Ernst Rank,et al.  On the accuracy of p-version elements for the Reissner-Mindlin plate problem , 1998 .

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

[28]  J. C. Simo,et al.  A CLASS OF MIXED ASSUMED STRAIN METHODS AND THE METHOD OF INCOMPATIBLE MODES , 1990 .

[29]  K. S. Lo,et al.  Computer analysis in cylindrical shells , 1964 .

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

[31]  L. Donnell,et al.  Stability of Thin-Walled Tubes Under Torsion , 1934, Journal of Fluids Engineering.

[32]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[33]  Wing Kam Liu,et al.  Stress projection for membrane and shear locking in shell finite elements , 1985 .

[34]  B. Simeon,et al.  Isogeometric Reissner–Mindlin shell analysis with exactly calculated director vectors , 2013 .

[35]  Gangan Prathap,et al.  A linear thick curved beam element , 1986 .

[36]  Theodore H. H. Pian,et al.  Alternative ways for formulation of hybrid stress elements , 1982 .

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

[38]  John A. Evans,et al.  Robustness of isogeometric structural discretizations under severe mesh distortion , 2010 .

[39]  Yuri Bazilevs,et al.  The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches , 2010 .

[40]  Alessandro Reali,et al.  Studies of Refinement and Continuity in Isogeometric Structural Analysis (Preprint) , 2007 .

[41]  J. M. Kennedy,et al.  Hourglass control in linear and nonlinear problems , 1983 .

[42]  Habibou Maitournam,et al.  Improved numerical integration for locking treatment in isogeometric structural elements, Part I: Beams , 2014 .

[43]  Medhat A. Haroun,et al.  Reduced and selective integration techniques in the finite element analysis of plates , 1978 .

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

[45]  D. Malkus,et al.  Mixed finite element methods—reduced and selective integration techniques: a unification of concepts , 1990 .

[46]  A. Combescure,et al.  Efficient isogeometric NURBS-based solid-shell elements: Mixed formulation and B-method , 2013 .

[47]  E. Reissner.  On a Variational Theorem in Elasticity , 1950 .

[48]  Yuri Bazilevs,et al.  Rotation free isogeometric thin shell analysis using PHT-splines , 2011 .

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