A locking-free model for Reissner-Mindlin plates: Analysis and isogeometric implementation via NURBS and triangular NURPS

We study a reformulated version of Reissner–Mindlin plate theory in which rotation variables are eliminated in favor of transverse shear strains. Upon discretization, this theory has the advantage that the "shear locking" phenomenon is completely precluded, independent of the basis functions used for displacement and shear strains. Any combination works, but due to the appearance of second derivatives in the strain energy expression, smooth basis functions are required. These are provided by Isogeometric Analysis, in particular, NURBS of various degrees and quadratic triangular NURPS. We present a mathematical analysis of the formulation proving convergence and error estimates for all physically interesting quantities, and provide numerical results that corroborate the theory.

[1]  H. Saunders,et al.  Finite element procedures in engineering analysis , 1982 .

[2]  Paul Dierckx,et al.  On calculating normalized Powell-Sabin B-splines , 1997, Comput. Aided Geom. Des..

[3]  M. Fortin,et al.  ERROR ANALYSIS OF MIXED-INTERPOLATED ELEMENTS FOR REISSNER-MINDLIN PLATES , 1991 .

[4]  Hendrik Speleers,et al.  Isogeometric analysis with Powell–Sabin splines for advection–diffusion–reaction problems , 2012 .

[5]  J. Lions,et al.  Inequalities in mechanics and physics , 1976 .

[6]  Hendrik Speleers,et al.  Construction of Normalized B-Splines for a Family of Smooth Spline Spaces Over Powell–Sabin Triangulations , 2013 .

[7]  T. Hughes,et al.  ISOGEOMETRIC ANALYSIS: APPROXIMATION, STABILITY AND ERROR ESTIMATES FOR h-REFINED MESHES , 2006 .

[8]  C. Chinosi,et al.  Numerical analysis of some mixed finite element methods for Reissner-Mindlin plates , 1995 .

[9]  Dominique Chapelle,et al.  An optimal low-order locking-free finite element method for Reissner-Mindlin plates , 1998 .

[10]  Fehmi Cirak,et al.  Shear‐flexible subdivision shells , 2012 .

[11]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

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

[13]  C. Ross,et al.  Advanced finite element methods , 1998 .

[14]  Hendrik Speleers,et al.  From NURBS to NURPS geometries , 2013 .

[15]  Malcolm A. Sabin,et al.  Piecewise Quadratic Approximations on Triangles , 1977, TOMS.

[16]  D. Arnold,et al.  A uniformly accurate finite element method for the Reissner-Mindlin plate , 1989 .

[17]  Hendrik Speleers,et al.  Numerical solution of partial differential equations with Powell-Sabin splines , 2006 .

[18]  R. Echter,et al.  A hierarchic family of isogeometric shell finite elements , 2013 .

[19]  Larry L. Schumaker,et al.  Spline functions on triangulations , 2007, Encyclopedia of mathematics and its applications.

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

[21]  Hendrik Speleers,et al.  Multivariate normalized Powell-Sabin B-splines and quasi-interpolants , 2013, Comput. Aided Geom. Des..

[22]  Juhani Pitkäranta,et al.  Boundary subspaces for the finite element method with Lagrange multipliers , 1979 .

[23]  Giancarlo Sangalli,et al.  Anisotropic NURBS approximation in isogeometric analysis , 2012 .

[24]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

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

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

[27]  Hendrik Speleers,et al.  A normalized basis for quintic Powell-Sabin splines , 2010, Comput. Aided Geom. Des..

[28]  S. Agmon.  Lectures on Elliptic Boundary Value Problems , 1965 .

[29]  L. Beirao da Veiga,et al.  An isogeometric method for the Reissner-Mindlin plate bending problem , 2011, 1106.4436.

[30]  Rüdiger Verfürth,et al.  Error estimates for a mixed finite element approximation of the Stokes equations , 1984 .

[31]  Rolf Stenberg,et al.  A Refined Error Analysis of MITC Plate Elements , 2006 .