NURBS-based three-dimensional analysis of geometrically nonlinear elastic structures

Abstract A comprehensive kinematical description with a corotational approach, which is based on exact polar decomposition for solid elements using isogeometric analysis is the main objective of this work. A numerical model using NURBS-based FEM is developed to deal with nonlinear problems in solid mechanics involving large rotations. From the theoretical viewpoint, the rate type form of state equation is formulated within the framework of isotropic infinitesimal elasticity. It is shown in particular that the state equation relates the Jaumann derivative of the Cauchy stress to the Eulerian strain rate. Illustrative examples are analyzed to verify the efficiency in solving a wide variety of strong nonlinear problems.

[1]  Valente Robertt Angelo Fontes.  Developments on Shell and Solid-Shell finite elements technology in nonlinear continuum mechanics , 2013 .

[2]  Gerald Wempner,et al.  Finite elements, finite rotations and small strains of flexible shells , 1969 .

[3]  Yu-Kan Hu,et al.  A one-point quadrature eight-node brick element with hourglass control , 1997 .

[4]  P. Bézier.  Numerical control : mathematics and applications , 1972 .

[5]  T. Hughes,et al.  Finite rotation effects in numerical integration of rate constitutive equations arising in large‐deformation analysis , 1980 .

[6]  Jeong Whan Yoon,et al.  Enhanced assumed strain (EAS) and assumed natural strain (ANS) methods for one‐point quadrature solid‐shell elements , 2008 .

[7]  C. D. Boor,et al.  On Calculating B-splines , 1972 .

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

[9]  K. Y. Sze,et al.  Popular benchmark problems for geometric nonlinear analysis of shells , 2004 .

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

[11]  Riesenfeld,et al.  Homogeneous Coordinates and Projective Planes in Computer Graphics , 1981, IEEE Computer Graphics and Applications.

[12]  I. J. Schoenberg.  Contributions to the problem of approximation of equidistant data by analytic functions. Part A. On the problem of smoothing or graduation. A first class of analytic approximation formulae , 1946 .

[13]  B. J. Hsieh,et al.  Non-Linear Transient Finite Element Analysis with Convected Co--ordinates , 1973 .

[14]  M. Cox.  The Numerical Evaluation of B-Splines , 1972 .

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

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

[17]  K. Bathe.  Finite Element Procedures , 1995 .

[18]  Dongdong Wang,et al.  An improved NURBS-based isogeometric analysis with enhanced treatment of essential boundary conditions , 2010 .

[19]  Armando Miguel Awruch,et al.  GEOMETRICALLY NON-LINEAR ANALYSIS IN ELASTODYNAMICS USING THE EIGHT-NODE FINITE ELEMENT WITH ONE-POINT QUADRATURE AND THE GENERALIZED-ALPHA METHOD , 2008 .

[20]  L. E. Malvern.  Introduction to the mechanics of a continuous medium , 1969 .

[21]  Carlos A. Felippa,et al.  A unified formulation of small-strain corotational finite elements: I. Theory , 2005 .

[22]  Yeong-bin Yang,et al.  Solution method for nonlinear problems with multiple critical points , 1990 .

[23]  T. Hughes,et al.  Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows , 2007 .

[24]  Wing Kam Liu,et al.  A multiple-quadrature eight-node hexahedral finite element for large deformation elastoplastic analysis , 1998 .

[25]  Richard R. Patterson,et al.  Projective transformations of the parameter of a Bernstein-Bézier curve , 1985, TOGS.

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

[27]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

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

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

[30]  I. J. Schoenberg.  Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions , 1988 .

[31]  Armando Miguel Awruch,et al.  Geometrically nonlinear static and dynamic analysis of shells and plates using the eight-node hexahedral element with one-point quadrature , 2004 .