Contact mechanics with maximum-entropy meshfree approximants blended with isogeometric analysis on the boundary

A coupled isogeometric-meshless approach is applied to the simulation of contact.A new strategy is proposed for the numerical integration.Some numerical examples are studied, including a gear contact simulation. This paper explores the simulation of small deformation contact problems with a coupled isogeometric-meshless formulation, based on local maximum-entropy (LME) approximants. The framework of isogeometric analysis (IGA) is employed on a thin region close to the boundary of the domain and the meshless formulation in its interior. The method maintains the advantages of IGA in the simulation of contact and introduces a higher flexibility in the discretization of complex shapes. In addition, a new integration strategy is presented for the IGA-LME blending and its advantages respect to the original approach are discussed.

[1]  Nicholas M. Patrikalakis,et al.  Approximation of involute curves for CAD-system processing , 2007, Engineering with Computers.

[2]  Jia Lu,et al.  Isogeometric contact analysis: Geometric basis and formulation for frictionless contact , 2011 .

[3]  Magdalena Ortiz,et al.  Local maximum‐entropy approximation schemes: a seamless bridge between finite elements and meshfree methods , 2006 .

[4]  Stephen Demko,et al.  On the existence of interpolating projections onto spline spaces , 1985 .

[5]  Marino Arroyo,et al.  Blending isogeometric analysis and local maximum entropy meshfree approximants , 2013 .

[6]  Bo Li,et al.  Optimal transportation meshfree approximation schemes for fluid and plastic flows , 2010 .

[7]  Dongdong Wang,et al.  A consistently coupled isogeometric-meshfree method , 2014 .

[8]  Ruben Sevilla,et al.  Numerical integration over 2D NURBS-shaped domains with applications to NURBS-enhanced FEM , 2011 .

[9]  Bernd Hamann,et al.  Iso‐geometric Finite Element Analysis Based on Catmull‐Clark : ubdivision Solids , 2010, Comput. Graph. Forum.

[10]  Marino Arroyo,et al.  Second‐order convex maximum entropy approximants with applications to high‐order PDE , 2013 .

[11]  Marino Arroyo,et al.  On the optimum support size in meshfree methods: A variational adaptivity approach with maximum‐entropy approximants , 2010 .

[12]  Antonio Huerta,et al.  3D NURBS‐enhanced finite element method (NEFEM) , 2008 .

[13]  O. Botella,et al.  On a collocation B-spline method for the solution of the Navier-Stokes equations , 2002 .

[14]  Jiansong Deng,et al.  Polynomial splines over hierarchical T-meshes , 2008, Graph. Model..

[15]  Marino Arroyo,et al.  Efficient implementation of Galerkin meshfree methods for large-scale problems with an emphasis on maximum entropy approximants , 2015 .

[16]  John A. Evans,et al.  Isogeometric analysis using T-splines , 2010 .

[17]  M. Arroyo,et al.  Thin shell analysis from scattered points with maximum‐entropy approximants , 2010, International Journal for Numerical Methods in Engineering.

[18]  Peter Wriggers,et al.  Computational Contact Mechanics , 2002 .

[19]  M. Arroyo,et al.  Nonlinear manifold learning for meshfree finite deformation thin‐shell analysis , 2013 .

[20]  Per-Olof Persson,et al.  A Simple Mesh Generator in MATLAB , 2004, SIAM Rev..

[21]  Yuri Bazilevs,et al.  A coupled IGA–Meshfree discretization of arbitrary order of accuracy and without global geometry parameterization , 2015 .

[22]  T. Belytschko,et al.  THE NATURAL ELEMENT METHOD IN SOLID MECHANICS , 1998 .

[23]  Antonio Huerta,et al.  Imposing essential boundary conditions in mesh-free methods , 2004 .

[24]  Christophe Schlick,et al.  Accurate parametrization of conics by NURBS , 1996, IEEE Computer Graphics and Applications.

[25]  B. Simeon,et al.  A hierarchical approach to adaptive local refinement in isogeometric analysis , 2011 .

[26]  Antonio Huerta,et al.  NURBS-enhanced finite element method , 2006 .

[27]  Oubay Hassan,et al.  The generation of triangular meshes for NURBS‐enhanced FEM , 2016 .

[28]  Wing Kam Liu,et al.  Meshfree and particle methods and their applications , 2002 .

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

[30]  P. Lancaster,et al.  Surfaces generated by moving least squares methods , 1981 .

[31]  Marino Arroyo,et al.  Nonlinear manifold learning for model reduction in finite elastodynamics , 2013 .

[32]  A. Rosolen,et al.  An adaptive meshfree method for phase-field models of biomembranes. Part I: Approximation with maximum-entropy basis functions , 2013, J. Comput. Phys..

[33]  Peter Wriggers,et al.  Three-dimensional mortar-based frictional contact treatment in isogeometric analysis with NURBS , 2012 .

[34]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

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

[36]  M. Ortiz,et al.  Subdivision surfaces: a new paradigm for thin‐shell finite‐element analysis , 2000 .

[37]  Trond Kvamsdal,et al.  Isogeometric analysis using LR B-splines , 2014 .

[38]  Manfred Bischoff,et al.  A point to segment contact formulation for isogeometric, NURBS based finite elements , 2013 .

[39]  Peter Wriggers,et al.  Contact treatment in isogeometric analysis with NURBS , 2011 .

[40]  Michael Ortiz,et al.  Convergent meshfree approximation schemes of arbitrary order and smoothness , 2012 .

[41]  Michael Ortiz,et al.  Smooth, second order, non‐negative meshfree approximants selected by maximum entropy , 2009 .

[42]  D. A. Dunavant.  High degree efficient symmetrical Gaussian quadrature rules for the triangle , 1985 .

[43]  A. Rosolen,et al.  An adaptive meshfree method for phase-field models of biomembranes. Part II: A Lagrangian approach for membranes in viscous fluids , 2013, J. Comput. Phys..

[44]  Peter Wriggers,et al.  A large deformation frictional contact formulation using NURBS‐based isogeometric analysis , 2011 .

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

[46]  N. Sukumar,et al.  Derivatives of maximum‐entropy basis functions on the boundary: Theory and computations , 2013 .

[47]  N. Sukumar.  Construction of polygonal interpolants: a maximum entropy approach , 2004 .

[48]  Elías Cueto,et al.  A higher order method based on local maximum entropy approximation , 2010 .