Stable Isogeometric Analysis of Trimmed Geometries

Abstract We explore extended B-splines as a stable basis for isogeometric analysis with trimmed parameter spaces. The stabilization is accomplished by an appropriate substitution of B-splines that may lead to ill-conditioned system matrices. The construction for non-uniform knot vectors is presented. The properties of extended B-splines are examined in the context of interpolation, potential, and linear elasticity problems and excellent results are attained. The analysis is performed by an isogeometric boundary element formulation using collocation. It is argued that extended B-splines provide a flexible and simple stabilization scheme which ideally suits the isogeometric paradigm.

[1]  Joris C. G. Verschaeve,et al.  A weighted extended B-spline solver for bending and buckling of stiffened plates , 2015, 1512.04276.

[2]  Sung-Kie Youn,et al.  Isogeometric analysis with trimming technique for problems of arbitrary complex topology , 2010 .

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

[4]  Fehmi Cirak,et al.  Subdivision-stabilised immersed b-spline finite elements for moving boundary flows , 2012 .

[5]  Roland Wüchner,et al.  Isogeometric analysis of trimmed NURBS geometries , 2012 .

[6]  KLAUS HÖLLIG,et al.  B-SPLINE APPROXIMATION OF NEUMANN PROBLEMS , .

[7]  T. Fries,et al.  Higher‐order accurate integration of implicit geometries , 2016 .

[8]  Hyun-Jung Kim,et al.  Isogeometric analysis for trimmed CAD surfaces , 2009 .

[9]  David J. Benson,et al.  A multi-patch nonsingular isogeometric boundary element method using trimmed elements , 2015, Computational Mechanics.

[10]  Maharavo Randrianarivony,et al.  From Computer Aided Design to wavelet BEM , 2009, Comput. Vis. Sci..

[11]  Ulrich Reif,et al.  Collocation with WEB–Splines , 2016, Adv. Comput. Math..

[12]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[13]  K. Höllig.  Finite element methods with B-splines , 1987 .

[14]  Thomas-Peter Fries,et al.  Fast Isogeometric Boundary Element Method based on Independent Field Approximation , 2014, ArXiv.

[15]  Ulrich Reif,et al.  Nonuniform web-splines , 2003, Comput. Aided Geom. Des..

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

[17]  C. D. Boor,et al.  Spline approximation by quasiinterpolants , 1973 .

[18]  B. V. Rathish Kumar,et al.  Weighted extended B-spline method for the approximation of the stationary Stokes problem , 2006 .

[19]  Zhengdong Huang,et al.  Isogeometric analysis for compound B-spline surfaces , 2013 .

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

[21]  David J. Benson,et al.  On the numerical integration of trimmed isogeometric elements , 2015 .

[22]  C. Patterson,et al.  Interelement Continuity in the Boundary Element Method , 1984 .

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

[24]  Olaf Steinbach,et al.  Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements , 2007 .

[25]  Martin Kögl,et al.  Boundary Element Methods for Engineers and Scientists: An Introductory Course With Advanced Topics , 2004 .

[26]  Zafer Gürdal,et al.  On the variational formulation of stress constraints in isogeometric design , 2010 .

[27]  Gernot Beer,et al.  Advanced Numerical Simulation Methods: From CAD Data Directly to Simulation Results , 2015 .

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

[29]  Gokhan Apaydin,et al.  Application of web-spline method in electromagnetics , 2008 .

[30]  Kang Li,et al.  Isogeometric analysis and shape optimization via boundary integral , 2011, Comput. Aided Des..

[31]  Ulrich Reif,et al.  Weighted Extended B-Spline Approximation of Dirichlet Problems , 2001, SIAM J. Numer. Anal..

[32]  G. Farin.  Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[33]  Klaus Höllig,et al.  Finite Element Analysis with B-Splines: Weighted and Isogeometric Methods , 2010, Curves and Surfaces.

[34]  Gernot Beer,et al.  A simple approach to the numerical simulation with trimmed CAD surfaces , 2015, ArXiv.