A Versatile Strategy for the Implementation of Adaptive Splines

This paper presents an implementation framework for spline spaces over T-meshes (and their d-dimensional analogs). The aim is to share code between the implementations of several spline spaces. This is achieved by reducing evaluation to a generalized Bezier extraction.

[1]  Tom Lyche,et al.  Polynomial splines over locally refined box-partitions , 2013, Comput. Aided Geom. Des..

[2]  Giancarlo Sangalli,et al.  Nonlinear static isogeometric analysis of arbitrarily curved Kirchhoff-Love shells , 2015, International Journal of Mechanical Sciences.

[3]  Tom Lyche,et al.  Generalized spline spaces over T-meshes: Dimension formula and locally refined generalized B-splines , 2016, Appl. Math. Comput..

[4]  C. C. Law,et al.  ParaView: An End-User Tool for Large-Data Visualization , 2005, The Visualization Handbook.

[5]  Carlotta Giannelli,et al.  Adaptive isogeometric methods with hierarchical splines: error estimator and convergence , 2015, 1502.00565.

[6]  Bert Jüttler,et al.  Adaptively refined multi-patch B-splines with enhanced smoothness , 2016, Appl. Math. Comput..

[7]  Amy Henderson,et al.  The ParaView Guide: A Parallel Visualization Application , 2004 .

[8]  Bert Jüttler,et al.  Algorithms and Data Structures for Truncated Hierarchical B-splines , 2012, MMCS.

[9]  Giancarlo Sangalli,et al.  ANALYSIS-SUITABLE T-SPLINES OF ARBITRARY DEGREE: DEFINITION, LINEAR INDEPENDENCE AND APPROXIMATION PROPERTIES , 2013 .

[10]  Jiansong Deng,et al.  A new basis for PHT-splines , 2015, Graph. Model..

[11]  A. Buffa,et al.  New refinable spaces and local approximation estimates for hierarchical splines , 2015, 1507.06534.

[12]  Trond Kvamsdal,et al.  On the similarities and differences between Classical Hierarchical, Truncated Hierarchical and LR B-splines , 2015 .

[13]  Bert Jüttler,et al.  Geometry + Simulation Modules: Implementing Isogeometric Analysis , 2014 .

[14]  Andrea Bressan,et al.  Some properties of LR-splines , 2013, Comput. Aided Geom. Des..

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

[16]  Mario Kapl,et al.  Isogeometric analysis with geometrically continuous functions on two-patch geometries , 2015, Comput. Math. Appl..

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

[18]  M. Brovka,et al.  A simple strategy for defining polynomial spline spaces over hierarchical T-meshes , 2016, Comput. Aided Des..

[19]  Philipp Morgenstern.  3D Analysis-suitable T-splines: definition, linear independence and m-graded local refinement , 2015 .

[20]  Hendrik Speleers,et al.  THB-splines: The truncated basis for hierarchical splines , 2012, Comput. Aided Geom. Des..

[21]  John A. Evans,et al.  Isogeometric finite element data structures based on Bézier extraction of NURBS , 2011 .

[22]  Daniel Peterseim,et al.  Analysis-suitable adaptive T-mesh refinement with linear complexity , 2014, Comput. Aided Geom. Des..

[23]  Mark Ainsworth,et al.  Bernstein-Bézier Finite Elements of Arbitrary Order and Optimal Assembly Procedures , 2011, SIAM J. Sci. Comput..

[24]  Markus Kästner,et al.  Bézier extraction and adaptive refinement of truncated hierarchical NURBS , 2016 .

[25]  G. Sangalli,et al.  Approximation properties of multi-patch $C^1$ isogeometric spaces , 2015 .

[26]  Bert Jüttler,et al.  On the completeness of hierarchical tensor-product B-splines , 2014, J. Comput. Appl. Math..

[27]  Christophe Rabut.  Locally tensor product functions , 2004, Numerical Algorithms.

[28]  Ahmad H. Nasri,et al.  T-splines and T-NURCCs , 2003, ACM Trans. Graph..

[29]  Joseph O'Rourke,et al.  Handbook of Discrete and Computational Geometry, Second Edition , 1997 .

[30]  Philipp Morgenstern,et al.  Globally Structured Three-Dimensional Analysis-Suitable T-Splines: Definition, Linear Independence and m-graded local refinement , 2015, SIAM J. Numer. Anal..

[31]  Daniel Peterseim,et al.  Adaptive mesh refinement strategies in isogeometric analysis— A computational comparison , 2016, 1605.00825.

[32]  Bert Jüttler,et al.  A hierarchical construction of LR meshes in 2D , 2015, Comput. Aided Geom. Des..

[33]  Hendrik Speleers,et al.  Strongly stable bases for adaptively refined multilevel spline spaces , 2014, Adv. Comput. Math..

[34]  Thomas J. R. Hughes,et al.  On linear independence of T-spline blending functions , 2012, Comput. Aided Geom. Des..

[35]  David R. Forsey,et al.  Hierarchical B-spline refinement , 1988, SIGGRAPH.

[36]  Giancarlo Sangalli,et al.  Analysis-suitable G1 multi-patch parametrizations for C1 isogeometric spaces , 2016, Comput. Aided Geom. Des..

[37]  Xin Li,et al.  Hierarchical T-splines: Analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis , 2014, 1404.4346.

[38]  Gert Vegter,et al.  In handbook of discrete and computational geometry , 1997 .

[39]  Nicholas S. North,et al.  T-spline simplification and local refinement , 2004, SIGGRAPH 2004.

[40]  Rafael Vázquez,et al.  Algorithms for the implementation of adaptive isogeometric methods using hierarchical splines , 2016 .

[41]  Bert Jüttler,et al.  TDHB-splines: The truncated decoupled basis of hierarchical tensor-product splines , 2014, Comput. Aided Geom. Des..

[42]  Bruce F. Naylor,et al.  Set operations on polyhedra using binary space partitioning trees , 1987, SIGGRAPH.