Blended B-spline construction on unstructured quadrilateral and hexahedral meshes with optimal convergence rates in isogeometric analysis

Abstract We present a novel blended B-spline method to construct bicubic/tricubic splines over unstructured quadrilateral and hexahedral meshes for isogeometric analysis. C 1 and (truncated) C 2 B-spline functions are used in regular elements, whereas C 0 and (truncated) C 1 B-spline functions are adopted in boundary elements and interior irregular elements around extraordinary edges/vertices. The truncation mechanism is employed for a seamless transition from irregular to regular elements. The resulting smoothness of the blended construction is C 2 -continuous everywhere except C 0 -continuous around extraordinary edges and C 1 -continuous across the interface between irregular and regular elements. The blended B-spline construction yields consistent parameterization during refinement and exhibits optimal convergence rates. Spline functions in the blended construction form a non-negative partition of unity, are linearly independent, and support Bezier extraction such that the construction can be used in existing finite element frameworks. Several examples provide numerical evidence of optimal convergence rates.

[1]  Jörg Peters,et al.  A Comparative Study of Several Classical, Discrete Differential and Isogeometric Methods for Solving Poisson's Equation on the Disk , 2014, Axioms.

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

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

[4]  Thomas J. R. Hughes,et al.  Trivariate solid T-spline construction from boundary triangulations with arbitrary genus topology , 2012, Comput. Aided Des..

[5]  Jörg Peters,et al.  Refinable C1 spline elements for irregular quad layout , 2016, Comput. Aided Geom. Des..

[6]  Thomas J. R. Hughes,et al.  Truncated hierarchical tricubic spline construction on unstructured hexahedral meshes for isogeometric analysis applications , 2017, Comput. Math. Appl..

[7]  Fehmi Cirak,et al.  Isogeometric analysis using manifold-based smooth basis functions , 2016, ArXiv.

[8]  Yongjie Jessica Zhang,et al.  Geometric Modeling and Mesh Generation from Scanned Images , 2016 .

[9]  W. Boehm.  Inserting New Knots into B-spline Curves , 1980 .

[10]  Nancy Wilkins-Diehr,et al.  XSEDE: Accelerating Scientific Discovery , 2014, Computing in Science & Engineering.

[11]  F. Cirak,et al.  A subdivision-based implementation of the hierarchical b-spline finite element method , 2013 .

[12]  Les A. Piegl,et al.  The NURBS book (2nd ed.) , 1997 .

[13]  Thomas J. R. Hughes,et al.  Extended Truncated Hierarchical Catmull–Clark Subdivision , 2016 .

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

[15]  T. Hughes,et al.  Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: Geometric design and isogeometric analysis considerations , 2017 .

[16]  J. Dolbow,et al.  Imposing Dirichlet boundary conditions with Nitsche's method and spline‐based finite elements , 2010 .

[17]  Thomas J. R. Hughes,et al.  Truncated T-splines: Fundamentals and methods , 2017 .

[18]  Hendrik Speleers,et al.  Analysis-suitable spline spaces of arbitrary degree on unstructured quadrilateral meshes by , 2017 .

[19]  Michael A. Scott,et al.  Isogeometric spline forests , 2014 .

[20]  Hector Gomez,et al.  Arbitrary-degree T-splines for isogeometric analysis of fully nonlinear Kirchhoff-Love shells , 2017, Comput. Aided Des..

[21]  Ulrich Reif,et al.  A Refineable Space of Smooth Spline Surfaces of Arbitrary Topological Genus , 1997 .

[22]  Ming Li,et al.  Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization , 2017, ArXiv.

[23]  Michael A. Scott,et al.  T-splines as a design-through-analysis technology , 2011 .

[24]  K. Tang,et al.  Rectified unstructured T-splines with dynamic weighted refinement for improvement in geometric consistency and approximation convergence , 2017 .

[25]  T. Hughes,et al.  Local refinement of analysis-suitable T-splines , 2012 .

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

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

[28]  Michel Bercovier,et al.  Smooth Bézier Surfaces over Unstructured Quadrilateral Meshes , 2014, 1412.1125.

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

[30]  Wolfgang Böhm,et al.  On de Casteljau's algorithm , 1999, Comput. Aided Geom. Des..

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

[32]  Bernard Mourrain,et al.  Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology , 2016, Comput. Aided Geom. Des..

[33]  Ralph Roskies,et al.  Bridges: a uniquely flexible HPC resource for new communities and data analytics , 2015, XSEDE.

[34]  Hendrik Speleers,et al.  Effortless quasi-interpolation in hierarchical spaces , 2016, Numerische Mathematik.

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

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

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

[38]  Giancarlo Sangalli,et al.  Unstructured spline spaces for isogeometric analysis based on spline manifolds , 2015, Comput. Aided Geom. Des..

[39]  John A. Evans,et al.  Isogeometric boundary element analysis using unstructured T-splines , 2013 .

[40]  Régis Duvigneau,et al.  Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications , 2013, Comput. Aided Des..

[41]  André Galligo,et al.  Hermite type Spline spaces over rectangular meshes with complex topological structures , 2017 .

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

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

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

[45]  T. Hughes,et al.  Converting an unstructured quadrilateral/hexahedral mesh to a rational T-spline , 2012 .

[46]  Thomas J. R. Hughes,et al.  Truncated hierarchical Catmull–Clark subdivision with local refinement , 2015 .