A sharp degree bound on G2-refinable multi-sided surfaces

Refinement of a space of splines should yield additional degrees of freedom for modeling and engineering analysis, both along boundaries and in the interior. Yet such additional flexibility fails to materialize for multi-sided G 2 surface constructions when the polynomial degree is too low. This paper establishes a tight lower bound on the polynomial degree of flexibility-increasing refinable multi-sided G 2 surface constructions within a C 2 spline complex - by ruling out bi-5 constructions and by exhibiting a multi-sided bi-6 construction that yields good highlight line and curvature distributions. The bi-6 construction consists of one 2 × 2 macro-patch for each of the n sectors that join to form the multi-sided surface.

[1]  Ronald Maier,et al.  Integrated Modeling , 2011, Encyclopedia of Knowledge Management.

[2]  Jörg Peters,et al.  Finite Curvature Continuous Polar Patchworks , 2009, IMA Conference on the Mathematics of Surfaces.

[3]  Jörg Peters,et al.  Matched Gk-constructions always yield Ck-continuous isogeometric elements , 2015, Comput. Aided Geom. Des..

[4]  Jörg Peters,et al.  Fair free-form surfaces that are almost everywhere parametrically C2 , 2019, J. Comput. Appl. Math..

[5]  Jörg Peters,et al.  Rapidly contracting subdivision yields finite, effectively C2 surfaces , 2018, Comput. Graph..

[6]  Jörg Peters,et al.  Subdivision Surfaces , 2002, Handbook of Computer Aided Geometric Design.

[7]  Hartmut Prautzsch,et al.  Freeform splines , 1997, Computer Aided Geometric Design.

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

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

[10]  C. Fleury,et al.  Shape optimal design using high-order elements , 1988 .

[11]  Peter Schröder,et al.  Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision , 2002, Comput. Aided Des..

[12]  Jörg Peters,et al.  Refinable bi-quartics for design and analysis , 2018, Comput. Aided Des..

[13]  Hendrik Speleers,et al.  Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis , 2017 .

[14]  Jörg Peters,et al.  Minimal bi-6 G2 completion of bicubic spline surfaces , 2016, Comput. Aided Geom. Des..

[15]  V. Braibant,et al.  Shape optimal design using B-splines , 1984 .

[16]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[17]  Andrew Y. T. Leung,et al.  Spline finite elements for beam and plate , 1990 .

[18]  Jörg Peters,et al.  Generalizing bicubic splines for modeling and IGA with irregular layout , 2016, Comput. Aided Des..

[19]  Y. K. Cheung,et al.  Isoparametric spline finite strip for plane structures , 1993 .

[20]  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.

[21]  Mario Kapl,et al.  Isogeometric analysis with geometrically continuous functions on planar multi-patch geometries , 2017 .

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

[23]  Mario Kapl,et al.  Space of C2-smooth geometrically continuous isogeometric functions on two-patch geometries , 2017, Comput. Math. Appl..

[24]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[25]  Jörg Peters,et al.  Refinable G1 functions on G1 free-form surfaces , 2017, Comput. Aided Geom. Des..

[26]  Charles T. Loop,et al.  G2 Tensor Product Splines over Extraordinary Vertices , 2008, Comput. Graph. Forum.

[27]  Jörg Peters,et al.  On the complexity of smooth spline surfaces from quad meshes , 2009, Comput. Aided Geom. Des..

[28]  Walter D. Pilkey,et al.  The coupling of geometric descriptions and finite elements using NURBs: a study in shape optimization , 1993 .