Analysis-suitable G1 multi-patch parametrizations for C1 isogeometric spaces

One key feature of isogeometric analysis is that it allows smooth shape functions. Indeed, when isogeometric spaces are constructed from p-degree splines (and extensions, such as NURBS), they enjoy up to C p - 1 continuity within each patch. However, global continuity beyond C 0 on so-called multi-patch geometries poses some significant difficulties. In this work, we consider planar multi-patch domains that have a parametrization which is only C 0 at the patch interface. On such domains we study the h-refinement of C 1 -continuous isogeometric spaces. These spaces in general do not have optimal approximation properties. The reason is that the C 1 -continuity condition easily over-constrains the solution which is, in the worst cases, fully locked to linears at the patch interface. However, recently (Kapl et al., 2015b) has given numerical evidence that optimal convergence occurs for bilinear two-patch geometries and cubic (or higher degree) C 1 splines. This is the starting point of our study. We introduce the class of analysis-suitable G 1 geometry parametrizations, which includes piecewise bilinear parametrizations. We then analyze the structure of C 1 isogeometric spaces over analysis-suitable G 1 parametrizations and, by theoretical results and numerical testing, discuss their approximation properties. We also consider examples of geometry parametrizations that are not analysis-suitable, showing that in this case optimal convergence of C 1 isogeometric spaces is prevented. We study h-refinement for C 1 continuous isogeometric spaces over multi-patch domains.We introduce analysis-suitable G 1 (AS G 1 ) geometry parametrizations.AS G 1 parametrizations allow optimal approximation properties.For non-AS G 1 geometries the solution may be locked and convergence is prevented.

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

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

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

[4]  Giancarlo Sangalli,et al.  Some estimates for h–p–k-refinement in Isogeometric Analysis , 2011, Numerische Mathematik.

[5]  J. Peters Smooth interpolation of a mesh of curves , 1991 .

[6]  Tom Lyche,et al.  A Hermite interpolatory subdivision scheme for C2-quintics on the Powell-Sabin 12-split , 2014, Comput. Aided Geom. Des..

[7]  Jörg Peters Smooth mesh interpolation with cubic patches , 1990, Comput. Aided Des..

[8]  M. Kapl,et al.  Isogeometric Analysis with Geometrically Continuous Functions , 2015 .

[9]  André Galligo,et al.  Hermite type Spline spaces over rectangular meshes with arbitrary topology , 2015 .

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

[11]  Giancarlo Sangalli,et al.  Anisotropic NURBS approximation in isogeometric analysis , 2012 .

[12]  M. Pauletti,et al.  Istituto di Matematica Applicata e Tecnologie Informatiche “ Enrico Magenes ” , 2014 .

[13]  Joe D. Warren,et al.  Geometric continuity , 1991, Comput. Aided Geom. Des..

[14]  Thomas J. R. Hughes,et al.  n-Widths, sup–infs, and optimality ratios for the k-version of the isogeometric finite element method , 2009 .

[15]  T. Hughes,et al.  Isogeometric analysis of the Cahn–Hilliard phase-field model , 2008 .

[16]  Angelos Mantzaflaris,et al.  On Isogeometric Subdivision Methods for PDEs on Surfaces , 2015, 1503.03730.

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

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

[19]  L. Beirao da Veiga,et al.  An isogeometric method for the Reissner-Mindlin plate bending problem , 2011, 1106.4436.

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

[21]  Stefan Takacs,et al.  Approximation error estimates and inverse inequalities for B-splines of maximum smoothness , 2015, 1502.03733.

[22]  Bert Jüttler,et al.  Derivatives of isogeometric functions on n-dimensional rational patches in Rd , 2014, Comput. Aided Geom. Des..

[23]  Barbara Wohlmuth,et al.  Isogeometric mortar methods , 2014, 1407.8313.

[24]  Yuri Bazilevs,et al.  The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches , 2010 .

[25]  T. Hughes,et al.  Isogeometric analysis of the isothermal Navier-Stokes-Korteweg equations , 2010 .

[26]  Jörg Peters,et al.  C1 finite elements on non-tensor-product 2d and 3d manifolds , 2016, Appl. Math. Comput..

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

[28]  Giancarlo Sangalli,et al.  Mathematical analysis of variational isogeometric methods* , 2014, Acta Numerica.

[29]  Hendrik Speleers,et al.  Isogeometric analysis with Powell–Sabin splines for advection–diffusion–reaction problems , 2012 .

[30]  J. A. Gregory Geometric continuity , 1989 .

[31]  Etienne Beeker Smoothing of shapes designed with free-form surfaces , 1986 .

[32]  Thomas J. R. Hughes,et al.  A large deformation, rotation-free, isogeometric shell , 2011 .

[33]  T. Hughes,et al.  ISOGEOMETRIC ANALYSIS: APPROXIMATION, STABILITY AND ERROR ESTIMATES FOR h-REFINED MESHES , 2006 .

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

[35]  B. Mourrain,et al.  Geometrically continuous splines for surfaces of arbitrary topology , 2015, 1509.03274.

[36]  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.

[37]  Alessandro Reali,et al.  Duality and unified analysis of discrete approximations in structural dynamics and wave propagation : Comparison of p-method finite elements with k-method NURBS , 2008 .

[38]  Roland Wüchner,et al.  Isogeometric shell analysis with Kirchhoff–Love elements , 2009 .

[39]  André Galligo,et al.  Spline spaces over rectangular meshes with arbitrary topologies and its application to the Grad-Shafranov equation , 2015 .

[40]  Bert Jüttler,et al.  IETI – Isogeometric Tearing and Interconnecting , 2012, Computer methods in applied mechanics and engineering.

[41]  Josef Hoschek,et al.  GC 1 continuity conditions between adjacent rectangular and triangular Bézier surface patches , 1989 .

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