Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations

In this paper we introduce methods for electromagnetic wave propagation, based on splines and on T-splines. We define spline spaces which form a De Rham complex and following the isogeometric paradigm, we map them on domains which are (piecewise) spline or NURBS geometries. We analyze their geometric and topological structure, as related to the connectivity of the underlying mesh, and we present degrees of freedom together with their physical interpretation. The theory is then extended to the case of meshes with T-junctions, leveraging on the recent theory of T-splines. The use of T-splines enhance our spline methods with local refinement capability and numerical tests show the efficiency and the accuracy of the techniques we propose.

[1]  R. Hiptmair.  Finite elements in computational electromagnetism , 2002, Acta Numerica.

[2]  Alessandro Reali,et al.  GeoPDEs: A research tool for Isogeometric Analysis of PDEs , 2011, Adv. Eng. Softw..

[3]  L. Beirão da Veiga,et al.  Analysis-suitable T-splines of arbitrary degree : definition and properties , 2012 .

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

[5]  Ronald H. W. Hoppe,et al.  Finite element methods for Maxwell's equations , 2005, Math. Comput..

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

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

[8]  T. Hughes,et al.  Solid T-spline construction from boundary representations for genus-zero geometry , 2012 .

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

[10]  Dietrich Braess,et al.  Equilibrated residual error estimator for edge elements , 2007, Math. Comput..

[11]  Annalisa Buffa,et al.  Algebraic convergence for anisotropic edge elements in polyhedral domains , 2005, Numerische Mathematik.

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

[13]  Joachim Schöberl,et al.  Exact Sequences of High Order Hardy Space Infinite Elements for Exterior Maxwell Problems , 2013, SIAM J. Sci. Comput..

[14]  Falai Chen,et al.  On the instability in the dimension of splines spaces over T-meshes , 2011, Comput. Aided Geom. Des..

[15]  Tom Lyche,et al.  Analysis-aware modeling: Understanding quality considerations in modeling for isogeometric analysis , 2010 .

[16]  John A. Evans,et al.  Isogeometric divergence-conforming b-splines for the darcy-stokes-brinkman equations , 2013 .

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

[18]  B. Simeon,et al.  A hierarchical approach to adaptive local refinement in isogeometric analysis , 2011 .

[19]  Daniele Boffi.  Approximation of eigenvalues in mixed form, Discrete Compactness Property, and application to hp mixed finite elements , 2007 .

[20]  M. Costabel,et al.  Singularities of Electromagnetic Fields¶in Polyhedral Domains , 2000 .

[21]  G. Sangalli,et al.  IsoGeometric analysis using T-splines on two-patch geometries , 2011 .

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

[23]  T. Hughes,et al.  Converting an unstructured quadrilateral mesh to a standard T-spline surface , 2011 .

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

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

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

[27]  M. Clemens,et al.  Computational electromagnetic-field calculation with the finite-integration method , 1999 .

[28]  Bernard Mourrain,et al.  On the dimension of spline spaces on planar T-meshes , 2010, Math. Comput..

[29]  Elaine Cohen,et al.  Volumetric parameterization and trivariate B-spline fitting using harmonic functions , 2009, Comput. Aided Geom. Des..

[30]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

[31]  Luca F. Pavarino,et al.  Overlapping Schwarz Methods for Isogeometric Analysis , 2012, SIAM J. Numer. Anal..

[32]  Francesca Rapetti,et al.  Whitney Forms of Higher Degree , 2009, SIAM J. Numer. Anal..

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

[34]  Ahmed Ratnani,et al.  An Arbitrary High-Order Spline Finite Element Solver for the Time Domain Maxwell Equations , 2012, J. Sci. Comput..

[35]  D. Boffi,et al.  Computational Models of Electromagnetic Resonators: Analysis of Edge Element Approximation , 1999 .

[36]  Thomas J. R. Hughes,et al.  Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations , 2013, J. Comput. Phys..

[37]  Ralf Hiptmair,et al.  Discrete Hodge operators , 2001, Numerische Mathematik.

[38]  Carretera de Valencia,et al.  The finite element method in electromagnetics , 2000 .

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

[40]  D. Arnold,et al.  Finite element exterior calculus: From hodge theory to numerical stability , 2009, 0906.4325.

[41]  Giancarlo Sangalli,et al.  Analysis-Suitable T-splines are Dual-Compatible , 2012 .

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

[43]  Victor M. Calo,et al.  The Cost of Continuity: Performance of Iterative Solvers on Isogeometric Finite Elements , 2012, SIAM J. Sci. Comput..

[44]  Serge Nicaise,et al.  Edge Elements on Anisotropic Meshes and Approximation of the Maxwell Equations , 2001, SIAM J. Numer. Anal..

[45]  Tom Lyche,et al.  T-spline Simplication and Local Renement , 2004 .

[46]  Annalisa Buffa,et al.  Isogeometric Analysis for Electromagnetic Problems , 2010, IEEE Transactions on Magnetics.

[47]  T. Apel.  Anisotropic Finite Elements: Local Estimates and Applications , 1999 .

[48]  John A. Evans,et al.  ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE STEADY NAVIER–STOKES EQUATIONS , 2013 .

[49]  M. Scott,et al.  On the Nesting Behavior of T-splines , 2011 .

[50]  Ernst Rank,et al.  Geometric modeling, isogeometric analysis and the finite cell method , 2012 .

[51]  A. Bossavit.  Discretization of Electromagnetic Problems: The “Generalized Finite Differences” Approach , 2005 .

[52]  D. Arnold,et al.  Finite element exterior calculus, homological techniques, and applications , 2006, Acta Numerica.

[53]  Thomas Weiland,et al.  Efficient modelling techniques for complicated boundary conditions applied to structured grids , 2004 .

[54]  Giancarlo Sangalli,et al.  Characterization of analysis-suitable T-splines , 2015, Comput. Aided Geom. Des..

[55]  G. Sangalli,et al.  Linear independence of the T-spline blending functions associated with some particular T-meshes , 2010 .

[56]  Alessandro Reali,et al.  Isogeometric collocation: Cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations , 2013 .

[57]  Snorre H. Christiansen,et al.  A dual finite element complex on the barycentric refinement , 2005, Math. Comput..

[58]  G. Sangalli,et al.  Isogeometric analysis in electromagnetics: B-splines approximation , 2010 .

[59]  Michael E. Mortenson,et al.  Geometric Modeling , 2008, Encyclopedia of GIS.

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

[61]  John A. Evans,et al.  Discrete spectrum analyses for various mixed discretizations of the Stokes eigenproblem , 2012 .

[62]  T. Hughes,et al.  Isogeometric fluid-structure interaction: theory, algorithms, and computations , 2008 .

[63]  Daniele Boffi,et al.  Finite element approximation of eigenvalue problems , 2010, Acta Numerica.

[64]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

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

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

[67]  Peter Wriggers,et al.  A large deformation frictional contact formulation using NURBS‐based isogeometric analysis , 2011 .

[68]  G. Sangalli,et al.  IsoGeometric Analysis using T-splines , 2012 .

[69]  Victor M. Calo,et al.  Isogeometric Variational Multiscale Large-Eddy Simulation of Fully-developed Turbulent Flow over a Wavy Wall , 2012 .

[70]  John A. Evans,et al.  Isogeometric analysis using T-splines , 2010 .

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

[72]  Alessandro Reali,et al.  Isogeometric Analysis of Structural Vibrations , 2006 .

[73]  Giancarlo Sangalli,et al.  Isogeometric Discrete Differential Forms in Three Dimensions , 2011, SIAM J. Numer. Anal..

[74]  Manfred Bischoff,et al.  Numerical efficiency, locking and unlocking of NURBS finite elements , 2010 .

[75]  Gershon Elber,et al.  Geometric modeling with splines - an introduction , 2001 .

[76]  John A. Evans,et al.  Robustness of isogeometric structural discretizations under severe mesh distortion , 2010 .

[77]  Victor M. Calo,et al.  The cost of continuity: A study of the performance of isogeometric finite elements using direct solvers , 2012 .

[78]  Giancarlo Sangalli,et al.  IsoGeometric Analysis: Stable elements for the 2D Stokes equation , 2011 .