Hierarchical B-spline complexes of discrete differential forms

In this paper we introduce the hierarchical B-spline complex of discrete differential forms for arbitrary spatial dimension. This complex may be applied to the adaptive isogeometric solution of problems arising in electromagnetics and fluid mechanics. We derive a sufficient and necessary condition guaranteeing exactness of the hierarchical B-spline complex for arbitrary spatial dimension, and we derive a set of local, easy-to-compute and sufficient exactness conditions for the two-dimensional setting. We examine the stability properties of the hierarchical B-spline complex, and we find that it yields stable approximations of both the Maxwell eigenproblem and Stokes problem provided that the local exactness conditions are satisfied. We conclude by providing numerical results showing the promise of the hierarchical B-spline complex in an adaptive isogeometric solution framework.

[1]  Martin Costabel,et al.  Computation of resonance frequencies for Maxwell equations in non-smooth domains , 2003 .

[2]  Thomas de Quincey.  [C] , 2000, The Works of Thomas De Quincey, Vol. 1: Writings, 1799–1820.

[3]  John A. Evans,et al.  Immersogeometric cardiovascular fluid-structure interaction analysis with divergence-conforming B-splines. , 2017, Computer methods in applied mechanics and engineering.

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

[5]  Salvatore Caorsi,et al.  Mathematical Modelling and Numerical Analysis Spurious-free Approximations of Electromagnetic Eigenproblems by Means of Nedelec-type Elements , 2022 .

[6]  D. Arnold.  Finite Element Exterior Calculus , 2018 .

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

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

[9]  R. Ho.  Algebraic Topology , 2022 .

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

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

[12]  K. Berenhaut,et al.  Applied Mathematical Sciences , 2012 .

[13]  John A. Evans,et al.  Isogeometric Analysis , 2010 .

[14]  Sebastian Schöps,et al.  Isogeometric simulation of Lorentz detuning in superconducting accelerator cavities , 2014, Comput. Phys. Commun..

[15]  Paul W. Gross,et al.  Electromagnetic Theory and Computation: Examples and Tables , 2004 .

[16]  Wulf G. Dettmer,et al.  A stabilised immersed boundary method on hierarchical b-spline grids , 2016 .

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

[18]  Alan Demlow,et al.  A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex , 2012, Foundations of Computational Mathematics.

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

[20]  A. Bossavit.  Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism , 1988 .

[21]  Gunar Matthies,et al.  Mass conservation of finite element methods for coupled flow-transport problems , 2007, Int. J. Comput. Sci. Math..

[22]  Trond Kvamsdal,et al.  Divergence-conforming discretization for Stokes problem on locally refined meshes using LR B-splines , 2015 .

[23]  Carlotta Giannelli,et al.  Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates , 2017 .

[24]  Chennakesava Kadapa,et al.  A fictitious domain/distributed Lagrange multiplier based fluid–structure interaction scheme with hierarchical B-Spline grids , 2016 .

[25]  Anil N. Hirani,et al.  Discrete exterior calculus , 2005, math/0508341.

[26]  Cv Clemens Verhoosel,et al.  Goal-adaptive Isogeometric Analysis with hierarchical splines , 2014 .

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

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

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

[30]  Luca de Meo,et al.  H1 , 2019, Springer Reference Medizin.

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

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

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

[34]  Leszek Demkowicz,et al.  H1, H(curl) and H(div)-conforming projection-based interpolation in three dimensionsQuasi-optimal p-interpolation estimates , 2005 .

[35]  Carlotta Giannelli,et al.  Complexity of hierarchical refinement for a class of admissible mesh configurations , 2015, Comput. Aided Geom. Des..

[36]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

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

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

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

[40]  Paul W. Gross,et al.  Electromagnetic Theory and Computation: A Topological Approach , 2004 .

[41]  Morton Brown.  Locally Flat Imbeddings of Topological Manifolds , 1962 .

[42]  B. Jüttler,et al.  Inf–sup stability of isogeometric Taylor–Hood and Sub-Grid methods for the Stokes problem with hierarchical splines , 2018 .

[43]  Eitan Grinspun,et al.  CHARMS: a simple framework for adaptive simulation , 2002, ACM Trans. Graph..

[44]  Yuri Bazilevs,et al.  Isogeometric divergence-conforming variational multiscale formulation of incompressible turbulent flows , 2017 .

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

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

[47]  John A. Evans,et al.  An Isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces , 2012 .

[48]  Bernard Mourrain,et al.  Dimensions and bases of hierarchical tensor-product splines , 2014, J. Comput. Appl. Math..

[49]  Daniel Boffi,et al.  A note on the deRham complex and a discrete compactness property , 1999, Appl. Math. Lett..

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

[51]  Anil N. Hirani,et al.  Numerical Method for Darcy Flow Derived Using Discrete Exterior Calculus , 2008, ArXiv.

[52]  O. Botella,et al.  BENCHMARK SPECTRAL RESULTS ON THE LID-DRIVEN CAVITY FLOW , 1998 .

[53]  Ana Alonso Rodríguez,et al.  Eddy Current Approximation of Maxwell Equations , 2010 .

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

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

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

[57]  K. Bathe,et al.  The inf-sup test , 1993 .

[58]  Giancarlo Sangalli,et al.  Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations , 2012, J. Comput. Phys..

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