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Isogeometric Methods for Computational Electromagnetics: B-spline and T-spline discretizations

A. Buffa, G. Sangalli, R. Vazquez

arXiv:1209.2048v1math.NAcs.NA

TL;DR

Electromagnetic discretizations must preserve geometric structure while remaining accurate and efficient, yet local refinement is also needed to capture strong field singularities. This paper develops spline and T-spline De Rham complexes, connects their bases to mesh structure and control fields, and reports numerical tests of the proposed techniques.

  • Problem

    Electromagnetic discretizations need to preserve the PDE's geometric structure to avoid spurious behavior while providing accuracy, efficiency, and local refinement for strong field singularities.

    Source: §1 · p. 1

  • Method

    The paper constructs spline and T-spline De Rham complexes on spline or NURBS geometries, with canonical bases, control-field interpretations, and T-junction meshes.

    Source: §1 · p. 2

  • Results

    Canonical bases make differential-operator matrices mesh incidence matrices, and the paper extends the spline complex to multi-patch geometries and a two-dimensional T-spline complex for symmetric three-dimensional problems.

    Source: §1 · p. 2

  • Takeaways & Limitations

    T-spline meshes with T-junctions provide a step toward local adaptivity in spline-based electromagnetic methods.

    Source: §1 · p. 2

    The authors describe the T-spline complex as less well understood than the spline complex, even in two dimensions.

    Source: §5.5 · p. 29

Abstract

from arXiv · show

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 analyse their geometric structure, as related to the connectivity of the underlying mesh, and we give a physical interpretation of the fields degrees-of-freedom through the concept of control fields. 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. Introduction

The paper develops spline-based electromagnetic discretizations that preserve the De Rham structure while addressing accuracy, efficiency, and local refinement. It analyzes mesh-related geometry and degrees of freedom, extends the construction to T-splines, and presents numerical tests.

1. Introduction

  • Electromagnetic discretizations should preserve the PDE's geometric structure to avoid spurious behavior, instability, or non-physical solutions, while remaining efficient and accurate.The paper identifies local refinement as useful for capturing strong field singularities, potentially guided by a posteriori error estimators.Source: §1 · p. 1
  • The paper presents spline-based methods for electromagnetic wave computations as a developing research area and examines their geometry and local-refinement strategies.The authors frame this work as showing the potential impact of isogeometric and spline-based techniques in electromagnetics.Source: §1 · p. 2
  • The spline complex admits canonical bases whose differential-operator matrices are mesh incidence matrices, clarifying the connection between spline spaces and mesh geometry.The paper also interprets field degrees of freedom through control fields and extends the results to multi-patch geometries.Source: §1 · p. 2
  • A two-dimensional T-spline complex uses meshes with T-junctions to add adaptivity to spline-based methods and is applied to three-dimensional problems with symmetry.Three-dimensional T-splines remain insufficiently understood for the authors to develop that case here.Source: §1 · p. 2
  • Spline spaces studied here can also be applied to Darcy flow and Hodge Laplacian problems, while cited work applies them to fluid equations.The cited fluid applications include Stokes, Brinkman, and Navier–Stokes equations.Source: §1 · p. 2

2. Notation

The notation section sets up the Maxwell problem on a physical domain and its parametric counterpart. Smooth geometric mappings induce pullbacks that preserve curl and divergence and yield a commuting De Rham diagram.

2. Notation

  • The framework uses H1, H(curl), H(div), and L2 spaces on both the parametric and physical domains, with subscript 0 denoting homogeneous boundary conditions.For H(curl) and H(div), the zero-boundary subspaces have vanishing tangential and normal components, respectively.Source: §2 · p. 3
  • The paper assumes a bounded, simply connected Lipschitz physical domain with connected boundary and a continuously differentiable parametrization with continuously differentiable inverse.The authors note that the sequences are exact under trivial topology; the arbitrary-topology case is not detailed.Source: §2 · pp. 3–4
  • Pullbacks induced by the smooth parametrization preserve curl and divergence and make the De Rham diagram commute between parametric and physical spaces.The pullback definitions use the Jacobian matrix of the mapping.Source: §2 · p. 3

3. Preliminaries on splines and NURBS

The preliminaries define spline bases, their refinement and geometric representations, then introduce the tensor-product and multi-patch settings used later. Anchors and Greville sites provide complementary ways to associate geometric locations with basis functions.

3.1.1. Knot vector and B-spline functions, refinement, spline derivatives

  • B-splines are recursively defined from knot vectors and form bases of piecewise-polynomial spaces whose continuity depends on knot multiplicity.A basis function is supported on its local knot interval, and its derivative belongs to a lower-degree spline space when internal knot multiplicities are at most the degree.Source: §3.1.1 · pp. 4–5
  • The paper distinguishes k-, h-, and p-refinement: degree and regularity elevation, knot insertion, and degree raising at fixed interelement regularity, respectively.K-refinement does not produce nested spaces, whereas p-refinement does.Source: §3.1.1 · p. 5

3.1.2. Anchors and Greville sites

  • Anchors and Greville sites are alternative locations associated with B-spline functions; anchors support T-mesh extensions, while Greville sites relate naturally to geometric degrees of freedom.Distinct anchors can occupy the same parametric location, while Greville sites are distinct when the knot multiplicities keep the B-splines continuous.Source: §3.1.2 · p. 6

3.1.3. B-spline curves

  • A B-spline curve has corresponding control points and a control polygon that converges to the curve under h-refinement.For sufficiently smooth parametrizations, the control polygon approximates the curve with error O(h^2).Source: §3.1.3 · p. 7

3.2. Multivariate B-splines

  • Multivariate B-splines are constructed by tensor products of univariate bases over tensor-product meshes in the parametric domain.The framework allows degrees and knot vectors to vary by coordinate direction.Source: §3.2 · p. 7, §3.2.1 · p. 8

3.2.1. Knot vectors, B-spline functions, anchors, Greville sites

  • In multiple dimensions, anchors are Cartesian products of univariate anchors, while the associated B-spline functions are tensor products and Greville sites form a positive-volume mesh.Depending on the parity of the degrees, anchors lie at mesh vertices, edges, faces, or element centers.Source: §3.2.1 · p. 8

3.2.2. Spline and NURBS geometries, multi-patch domains

  • Spline and NURBS mappings describe single-patch geometries, while multi-patch domains join disjoint mapped cubes under conformity assumptions at patch interfaces.The unknown fields are always constructed from splines, even when the geometry is NURBS-parametrized, so the isoparametric concept is not used in that case.Source: §3.2.2 · pp. 8, 10

4. The spline complex

This section develops spline spaces compatible with the De Rham complex, relating their degrees of freedom to mesh geometry and interpreting them through control fields. It also describes their mapping to physical domains and assembly across conforming multi-patch geometries.

4.1. Complex on the parametric domain bΩ

  • On the parametric domain, the spline spaces form an exact De Rham complex that is isomorphic to the mesh cochain complex for odd degree and the interior chain complex for even degree.The even-degree chain complex does not require explicitly constructing a dual mesh.Source: §4.1 · pp. 12, 15
  • The selected bases associate degrees of freedom with mesh entities, and their differential operators have incidence-matrix representations that are essentially independent of spline degree.For odd degree, the gradient is represented by the edge-vertex incidence matrix; for even degree, it is represented by the face-element incidence matrix.Source: §4.1 · pp. 14–15
  • The anchor and boundary-mesh structure treats boundaries canonically, while the spline complex provides higher-order approximation than low-order finite elements.For homogeneous boundary conditions on the whole boundary or a union of cube faces, the associated De Rham complex remains exact.Source: §4.1 · p. 15

4.2. Push-forward to the single-patch physical domain Ω

  • Commuting push-forward maps transfer the parametric spaces to the physical domain, preserving exactness and incidence-matrix operators; optimal approximation holds when the mapping meets the stated assumption and its inverse is sufficiently smooth.For odd degree the physical complex corresponds to the mesh cochain complex, and for even degree to the chain complex.Source: §4.2 · pp. 16–17

4.3. Control fields and degrees-of-freedom interpretation

  • Control fields interpret scalar spline degrees of freedom as values at Greville sites parametrically or control points physically, while splines achieve O(h^p) H1 accuracy versus O(h) for trilinear controls.The corresponding control fields approximate the spline fields, but do not inherit their higher-order accuracy.Source: §4.3 · p. 18
  • A commuting map identifies the spline complex’s geometric structure with the low-order finite-element complex on the control mesh, where vector-field degrees of freedom become edge circulations.For example, a Nédélec control field controls the corresponding spline field.Source: §4.3 · p. 18
  • The spline complex has O(h^p) approximation while its control complex is first order; when all degrees are even, the control mesh is dual to the primal mesh in the stated cochain sense.The control complex is isomorphic to the cochain complex of the control-mesh partition.Source: §4.3 · p. 19

4.4. Push-forward to the multi-patch physical domain

  • On each multi-patch domain, patchwise De Rham complexes are assembled by imposing scalar trace, tangential trace, and normal trace continuity for X0 h, X1 h, and X2 h, respectively.X3 h requires no continuity; geometric conformity requires matching interface spaces and bases.Source: §4.4 · p. 19
  • Geometric conformity makes patch meshes and control meshes match at interfaces, and continuity is implemented by merging coincident control-field degrees of freedom, accounting for orientation where needed.The resulting meshes are locally structured but globally unstructured.Source: §4.4 · pp. 19–20

5. Beyond the tensor product structure: T-splines

This section extends tensor-product B-splines to T-splines by developing a compatible two-dimensional complex and constructing a three-dimensional complex from it and one-dimensional B-splines. It also addresses the more limited understanding of T-spline complexes and notes that some properties of spline complexes do not hold for T-splines in general.

5.1. T-mesh

  • A T-mesh is a rectangular tiling of the unit square whose vertices lie on the knot grid, and it may contain T-junctions where only three edges meet.These T-junctions break the tensor-product structure of the mesh.Source: §5.1 · p. 20

5.2. Analysis suitable T-meshes and T-splines.

  • Analysis-suitable T-meshes use degree-dependent T-junction extensions that do not intersect; with additional extension conditions, their T-spline space equals piecewise bivariate polynomials on the extended mesh.The extended mesh is the minimal mesh on which basis functions restrict to polynomials of degree (p1, p2), and integration must use its elements. The stated characterization also requires extensions not to intersect mesh lines of multiplicity greater than one; the general-degree proof is described as work in progress.Source: §5.2 · pp. 21, 23

5.3. Two-dimensional De Rham complex with T-splines on the parametric domain (0, 1)2

  • Under Proposition 5.2’s assumptions, the constructed two-dimensional T-spline De Rham complexes are well-defined and exact.The construction assumes equal degrees in both directions for simplicity; the results also hold for unequal degrees, with more intricate proofs. The spaces use different meshes but share one extended mesh for integration and implementation.Source: §5.3 · pp. 23, 24, 27

5.4. Three-dimensional De Rham complex based on T-splines and B-splines

  • The three-dimensional parametric complex is formed by tensor products of the two-dimensional T-spline complexes and the one-dimensional B-spline complex.For a tensor-product single-patch spline or NURBS geometry map satisfying the stated assumption, push-forwards give the corresponding complex on the physical domain.Source: §5.4 · p. 29

5.5. Concluding remarks on the T-spline complex

  • For T-splines, operator matrices are no longer mesh incidence matrices, a limitation also seen in standard finite elements with hanging nodes, including p = 1.This means the incidence-matrix property does not generally extend to the T-spline complex.Source: §5.5 · p. 29
  • Defining control meshes and control fields for T-splines is nontrivial, especially for even p, and the spline analogue of Section 4.3 is unavailable.The authors identify this as an area requiring further study.Source: §5.5 · p. 29

6. Numerical results

Across the numerical tests, T-spline discretizations reproduce the reported eigenvalue and field behavior while matching B-spline accuracy with fewer degrees of freedom in two comparisons. The tests also show correct waveguide propagation through a twist, with reflection and transmission coefficients close to zero and one, respectively.

6. Numerical results

  • The numerical tests use GeoPDEs, which does not provide T-spline adaptivity in the sense of.The authors note that the library also lacks full T-spline capability.Source: §6 · p. 30

6.1. Maxwell eigenproblem in the square domain

  • The square-domain T-spline discretization has no spurious eigenvalues, converges well, and computes discrete eigenvalues with the right multiplicity.A mixed formulation gives the same non-null eigenvalues while filtering zero eigenvalues, confirming the T-spline De Rham complex construction.Source: §6.1 · pp. 30–31, Figure 14 · p. 31

6.2. Maxwell eigenproblem in the thick L-shaped domain

  • Dyadic T-spline refinement targets elements near the thick L-shaped domain’s reentrant edge, with anisotropic elements needed for optimal convergence.The two-dimensional refinement is extended through the three-dimensional domain, and refinement is made identical across patches to ensure conformity at interfaces.Source: §6.2 · p. 32
  • T-splines obtain the same first-eigenvalue error as same-degree B-splines in the refined tensor-product mesh, with an important reduction in degrees of freedom.The test uses degrees 4 and 5 in the thick L-shaped domain.Source: §6.2 · p. 32, Figure 16 · p. 33

6.4. Numerical simulation of a twisted waveguide

  • The twisted-waveguide test uses a 90-degree twist at 32 GHz, between the first and second mode cutoff frequencies.The computational domain is 24 cm long, with a 16 cm twisted region and 4 cm straight regions at each port.Source: §6.4 · pp. 34–35

6.3. Maxwell source problem in three quarters of the cylinder

  • T-splines obtain results similar to same-degree B-splines for the cylinder source problem, with fewer degrees of freedom.The exact solution is singular in the first two cylindrical directions and regular in the z direction.Source: §6.3 · p. 34, Figure 18 · p. 35

6.4. Numerical simulation of a twisted waveguide

  • |R| = 0.0025 and |T| = 0.9998, confirming that the 90-degree twist does not affect propagation of the mode, as expected.The computed field also shows the mode propagating correctly.Source: §6.4 · p. 36
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