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High-Order-Accurate Continuity Enforcing Nyström Discretization of 3D Maxwell Combined Field Integral Equations

Bernd Hofmann, Reza Molavi, Constantine Sideris

arXiv:2608.28876v1math.NAphysics.comp-ph

TL;DR

Nyström EFIE discretizations lose accuracy when surface densities jump across patch boundaries, and this issue also affects formulations and scattered-field computations containing the EFIE operator. The paper combines Chebyshev-based Nyström discretization with geometry-derived continuity mappings; across canonical and realistic geometries, continuity enforcement improves conditioning, iterations, and scattered-field accuracy while reducing unknowns.

  • Problem

    Discontinuous patch-local surface densities violate the divergence-conformity required by the EFIE, producing spurious line charges and degrading accuracy in formulations and field representations containing the EFIE operator.

  • Method

    The paper develops a high-order Nyström-collocation scheme using Chebyshev quadrature and sparse geometry-derived basis mappings to enforce continuity for EFIEs, MFIEs, CFIEs, and regularized CFIEs on smooth surfaces.

  • Results

    Continuity enforcement improves the considered formulations' conditioning, iterative-solver iteration counts, and accuracy, with the regularized CFIE achieving the lowest values while reducing unknowns by up to 30%.

  • Takeaways & Limitations

    Restored EFIE accuracy permits equal-weight EFIE–MFIE CFIEs that avoid interior-resonance breakdowns while inheriting the accuracy of both constituents.

Abstract

from arXiv · show

In Nyström-collocation discretizations of the electric field integral equation (EFIE), the surface divergence acts on surface densities that may be discontinuous across patch boundaries, which degrades accuracy and convergence. We show that this not only affects the EFIE but every formulation in which the operator occurs, either in the equation itself or in the scattered field computation, and propose a high-order-accurate continuity-enforcing scheme for smooth surfaces as a remedy for the direct and indirect EFIEs, magnetic field integral equations (MFIEs), and regularized combined field integral equations (CFIEs) alike. The scheme comprises two ingredients: i) We show how to discretize the equations via a Chebyshev-based Nyström scheme, which admits closed quadrature rules. ii) Since unknowns and test vectors are in terms of patch-local curvilinear bases, continuity is enforced by a change of basis: we construct sparse mapping matrices assembled solely from the curvilinear geometry description. In doing so, we restore the accuracy of the EFIE such that it can be combined with the MFIEs with equal weights to form CFIEs. Numerical studies for the scattering from canonical and realistic geometries show that all considered formulations individually and combined benefit from the continuity enforcement in terms of better conditioning, reduced iterations of an iterative solver, and several more digits of accuracy in the scattered fields, despite reducing the total number of unknowns.

I. Introduction T

The paper addresses accuracy loss in Nyström discretizations caused by discontinuous surface densities and develops a Chebyshev-based approach for integral-equation formulations and scattered-field computation.

  • Closed PEC structures cause direct and indirect EFIEs and MFIEs to break down at cavity resonance frequencies.
  • Unlike higher-order MoM schemes requiring costly 4D integrals, the proposed Nyström discretization uses Chebyshev interpolation, singularity cancellation, and essentially kernel evaluations for matrix entries.Alternative locally corrected Nyström and GP-DIM approaches may require an LSE or least-squares problem per patch.
  • Nyström schemes can be highly accurate for MFIEs but less accurate for EFIEs because patch-boundary jumps create spurious line charges under surface divergence.The EFIE operator also appears in scattered-field representation formulas, so the accuracy deficit extends beyond the EFIE itself.
  • Regularization makes the CFIE operator a sum of identity and compact operators while preserving unique solvability for k>0 with w=i k/2.The regularizer acts on different sides in direct and indirect formulations, potentially smoothing the density before surface divergence is applied in the indirect case.
  • The discretization represents patch-local tangential densities in covariant curvilinear bases and evaluates operators through closed quadrature rules at quadrature points.Lagrange interpolation and Fejér or Clenshaw-Curtis rules connect quadrature values to accurate integration of the interpolants.

B. Testing and Handling of Derivatives

The derivative treatment converts interpolated patch-local density components into charge-related quantities and supports covariant or contravariant testing choices for the discretized operators.

  • B. Testing and Handling of Derivatives: Surface-divergence derivatives are represented by dense patchwise derivative matrices D_u and D_v acting on quadrature-point density components.The derivative matrices are obtained by differentiating the interpolation representation numerically over each patch.
  • B. Testing and Handling of Derivatives: The charge counterparts are related to the density coefficients through the matrix relation σ=Dj.The charge components use σ^u,ij=∂_u j_u(u_i,v_j) and σ^v,ij=∂_v j_v(u_i,v_j).
  • B. Testing and Handling of Derivatives: Testing the EFIE with contravariant vectors D e_u and D e_v produces a formulation based on the derivative operators and freely selectable index mappings.
  • B. Testing and Handling of Derivatives: Covariant testing uses block-diagonal derivative matrices, whereas contravariant testing additionally introduces metric-tensor components through the geometry.The divergence contribution is naturally expressed in covariant components, which the metric converts to contravariant testing.
  • B. Testing and Handling of Derivatives: The MFIE term combined with the identity operator is tested contravariantly so its discrete identity contribution is a true identity matrix.The resulting operator representation includes the corresponding tested contributions and geometric sign or permutation structure.

C. Operator Compositions

Operator compositions are discretized by matrix products because the chosen representation remains closed under mappings between tangential vector fields.

  • The discretization strategy maps a tangential vector field in patch-local covariant components to another field in the same representation after contravariant testing.This closure permits compositions to be formed without intermediate changes of representation.
  • For y=n×Tj, the components are obtained by contravariant testing, giving the discrete relation y=nTj.Applying n×R_w and testing again yields the corresponding matrix-product discretization.
  • The indirect CFIE-R composition uses nT nR_c, with R_w acting as a right regularizer that smooths the density before surface divergence.In the direct formulation, the same regularizer acts on the already differentiated density as a left regularizer.
  • The indirect CFIE term involving n×T(n×b) is discretized as nTGb, where G collects contravariant metric-tensor components.
  • Continuity-enforcing matrices are applied only at the outer ends of operator products because intermediate fields remain in the redundant full-quadrature-point representation.Numerical comparisons found no advantage from enforcing continuity between operator compositions.

D. Excitation Vectors

Excitation vectors are evaluated at quadrature points and tested with the same covariant or contravariant vectors used for their corresponding operators.

  • Right-hand sides are formed by evaluating excitation fields at quadrature points and applying the matching operator test vectors.
  • Testing the tangential incident electric field with covariant or contravariant vectors yields the corresponding excitation representations.
  • Rotated excitations are tested with contravariant vectors using the relation between covariant and contravariant basis vectors.The complementary covariant-vector index determines the associated component.
  • Figure 1 depicts covariant basis vectors at 3 x 3 quadrature points on three patches sharing edges and a vertex.

IV. Proposed Continuity Enforcement

The formulations require divergence-conforming surface densities, but the equations do not enforce continuity at patch boundaries. The proposed remedy explicitly enforces density continuity across all considered formulations.

  • The integral equations are valid only when surface densities are divergence-conforming, preventing line charges along patch boundaries.This requirement follows from the need for finite-energy electric and magnetic fields.
  • Continuity is enforced explicitly for all formulations instead of accounting for discontinuities with additional line integrals.
  • The Nyström representation is smooth within each patch, so discontinuities can occur only at patch boundaries.Closed quadrature rules place points on those boundaries, exposing duplicated unknowns at coinciding physical points.

A. Surface Density

Surface-density continuity is imposed by relating duplicated boundary unknowns in patch-local curvilinear bases to unique unknowns through sparse geometry-derived mappings. The construction reduces the unknown count while preserving exact coordinate transformations and leaves the divergence patch-wise.

  • Surface Density: The mapping coefficients are determined solely by the different patch parametrizations, making the resulting density representation independent of the reference patch.
  • Surface Density: Patch connectivity and index reversals or transpositions are obtained from CAD connectivity or boundary-point proximity searches.The index mapping is determined once per shared edge.
  • Surface Density: Coincident boundary and vertex quadrature points create duplicated unknowns that are related to a set of unique unknowns.The duplication enables continuity enforcement rather than making the system intrinsically unusable.
  • Surface Density: The continuity-enforced system uses fewer unknowns, with N′ = 2P(N_u−2)(N_v−2) + 2U < N.Here P is the number of patches and U is the number of unique patch-boundary points.
  • Surface Density: With continuity enforcement, the discretization uses closed Clenshaw-Curtis quadrature; without it, Table IV specifies open Fejér quadrature and identity mappings.
  • Surface Density: The sparse mapping has O(N) nonzero entries, all obtained from the curvilinear geometry description alone.
  • Surface Density: Continuity applies to the density itself, while its surface divergence remains patch-wise and may jump across patch boundaries.This is consistent with requiring a square-integrable rather than continuous surface divergence.

B. Collocation Scheme

Test vectors are mapped from duplicated patch-local representations to unique vectors by relating curvilinear coordinate systems and averaging contributions at coinciding points. This averaging improves robustness to small geometry inconsistencies while preserving exact representation properties.

  • Collocation Scheme: Contravariant test vectors on adjacent patches are constructed by relating their curvilinear coordinate systems.
  • Collocation Scheme: At a point shared by multiple patches, the actual test vector is chosen as the average of the patch-specific vectors.
  • Collocation Scheme: The averaging is exact and reference-patch independent because each contribution represents the same vector.It addresses only small inconsistencies caused by imperfectly matching curvilinear basis vectors in the geometry description.
  • Collocation Scheme: The test-vector mapping is expressed by a sparse, wide matrix with the same structure as the density mapping.
  • Collocation Scheme: A third continuity-enforcing matrix is required for the covariant test-vector construction and omits Jacobian determinants from its coefficients.
  • Collocation Scheme: Averaging contravariant tests reproduces the density components exactly when testing a covariantly expanded density.This preserves the identity contribution of MFIEs and CFIEs after continuity enforcement.

C. Continuity Enforced Systems

Continuity-enforced systems apply mapping matrices on the unknown and test sides of the discretized formulations. The construction preserves the second-kind structure of MFIEs and CFIEs and supports multiple EFIE and CFIE testing choices.

  • Continuity Enforced Systems: The continuity-enforced equations apply the transpose mapping on the unknown side of every equation and the corresponding test mapping on the test side.
  • Continuity Enforced Systems: Continuity enforcement preserves the MFIE and CFIE identity contributions as ±I/2, acting only on compact contributions.Thus, the second-kind structure of these formulations remains intact.
  • Continuity Enforced Systems: The direct EFIE is available with covariant or contravariant testing, while CFIEs include all-contravariant and mixed discretizations.In the mixed scheme, the EFIE uses covariant tests and the MFIE uses contravariant tests.

D. Implications

The continuity-enforcing scheme assumes conforming quadrilateral patch meshes and, generally, identical quadrature-point counts in the two parametric directions, while adding sparse mappings with negligible computational cost.

  • The explicit continuity-enforcing scheme assumes a conforming mesh of quadrilateral patches.Generalizing it to certain non-conforming meshes would require at least non-trivial bookkeeping.
  • For the sphere example, induced densities j, a, b, and c are evaluated for a 1 m sphere at 300 MHz under identical incidence and orientation.
  • For most geometries, the quadrature-point counts in the u and v directions should be identical.A torus is a special case allowing different counts along toroidal and poloidal lines, whereas a sphere is not.
  • The three continuity-enforcing matrices contain O(N) nonzero entries and are applied as sparse matrix-vector products around operator applications.They are constructed once from the curvilinear geometry description, so their cost is negligible compared with applying the operators.

V. Numerical Results

The numerical study compares direct and indirect formulations for plane-wave scattering from a sphere, varying frequency and quadrature resolution while solving with GMRES.

  • The study evaluates condition numbers, GMRES iteration counts, and error levels for scattering from canonical and realistic PEC objects.The objects include a sphere, superellipsoid, spaceplane, and car model, with unrestarted GMRES used for the corresponding linear systems.
  • Figures 3 and 4 compare direct and indirect formulations, respectively, with and without continuity enforcement across different frequencies.Both sphere discretizations use N=3072 unknowns without enforcement and N′=2704 with enforcement, with GMRES stopping criterion ε=1 × 10−12.
  • The numerical results also examine direct formulations at 300 MHz while refining quadrature points per patch from 10 x 10 to 24 x 24.

A. Scattering from a Sphere

For the sphere, continuity enforcement substantially improves EFIE accuracy and conditioning, while CFIE variants retain resonance robustness and benefit across conditioning, iterations, and field errors. Similar advantages extend to a smooth cube-like superellipsoid and remain visible in convergence studies, with fewer unknowns in enforced systems.

  • Sphere: Four orders of magnitude improve the EFIE condition number with continuity enforcement.Current-density error improves by more than two digits, while electric- and magnetic-field errors improve by about six digits.
  • Sphere: Continuity enforcement improves MFIE field errors despite little change in surface-density error, condition number, or iteration count.Magnetic-field error improves by about a digit, and electric-field error improves by up to three digits because the T operator enters electric-field computation.
  • Sphere: Continuity enforcement improves CFIE conditioning, iterations, and error levels while preserving the absence of resonance breakdowns.The continuity-enforced regularized CFIE attains the overall lowest condition number and iteration count, while matching the continuity-enforced CFIE error level after another three-digit improvement.
  • Sphere: For indirect formulations, the continuity-enforced regularized CFIE performs best overall, although regularization before surface divergence does not remove the need for continuity enforcement.The reported comparison lacks reference solutions for the nonphysical indirect surface densities.
  • Sphere: At 300 MHz, mixed CFIE discretization has no notable accuracy or iteration advantage over the non-mixed variant, while continuity enforcement remains effective during refinement.The mixed variant shows only a slightly lower condition number and demonstrates that the Nyström approach does not require a dual basis.
  • Superellipsoid: For the smooth superellipsoid, continuity enforcement improves conditioning, iterations, and error levels by up to four digits, making the EFIE accurate rather than virtually unusable.The continuity-enforced regularized CFIE performs best in convergence studies at 300 MHz, while the non-regularized continuity-enforced CFIE is slightly more accurate in one comparison.
  • Realistic geometries: The realistic spaceplane and car cases use fewer unknowns with continuity enforcement, and the enforced density on the car side mirror matches a high-order MoM reference.The spaceplane comparisons include both direct discretizations with different unknown counts; the car model contains 8676 NURBS patches.

C. Scattering from Realistic Geometries

Realistic-geometry tests show that continuity enforcement improves accuracy while reducing unknown counts, with the strongest evidence from the spaceship and car models.

  • Spaceship model: At higher frequencies, the MFIE does not produce accurate results across the whole spaceship range because virtually all frequencies are near resonance.This comparison uses 165 000 unknowns without continuity enforcement and 154 748 with it.
  • Spaceship model: 4 digits of accuracy improve for the continuity-enforced regularized CFIE across the higher-frequency spaceship range, despite 10 205 fewer unknowns.The unknown count reduction is about 6%, while the iteration count remains 60 throughout the frequency range.
  • Car model: The car model uses 624 672 unknowns without continuity enforcement and 433 804 with it, a reduction of 190 868 unknowns or about 30%.The model is 5 m in size, approximately 8.3 wavelengths, and is tested at 0.5 GHz.
  • Car model: Only the continuity-enforced Nyström solution agrees well with the high-order MoM reference for the car’s induced surface current on the side mirror.The reference uses 277 632 second-order B-spline basis functions.
  • Overall findings: Across sphere, superellipsoid, spaceplane, and car studies, continuity enforcement improves formulations beyond the EFIE, including MFIEs and regularized CFIEs.For the EFIE, the conclusion reports up to four orders of magnitude better conditioning and up to six additional digits in scattered-field accuracy.
  • Overall findings: The continuity-enforced regularized CFIE achieves the lowest condition numbers, iteration counts, and error levels while reducing unknowns by up to 30%.Equal-weight EFIE–MFIE CFIEs avoid interior-resonance breakdown while inheriting constituent accuracy.
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