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A High-order Nyström-based Scheme Explicitly Enforcing Surface Density Continuity for the Electric Field Integral Equation

Jin Hu, Constantine Sideris

TL;DR

The paper addresses the challenge of solving the EFIE with high-order accuracy while ensuring continuity of the surface current density $\,\mathbf{J}$ across patch boundaries in Nyström discretizations. It introduces a continuity-enforcing scheme within a Chebyshev-based Nyström (CBIE) framework, using endpoint-inclusive Clenshaw–Curtis quadrature and a mapping that aligns coincident boundary points to a single unique density; this avoids auxiliary line integrals and SVD-based constraints. A forward operator is implemented with a tall 0-1 projection and a subsequent compression, yielding a square, well-posed system suitable for GMRES. Numerical results on a PEC sphere, cube, a CAD-based NURBS model, and a dipole antenna demonstrate reduced GMRES iterations and excellent agreement with MFIE and HFSS, validating efficiency and robustness for high-order EFIE solvers in complex geometries. The approach offers a practical pathway to accurate, scalable electromagnetic scattering analyses in engineering applications.

Abstract

This paper introduces an efficient approach for solving the Electric Field Integral Equation (EFIE) with high-order accuracy by explicitly enforcing the continuity of the impressed current densities across boundaries of the surface patch discretization. The integral operator involved is discretized via a Nyström-collocation approach based on Chebyshev polynomial expansion within each patch and a closed quadrature rule is utilized such that the discretization points inside one patch coincide with those inside another patch on the shared boundary of those two patches. The continuity enforcement is achieved by constructing a mapping from those coninciding points to a vector containing unique discretization points used in the GMRES iterative solver. The proposed approach is applied to the scattering of several different geometries including a sphere, a cube, a NURBS model imported from CAD software, and a dipole structure and results are compared with the Magnetic Field Integral Equation (MFIE) and the EFIE without enforcing continuity to illustrate the effectiveness of the approach.

A High-order Nyström-based Scheme Explicitly Enforcing Surface Density Continuity for the Electric Field Integral Equation

TL;DR

The paper addresses the challenge of solving the EFIE with high-order accuracy while ensuring continuity of the surface current density across patch boundaries in Nyström discretizations. It introduces a continuity-enforcing scheme within a Chebyshev-based Nyström (CBIE) framework, using endpoint-inclusive Clenshaw–Curtis quadrature and a mapping that aligns coincident boundary points to a single unique density; this avoids auxiliary line integrals and SVD-based constraints. A forward operator is implemented with a tall 0-1 projection and a subsequent compression, yielding a square, well-posed system suitable for GMRES. Numerical results on a PEC sphere, cube, a CAD-based NURBS model, and a dipole antenna demonstrate reduced GMRES iterations and excellent agreement with MFIE and HFSS, validating efficiency and robustness for high-order EFIE solvers in complex geometries. The approach offers a practical pathway to accurate, scalable electromagnetic scattering analyses in engineering applications.

Abstract

This paper introduces an efficient approach for solving the Electric Field Integral Equation (EFIE) with high-order accuracy by explicitly enforcing the continuity of the impressed current densities across boundaries of the surface patch discretization. The integral operator involved is discretized via a Nyström-collocation approach based on Chebyshev polynomial expansion within each patch and a closed quadrature rule is utilized such that the discretization points inside one patch coincide with those inside another patch on the shared boundary of those two patches. The continuity enforcement is achieved by constructing a mapping from those coninciding points to a vector containing unique discretization points used in the GMRES iterative solver. The proposed approach is applied to the scattering of several different geometries including a sphere, a cube, a NURBS model imported from CAD software, and a dipole structure and results are compared with the Magnetic Field Integral Equation (MFIE) and the EFIE without enforcing continuity to illustrate the effectiveness of the approach.
Paper Structure (8 sections, 7 equations, 4 figures, 1 table)

This paper contains 8 sections, 7 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. EFIE with Explicit Continuity Enforcement
  3. Numerical Results
  4. PEC sphere
  5. PEC cube
  6. PEC CAD model
  7. PEC dipole antenna
  8. Conclusion

Figures (4)

  • Figure 1: Illustration for the whole forward map used in GMRES iterative solver with $\varphi$ and $\phi$ representing either one of the contravariant component of $\mathbf{J}$ and $\mathcal{T}\mathbf{J}$ in equation \ref{['eq:efie']}.
  • Figure 2: (Left) Current magnitude solved with MFIE. (Middle) Current magnitude solved with EFIE without continuity enforcement. (Right) Current magnitude solved with EFIE with explicit continuity enforcement.
  • Figure 3: Surface electric current density induced on $16\lambda$ tall PEC CAD humanoid bunny model by incident plane wave solved with (a) EFIE with explicit continuity enforcement (b) EFIE without continuity enforcement
  • Figure 4: (a) Surface electric current density induced on the surface of a dipole structure by an incident plane wave. The total length of the dipole is $0.54\lambda$. (b) RCS at $\phi=0^{\circ}$ corresponding to plane wave scattering obtained with MFIE, EFIE with continuity enforced and HFSS.