Fully Automated Adaptive Parameter Selection for 3-D High-order Nyström Boundary Integral Equation Methods

TL;DR

This work develops a fully automated adaptive CBIE solver for 3-D electromagnetic scattering by PEC objects, eliminating manual parameter tuning. It combines per-patch automatic selection of the near-singular interaction distance $\Delta_{near}$ with adaptive precomputation of kernel integrals using either $\text{GK}$ (h-adaptive) or $\text{CC}$ (p-adaptive) quadrature, augmented by singularity-resolving changes of variables. The approach yields high-order accuracy and robustness across complex geometries, matching or exceeding the performance of optimally tuned fixed-grid CBIE implementations while reducing precomputation time and memory and enabling scalability to electrically large problems. Results on canonical and CAD geometries (sphere, toroid, glider) demonstrate automated convergence and good agreement with commercial solvers, highlighting the method’s practical impact in efficient, parameter-free BIE-based EM simulations.

Abstract

We present an adaptive Chebyshev-based Boundary Integral Equation (CBIE) solver for electromagnetic scattering from smooth perfect electric conductor (PEC) objects. The proposed approach eliminates manual parameter tuning by introducing (i) a unified adaptive quadrature strategy for automatic selection of the near-singular interaction distance and (ii) an adaptive computation of all self- and near-singular precomputation integrals to a prescribed accuracy using Gauss-Kronrod (h-adaptive) or Clenshaw-Curtis (p-adaptive) rules and singularity-resolving changes of variables. Both h-adaptive and p-adaptive schemes are explored within this framework, ensuring high-order accuracy and robustness across a broad range of geometries without loss of efficiency. Numerical results for canonical and complex CAD geometries demonstrate that the adaptive solver achieves accuracy and convergence rates comparable to optimally tuned fixed-grid CBIE implementations, while offering automation and scalability to electrically large, geometrically complex problems.

Figures (13)

• Figure 1: Plane wave incident upon a PEC object.
• Figure 2: Parametric mapping and surface partitioning.
• Figure 3: Singular Change-of-Variables depending on the location of the "target" point and the "integration" patch. (a) Integration patch not subdivided: CoV applied on a single subpatch, (b) Integration patch divided in $v$-direction: CoV applied on two subpatches, (c) Integration patch divided in both $u$ and $v$ directions: CoV applied on four subpatches. Note that in this case, the "projection" point coincides with the "target" point.
• Figure 4: An example of the interval splitting used by the Gauss-Kronrod Quadrature for calculating precomputation integrals
• Figure 5: The real part of $J_x$ solved surface current density on a PEC Sphere of $D = 4\lambda$ diameter.