High order-accurate solution of scattering integral equations with unbounded solutions at corners

Abstract

Although high-order Maxwell integral equation solvers provide significant advantages in terms of speed and accuracy over corresponding low-order integral methods, their performance significantly degrades in presence of non-smooth geometries--owing to field enhancement and singularities that arise at sharp edges and corners which, if left untreated, give rise to significant accuracy losses. The problem is particularly challenging in cases in which the "density" (i.e., the solution of the integral equation) tends to infinity at corners and edges--a difficulty that can be bypassed for 2D configurations, but which is unavoidable in 3D Maxwell integral formulations, wherein the component tangential to an edge of the electrical-current integral density vector tends to infinity at the edge. In order to tackle the problem this paper restricts attention to the simplest context in which the unbounded-density difficulty arises, namely, integral formulations in 2D space whose integral density blows up at corners; the strategies proposed, however, generalize directly to the 3D context. The novel methodologies presented in this paper yield high-order convergence for such challenging equations and achieve highly accurate solutions (even near edges and corners) without requiring a priori analysis of the geometry or use of singular bases.

Figures (16)

• Figure 1: Partitioning of the domain boundary $\Gamma$ as the union \ref{['partition']} of nonoverlapping patches $\Gamma^q$ ($1\leq q\leq M$).
• Figure 2: Illustration of the graded-mesh Changes of Variables (CoV) in equations \ref{['eq:CoVCK']} and \ref{['eq:power_cov']} of order $p=6$.
• Figure 3: The blue crosses display the magnitude of the kernel $\frac{\partial G_{k}(\widetilde{r}_{q}(\theta), \widetilde{r}_{q'}(\theta')}{\partial n(\widetilde{r}_{q}(\theta))}$ in the first integral in \ref{['absorbed_ricfie']}, as a function of $\theta'$, for values of $\theta$ corresponding to target points at distances $1.98\times10^{-12}$ (left) and $1.23\times10^{-16}$ (right) away from the corner (target point and source patch are on different sides of a corner). The vertical lines correspond to abscissas selected by the Gauss-Kronrod adaptive quadrature algorithm.
• Figure 4: The real and imaginary parts of the quantities $\widetilde{L}_q(\theta)\widetilde{\beta}_m^{q'}[H](r)$ are shown, at points $r$ located at various distances $d$ from a corner $C$ of a square scatterer, where these quantities are obtained as the product of the line element $\widetilde{L}_q(\theta)$ and the precomputed weights \ref{['precomp_def2']} for the kernel $H = \frac{\partial G_k(r, r')}{\partial n_t(r)}$ associated with the corner-regularized formulation. These values are relevant for implementing the integral formulation \ref{['absorbed_ricfie']}. The results are shown for the $q'$-th source patch and for target points $r$ on the $q$-th observation patch ($q \neq q'$), where the patches are positioned on opposite sides of the corner $C$. As illustrated, the precomputed values remain bounded throughout the $q$-th patch for all Chebyshev polynomial orders $m$.
• Figure 5: Plane wave incident upon a PEC cylinder of square cross-section.

Theorems & Definitions (1)

• Remark 1