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A Generalized Source Integral Equation for Homogeneous Penetrable Scatterers
Boris Diner, Yaniv Brick
TL;DR
Conventional SIEs for electrically large objects produce dense matrix blocks that are expensive to store and solve, while broadside interactions can limit low-rank compression. This paper extends GSIEs to homogeneous penetrable scatterers using generalized sources for scattered fields and conventional sources for interior fields. The formulation exhibits enhanced low-rank compressibility and can yield solver savings and asymptotically superior performance for certain scatterer classes.
Problem
Conventional SIE discretizations create large dense matrices that are prohibitively expensive for electrically large objects, and general-surface broadside interactions can produce O(N)-rank blocks.
Source: §I · p. 1Method
The indirect GSIE uses generalized sources for external scattered fields and conventional sources for internal fields, with one source distribution per region.
Source: §I · p. 1Results
The formulation exhibits enhanced low-rank compressibility compared with conventional formulations and shows similar accuracy across ranges of scatterer material and geometric parameters.
Source: §IV · p. 4Takeaways & Limitations
Fast direct and iterative solvers can translate the enhanced compressibility into savings and asymptotically superior performance for certain classes of scatterers.
Source: §IV · p. 4Well-posedness of GSIEs for impenetrable and penetrable objects remains under investigation.
Source: §IV · p. 4
Abstract
from arXiv · showhide
The generalized source integral equation (GSIE)approach is extended to enhance the compressibility of moment matrices for essentially-convex homogeneous penetrable scatterers. In the proposed indirect surface integral equation (SIE) formulation, the scattered field representations use a single distribution of generalized sources that radiate only weakly into the scatterer. A separate distribution of conventional sources is used for representing the interior fields. The wave interactions between surface subdomains via the generalized kernel are of a reduced effective dimensionality. The corresponding exterior integral operator moment matrix blocks exhibit enhanced rank deficiency. For lossy scatterers, this secures the low-rank (LR) compressibility of hierarchical matrix structures for the full system. For scatterers in a much slower background medium, significant savings with very delayed asymptotic scaling of the costs can be achieved. The GSIE is formulated for the transverse magnetic (TM) scattering problem. It is validated for representative examples and its superior compressibility is demonstrated.
I. INTRODUCTION
Conventional SIE discretizations produce large dense matrix blocks whose storage and solution are prohibitively expensive for electrically large objects. This work extends GSIEs to homogeneous penetrable scatterers, using generalized sources externally and conventional sources internally to improve exterior-block compressibility.
I. INTRODUCTION
- Conventional SIE discretizations produce large, dense matrix blocks that make storage and solution prohibitively expensive for electrically large objects.Kernel-independent algebraic compression can represent potentially compressible hierarchical blocks economically.Source: §I · p. 1
- For general surfaces, broadside interactions can give blocks O(N) rank, preventing low-rank compression from reducing asymptotic costs.Reduced-dimensional geometries can instead have ranks that scale sublinearly with the number of unknowns.Source: §I · p. 1
- Earlier GSIEs,,,,,,, were introduced for impenetrable essentially-convex geometries and reduce effective interaction dimensionality.Their modified Green’s function kernels suppress broadside and accentuate end-fire interactions, with block ranks and solver costs scaling more slowly with kR.Source: §I · p. 1
- The proposed indirect GSIE represents scattered fields with generalized sources and interior fields with conventional sources, using one distribution per region.Tangential-field continuity yields an SIE whose external-region blocks exhibit enhanced compressibility.Source: §I · p. 1
- If internal blocks represent a lossy or much faster material, they are highly compressible; only the latter may govern costs at very large kR.Conventional formulations combine internal and external blocks into structures whose fast-scaling ranks are dictated by the conventional kernel in the lossless slow medium.Source: §I · p. 1
II. GENERALIZED SIE FORMULATIONS
The indirect GSIE separates generalized exterior-field sources from conventional interior-field sources, producing more compressible exterior operator blocks. Interior-block compressibility extends to lossy scatterers and may persist for lossless scatterers when the exterior medium is much slower.
II. GENERALIZED SIE FORMULATIONS
- The formulation considers a z-invariant homogeneous scatterer bounded by an essentially convex surface, with exterior and interior media specified by wavenumber and impedance.The exterior wavenumber is denoted k, with wavelength λ = 2π/k.Source: §II · p. 2
- Conventional exterior-interior block combinations are incompressible when the exterior is the slower medium because broadside interactions govern their ranks.A related large-subdomain exterior block has asymptotic rank O(kR) = O(N).Source: §II · p. 2
- The indirect formulation uses generalized sources for exterior scattered fields and conventional sources for interior fields, with one source distribution per region.The generalized exterior sources radiate weakly into the scatterer; tangential-field continuity across S yields the SIE system.Source: §II · p. 2, Figure 1 · p. 2
- The generalized exterior operator blocks are asymptotically compressible, with large-subdomain blocks growing effectively as O(1) with kR.These blocks are associated only with the modified Green’s function kernel.Source: §II · p. 2
- For lossy scatterers, the interior blocks are also compressible, making the full system matrix compressible.For lossless scatterers, their ranks may grow slowly until very large kR when kext ≫ kint.Source: §II · p. 2
III. NUMERICAL RESULTS
The numerical examples demonstrate rank compression for the proposed kernels and validate the formulation’s scattered-field accuracy across dielectric and lossy scatterers. The examples also tie useful accuracy in the corrugated case to auxiliary sources emanating within or close to the convex skull.
Numerical setup
- The tested generalized kernel uses multipole-based auxiliary contributions centered at r′′, with amplitudes selected to create a deep far-field shadow.The examples use a coarsest discretization length δ0 = λmin/10.Source: §III · p. 3
Rank deficiency
- GSIE ranks remain roughly constant across broad kR ranges, while conventional exterior ranks approach O(kR)=O(N) sooner.For the studied wavenumber ratio, GSIE ranks govern solver costs over a broad range; for lossy scatterers, ranks converge to a low constant value.Source: Figure 2 · p. 3, §III · p. 3
- For lossless domains, the GSIE rank-constant range broadens with Δφ, and internal-domain ranks are much smaller than external ranks beyond the crossover.The broader range comes with a slight error increase that can be mitigated through discretization parameters.Source: §III · p. 3
Scattered-field validation
- For the circular dielectric cylinder, error is controlled by mesh size against the analytical series solution; the ogive shows similar error levels across three incident angles versus PMCHWT.For the ogive, auxiliary contributions were deactivated within λ/3 of the tips, where the normal is ill-defined.Source: Figure 3 · p. 3, §III · p. 4
Lossy scatterers
- For lossy polyurethane, the ellipse error remains controllable with mesh size, while the corrugated scatterer has error levels similar to Fig. 5 for υ > 0.8λ.The formulation remains useful as long as auxiliary contributions emanate within or close to the convex skull.Source: §III · p. 4
IV. CONCLUSION
The GSIE formulation has enhanced low-rank compressibility over conventional formulations while retaining similar accuracy across ranges of scatterer materials and geometries.
IV. CONCLUSION
- Enhanced low-rank compressibility can yield savings and asymptotically superior performance for certain classes of scatterers.The comparison is with conventional formulations.Source: §IV · p. 4
- The formulation shows similar accuracy across ranges of scatterer material and geometric parameters.The conclusion’s examples include a dielectric ogive evaluated across incident angles and a lossy ellipse evaluated across mesh sizes.Source: §IV · p. 4, Figure 4 · p. 4, Figure 5 · p. 4
- The formulation can be integrated into multi-trace domain-decomposition solvers for more complex scatterers.TE and 3-D acoustic and vector extensions are prospective directions.Source: §IV · p. 4
- Well-posedness of GSIEs for both impenetrable and penetrable objects remains under investigation.Source: §IV · p. 4