Abstract
Integral equation formulations are a competitive strategy in computational electromagnetics but, lamentably, are often plagued by ill-conditioning and by related numerical instabilities that can jeopardize their effectiveness in several real case scenarios. Luckily, however, it is possible to leverage effective preconditioning and regularization strategies that can cure a large majority of these problems. Not surprisingly, integral equation preconditioning is currently a quite active field of research. To give the reader a propositive overview of the state of the art, this paper will review and discuss the main advancements in the field of integral equation preconditioning in electromagnetics summarizing strengths and weaknesses of each technique. The contribution will guide the reader through the choices of the right preconditioner for a given application scenario. This will be complemented by new analyses and discussions which will provide a further and more intuitive understanding of the ill-conditioning of the electric field (EFIE), magnetic field (MFIE), and combined field integral equation (CFIE) and of the associated remedies.
Highlights
Integral equation formulations, solved by the boundary element method (BEM), have become a well established tool to solve scattering and radiation problems in electromagnetics [1]–[4]. What makes these schemes so suitable for electromagnetic analyses is that, differently from approaches based on differential equations such as the finite element method (FEM) or the finite-difference time-domain method (FDTD), they naturally incorporate radiation conditions without the need for artificial absorbing boundary conditions, they only set unknowns on boundary surfaces instead of discretizing the entire volume, and they are mostly free from numerical dispersion
This effect is related to the high-frequency breakdown in the scalar Helmholtz equation in acoustics, where both the Dirichlet problem solved with a standard combined field integral equation (CFIE) and the Neumann problem solved with a Calderón-Yukawa CFIE have condition numbers that grow as O ( 1/3) on a sphere [163], [164]
Among the solutions found in the literature, [162] proposes a Calderón-like preconditioning with a modified wavenumber m in the preconditioner of the electric field (EFIE) operator, an unchanged magnetic field (MFIE) is added in a CFIE fashion
Summary
To put into light the low-frequency challenges that plague the EFIE, its behavior on both the solenoidal and the nonsolenoidal subspaces must be analyzed. Quasi-Helmholtz projectors can be used to cure the different deleterious effects of the low-frequency breakdown by isolating the solenoidal and non-solenoidal parts of the system matrix, unknowns, and right-hand side and rescaling them appropriately They are an alternative to loop-star/tree decompositions that presents several advantages when compared to these schemes. From previous sections it is clear that the main drawbacks of loop-star/tree decompositions reside in their constant-infrequency, but still high, condition number and in the need to be enriched with global loop functions [67], [68] Both of these drawbacks can be overcome by the use of quasi-Helmholtz projectors, as explained above, but other schemes can alternatively be used as effective cures for one or both of the drawbacks above. For the near-field computation, the separation in jsol-qhar and in jnsol must be maintained, the static contribution removed from the Green’s function, and, in addition, the divergence of the scalar potential explicitly enforced by omitting it
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