Yves Beghein

A Space-Time Mixed Galerkin Marching-on-in-Time Scheme for the Time-Domain Combined Field Integral Equation

The time domain combined field integral equation (TD-CFIE), which is constructed from a weighted sum of the time domain electric and magnetic field integral equations (TD-EFIE and TD-MFIE) for analyzing transient scattering from closed perfect electrically conducting bodies, is free from spurious resonances. The standard marching-on-in-time technique for discretizing the TD-CFIE uses Galerkin and collocation schemes in space and time, respectively. Unfortunately, the standard scheme is theoretically not well understood: stability and convergence have been proven for only one class of space-time Galerkin discretizations. Moreover, existing discretization schemes are nonconforming, i.e., the TD-MFIE contribution is tested with divergence conforming functions instead of curl conforming functions. We therefore introduce a novel space-time mixed Galerkin discretization for the TD-CFIE. A family of temporal basis and testing functions with arbitrary order is introduced. It is explained how the corresponding interactions can be computed efficiently by existing collocation-in-time codes. The spatial mixed discretization is made fully conforming and consistent by leveraging both Rao-Wilton-Glisson and Buffa-Christiansen basis functions and by applying the appropriate bi-orthogonalization procedures. The combination of both techniques is essential when high accuracy over a broad frequency band is required.

A Calderon Multiplicative Preconditioner for the PMCHWT Equation for Scattering by Chiral Objects

Scattering of time-harmonic electromagnetic waves by chiral structures can be modeled via an extension of the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) boundary integral equation for analyzing scattering by dielectric objects. The classical PMCHWT equation however suffers from dense discretization breakdown: the matrices resulting from its discretization become increasingly ill-conditioned when the mesh density increases. This contribution revisits the PMCHWT equation for chiral media. It is demonstrated that it also suffers from dense discretization breakdown. This dense discretization breakdown is mitigated by the construction of a Calderón multiplicative preconditioner. A stable discretization scheme is introduced, and the resulting algorithms accuracy and efficiency are corroborated by numerical examples.

A Higher Order Space-Time Galerkin Scheme for Time Domain Integral Equations

Stability of time domain integral equation (TDIE) solvers has remained an elusive goal for many years. Advancement of this research has largely progressed on four fronts: 1) Exact integration, 2) Lubich quadrature, 3) smooth temporal basis functions, and 4) space-time separation of convolutions with the retarded potential. The latter methods efficacy in stabilizing solutions to the time domain electric field integral equation (TD-EFIE) was previously reported for first-order surface descriptions (flat elements) and zeroth-order functions as the temporal basis. In this work, we develop the methodology necessary to extend the scheme to higher order surface descriptions as well as to enable its use with higher order basis functions in both space and time. These basis functions are then used in a space-time Galerkin framework. A number of results are presented that demonstrate convergence in time. The viability of the space-time separation method in producing stable results is demonstrated experimentally for these examples.

A DC Stable and Large-Time Step Well-Balanced TD-EFIE Based on Quasi-Helmholtz Projectors

The marching-on-in-time (MOT) solution of the time-domain electric field integral equation (TD-EFIE) has traditionally suffered from a number of issues, including the emergence of spurious static currents (dc instability) and ill-conditioning at large-time steps (low frequencies). In this contribution, a space-time Galerkin discretization of the TD-EFIE is proposed, which separates the loop and star components of both the equation and the unknown. Judiciously integrating or differentiating these components with respect to time leads to an equation which is free from dc instability. By choosing the correct temporal basis and testing functions for each of the components, a stable MOT system is obtained. Furthermore, the scaling of these basis and testing functions ensure that the system remains well conditioned for large-time steps. The loop-star decomposition is performed using quasi-Helmholtz projectors to avoid the explicit transformation to the unstable bases of loops and stars (or trees), and to avoid the search for global loops, which is a computationally expensive operation.

Calderon multiplicative preconditioner for the PMCHWT equation applied to chiral media

In this contribution, a Calderón preconditioned algorithm for the modeling of scattering of time harmonic electromagnetic waves by a chiral body is introduced. The construction of the PMCHWT in the presence of chiral media is revisited. Since this equation reduces to the classic PMCHWT equation when the chirality parameter tends to zero, it shares its spectral properties. More in particular, it suffers from dense grid breakdown. Based on the work in [1], [2], a regularized version of the PMCHWT equation is introduced. A discretization scheme is described. Finally, the validity and spectral properties are studied numerically. More in particular, it is proven that linear systems arising in the novel scheme can be solved in a small number of iterations, regardless the mesh parameter.

On the Hierarchical Preconditioning of the PMCHWT Integral Equation on Simply and Multiply Connected Geometries

We present a hierarchical basis preconditioning strategy for the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) integral equation considering both simply and multiply connected geometries. To this end, we first consider the direct application of hierarchical basis preconditioners, developed for the electric field integral equation (EFIE), to the PMCHWT. It is notably found that, whereas for the EFIE a diagonal preconditioner can be used for obtaining the hierarchical basis scaling factors, this strategy is catastrophic in the case of the PMCHWT since it leads to a severely ill-conditioned PMCHWT system in the case of multiply connected geometries. We then proceed to a theoretical analysis of the effect of hierarchical bases on the PMCHWT operator for which we obtain the correct scaling factors and a provably effective preconditioner for both low frequencies and mesh refinements. Numerical results will corroborate the theory and show the effectiveness of our approach.

Handling the low-frequency breakdown of the PMCHWT integral equation with the quasi-Helmholtz projectors

This contribution presents a quasi-Helmholtz projectors based regularization of the low frequency breakdown of the PMCHWT integral equation. The PMCHWT equation in the low-frequency regime shows an ill-conditioned behavior inherited from the Electric Field Integral Operators it contains. The stabilization via quasi-Helmholtz projectors, differently from the use of standard Loop-Star/Tree decompositions, does not introduce an additional mesh-size-related ill-conditioning and it applies smoothly to both simply and non-simply connected geometries. The presentation of the main formulation will be complemented by numerical results demonstrating the effectiveness and accuracy of the proposed scheme.

A temporal Galerkin discretization of the charge-current continuity equation

In time domain boundary integral equations, scattered fields are computed from a priori unknown electric current and charge densities. In many implementations, the charge density is eliminated from the integral equation prior to discretization, using the charge-current continuity equation. In this contribution, the charge density is explicitly discretized, and the continuity equation is weakly enforced by a space-time Galerkin procedure, leading to a simpler and more consistent implementation. The effects of this discretization scheme on the stability and the accuracy of the resulting solution method are discussed.

Accurate and conforming mixed discretization of the chiral müller equation

Scattering of time-harmonic fields by chiral objects can be modeled by a second kind boundary integral equation, similar to Müllers equation for scattering by nonchiral penetrable objects. In this contribution, a mixed discretization scheme for the chiral Müller equation is introduced using both Rao-Wilton-Glisson and Buffa-Christiansen funtions. It is shown that this mixed discretization yields more accurate solutions than classical discretizations, and that they can be computed in a limited number of iterations using Krylov type solvers.

Spectral and algorithmic strategies for penetrable scatterers on simply and non-simply connected geometries

The Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) is a widely used integral equation for simulating radiation and scattering from penetrable objects. This formulation, however, is plagued from mesh refinement and low-frequency ill-conditioning. Existing techniques for handling these problems, however, suffer from very low-frequency numerical cancellations or they require the detection of global loops. This work presents a new Calderon-like strategy for the PMCHWT which, leveraging on the quasi-Helmholtz projectors, solves both frequency and refinement ill-conditioning without detecting global loops. Moreover the technique is immune from very low-frequency numerical cancellations. Numerical results confirms all theoretical developments and show the practical impact of the new scheme.