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 severly 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

Low frequency stability of the mixed discretization of the MFIE

Recently, a novel discretization for the magnetic field integral equation (MFIE) was presented. This discretization involves both Rao-Wilton-Glisson (RWG) basis functions and Buffa-Christiansen (BC) basis functions and is dubbed `mixed'. The scheme conforms to the functional spaces most natural to electromagnetics and thus can be expected to yield more accurate results. In this contribution, this intuition is corroborated by an analysis of the low frequency behavior of the classical and mixed discretizations of the MFIE. It is proved that the mixed discretization of the MFIE yields accurate results at very low frequencies whereas the classical discretization breaks down, as was already discussed extensively in literature

Using an LU Recombination Method to Improve the Performance of the Boundary Element Method at Very Low Frequencies

Many numerical electromagnetic modeling techniques that work very well at high frequencies do not work well at lower frequencies. This is directly or indirectly due to the weak coupling between the electric and magnetic fields at low frequencies. One technique for improving the performance of boundary element techniques at low frequencies is through the use of loop-tree basis functions, which decouple the contributions from the vector and scalar electric potential. However, loop-tree basis functions can be difficult to define for large, complex geometries. This paper describes a new method for improving the low-frequency performance of boundary element techniques. The proposed method does not require special basis functions and is relatively easy to implement. Numerical errors introduced by the great difference in scale between the vector and scalar electric potential are corrected automatically during the LU decomposition of the impedance matrix

Tomographie par induction magnétique : une formulation par équation intégrale volumique

RÉSUMÉ La tomographie par induction magnétique (TIM) est une technique d’imagerie qui cherche à reconstruire une carte des propriétés électriques passives d’un objet. Ses principaux domaines d’application sont le contrôle non-destructif, le sondage géologique et l’imagerie biomédicale. Des projets de recherche récents ont démontré que la tomographie par induction magnétique biomédicale est applicable, par exemple, à la détection d’accidents cérébraux vasculaires, à la mesure inductive de la conductivité de plaies et à l’imagerie pulmonaire. En comparaison avec les autres techniques d’imagerie biomédicale, la TIM a l’avantage d’utiliser un champ nonionisant, de ne pas nécessiter de contact direct avec les tissus et d’être très peu dispendieuse, ne nécessitant essentiellement qu’une bobine d’induction et une de mesure. À cause de la simplicité du matériel nécessaire et de son très faible coût d’utilisation, le développement de systèmes de TIM performants a le potentiel de grandement améliorer la détection des pathologies mentionnées précédemment en permettant à davantage de gens de bénéficier d’une technologie d’imagerie biomédicale. Ces aspects techniques la rendent particulièrement profitable aux populations en régions rurales où l’accès à des hôpitaux possédant des appareils d’imagerie plus sophistiqués est difficile. Les principaux inconvénients de la TIM sont sa faible résolution (par exemple par rapport à l’IRM), son champ diffusé très faible et son coût de calcul très élevé pour la reconstruction d’images. Il a été montré que la faible résolution n’est pas problématique pour certaines applications telles que la détection d’accidents cérébraux vasculaires. Dans les dernières années, le développement de capteurs ultrasensibles, par exemple les capteurs à magnéto-impédance géante, a permis la détection de variations très faibles du champ magnétique, ce qui permet de traiter le problème du faible champ diffusé. Le dernier aspect, le coût de calcul élevé pour la reconstruction d’images, est celui traité dans ce projet. Puisque la TIM opère à basse fréquence (de quelques kHz à quelques dizaines de MHz), le patron de diffraction est plus complexe à évaluer que pour d’autres méthodes d’imagerie, par exemple la tomographie par rayons X. Pour évaluer le champ magnétique diffusé par l’objet d’intérêt aux détecteurs, les équations de Maxwell qui décrivent l’interaction des ondes électromagnétiques avec la matière, doivent être résolues. Des solutions analytiques au problème de diffusion existent pour des géométries très simples, par exemple pour une sphère illuminée par une onde plane, mais pour des géométrie plus complexes des méthodes numériques doivent être utilisées afin de trouver une solution approximative.----------ABSTRACT Magnetic induction tomography (MIT) is a biomedical imaging method that aims to reconstruct the passive electrical properties of an object. The main areas of applications are non-destructive evaluation, geological surveys and biomedical imaging. Recent research has shown that biomedical MIT is applicable for example to stroke detection, inductive measurement of wound conductivity and lung imaging. In comparison with other imaging methods, MIT has the benefits of using a non-ionizing field, of being contactless and of being very cheap, basically requiring only an inducing and a measuring coil. Because of the simplicity of its apparatus and its low operating cost, the development of efficient MIT systems could greatly increase the detection of the aforementioned pathologies by allowing more people to have access to a biomedical imaging system. These technical aspects make this imaging technique especially useful to rural populations, where the access to hospitals with sophisticated apparatus is not available. The main drawbacks of MIT are its low resolution (e.g. compared to MRI), it’s weak scattered field and the high computation cost implied in image reconstruction from the magnetic field data. It was shown that the low resolution still allows the detection of certain pathologies such as ischemic strokes. In the last years, the development of very sensitive magnetic field sensors, such as giant magnetoimpendance sensors, has allowed the detection of very small perturbations in the magnetic field. The last issue, the high computation cost in image reconstruction, is the one being tackled by this project. Because of the low frequency range of MIT, the diffraction pattern is more complex than other imaging methods such as X-ray tomography. In order to evaluate the scattered magnetic field at the sensors, one has to solve the Maxwell equations, which describe the interaction between electromagnetic waves and matter. Analytical solutions exist for very simple cases, such as a sphere under a plane wave illumination. For more complex geometries, numerical methods must be employed to find an approximate solution to the problem. State of the art numerical methods either apply approximations in order to reduce the complexity of the equations, or use differential methods or surface integrals. Approximate methods include either the “quasistatic” approximation, the “low conductivity” approximation or a combination of both. This allows for a reduction of the full wave equation, which is complex to solve, to a simpler case such as a Poisson equation. This greatly reduces the computation cost, but it is done at the expense of numerical accuracy

Analysis of low frequency scattering from penetrable scatterers

In this paper, we present a method for solving the surface integral equation using the method of moments (MoM) at very low frequencies, which finds applications in geoscience. The nature of the Helmholtz decomposition leads us to choose loop-tree basis functions to represent the surface current. Careful analysis of the frequency scaling property of each operator allows us to introduce a frequency normalization scheme to reduce the condition number of the MoM matrix. After frequency normalization, the MoM matrix can be solved using LU decomposition. The poor spectral properties of the matrix, however, makes it ill-suited for an iterative solver. A basis rearrangement is used to improve this property of the MoM matrix. The basis function rearrangement(BFR), which involves inverting the connection matrix, can be viewed as a pre-conditioner. The complexity of BFR is reduced to O(N), allowing this method to be combined with iterative solvers. Both rectilinear and curvilinear patches have been used in the simulations. The use of curvilinear patches reduces the number of unknowns significantly, thereby making the algorithm more efficient. This method is capable of solving Maxwell's equations from quasistatic to electrodynamic frequency range. This capability is of great importance in geophysical applications because the sizes of the simulated objects can range from a small fraction of a wavelength to several wavelengths.link_to_subscribed_fulltex