A Regularized Boundary Element Formulation for Contactless SAR Evaluations within Homogeneous and Inhomogeneous Head Phantoms

This work presents a Boundary Element Method (BEM) formulation for contactless electromagnetic field assessments. The new scheme is based on a regularized BEM approach that requires the use of electric measurements only. The regularization is obtained by leveraging on an extension of Calderon techniques to rectangular systems leading to well-conditioned problems independent of the discretization density. This enables the use of highly discretized Huygens surfaces that can be consequently placed very near to the radiating source. In addition, the new regularized scheme is hybridized with both surfacic homogeneous and volumetric inhomogeneous forward BEM solvers accelerated with fast matrix-vector multiplication schemes. This allows for rapid and effective dosimetric assessments and permits the use of inhomogeneous and realistic head phantoms. Numerical results corroborate the theory and confirms the practical effectiveness of all newly proposed formulations

On a low-frequency and refinement stable PMCHWT integral equation leveraging the quasi-Helmholtz projectors

Classical Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) formulations for modeling radiation and scattering from penetrable objects suffer from ill-conditioning when the frequency is low or when the mesh density is high. The most effective techniques to solve these problems, unfortunately, either require the explicit detection of the so-called global loops of the structure, or suffer from numerical cancellation at extremely low frequency. In this contribution, a novel regularization method for the PMCHWT equation is proposed, which is based on the quasi-Helmholtz projectors. This method not only solves both the low frequency and the dense mesh ill-conditioning problems of the PMCHWT, but it is immune from low-frequency numerical cancellations and it does not require the detection of global loops. This is obtained by projecting the range space of the PMCHWT operator onto a dual basis, by rescaling the resulting quasi- Helmholtz components, by replicating the strategy in the dual space, and finally by combining the primal and the dual equations in a Calderón like fashion. Implementation-related treatments and details alternate the theoretical developments in order to maximize impact and practical applicability of the approach. Finally, numerical results corroborate the theory and show the effectiveness of the new schemes in real case scenarios

A Two-Dimensional Numerical Simulation of Plasma Wake Structure Around a CubeSat

A numerical model was developed to understand the time evolution of a wake structure around a CubeSat moving in a plasma with transonic speed. A cubeSat operates in the F2 layer of ionosphere with an altitude of 300 − 600 Km. The average plasma density varies between 10−6cm−3 − 10−9cm−3 and the temperature of ions and electrons is found between 0.1−0.2 eV. The study of a wake structure can provide insights for its effects on the measurements obtained from space instruments. The CubeSat is modeled to have a metal surface, which is a realistic assumption, with a negative electric potential. To solve the equations of plasma, the numerical difference equations were obtained by discretizing the fluid equations of the plasma along with nonlinear Poisson’s equation. The electrons were assumed to follow the Boltzmann’s relation and the dynamics of ions was followed using the fluid equations. The initial and boundary conditions for the evolution of the structure are discussed. The computation was compared to the analytical solution for a 1D problem before being applied to the 2D model. There was a good agreement between the numerical and analytical solution. In the 2D simulation, we observe the formation of plasma wake structure around the CubeSat. The plasma wake structure consists of rarefaction region where ion density and ion velocity decreases compared to the initial density and velocity

On the Multiplicative Regularization of Graph Laplacians on Closed and Open Structures with Applications to Spectral Partitioning

International audienceA new regularization technique for graph Laplacians arising from triangular meshes of closed and open structures is presented. The new technique is based on the analysis of graph Laplacian spectrally equivalent operators in terms of Sobolev norms and on the appropriate selection of operators of opposite differential strength to achieve a multiplicative regularization. Moreover, a new 3D/2D nested regularization strategy is presented to deal with open geometries. Numerical results shows the advantages of the proposed regularization as well as its effectiveness when used in spectral partitioning applications

Regularized Graph Laplacians for Hierarchical Spectral Partitioning

International audienceThis contribution will introduce an innovative approach to the Hierarchical Spectral Partitioning problem. The graph Laplacian will be regularized, by the scheme we are proposing, in an analytic way, by leveraging on fractional Sobolev norm realizations obtained with integral operators and by linking the Laplacian and the operators with suitably chosen mapping matrices that would connect, in the correct way, the underlying discretization spaces. In practice this will result in a multiplicative preconditioner for the graph Laplacian that, as will be detailed in the talk, will be easily obtained from matrices currently available in any standard EM boundary element implementation. Numerical results will corroborate our theoretical developments and will show the practical impact of our new technique on several realistic cases arising from applications

Regularized Formulations for Spectral Graph Partitioning

International audienceIn this paper, we introduce a novel regularization technique for the spectral partitioning of a mesh that relies on a efficient preconditioning of the associated graph Laplacian. The regularization is obtained by leveraging on fractional order Sobolev norms obtained with integral operators and by linking the Laplacian and the operators with suitably chosen Gram matrices that connect the underlying discretization spaces. The numerical results support the developed theory when applied to some of the realistic examples arising in Computational Electro-magnetics applications

Contactless Dosimetry with a Regularized Integral Equation Based Method

International audienceThis talk will present a source inversion algorithm that is capable to recover the field within a model of the human head in the presence of a radiating source (such as a mobile phone), starting from external field measurements and without a need for the numerical modeling of the source. The latter in fact will be replaced by an appropriately chosen simplified model that will avoid the detailed characterization of the electrically irrelevant details. This aspect of the method will be especially important for complex, hard to discretize and model, sources such as all the commercially available mobile phones. Moreover, differently from other inverse source formulations, the approach we are proposing relies on suitably regularized integral formulations both for modeling the dielectric parts and for discretizing the Helmholtz screens. The new regularized formulation is leveraging on a newly developed hierarchical spectral partitioning. The formulation has conditioning properties that are completely independent on the discretization density. This allows for the source to reside very near to the human head phantom under study

A novel volume integral equation for solving the Electroencephalography forward problem

International audienceIn this paper, a novel volume integral equation for solving the Electroencephalography forward problem is presented. Differently from other integral equation methods standardly used for the same purpose, the new formulation can handle inhomogeneous and fully anisotropic realistic head models. The new equation is obtained by a suitable use of Green's identities together with an appropriate handling of all boundary conditions for the EEG problem. The new equation is discretized with a consistent choice of volume and boundary elements. Numerical results shows validity and convergence of the approach, together with its applicability to real case models obtained from MRI data

Two volume integral equations for the inhomogeneous and anisotropic forward problem in electroencephalography

International audienceThis work presents two new volume integral equations for the Electroencephalography (EEG) forward problem which, differently from the standard integral approaches in the domain, can handle heterogeneities and anisotropies of the head/brain conductivity profiles. The new formulations translate to the quasi-static regime some volume integral equation strategies that have been successfully applied to high frequency electromagnetic scattering problems. This has been obtained by extending, to the volume case, the two classical surface integral formulations used in EEG imaging and by introducing an extra surface equation, in addition to the volume ones, to properly handle boundary conditions. Numerical results corroborate theoretical treatments, showing the competitiveness of our new schemes over existing techniques and qualifying them as a valid alternative to differential equation based methods

A mixed discretized surface-volume integral equation for solving EEG forward problems with inhomogeneous and anisotropic head models

International audienceThis work presents a new integral formulation to solve the EEG forward problem for potentially inhomogeneous and anisotropic head conductivity profiles. The formulation has been obtained from a surface/volume variational expression derived from Green's third identity and then solved in terms of both surface and volume unknowns. These unknowns are expanded with suitably chosen basis functions which systematically enforce transmission conditions. Finally, by leveraging on a mixed discretization, the equation is tested within the framework of a Petrov-Galerkin's scheme. Numerical results show the high level of accuracy of the proposed method, which compares very favourably with those obtained with existing, finite element, schemes