1,720,983 research outputs found
On the Use of Quasi-Helmholtz Projectors for Modeling Structures with Junctions at Arbitrarily Low Frequencies
Full text linkThis work investigates the usage of quasi-Helmholtz projectors to cure the low frequency breakdown of the electric field integral equation (EFIE) when modeling structures containing junctions. We show that the quasi-Helmholtz projectors can still efficiently perform projection into the solenoidal and non-solenoidal subspaces in the presence of junctions. Preconditioning strategies leveraging these projectors are capable of curing the low-frequency breakdown that severely compromises the accuracy and conditioning of the EFIE. Numerical studies validate the use of algebraic multigrid approaches for achieving fast and efficient implementation of the proposed scheme for problems with multiple junctions. The effectiveness of the scheme and its applicability to problems of industrial relevance are illustrated through series of numerical results
A Fast Quasi-Conformal Mapping Preconditioner for Electromagnetic Integral Equations
Full text linkBoundary Element Methods (BEMs) are efficient strategies to numerically solve electromagnetic radiation and scattering problems. Unfortunately, however, classical BEM formulations suffer from ill-conditioning when the frequency is low, or the discretization density is high. In the past, several remedies have been presented for these ill-conditioning problems including preconditioners based on Calderón identities, hierarchical bases, and current decompositions. While effective, these strategies however require ad-hoc procedures including mesh-refinements, new basis function definitions, and adapted fast methods that, if not implemented properly, can become computationally cumbersome
On a Frequency-Stabilized Single Current Inverse Source Formulation
Full text linkSeveral strategies are available for solving the inverse source problem in electromagnetics. Among them, many have been focusing in retrieving Love currents by solving, after regularization, for Love’s electric and magnetic currents. In this work we present a dual-element discretization, analysis, and stabilization of an inverse source formulation providing Love data by solving for only one current. This results in substantial savings and allows for an effective quasi-Helmholtz projector stabilization of the resulting operator. Theoretical considerations are complemented by numerical tests showing effectiveness and efficiency of the newly proposed method
On preconditioning electromagnetic integral equations in the high frequency regime via helmholtz operators and quasi-helmholtz projectors
Full text linkFast and accurate resolution of electromagnetic problems via the boundary element method (BEM) is oftentimes challenged by conditioning issues occurring in three distinct regimes: (i) when the frequency decreases and the discretization density remains constant, (ii) when the frequency is kept constant while the discretization is refined and (iii) when the frequency increases along with the discretization density. While satisfactory remedies to the problems arising in regimes (i) and (ii), respectively based on Helmholtz decompositions and Calderon-like techniques have been presented, the last regime is still challenging. In fact, this last regime is plagued by both spurious resonances and ill-conditioning, the former can be tackled via combined field strategies and is not the topic of this work. In this contribution new symmetric scalar and vectorial electric type formulations that remain well-conditioned in all of the aforementioned regimes and that do not require barycentric discretization of the dense electromagnetic potential operators are presented along with a spherical harmonics analysis illustrating their key properties
On a Low-Frequency and Contrast Stabilized Full-Wave Volume Integral Equation Solver for Lossy Media
Full text linkIn this article, we present a new regularized electric flux volume integral equation (D-VIE) for modeling high-contrast conductive dielectric objects in a broad frequency range. This new formulation is particularly suitable for modeling biological tissues at low frequencies, as it is required by brain epileptogenic area imaging, but also at higher ones, as it is required by several applications, including, but not limited to, deep brain stimulation (DBS). When modeling inhomogeneous objects with high complex permittivities at low frequencies, the traditional D-VIE is ill-conditioned and suffers from numerical instabilities that result in slower convergence and less accurate solutions. In this work, we address these shortcomings by leveraging a new set of volume quasi-Helmholtz projectors. Their scaling by the material permittivity matrix allows for the rebalancing of the equation when applied to inhomogeneous scatterers and, thereby, makes the proposed method accurate and stable even for high complex permittivity objects until arbitrarily low frequencies. Numerical results, canonical and realistic, corroborate the theory and confirm the stability and the accuracy of this new method both in the quasi-static regime and at higher frequencies
Electromagnetic modelling at arbitrarily low frequency via the quasi-Helmholtz projectors
No full textWe have identified the sources of the different problems plaguing the EFIE at low frequencies in both the frequency and the TD, as well as their traditional cures. Despite their apparent effectiveness, these techniques have been shown to have a limited applicability because they introduce their own set of problems which include the high computational burden of the LS decomposition and its effect on the high-refinement conditioning of the FD-EFIE and the numerical instabilities introduced by the treatment of the TD-EFIE. Techniques leveraging qH projectors, immune from the aforementioned side-effects, have been introduced to address the different aspects of the low-frequency breakdown of the FD formulation and of the large time step breakdown of its TD counterpart. In case of the FD, using projectors allows the same re-scaling of the solenoidal and non-solenoidal parts of the RWG space as traditional LS, but it has the added benefits of not requiring identification of the global loops of the structure as well as not introducing any further high-refinement ill-conditioning. In the TD case, the projectors are still used to separate the loop and star parts of the discretized space, but this separation is used to apply the correct derivative and integrative terms to the different parts of the operators. Coupled with an adequate mixed time-discretization scheme, this technique fully addresses the low-frequency limitations of the TD-EFIE. Along with presenting these purely theoretical concepts, we have provided implemen-tation related hints, allowing the techniques presented in this chapter to be reliably and readily implemented into existing solvers. Finally, while we have addressed their low-frequency breakdown, both EFIE formulations still suffer from a high-refinement breakdown. While in standard low-frequency scenarios, a curing of low-frequency issues may suffice, for more pathological cases techniques addressing both break-downs may be required. Strategies based on qH projectors and Calderon identities have recently been introduced for the frequency and TD formulations [23, 40] and should be used in this case
A New Refinement-Free Preconditioner for the Symmetric Formulation in Electroencephalography
Full text linkWidely employed for the accurate solution of the electroencephalography forward problem, the symmetric formulation gives rise to a first kind, ill-conditioned operator illsuited for complex modelling scenarios. This work presents a novel preconditioning strategy based on an accurate spectral analysis of the operators involved which, differently from other Calderón-based approaches, does not necessitate the barycentric refinement of the primal mesh (i.e., no dual matrix is required). The discretization of the new formulation gives rise to a well-conditioned, symmetric, positive-definite system matrix, which can be efficiently solved via fast iterative techniques. Numerical results for both canonical and realistic head models validate the effectiveness of the proposed formulation
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