An official website of the United States government
We study an inverse problem for the time-dependent Maxwell system in an inhomogeneous and anisotropic medium. The objective is to recover the initial electric field $$\mathbf{E}_0$$ in a bounded domain $$\Omega \subset \mathbb{R}^3$$, using boundary measurements of the electric field and its normal derivative over a finite time interval. Informed by practical constraints, we adopt an under-determined formulation of Maxwell's equations that avoids the need for initial magnetic field data and charge density information. To address this inverse problem, we develop a time-dimension reduction approach by projecting the electric field onto a finite-dimensional Legendre polynomial-exponential basis in time. This reformulates the original space-time problem into a sequence of spatial systems for the projection coefficients. The reconstruction is carried out using the quasi-reversibility method within a minimum-norm framework, which accommodates the inherent non-uniqueness of the under-determined setting. We prove a convergence theorem that ensures the quasi-reversibility solution approximates the true solution as the noise and regularization parameters vanish. Numerical experiments in a fully three-dimensional setting validate the method's performance. The reconstructed initial electric field remains accurate even with $$10\%$$ noise in the data, demonstrating the robustness and applicability of the proposed approach to realistic inverse electromagnetic problems.
more »
« less
A note on transmission eigenvalues in electromagnetic scattering theory
This short note was motivated by our e orts to investigate whether there exists a half plane free of transmission eigenvalues for Maxwell's equations. This question is related to solvability of the time domain interior transmission problem which plays a fundamental role in the justi cation of linear sampling and factorization methods with time dependent data. Our original goal was to adapt semiclassical analysis techniques developed in [21, 23] to prove that for some combination of electromagnetic parameters, the transmission eigenvalues lie in a strip around the real axis. Unfortunately we failed. To try to understand why, we looked at the particular example of spherically symmetric media, which provided us with some insight on why we couldn't prove the above result. Hence this paper reports our ndings on the location of all transmission eigenvalues and the existence of complex transmission eigenvalues for Maxwell's equations for spherically strati ed media. We hope that these results can provide reasonable conjectures for general electromagnetic media.
more »
« less
- Award ID(s):
- 1813492
- PAR ID:
- 10253706
- Date Published:
- Journal Name:
- Inverse problems and imaging
- ISSN:
- 1930-8337
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)By using an asymptotic analysis and numerical simulations, we derive and investigate a system of homogenized Maxwell's equations for conducting material sheets that are periodically arranged and embedded in a heterogeneous and anisotropic dielectric host. This structure is motivated by the need to design plasmonic crystals that enable the propagation of electromagnetic waves with no phase delay (epsilon-near-zero effect). Our microscopic model incorporates the surface conductivity of the two-dimensional (2D) material of each sheet and a corresponding line charge density through a line conductivity along possible edges of the sheets. Our analysis generalizes averaging principles inherent in previous Bloch-wave approaches. We investigate physical implications of our findings. In particular, we emphasize the role of the vector-valued corrector field, which expresses microscopic modes of surface waves on the 2D material. We demonstrate how our homogenization procedure may set the foundation for computational investigations of: effective optical responses of reasonably general geometries, and complicated design problems in the plasmonics of 2D materials.more » « less
-
In recent years, a new approach has been proposed in the study of the inverse scattering problem for electromagnetic waves. In particular, a study is made of the analytic properties of the scattering operator, and the results of this study are used to design target signatures that respond to changes in the electromagnetic parameters of the scattering medium. These target signatures are characterized by novel eigenvalue problems such that the eigenvalues can be determined from measured scattering data. Changes in the structural properties of the material or the presence of flaws cause changes in the measured eigenvalues. In this article, we provide a general framework for developing target signatures and numerical evidence of the efficacy of new target signatures based on recently introduced eigenvalue problems arising in electromagnetic scattering theory for anisotropic media.more » « less
-
Induction heating is one of the cleanest and most efficient methods for heating materials, utilizing electromagnetic fields induced through AC electric current. This article reports an analytical solution for transient heat transfer in a three‐dimensional (3D) cylindrical object under induction heating. A simplified form of Maxwell's equations is solved to determine the heat generation inside the cylinder by calculating the current density distribution within the body. The temperature within the solid is found from the solution of the unsteady heat equation based on Green's function. Owing to multiple spatial dimensions and time, a separation of variables technique is used to find Green's function. In addition, an innovative algorithm is proposed to take care of the variable material properties in analytical treatment. The analytical solution for temperature is verified with the data obtained from experiments for identical operating conditions. The analytical solution is used to study the impact of heat transfer coefficient and input AC current frequency and amplitude during transient heat diffusion. Our analytical solution suggests that the temperature‐dependent material properties significantly affect the thermal response within the solid. Unlike many other conventional heating methods, the thermal boundary condition changes with time in induction heating, which makes our solution much more challenging.more » « less
-
Abstract Positive lightning leaders are a ubiquitous, yet poorly understood, component of lightning flashes. Upward lightning started by positive leaders may be formed when nearby storm activity induces electrical charges in a tall structure, such as communications towers or wind turbines. Alternatively, upward lightning can be triggered with the rocket‐and‐wire technique. In this paper, we introduce a new self‐consistent model for this important discharge mode, one which solves Maxwell's equations under the quasi‐electrostatic approximation. The model also includes a realistic treatment of the nonlinear plasma conductivity within the leader channel. This new computational tool explains the origin of the positive leader speed, of 10s of km/s, as well as why it displays a steady behavior over time. The model also explains the temporal evolution of current to ground measured during the early stages of rocket‐triggered lightning, where the current exhibits a series of small‐amplitude pulses, which disappear over time. The article also outlines straightforward criteria for leader inception, which may have practical applications for lightning protection.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- The DOI is not currently available.
