Article
Mathematics, Applied
Yernat M. Assylbekov, Ting Zhou
Summary: This paper explores an inverse boundary value problem of electromagnetism in a nonlinear Kerr medium, demonstrating the unique determination of the electromagnetic material parameters and nonlinear susceptibility parameters of the medium through electromagnetic measurements on the boundary. The study focuses on the case of the time-harmonic Maxwell equations.
JOURNAL OF SPECTRAL THEORY
(2021)
Article
Mathematics, Applied
Long Yuan, Qiya Hu
Summary: This paper focuses on the plane wave discontinuous Galerkin (PWDG) methods for Helmholtz equation and time-harmonic Maxwell equations in three-dimensional anisotropic media, where the coefficients are matrices. The study defines novel plane wave basis functions and derives error estimates of the approximate solutions with respect to the condition number of the coefficient matrices. Numerical results confirm the theoretical validity and the high accuracy of the solutions obtained by the proposed PWDG method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Martin Halla
Summary: In recent decades, qualitative inverse scattering methods with eigenvalues as target signatures have attracted much attention. Understanding the properties of related eigenvalue problems is crucial for comprehending those methods. However, the existence of eigenvalues for such (nonselfadjoint) problems is challenging, and existing results for absorbing media are often established under unrealistic assumptions or a smoothing of the eigenvalue problem. This article presents a technique to prove the existence of infinitely many eigenvalues for such problems under realistic assumptions.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Long Yuan
Summary: This paper considers time-harmonic Maxwell equations in three-dimensional anisotropic media with Dirichlet boundary conditions. It establishes stability estimates between the original electric field and transformed nonphysical field, and proves that approximate solutions generated by the PWLS method have nearly optimal L(2) error estimates. Numerical results confirm the theoretical validity, and comparisons with the existing PWDG method are provided.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Damien Chicaud, Patrick Ciarlet
Summary: This study focuses on the time-harmonic Maxwell's equations in anisotropic media. The problem addressed is an approximation of the diffraction problem or scattering from bounded objects that are usually located in an exterior domain in R3. The study considers perfectly conducting objects and imposes a Dirichlet boundary condition on those objects and an impedance condition on an artificial boundary to model an approximate radiation condition. The research examines the mathematical meaning of the impedance condition and determines the well-posedness of the model, as well as the a priori regularity of the fields in the domain and on the boundaries.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2023)
Article
Physics, Multidisciplinary
Almas A. Kurmanov, Nurlybek A. Ispulov, Abdul Qadir, Almar Zh Zhumabekov, Sholpan N. Sarymova, Kairat R. Dossumbekov
Summary: This paper investigates the fundamental properties of solutions of Maxwell's equations in stationary anisotropic media, with tensor characteristics along the Z axis. Coefficients matrix, matrix structure of Maxwell's equations, dispersion equations, and indicatrix curves for a transparent anisotropic one-dimensional inhomogeneous conductive medium are obtained. Exact analytical solutions in case of homogeneous anisotropic media are constructed based on the matrix structure.
Article
Mathematics, Applied
Tielei Zhu, Jiaqing Yang, Bo Zhang
Summary: This paper deals with the scattering of a time-harmonic electromagnetic wave by a three-dimensional elastic body, considering the general transmission conditions with Voigt's model to describe the interaction between the electromagnetic field and the elastic body on the interface. The existence of a unique solution is proved in an appropriate Sobolev space using the variational method with the classical Fredholm alternative. The inverse problem of recovering the elastic body by the scattered wave-field data is also investigated, showing that the shape and location of the body can be uniquely determined by fixed energy magnetic (or electric) far-field measurements corresponding to incident plane waves with all polarizations.
INVERSE PROBLEMS AND IMAGING
(2022)
Article
Mathematics, Applied
Meng Chen, Rong Gao, Yan He, Linghua Kong
Summary: Recently, researchers have developed a direct method for solving 2D Maxwell's equations in Kerr-type nonlinear media. This method eliminates iteration error and is more efficient than the classical iterative method. The stability of 3D Maxwell's equations has been theoretically investigated, and numerical results have confirmed a second-order convergence rate in both time and space.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics
Aleksandr Belov, Zhanna Dombrovskaya
Summary: This study proposes an approximate method of integrating the Maxwell equations along the beam propagation direction to reduce the computational costs of integrated photonics problems. By using one-dimensional bicompact schemes, the original two-dimensional problem is simplified to a one-dimensional one. The proposed method is validated through calculations of test and applied problems with known exact reflection spectra.
Article
Mathematics, Applied
Damien Chicaud, Patrick Ciarlet, Axel Modave
Summary: This study proves the well-posedness of formulations for the Dirichlet and Neumann problems using suitable function spaces and Helmholtz decompositions, and analyzes the a priori regularity of the solution and the solution's curl. The discretization of the formulations with a H(curl)-conforming approximation based on edge finite elements is considered, and an a priori error estimate is derived and verified with numerical results.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2021)
Article
Materials Science, Multidisciplinary
Zhong Lin Wang
Summary: This study derived the expanded Maxwell's equations for moving media and time-dependent configurations from the integral form of the equations. These expanded equations are the most comprehensive governing equations including both electromagnetic interaction and power generation. The study also developed a first principle theory for triboelectric nanogenerators based on these expanded equations. Additionally, general approaches for solving the expanded Maxwell's equations and calculating the displacement current for the output power of the nanogenerators were presented. The impact of this theory extends to electromagnetic wave generation and interaction with moving objects such as trains, cars, jets, missiles, comets, and galaxy stars if observed from earth.
Article
Mathematics, Applied
Martin Halla
Summary: This article discusses the time-harmonic electromagnetic wave equations in composites of dispersive materials surrounded by classical materials. The authors generalize previous meshing rules to the electromagnetic two-dimensional vectorial equations and the related holomorphic eigenvalue problems. They confirm their theoretical results through computational studies.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics
Yernat M. Assylbekov, Ting Zhou
Summary: In this paper, an inverse boundary value problem of electromagnetism with nonlinear Second Harmonic Generation (SHG) process is considered. The unique determination of the electromagnetic material parameters and the SHG susceptibility parameter of the medium is shown by making electromagnetic measurements on the boundary. The case of interest is when a frequency is fixed.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Thu Le, Dinh-Liem Nguyen, Hayden Schmidt, Trung Truong
Summary: This paper focuses on the imaging of 3D scattering objects using experimental data from the Fresnel database. The modified version of the orthogonality sampling method (OSM) is investigated for the imaging problem. The modified OSM is shown to be applicable to more types of polarization vectors and performs better than its original version and the factorization method in the verification using the 3D Fresnel database.
Article
Mathematics, Applied
Dinh-Liem Nguyen, Trung Truong
Summary: This paper investigates the inverse scattering problem for the three-dimensional Maxwell equations in bi-anisotropic periodic structures. The factorization method is used as an analytical and numerical tool, providing a rigorous justification and a fast imaging algorithm for the periodic scatterer. Numerical examples are presented to demonstrate the efficiency of the method.
JOURNAL OF INVERSE AND ILL-POSED PROBLEMS
(2022)
Article
Engineering, Multidisciplinary
Zixian Jiang, Houssem Haddar, Armin Lechleiter, Mabrouka El-Guedri
INVERSE PROBLEMS IN SCIENCE AND ENGINEERING
(2016)
Article
Mathematics, Applied
Houssem Haddar, Rainer Kress
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2016)
Article
Mathematics, Applied
Houssem Haddar, Thi-Phong Nguyen
APPLICABLE ANALYSIS
(2017)
Article
Mathematics, Applied
Houssem Haddar, Zixian Jiang, Mohamed Kamel Riahi
JOURNAL OF SCIENTIFIC COMPUTING
(2017)
Article
Mathematics, Applied
Fabien Caubet, Houssem Haddar, Jing-Rebecca Li, Dang Van Nguyen
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2017)
Article
Mathematics, Applied
Lorenzo Audibert, Fioralba Cakoni, Houssem Haddar
Article
Computer Science, Artificial Intelligence
Lorenzo Audibert, Houssem Haddar
SIAM JOURNAL ON IMAGING SCIENCES
(2017)
Article
Mathematics
Lorenzo Audiberta, Lucas Chesnel, Houssem Haddar
COMPTES RENDUS MATHEMATIQUE
(2018)
Article
Mathematics, Applied
H. Boujlida, H. Haddar, M. Khenissi
SIAM JOURNAL ON APPLIED MATHEMATICS
(2018)
Article
Mathematics, Applied
Houssem Haddar, Shixu Meng
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2018)
Article
Mathematics, Applied
Lorenzo Audibert, Lucas Chesnel, Houssem Haddar
Article
Chemistry, Multidisciplinary
M. Bakry, H. Haddar, O. Bunau
JOURNAL OF APPLIED CRYSTALLOGRAPHY
(2019)
Proceedings Paper
Acoustics
Lorenzo Audibert, Lucas Chesnel, Houssem Haddar, Kevish Napal
ADVANCES IN ACOUSTICS AND VIBRATION II (ICAV2018)
(2019)
Article
Mathematics, Applied
Houssem Haddar, Thi-Phong Nguyen
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2017)
Article
Engineering, Multidisciplinary
Fatemeh Pourahmadian, Bojan B. Guzina, Houssem Haddar
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2017)
