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Author: Hu, Guanghui × End date: 2019 × Start date: 2010 × DDC: 510 × Type: report ×

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Now showing 1 - 10 of 32
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    Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Elschner, Johannes; Hu, Guanghui
    In this paper, we investigate the inverse problem of recovering a two-dimensional perfectly reflecting diffraction grating from the scattered waves measured above the structure. Inspired by a novel idea developed by Bao, Zhang and Zou [to appear in Trans. Amer. Math. Soc.], we present a complete characterization of the global uniqueness in determining polygonal periodic structures using a minimal number of incident plane waves. The idea in this paper combines the reflection principle for the Helmholtz equation and the dihedral group theory. We characterize all periodic polygonal structures that cannot be identified by one incident plane wave, including the resonance case where a Rayleigh frequency is allowed. Furthermore, we show that those unidentifiable gratings provide non-uniqueness examples for appropriately chosen wave number and incident angles. We also indicate and fix a gap in the proof of the main theorem of Elschner and Yamamoto [Z. Anal. Anwend., 26 (2007), 165-177], and generalize the uniqueness results of that paper.
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    Inverse scattering of electromagnetic waves by multilayered structures : uniqueness in TM mode
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Elschner, Johannes; Hu, Guanghui
    Assume a time-harmonic electromagnetic wave is scattered by an infinitely long cylindrical conductor surrounded by an unknown piecewise homogenous medium remaining invariant along the cylinder axis. We prove that, in TM mode, the far field patterns for all observation directions at a fixed frequency uniquely determine the unknown surrounding medium as well as the shape of the cylindrical conductor. A similar uniqueness result is obtained for the scattering by multilayered penetrable periodic structures in a piecewise homogenous medium. The periodic interfaces and refractive indices can be uniquely identified from the near field data measured only above (or below) the structure for all quasi-periodic incident waves with a fixed phase-shift. The proofs are based on the singularity of the Green function to a two dimensional elliptic equation with piecewise constant leading coefficients
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    Inverse scattering of elastic waves by periodic structures : uniqueness under the third or fourth kind boundary conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Elschner, Johannes; Hu, Guanghui
    The inverse scattering of a time-harmonic elastic wave by a two-dimensional periodic structure in R 2 is investigated. The grating profile is assumed to be a graph given by a piecewise linear function on which the third or fourth kind boundary conditions are satisfied. Via an equivalent variational formulation, existence of quasi-periodic solutions for general Lipschitz grating profiles is proved by applying the Fredholm alternative. However, uniqueness of solution to the direct problem does not hold in general. For the inverse problem, we determine and classify all the unidentifiable grating profiles corresponding to a given incident elastic field, relying on the reflection principle for the Navier equation and the rotational invariance of propagating directions of the total field. Moreover, global uniqueness for the inverse problem is established with a minimal number of incident pressure or shear waves, including the resonance case where a Rayleigh frequency is allowed. The gratings that are unidentifiable by one incident elastic wave provide non-uniqueness examples for appropriately chosen wave number and incident angles
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    Single logarithmic conditional stability in determining unknown boundaries
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Elschner, Johannes; Hu, Guanghui; Yamamoto, Masahiro
    We prove a conditional stability estimate of log-type for determining unknown boundaries from a single Cauchy data taken on an accessible subboundary. Our approach relies on new interior and boundary estimates derived from the Carleman estimate for elliptic equations. A local stability result for target identification of an acoustic sound-soft scatterer from a single far-field pattern is also obtained.
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    Elastic scattering coefficients and enhancement of nearly elastic cloaking
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Abbas, Tasawar; Ammari, Habib; Hu, Guanghui; Wahab, Abdul; Ye, Jong Chul
    The concept of scattering coefficients has played a pivotal role in a broad range of inverse scattering and imaging problems in acoustic and electromagnetic media. In view of their promising applications, we introduce the notion of scattering coefficients of an elastic inclusion in this article. First, we define elastic scattering coefficients and substantiate that they naturally appear in the expansions of elastic scattered field and far field scattering amplitudes corresponding to a plane wave incidence. Then an algorithm is developed and analyzed for extracting the elastic scattering coefficients from multi-static response measurements of the scattered field. Moreover, the estimate of the maximal resolving order is provided in terms of the signal-to-noise ratio. The decay rate and symmetry of the elastic scattering coefficients are also discussed. Finally, we design scattering-coefficients-vanishing structures and elucidate their utility for enhancement of nearly elastic cloaking.
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    Direct and inverse elastic scattering problems for diffraction gratings
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Elschner, Johannes; Hu, Guanghui
    This paper is concerned with the direct and inverse scattering of time-harmonic plane elastic waves by unbounded periodic structures (diffraction gratings). We present a variational approach to the forward scattering problems with Lipschitz grating profiles and give a survey of recent uniqueness and existence results. We also report on recent global uniqueness results within the class of piecewise linear grating profiles for the corresponding inverse elastic scattering problems. Moreover, a discrete Galerkin method is presented to efficiently approximate solutions of direct scattering problems via an integral equation approach. Finally, an optimization method for solving the inverse problem of recovering a 2D periodic structure from scattered elastic waves measured above the structure is discussed.
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    A time domain sampling method for inverse acoustic scattering problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Guo, Yukun; Hömberg, Dietmar; Hu, Guanghui; Li, Jingzhi; Liu, Hongyu
    This work concerns the inverse scattering problems of imaging unknown/inaccessible scatterers by transient acoustic near-field measurements. Based on the analysis of the migration method, we propose efficient and effective sampling schemes for imaging small and extended scatterers from knowledge of time-dependent scattered data due to incident impulsive point sources. Though the inverse scattering problems are known to be nonlinear and ill-posed, the proposed imaging algorithms are totally direct involving only integral calculations on the measurement surface. Theoretical justifications are presented and numerical experiments are conducted to demonstrate the effectiveness and robustness of our methods. In particular, the proposed static imaging functionals enhance the performance of the total focusing method (TFM) and the dynamic imaging functionals show analogous behavior to the time reversal inversion but without solving time-dependent wave equations.
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    Near-field imaging of scattering obstacles with the factorization method: Fluid-solid interaction
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Yin, Tao; Hu, Guanghui; Xu, Liwei; Zhang, Bo
    Consider a time-harmonic acoustic point source incident on a bounded isotropic linearly elastic body immersed in a homogeneous compressible inviscid fluid. This paper is concerned with the inverse fluid-solid interaction (FSI) problem of recovering the elastic body from near-field data generated by infinitely many incident point source waves at a fixed energy. The incident point sources and the receivers for recording scattered signals are both located on a non-spherical closed surface, on which an outgoing-to-incoming (OtI) operator is appropriately defined. We provide a theoretical justification of the factorization method for precisely characterizing the scatterer by utilizing the spectrum of the near-field operator. This generalizes the imaging scheme developed in [G. Hu, J. Yang, B. Zhang, H. Zhang, Inverse Problems 30 (2014): 095005] to the case when near-field data are measured on non-spherical surfaces. Numerical examples in 2D are demonstrated to show the validity and accuracy of the inversion algorithm, even if limited aperture data are available on one or several line segments.
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    Finite element method to fluid-solid interaction problems with unbounded periodic interfaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Hu, Guanghui; Rathsfeld, Andreas; Yin, Tao
    Consider a time-harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This paper is concerned with a variational approach to the fluid-solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasi-periodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet-to-Neumann mappings is proposed. The Dirichlet-to-Neumann mappings are approximated by truncated Rayleigh series expansions, and, finally, numerical tests in 2D are performed.
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    Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Elschner, Johannes; Hu, Guanghui; Yamamoto, Masahiro
    Consider the two-dimensional inverse elastic scattering problem of recovering a piecewise linear rigid rough or periodic surface of rectangular type for which the neighboring line segments are always perpendicular.We prove the global uniqueness with at most two incident elastic plane waves by using near-field data. If the Lamé constants satisfy a certain condition, then the data of a single plane wave is sufficient to imply the uniqueness. Our proof is based on a transcendental equation for the Navier equation, which is derived from the expansion of analytic solutions to the Helmholtz equation. The uniqueness results apply also to an inverse scattering problem for non-convex bounded rigid bodies of rectangular type.