Browsing by Author "Hu, Guanghui"
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- ItemConvergence analysis of the FEM coupled with Fourier-mode expansion for the electromagnetic scattering by biperiodic structures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Hu, Guanghui; Rathsfeld, AndreasScattering of time-harmonic electromagnetic plane waves by a doubly periodic surface structure in R3 can be simulated by a boundary value problem of the time-harmonic curl-curl equation. For a truncated FEM domain, non-local boundary value conditions are required in order to satisfy the radiation conditions for the upper and lower half spaces. Alternatively to boundary integral formulations, to approximate radiation conditions and absorbing boundary methods, Huber et al. [11] have proposed a coupling method based on an idea of Nitsche. In the case of profile gratings with perfectly conducting substrate, the authors have shown previously that a slightly modified variational equation can be proven to be equivalent to the boundary value problem and to be uniquely solvable. Now it is shown that this result can be used to prove convergence for the FEM coupled by truncated wave mode expansion. This result covers transmission gratings and gratings bounded by additional multi-layer systems.
- ItemCorners and edges always scatter(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Elschner, Johannes; Hu, GuanghuiConsider time-harmonic acoustic scattering problems governed by the Helmholtz equation in two and three dimensions. We prove that bounded penetrable obstacles with corners or edges scatter every incident wave nontrivially, provided the function of refractive index is real-analytic. Moreover, if such a penetrable obstacle is a convex polyhedron or polygon, then its shape can be uniquely determined by the far-field pattern over all observation directions incited by a single incident wave. Our arguments are elementary and rely on the expansion of solutions to the Helmholtz equation.
- ItemDirect and inverse acoustic scattering by a collection of extended and point-like scatterers(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Hu, Guanghui; Mantile, Andreas; Sini, MouradWe are concerned with the acoustic scattering by an extended obstacle surrounded by point-like obstacles. The extended obstacle is supposed to be rigid while the point-like obstacles are modeled by point perturbations of the exterior Laplacian. In the first part, we consider the forward problem. Following two equivalent approaches (the Foldy formal method and the Krein resolvent method), we show that the scattered field is a sum of two contributions: one is due to the diffusion by the extended obstacle and the other arises from the linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one. In the second part, we deal with the inverse problem. It consists in reconstructing both the extended and point-like scatterers from the corresponding far-field pattern. To solve this problem, we describe and justify the factorization method of Kirsch. Using this method, we provide several numerical results and discuss the multiple scattering effect concerning both the interactions between the point-like obstacles and between these obstacles and the extended one.
- ItemDirect and inverse elastic scattering from anisotropic media(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Bao, Gang; Hu, Guanghui; Yin, Tao; Sun, JiguangAssume a time-harmonic elastic wave is incident onto a penetrable anisotropic body embedded into a homogeneous isotropic background medium. We propose an equivalent variational formulation in a truncated bounded domain and show uniqueness and existence of weak solutions by applying the Fredholm alternative and using properties of the Dirichlet-to-Neumann map in both two and three dimensions. The Fréchet derivative of the near-field solution operator with respect to the scattering interface is derived. As an application, we design a descent algorithm for recovering the interface from the near-field data of one or several incident directions and frequencies. Numerical examples in 2D are demonstrated to show the validity and accuracy of our methods.
- ItemDirect and inverse elastic scattering problems for diffraction gratings(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Elschner, Johannes; Hu, GuanghuiThis 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.
- ItemDirect and inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Hu, Guanghui; Kirsch, AndreasConsider 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 homogeneous compressible inviscid fluid with a constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by the Lamé constants. This paper is concerned with direct (or forward) and inverse fluid-solid interaction (FSI) problems with unbounded bi-periodic interfaces between acoustic and elastic waves. We present a variational approach to the forward interaction problem with Lipschitz interfaces. 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. Concerning the inverse problem, we show that the factorization method by Kirsch (1998) is applicable to the FSI problem in periodic structures. A computational criterion and a uniqueness result are justified for precisely characterizing the elastic body by utilizing the scattered acoustic near field measured in the fluid.
- ItemElastic scattering by finitely many point-like obstacles(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Hu, Guanghui; Sini, MouradThis paper is concerned with the time-harmonic elastic scattering by a finite number N of point-like obstacles in Rn (n = 2, 3). We analyze the N-point interactions model in elasticity and derive the associated Green's tensor (integral kernel) in terms of the point positions and the scattering coefficients attached to them, following the approach in quantum mechanics for modeling N-particle interactions. In particular, explicit expressions are given for the scattered near and far fields corresponding to elastic plane waves or point-source incidences. As a result, we rigorously justify the Foldy method for modeling the multiple scattering by finitely many point-like obstacles for the Lame model. The arguments are based on the Fourier analysis and the Weinstein-Aronszajn inversion formula of the resolvent for the finite rank perturbations of closed operators in Hilbert spaces.
- ItemElastic scattering by unbounded rough surfaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Elschner, Johannes; Hu, GuanghuiWe consider the two-dimensional time-harmonic elastic wave scattering problem for an unbounded rough surface, due to an inhomogeneous source term whose support lies within a finite distance above the surface. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the elastic displacement vanishes. We propose an upward propagating radiation condition (angular spectrum representation) for solutions of the Navier equation in the upper half-space above the rough surface, and establish an equivalent variational formulation. Existence and uniqueness of solutions at arbitrary frequency is proved by applying a priori estimates for the Navier equation and perturbation arguments for semi-Fredholm operators.
- ItemElastic scattering by unbounded rough surfaces : solvability in weighted Sobolev spaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Elschner, Johannes; Hu, GuanghuiThis paper is concerned with the variational approach in weighted Sobolev spaces to timeharmonic elastic scattering by two-dimensional unbounded rough surfaces. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the total elastic displacement satisfies either the Dirichlet or impedance boundary condition. We establish uniqueness and existence results for both elastic plane and point source (spherical) wave incidence, following the recently developed variational approach in [SIAM J. Math. Anal., 42: 6 (2010), pp. 2554 2580] for the Helmholtz equation. This paper extends our previous solvability results [SIAM J. Math. Anal., 44: 6 (2012), pp. 4101-4127] in the standard Sobolev space to the weighted Sobolev spaces.
- ItemElastic 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 ChulThe 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.
- ItemThe factorization method for inverse elastic scattering from periodic structures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Hu, Guanghui; Lu, Yulong; Zhang, BoThis paper is concerned with the inverse scattering of time-harmonic elastic waves from rigid periodic structures. We establish the factorization method to identify an unknown grating surface from knowledge of the scattered compressional or shear waves measured on a line above the scattering surface. Near-field operators are factorized by selecting appropriate incident waves derived from quasi-periodic half-space Green’s tensor to the Navier equation. The factorization method gives rise to a uniqueness result for the inverse scattering problem by utilizing only the compressional or shear components of the scattered field corresponding to all quasi-periodic incident plane waves with a common phase-shift. A number of computational examples are provided to show the accuracy of the inversion algorithms, with an emphasis placed on comparing reconstructions from the scattered near-field and those from its compressional and shear components.
- ItemFinite 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, TaoConsider 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.
- ItemGlobal 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, GuanghuiIn 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.
- ItemInverse elastic scattering from rigid scatterers with a single incoming wave : this paper is dedicated to the memory of Armin Lechleiter(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Elschner, Johannes; Hu, GuanghuiThe @first part of this paper is concerned with uniqueness to inverse time-harmonic elastic scattering from bounded rigid obstacles in two dimensions. It is proved that a connected polygonal obstacle can be uniquely identified by the far-field pattern corresponding to a single elastic plane wave. Our approach is based on a new reflection principle for the first boundary value problem of the Navier equation. In the second part, we propose a revisited factorization method to recover a rigid elastic body with a single far-field pattern.
- ItemAn inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Hu, Guanghui; Yang, Jiaqing; Zhang, BoThis paper is concerned with uniqueness for reconstructing a periodic inhomogeneous medium covered on a perfectly conducting plate. We deal with the problem in the frame of time-harmonic Maxwell systems without TE or TM polarization. An orthogonal relation for two refractive indices is obtained, and then inspired by Kirsch's idea, the refractive index can be identified by utilizing the eigenvalues and eigenfunctions of a quasi-periodic Sturm-Liouville eigenvalue problem.
- ItemInverse 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, GuanghuiThe 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
- ItemInverse scattering of electromagnetic waves by multilayered structures : uniqueness in TM mode(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Elschner, Johannes; Hu, GuanghuiAssume 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
- ItemInverse wave scattering by unbounded obstacles : uniqueness for the two-dimensional Helmholtz equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Hu, GuanghuiIn this paper we present some uniqueness results on inverse wave scattering by unbounded obstacles for the two-dimensional Helmholtz equation. We prove that an impenetrable one-dimensional rough surface can be uniquely determined by the values of the scattered field taken on a line segment above the surface that correspond to the incident waves generated by a countable number of point sources. For penetrable rough layers in a piecewise constant medium, the refractive indices together with the rough interfaces (on which the TM transmission conditions are imposed) can be uniquely identified using the same measurements and the same incident point source waves. Moreover, a Dirichlet polygonal rough surface can be uniquely determined by a single incident point source wave provided a certain condition is imposed on it.
- ItemMultiple scattering of electromagnetic waves by a finite number of point-like obstacles(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Challa, Durga Prasad; Hu, Guanghui; Sini, MouradThis paper is concerned with the time-harmonic electromagnetic scattering problem for a finite number M of point-like obstacles in R^3. First, we give a rigorous justification of the Foldy method and describe the intermediate levels of scattering between the Born and Foldy models. Second, we study the problem of detecting the scatterers and the scattering strengths from the far-field measurements and discuss the effect of multiple scattering related to each of these models.
- ItemNear-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, BoConsider 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.