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Author: Hu, Guanghui × End date: 2019 × Start date: 2010 × DDC: 510 × Subject: variational formulation × WGL Institute: WIAS ×

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    Scattering of plane elastic waves by three-dimensional diffraction gratings
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Elschner, Johannes; Hu, Guanghui
    The reflection and transmission of a time-harmonic plane wave in an isotropic elastic medium by a three-dimensional diffraction grating is investigated. If the diffractive structure involves an impenetrable surface, we study the first, second, third and fourth kind boundary value problems for the Navier equation in an unbounded domain by the variational approach. Based on the Rayleigh expansions, a radiation condition for quasi-periodic solutions is proposed. Existence of solutions in Sobolev spaces is established if the grating profile is a two dimensional Lipschitz surface, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. Similar solvability results are obtained for multilayered transmission gratings in the case of an incident pressure wave. Moreover, by a periodic Rellich identity, uniqueness of the solution to the first kind (Dirichlet) boundary value problem is established for all frequencies under the assumption that the impenetrable surface is given by the graph of a Lipschitz function
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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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    Elastic scattering by unbounded rough surfaces : solvability in weighted Sobolev spaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Elschner, Johannes; Hu, Guanghui
    This 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.
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    Elastic scattering by unbounded rough surfaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Elschner, Johannes; Hu, Guanghui
    We 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.