On Numerical Methods For Direct And Inverse Fluid-solid Interaction Problems

The scattering of time-harmonic acoustic, elastic and electromagnetic waves has wide applications in many practical scientific and engineering fields. This thesis is concentrate on the numerical methods for a class of direct and inverse fluid-solid interaction problems. The so called fluid-solid interaction is the scattering of acoustic wave by a penetrable elastic body immersed in a fluid. The acoustic scattered field in the fluid and the elastic displacement field in the solid satisfies the Helmholtz equation and time-harmonic Navier equation, respectively, together with some transmission conditions on the fluid-solid interface. This thesis consists of two parts. The first part is the numerical methods for direct fluid-solid interaction problems. The second part is the reconstruction algorithms for inverse fluid-solid interaction problems. Both for these two parts, we consider two different structures, bounded and unbounded periodic(or bi-periodic) fluid-solid interfaces. For the scattering by bounded interface, only acoustic wave propagates back into unbounded domain. However, elastic wave propagating in the unbounded solid domain under the interface is incited due to periodic fluid-solid interface.For the direct fluid-solid interaction problem, a popular way to treat the infinite domain is applying the boundary integral equation methods. For bounded fluid-solid interface, we reduce the original problem to coupling systems of boundary integral equations based on direct method, indirect method and Burton-Miller formulation. Mathematical analysis on existence and uniqueness of the corresponding weak solutions is given. In addition, we derive a regularization formulation for the hyper-singular boundary integral operator corresponding to time-harmonic Navier equation. Another popular way to treat the infinite domain is introducing an artificial boundary to decompose the infinite domain into a bounded domain and an unbounded domain. When considering bounded fluid-solid interface, we can define two different Dirichlet-to-Neumann(Dt N) operators by using Fourier series and boundary integral operators, respectively due to the exterior acoustic scattering problem outside the artificial boundary. When focusing on periodic fluid-solid interface with quasi-periodic plane incident wave, we can investigate wave fields in a single periodic cell. DtN operators for acoustic and elastic waves can be derived by Rayleigh expansions of acoustic scattered field and elastic displacement field, respectively. Therefore, the original unbounded problems can be reduced to equivalent nonlocal boundary value problems by introducing DtN operators on artificial boundaries. We show the existence and uniqueness of solutions in appropriate Sobolev spaces for the corresponding variational problems.The inverse fluid-solid interaction problem we consider here is to determine the shape and position of elastic bodies by factorization method based on near-field measurements. Factorization method provides a sufficient and necessary condition for recovering the shape of an obstacle, which can be also used as an efficient computational criterion. In general, this method is based on far-field data due to incident plane waves and we are going to analyze the near-field operator within the same functional framework as in the far-field case. For inverse fluid-solid interaction with bounded structure, we introduce an Outgoing-to-Incoming(OtI) operator on sphere or non-sphere measurement surface containing the elastic body inside. Then we obtain a modified near-field operator by the product of Ot I operator and near-field operator which is then factorized by some solution operator and middle operator. For inverse fluid-solid interaction with periodic structure, in order to reconstruct the interface from above we measure the scattered field on a line segment in a single periodic cell. We utilize a set of incident acoustic waves with common quasi-periodicity parameters to factorize the near-field operator. Finally, we can obtain reconstruction algorithms by applying proper Range Identity for the considered inverse fluid-solid interaction problem with bounded and periodic structures.On the other hand, we consider the inverse problem to determine an extended elastic body surrounded by finitely many point-like scatterers. We construct a semi-explicit form for the unique analytic solution of the direct problem and a reconstruction algorithm based on near-field measurements for the corresponding inverse obstacle problem. Particularly, when there is no extended elastic body, we consider an inverse acoustic scattering by point-like scatterers. Compared with the traditional Multiple Signal Classification(MUSCI) algorithm, we investigate a new MUSIC algorithm based on near-field measurements by applying an OtI operator onto the near-field response matrix.At the end of each chapter, we present several numerical examples to illustrate the efficiency and accuracy of the proposed algorithms for the considered direct and inverse fluid-solid interaction problems.