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An incoming elastodynamic wave impinges on an elastic obstacle is embedded in an infinite elastic medium. The objective of the paper is to examine the subsequent elastic fields scattered by and transmitted into the elastic obstacle. By applying a boundary-field equation method, we are able to formulate a nonlocal boundary problem (NBP) in the Laplace transformed domain, using the field equations inside the obstacle and boundary integral equations in the exterior elastic medium. Existence, uniqueness and stability of the solutions to the NBP are established in Sobolev spaces for two different integral representations. The corresponding results in the time domain are obtained. The stability bounds are translated into time domain estimates that can serve as the starting point for a numerical discretization based on Convolution Quadrature. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a time-dependent transmission problem for a thermo-electric-magneto-elastic solid immersed in an inviscid and compressible fluid. It is shown that the problem can be treated by the boundary-field equation method, provided an appropriate scaling factor is employed. Based on estimates of variational solutions in the Laplace-transformed domain, we obtain the properties of corresponding solutions in the time-domain without having to perform explicit inversions of the variational solutions in the Laplace-transformed domain. [ABSTRACT FROM AUTHOR]

The problem of determining the manner in which an incoming acoustic wave is scattered by an elastic body immersed in a fluid is one of the central importance in detecting and identifying submerged objects. The problem is generally referred to as a fluid-structure interaction and is mathematically formulated as a time-dependent transmission problem. In this paper, we consider a typical fluid-structure interaction problem by using a coupling procedure that reduces the problem to a nonlocal initial-boundary problem in the elastic body with a system of integral equations on the interface between the domains occupied by the elastic body and the fluid. We analyze this nonlocal problem by the Lubich approach via the Laplace transform, an essential feature of which is that it works directly on data in the time domain rather than in the transformed domain. Our results may serve as a mathematical foundation for treating time-dependent fluid-structure interaction problems by convolution quadrature coupling of FEM and BEM. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this article, a variational formulation for the transmission problem of the fluid-bone interaction is formulated. The formulation is based on a modified Biot system of equations for the cancellous bone together with a boundary integral equation formulation of the pressure in the water. Existence and uniqueness for the weak solution of the interaction problem are established in appropriate Sobolev spaces. [ABSTRACT FROM AUTHOR]

Abstract: Variational methods for boundary integral equations deal with the weak formulations of boundary integral equations. Their numerical discretizations are known as the boundary element methods. This paper gives an overview of the method from both theoretical and numerical point of view. It summarizes the main results obtained by the author and his collaborators over the last 30 years. Fundamental theory and various applications will be illustrated through simple examples. Some numerical experiments in elasticity as well as in fluid mechanics will be included to demonstrate the efficiency of the methods. [Copyright &y& Elsevier]

This article is concerned with the application of the Newton-imbedding iteration procedure to nonlinear boundary value problems in Sobolev spaces. A simple model problem for the second-order semilinear elliptic equations is considered to illustrate the main idea. The essence of this method hinges on the a priori estimates of solutions of the associated linear problem in appropriate Sobolev spaces. It is to our surprise that H 1 (Ω)-solution is not smooth enough to guarantee the convergence of the sequence generated by the procedure. Existence and uniqueness of solution to the original nonlinear problem are established constructively. An application of this approach to the Lamé system with nonlinear body force and its generalization to contain a nonlinear surface traction in elasticity is also discussed. [ABSTRACT FROM AUTHOR]

This article is concerned with weak solution of a mixed boundary value problem for the biharmonic equation in the plane. Using Green's formula, the problem is converted into a system of Fredholm integral equations for the unknown data on different parts of the boundary. Existence and uniqueness of the solutions of the system of boundary integral equations are established in appropriate Sobolev spaces. [ABSTRACT FROM AUTHOR]