Boundary integral method for thermoelastic screen scattering problem in ?3

Abstract

We investigate a three‐dimensional mathematical thermoelastic scattering problem from an open surface which will be referred to as a screen. Under the assumption of the local finite energy of the unified thermoelastic scattered field, we give a weak model on the appropriate Sobolev spaces and derive equivalent integral equations of the first kind for the jump of some trace operators on the open surface. Uniqueness and existence theorems are proved, the regularity and the singular behaviour of the solution near the edge are established with the help of the Wiener–Hopf method in the halfspace, the calculus of pseudodifferential operators on the basis of the strong ellipticity property and Gårding's inequality. An improved Galerkin scheme is provided by simulating the singular behaviour of the exact solution at the edge of the screen. Copyright © 2000 John Wiley & Sons, Ltd.

Boundary integral equation methods for the elastic and thermoelastic waves in three dimensions

“…The direct TESP to be considered in this paper is to determine the displacement field u and the temperature variation field p satisfying (2.6), the boundary condition (2.7) and the Kupradze radiation conditions. For given F ∈ (H −1/2 (Γ)) 4 , we refer to [7,22] for the uniqueness of the direct problem.…”

“…For the TESP, it follows from the potential theory ( [7,22]) that the unknown function U can be represented as…”

Scattering theorems of elastic waves for a thermoelastic body

“…where the upper left 3 3 block of the matrix (2.6) is an elliptic differential operator, and Q I the identity dyadic in R 3 (the from now on in this paper will denote dyadics). Also in (2.5), the differential operator L is given by [3] L D…”

“…In the sequel, we will use a Betti's theorem arising from thermoelasticity theory [3]. In particular, for every U, We now employ the following Twersky's notation: for every U, V 2 C 2 .D/ Using asymptotic analysis of the fundamental solution of the Navier equation [9] expressions of the far-field patterns are obtained:…”

“…This problem has been initially studied for the case where an approximate solution has been found applying the low frequency theory [1,2]. In [3], uniqueness and existence theorems for the thermoelastic scattering problem from an open surface, using the boundary integral method have been also established. Scattering relations for plane elastic waves for an elastic body have been proved in [4], whereas corresponding results for spherical elastic waves in two and three dimensions can be found in [5,6].…”