Boundary Integral Equation Method in the Steady State Oscillation Problems for Anisotropic Bodies

Abstract

The three‐dimensional steady state oscillation problems of the elasticity theory for homogeneous anisotropic bodies are studied. By means of the limiting absortion principle the fundamental matrices maximally decaying at infinity are constructed and the generalized Sommerfeld–Kupradze type radiation conditions are formulated. Special functional spaces are introduced in which the basic and mixed exterior boundary value problems of the steady state oscillation theory have unique solutions for arbitrary values of the oscillation parameter. Existence theorems are proved by reduction of the original boundary value problems to equivalent boundary integral (pseudodifferential) equations. © 1997 by B.G. Teubner Stuttgart‐John Wiley & Sons, Ltd.

A modified volume integral equation for anisotropic elastic or conducting inhomogeneities: Unconditional solvability by Neumann series

“…(referring the reader to [8] and [13] for details on G ω and the meaning of "radiating" in the cases of isotropic and anisotropic background, respectively). The (now complex-valued) displacement field u B verifies the VIE…”

Reconstruction of cracks in an inhomogeneous anisotropic medium using point sources

“…, S n are convex. (2.6) Notice that the convexity of S 1 is also assumed in [19] and [20]. However, this assumption is redundant.…”

“…Here we only consider the outgoing fundamental solution, denoted by Γ (x, ω), corresponding to the limit of Γ (x, τ ε ) as +ε → 0. It was proved in [19] and [20] that the limit…”

“…Since the medium is anisotropic outside of a large domain, the classical Sommerfeld-Kupradze radiation conditions are not applicable in this case. For the homogeneous anisotropic elasticity system in two and three dimensions, Natroshvili [19,20] established the generalized Sommerfeld-Kupradze radiation conditions by analyzing the radiation pattern of the oscillation equations under some restrictions on the slowness curves. Here we will impose the same restrictions on these curves and adopt the radiation conditions derived in [19] and [20].…”

“…In Section 2, we will describe the radiation conditions derived in Natroshvili's papers [19] and [20]. In Section 3, we prove the limiting absorption principle for the inhomogeneous anisotropic elasticity system in R n based on the recent work [18].…”