Non-local approach in mathematical problems of fluid-structure interaction

Abstract

Three‐dimensional mathematical problems of interaction between elastic and scalar oscillation fields are investigated. An elastic field is to be defined in a bounded inhomogeneous anisotropic body occupying the domain Ω¯1⊂ℝ3 while a physical (acoustic) scalar field is to be defined in the exterior domain Ω¯2=ℝ3\Ω1 which is filled up also by an anisotropic (fluid) medium. These two fields satisfy the governing equations of steady‐state oscillations in the corresponding domains together with special kinematic and dynamic transmission conditions on the interface ∂Ω1. The problems are studied by the so‐called non‐local approach, which is the coupling of the boundary integral equation method (in the unbounded domain) and the functional‐variational method (in the bounded domain). The uniqueness and existence theorems are proved and the regularity of solutions are established with the help of the corresponding Steklov–Poincaré type operators and on the basis of the Gårding inequality and the Lax–Milgram theorem. In particular, it is shown that the physical fluid–solid acoustic interaction problem is solvable for arbitrary values of the frequency parameter. Copyright © 1999 John Wiley & Sons, Ltd.

Interaction of acoustic waves and piezoelectric structures

“…Similar interaction problems for the classical model of elasticity are studied in the references . Evidently, in this case, in the bounded domain Ω + , one has a three‐dimensional elastic field, the displacement vector with three components, and a scalar pressure field in the unbounded domain Ω − .…”

Computational Vibroacoustics in Low- and Medium- Frequency Bands: Damping, ROM, and UQ Modeling

“…A major drawback of the classical boundary integral formulations for the exterior Neumann problem related to the Helmholtz equation is related to the uniqueness problem, although the boundary value problem has a unique solution for all real frequencies [97,98]. Precisely, there is not a unique solution of the physical problem for a sequence of real frequencies called spurious or irregular frequencies, also called Jones eigenfrequencies [31,[128][129][130][131], and various methods are proposed in the literature to overcome this mathematical difficulty arising in the boundary element method [128,[132][133][134][135][136]. In this appendix, we present a boundary element method that was initially developed in [137] and detailed in [41].…”

Advanced Computational Dissipative Structural Acoustics and Fluid-Structure Interaction in Low-and Medium-Frequency Domains. Reduced-Order Models and Uncertainty Quantification

“…This is related to the uniqueness problem although the boundary value problem has a unique solution for all real frequencies [18,127]. Precisely, there is no unique solution of the physical problem for a sequence of real frequencies called as spurious or irregular frequencies and they are also called as Jones eigenfrequencies [112,[128][129][130][131]. Various methods are proposed in the literature to overcome this mathematical difficulty that arises in the boundary element method [3,129,[132][133][134][135][136][137].…”