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Domain decomposition methods for a coupled vibration between an acoustic field and a plate

The coupled vibration between an acoustic field and a plate is encountered in many engineering and industrial applications. The interaction between the wind and a windshield of a car is an interesting example found in the automobile industry. Mathematically, such an interaction is described by the coupled system of the second order scalar wave equation and the fourth order plate vibration equation. Since the thickness of the plate is negligible, the plate serves a dual role in the model. It is the solid medium and in the same time it is the interface between the acoustic field and the solid (so it is a part of the boundary of the acoustic field), where they interact each other. Let Ω ⊂ R be a three-dimensional acoustic field and Γ0 ⊂ R, a part of the boundary ∂Ω, denote the domain of the plate. Let Γ1 = ∂Ω\Γ0 be the remaining portion of the boundary of Ω. Let p = p(x1, x2, x3) denote the pressure function of the fluid in the acoustic field Ω and u = u(x′) (x′ = (x1, x2)) denote the vertical displacement of the plate Γ0. Then the governing partial differential equations of the fluid–plate interaction is given by [CS76]