Interaction of Elastic and Scalar Fields

“…In the present paper for boundary-transmission problems generated by the above mentioned mathematical model, we prove the existence and uniqueness theorems in Sobolev-Slobodetski spaces. Same type problems for the classical model of elasticity (elastic body is embedded in fluid) are considered and studied in the papers: [1][2][3][4][5], [7][8][9], [11], [12], [14], [17], [18]. In these papers in the bounded domain Ω + is three dimensional elastic field (displacement vector with three components) and scalar wave field in unbounded domain Ω − .…”

Section: Introductionmentioning

confidence: 77%

Mathematical problems of interaction of different dimensional physical fields

Chkadua

2013

J. Phys.: Conf. Ser.

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Abstract. In the paper we consider mathematical problem of interaction of different dimensional physical fields for complex composite structures. We investigate the mixed transmission problem arising in the model of fluid-solid acoustic interaction when a piezo-ceramic elastic body (Ω + ) is embedded in an unbounded fluid (Ω − ). The corresponding physical process is described by boundary-transmission problems for second order partial differential equations. In particular, in the bounded domain Ω + we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω − we have a scalar Helmholtz equation describing an acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations the uniqueness and existence theorems are proved in Sobolev-Slobodetski spaces.

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“…The cases when the elastic body under consideration is homogeneous and anisotropic, and the fluid is isotropic, has been considered in [49]. In [30,31] the interface problems of the interaction between a four-dimensional thermoelastic field and a scalar oscillation field have been studied in general homogeneous anisotropic case.…”

Section: Communicated By E Meistermentioning

confidence: 99%

“…The analysis of these PsDEs (uniqueness, index problem and existence), besides the radiation conditions for the scalar acoustic field, in addition, needs special, the so-called generalized Sommerfeld-Kupradze type radiation conditions for elastic (thermoelastic) field (for details see [46,49,30,31]). These latter conditions are too complicated in anisotropic case.…”

Section: Communicated By E Meistermentioning

confidence: 99%

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

Jentsch

Natroshvili

1999

Math. Meth. Appl. Sci.

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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.

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Interaction of Elastic and Scalar Fields

Cited by 11 publications

References 9 publications

Cylindrical vibration of an elastic cusped plate under the action of an incompressible fluid in case ofN = 0 approximation of I. Vekuas hierarchical models

Cylindrical vibration of an elastic cusped plate under the action of an incompressible fluid in case ofN = 0 approximation of I. Vekuas hierarchical models

Mathematical problems of interaction of different dimensional physical fields

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

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