Solvability, asymptotic analysis and regularity results for a mixed type interaction problem of acoustic waves and piezoelectric structures

Abstract

In the paper, we investigate the mixed type transmission problem arising in the model of fluid-solid acoustic interaction when a piezoceramic elastic body ( C ) is embedded in an unbounded fluid domain ( ). The corresponding physical process is described by the boundary-transmission problem for second-order partial differential equations. In particular, in the bounded domain C , we have a 4 4 dimensional matrix strongly elliptic second-order partial differential equation, while in the unbounded complement domain , we have a scalar Helmholtz equation describing 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 based on the Wiener-Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev-Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Hölder smoothness results for solutions.The present paper is devoted to the investigation of a mixed type .M ! / interaction boundary-transmission problem associated with the previously described mathematical model with a piezoelectric solid component. The Dirichlet type .D ! / and Neumann type .N ! / problems are studied in [7].Similar interaction problems for the classical model of elasticity are studied in the references [8][9][10][11][12][13][14][15][16][17][18][19][20]. In this case, one has a threedimensional elastic field, the displacement vector with three components in the bounded domain C and a scalar pressure field in the unbounded domain . In our case, in the domain C , we have an additional electric field that essentially complicates investigation of the transmission problems in question. In particular, except transmission conditions, electric potential is given on one part of the boundary of C (the Dirichlet type condition), while on the other part, normal component of electric displacement is given (the Neumann type condition).In the paper [21], uniqueness and existence theorems for the mixed type interaction problem of acoustic waves and piezoelectric structures are stated without proof.We investigate the aforementioned problem with the use of the potential method and the theory of pseudodifferential equations on manifolds with boundary and prove existence and uniqueness theorems in Sobolev-Slobodetskii spaces.The paper is organized as follows. In Section 2, we formulate mixed type .M ! / boundary-transmission problem for steady state oscillation equation in appropriate function spaces. In Section 3, we define Jones modes and Jones eigenfrequencies and prove the uniqueness theorem for the problem .M ! /. In Section 4, we describe properties of potentials related to the Helmholtz equation and the steady-state oscillation equations of piezoelectricity. In Section 5, mixed interaction boundary-transmission problem .M / is formulated for the pseudo-oscillation equations and the unique...

Mathematical aspects of fluid–multiferroic solid interaction problems

“…Using inequalities (31), positive definiteness of the matrix M defined by Equation (6), and the inequality (d 0 0 − h 0 ) > 0 (see Equation 4), we obtain that (see proof of Theorem 2.25 in Buchukuri et al 31 )…”

“…if not otherwise stated. Interaction Dirichlet Type Problem (ID): find a vector-function U = (u, u 4 , u 5 , u 6 ) ⊤ ∈ [H 1 (Ω + )] 6 and scalar function w ∈ H 1 (Ω − ) satisfying the differential equations…”

“…Interaction Neumann Type Problem (IN): find a vector-function U = (u, u 4 , u 5 , u 6 ) ⊤ ∈ [H 1 (Ω + )] 6 and scalar function w ∈ H 1 (Ω − ) satisfying the differential Equations ( 19) and ( 20) respectively, transmission conditions ( 21), (22), and the Neumann boundary condition…”

“…It is evident that b 1 and b 2 in Equation ( 33) remain arbitrary complex constants in the case of the homogeneous problem (IN), which completes the proof. 6 × H 1 (Ω − ) be a solution of the homogeneous boundary-transmission problem associated with the adjoint differential operators:…”

“…The Dirichlet type, Neumann type, and mixed type steady state oscillation interaction problems of acoustic waves and piezoelectric structures are studied in the papers Chkadua, 4 Chkadua and Natroshvili, 5 and Chkadua. 6 Similar interaction problems with the classical model of elasticity have been investigated by a number of authors. An exhaustive information concerning theoretical and numerical results, for the case when the both interacting media are isotropic, can be found in the references Bielak and MacCamy, 7 Bielak et al, 8 Boström, 9,10 Goswami et al, 11 Hsiao et al, 12 Hsiao, 13 Hsiao et al, 14 Junger and Fiet, 15 Kagawa and Yamabuchi, 16 Luke and Martin, 17 Natroshvili et al, 18 Natroshvili and Sadunishvili, 19 and Natroshvili et al 20 Interaction problems of steady state oscillations for homogeneous and anisotropic elastic solids are analyzed in Jentsch et al, 21,22 where the generalized Sommerfeld-Kupradze type radiation conditions for anisotropic solids are derived.…”