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

Abstract

Cylindrical vibration of an elastic cusped plate under the action of an incompressible fluid in case of N = 0 approximation of I. Vekuas hierarchical models,This article deals with the study of an interaction between an elastic plate and an incompressible fluid when in the elastic plate part, the N ¼ 0 approximation of Vekuas hierarchical models for cusped elastic plates is used.

Existence and uniqueness theorems for cusped prismatic shells in the Nth hierarchical model

“…In the case of the N = 1 approximation, the partial differential equations for the unknown vector v = (v 10 , v 20 , v 30 The corresponding bilinear and quadratic forms are 2 31 } d In accordance with the results in Sections 3 and 4, the variational problem…”

“…In the case of the N = 0 approximation, we have the following partial differential equations for the unknown vector v = (v 10 , v 20 The corresponding bilinear form reads as (see (25)) More detailed analysis based on Korn's inequality shows that the spaces X 0 and [ • W 1 2, ( )] 3 considered as sets coincide and the norms · X 0 and · Y 0 are equivalent for arbitrary = 1. For = 1 we have the following continuous embedding…”

“…The partial differential equations for the N = 2 approximation for the unknown vector v = (v 10 , v 20 , v 30 , v 11 , v 21 , v 31 , v 12 , v 22 , v 32 ) and the corresponding bilinear and quadratic forms can be expressed explicitly with the help of (21) and (25). Again, with the help of the results obtained in Sections 3 and 4 we conclude that the variational problem B (2) (v, v * ) = F, v * for all v * ∈ X 2 (69) APPENDIX A Let be as in Section 2 and let D( ) be a space of infinitely differentiable functions with compact support in .…”

“…The same questions for cusped prismatic shells when the shells occupy a 3D Lipschitz domain were investigated by Jaiani et al [8,9]. Various aspects of I. Vekua's models were studied by Vashakmadze [10], Meunargia [11], Zhgenti [12], Khoma [13], Guliaev et al [14], Khvoles [15], Jaiani [16][17][18][19], Chinchaladze and Gilbert [20], etc. A new stage for models in variational form began with the work of Vogelius and Babuška [21,22].…”

On Some Bending Problems of Prismatic Shell with the Thickness Vanishing at Infinity

“…Works of Babuska, Gordeziani, Guliaev, Khoma, Khvoles, Meunargia, Schwab, Vashakmadze, Zhgenti, Jaiani, Tsiskarishvili, M. and G. Avalishvili, Wendland, Natroshvili, Kharibegashvili, Chinchaladze, Gilbert, and others are devoted to further analysis of I.Vekua's models (rigorous estimation of the modeling error, numerical solutions, etc.) and their generalizations (see, e.g., [2], [3], [4], [5], [6], [7], [8], [9], [12], [16] …”