Shota Rustaveli National Science Foundation of GeorgiaStart Date: 2013-01-01 End Date: 2015-01-01
The present project is devoted to mathematical problems connected with pluri-dimensional problems for multi-structures consisting of elastic solid and fluid parts, in particular, to construction and investigation of the mathematical models of sea shores, neighborhoods of dams of hyd¬roe¬lec¬tric stations, and bridges. Both the problems are very important for Georgia at the present stage of development of the country, since they are connected with protection of Georgian Black See shore and settlement of crucial for Georgia energetic problems. The mathematical investigation of the above problems requires study of the boundary, initial boundary value, and contact problems on multi-structures, i. e., on domains with different geometrical peculiarities. Therefore, mathematical modeling and investigation of multi-structures is important as from the theoretical (properly, in sense of development of mathematics) as well as from practical points of view. Multi-structures consisting of plates, shells, rods and other substructures are interesting for practical applications. The mathematical study of multi-structures has hardly 20 years history. The first investigation was carried out by Ciarlet, Le Dret, Nzengwa [1]. Applying the asymptotic method they have constructed and investigated a problem defined on the product of three- and two-dimensional domains for multi-structure consisting of three-dimensional body with a plate inserted in it. The eigenvalue problem for multi-structures was considered by Bourquin, Ciarlet [1]. Multi-structures consisting of plates and rods were considered by Le Dret [1]. The method used in these works relies on scaling of different parts of the full structure independently of each other, then passing to the limit in the variational formulation and determining the limit (see also Ciarlet [1]). Later, the same approach was applied by Gruais [1] for static problems, by Kerdid [1] for eigenvalue problems and by Raoult [1] for elastoplastic multi-structures. Kozlov, Maz’ya, Movchan devoted their monograph [1] to the investigation of a mixed problem for three-dimensional elastic cap based on thin cylinders. Nazarov [1] proved weighted anisotropic Korn’s inequality for a junction of a plate and rods, when the non-adjacent to plate ends of the rods are clamped. The investigation of multi-structures consisting of nonlinear elastic and fluid parts is very topical direction in the practical point of view. Joint investigation of the corresponding problems is mathematically very complicated and enlarges the above problematic. The study of the problems in this direction and separation of sub-classes of problems which can be modeled and permit effective application of methods of mathematical physics and numerical analysis is a properly topical problem. Moreover, as it is well known (Ball [1], Ciarlet [2], Antman [1]) the questions of existence and uniqueness for three-dimensional problems of nonlinear elasticity and general Navier-Stokes systems aren’t fully investigated. The works of Bernadou, Fayolle, Léné [1] and others are devoted to numerical solution of problems for multi-structures consisting of rods and plates. The main difficulty for numerical solution of solid-fluid interaction problems for pluri-dimensional multi-structures consists in that, that the differential equations describing deformed state of different parts of the multi-structure may be pluri-dimensional. In this connection to use decomposition methods becomes topical. The essential difficulty arises on contact areas of the parts of the multi-structures, since the contact conditions don’t allow application of direct splitting. Therefore, it is necessary to use iterative methods combined with decomposition methods.
It should be pointed out that the above-mentioned methods of construction of models of multi-structures and the corresponding lower dimensional problems can be used only in the case of smallness of thickness or width of substructures with respect to other dimensions, while the above quantities for real bodies can not be sufficiently small. Hence, in order to study various complicated multi-structures by mathematical methods and to solve approximately the corresponding boundary value problems, it is convenient to use successive approximation method, when on each step we get lower-dimensional problems and, increasing the order of the approximation, the obtained approximate solution tends to the exact solution of the original three-dimensional problem. To the methods of these type belongs Vekua’s [1] method for elastic prismatic shells. This methods permits construction of hierarchy of two-dimensional models which, actually, contains Kirchhoff-Love and Reissner-Mindlin models, and by increasing order of the approximation the displacement vector-functions of three space variables, constructed by means of the solution of the reduced problem, converges to the solution of the original problem. Models constructed by I. Vekua’s dimension reduction method are mathematically consistently and do not need introduction of additional hypotheses for correct setting of boundary value problems.