V G Grigor’ev (Moscow Aviation Institute (National Research University)) :
Articles:
Grigor’ev V. G. (Moscow Aviation Institute (National Research University)), Kurakin V. V. (Moscow Aviation Institute (National Research University))
doi: 10.18698/2309-3684-2025-4-418
The application of the finite element method based on the mixed variational principle for solving a three-dimensional eigenvalue problem using the example of a cavity with an elastic bottom is considered. A brief review of the literature on the subject of the study is provided. A rigorous mathematical formulation of the problem for the described mechanical system is presented. A hexagonal finite element of the fluid is introduced, the techniques of numerical integration of the kinetic energy of the liquid volume is described in using the Gauss-Legendre quadrature. A quadrangular finite element of the free surface is introduced, the techniques of numerical integration of the potential energy of wave formation using the Gauss-Legendre quadrature is described. An analytical expression for integration of the potential of the bottom contact interaction forces with a liquid is obtained as applied to a finite element of the bottom. The techniques of integrating the terms of the functional of the total mechanical energy of the system, providing the conditions for conjugation of the introduced degrees of freedom is described. The conjugation condition of the free surface displacement and the displacement potential of the fluid volume is integrated numerically using the Gauss-Legendre quadrature. The conjugation condition of the elastic bottom bending displacement and the medium displacement potential is integrated analytically. The description of the algorithm of numerical solution of the frequency-modal problem is given. The calculation results for the case of a rigid bottom are presented. The analysis of convergence of finite element solutions to analytical ones for different variants of partitions by finite element mesh is performed. The calculation results for the case of an elastic bottom are presented. The convergence analysis of the solution results for the case of an elastic bottom to the case of a rigid bottom with an increase in its thickness is performed. The analysis of the first form of oscillations for the case of an elastic bottom is presented. Conclusions are made about the applicability of the implemented algorithms to mechanical engineering problems.
Григорьев В.Г., Куракин В.В. Моделирование собственных колебаний пространственной гидроупругой конструкции методом конечных элементов на основе смешанного вариационного принципа. Математическое моделирование и численные методы, 2025, № 4, с. 4–18.
Park S. (Moscow Aviation Institute (National Research University)), Grigor’ev V. G. (Moscow Aviation Institute (National Research University))
doi: 10.18698/2309-3684-2022-3-317
In this paper, we consider the problem of stability of a thin-walled shell structure with two hemispherical bottoms of the same thickness, partially filled with liquid, which is immersed in an external liquid medium and is under hydrostatic pressure. The dynamic characteristics of such a structure containing a limited volume of liquid under internal pressure and hydrostatic pressure are obtained. The developed program for calculating the dynamic characteristics of axisymmetric shell structures containing liquid is based on the finite element method. The finite elements have an annular shape when rotated around the axis of symmetry. The program is implemented in Excel spreadsheet using Visual Basic for Applications (VBA). It allows to calculate the natural frequencies of thin-walled elastic structures interacting with an arbitrary number of liquids, considering the influence of the static stress-strain state caused by hydrostatic and internal pressure and other external forces that do not violate the axial symmetry of the problem. At a fixed value of the internal pressure, the calculation of the lowest natural frequencies of vibrations with different numbers of waves along the circumference is performed. By successive refinement, the critical thickness of the shell is determined, at which at least one of the natural frequencies reaches zero. The internal pressure p varies from 0 to 1 atm. in increments of 0,1 atm. and the calculations are repeated to obtain each critical value. At each pressure value, curves are plotted on the graph "number of waves — natural frequency". On the coordinate plane "pressure — shell thickness" the boundary of the instability region is constructed.
Пак Сонги, Григорьев В.Г. Моделирование динамической устойчивости тонкостенных конструкций, частично заполненных жидкостью, при гидростатическом воздействии. Математическое моделирование и численные методы, 2022, № 3, с. 3–17.