Life is complex: it has both real and imaginary components.

Solution of geometrically nonlinear problems after the loss of stability

We propose the following algorithm for solving geometrically
nonlinear problems after the loss of stability.
A step-by-step method is used at first. If after a certain step a
loss of stability was established, then the corresponding dynamic
problem is solved with zero right part. The initial
conditions are set according to the first buckling mode.
The unconditionally stable implicit difference scheme is used.
We find the stable state with the same load at which the
stability was lost. A step-by-step method is applied again.

Bending of a multi-layered beam

For the plane stress state bending of multi-layered beam
with alternating layers is considered.
The layers differ in thickness and elastic properties.

Vibrations of a cylindrical shell stiffened
by rings with the "hat-type" cross-sectional areas.

A cylindrical shell stiffened by circular rings with the "hat-type"
cross-sectional areas is considered.
The parameters for stiffened shell with
the given mass corresponding to the maximum value of the first frequency
or the critical external pressure are found by means of asymptotic methods.
The advantage of using rings with the "hat-type" cross-sectional areas
as compared to rings with the rectangular cross-sections are shown.

Methods and Biotechnical Assessment System of the eye structures.

In the development process of data support of biotechnical
systems for biomechanical assessment of the eye structures,
an algorithm of investigation of displacements, stresses and
strains in the structures of the eyeball is proposed, which
allows to build computer models for it, taking into account
the anisotropy and inhomogeneity of their member structures.
Geometric modeling was conducted in application package
SolidWorks, VAT calculations and analysis of the number of
finite elements were conducted in the finite element package
CosmosWorks. Based on the results of calculations,
diagram differences of the values of the true IOP with
tonometric IOP values determined by the Maklakov’s method were
conducted, from parameters of the eyes, and plots stress-strain
state in the structures of lamina cribrosa and optic disk from
gradient IOP and ICP.

Vibrations and buckling of a cylindrical shell stiffened
by rings with various stiffnesses.

The influence of the law of distribution the ring’s stiffness along the
generator of the thin elastic cylindrical shell on the value of
critical pressure and fundamental vibration frequency is considered.
An approximate analytical solution of the problem of optimizing
the parameters in order to increase the critical pressure and the fundamental
vibration frequency of the system is obtained.

On mathematical models of tonography

Two existing mathematical models of eye's tonography are discussed.
Tonography is a research method of dynamics of aqueous humor. Main point of
the method is concluded in prolonged tonometry (usually 4 minutes) and determination
of coefficient of easiness outflow and volume of aqueous humor per minute.
Determination of inter-eye pressure during inter-sclera injections and different
parameters of anisotropy is considered, together with typical periods of relaxation
of inter-eye pressure after injection of additional volume of liquid into vitreous body.

On Pressure-volume dependence for a human eye

The dependence of pressure-volume for isotropic spherical
shells is studied. The results are obtained for shells of revolution,
which have equal initial volumes but different shapes
(different ratio of vertical and horizontal diameters).

Numerical solution of aerohydrodynamic problems using graphic processors.

The report is dedicated to the implementation notes of vector computation
algorithm of liquid and gas flows using graphic processors.
A number of test problems is analyzed. The modification of flow analysis with
low Mach numbers, incompressible flows and flows with free convection has been made.

Buckling of axisymmetric equilibrium states of circular plates under normal pressure

The stability of axisymmetric equilibrium states of an isotropic non-homogeneous
curcular and annular plate under uniform pressure is considered.
The unsymmetric part of the solution is sought in terms of multiples of the harmonics
of the angular coordinates. A numerical method is employed to obtain the lowest
load value, which leads to the appearance of waves in the circumferential direction.
Impact of the inner radius, degree of inhomogenenity on the critical load value and
buckling mode is studied.