Sessions 2012-2013

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.

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

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.

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.

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.

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.

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).

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.

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.