Vol 12, No 4 (2019)

A three-dimensional unsteady periodic flow of blood in xenogenic vascular bioprostheses is simulated by computational fluid dynamics methods. The geometry of the computational domain is based on microtomographic scanning of bioprostheses. To set a variable pressure gradient causing an unsteady flow in the prostheses, personal-specific data of the Doppler echography of the blood flow of a particular patient are used. A comparative analysis of the velocity fields in the flow areas corresponding to three real samples of bioprostheses with multiple stenoses is carried out. In the zones of stenosis and outside of them, the distribution of the near-wall shear stress, which affects the risk factors for thrombosis in the prostheses, is analyzed. An algorithm for predicting the hemodynamic effects arising in vascular bioprostheses is proposed; the algorithm is based on the numerical modeling of the blood flow in these prostheses.

This paper deals with the solution of the pure Neumann problem for the diffusion equation by a finite element method. First, an extended generalized formulation of the Neumann problem in the Sobolev space H¹(Ω) is derived and investigated. Then a discrete analog of this problem is formulated by using standard finite element approximations of the space H¹(Ω). An iterative method for solving the corresponding SLAE is proposed. Some examples of solving model problems are used to discuss the numerical properties of the algorithm proposed.

An approach to improving the stability of triangular decomposition of a dense positive definite matrix with a large condition number by using the Gauss and Cholesky methods is considered. It is proposed to introduce additions to standard computational schemes with an incomplete inner product of two vectors which is formed by truncating the lower digits of the sum of the products of two numbers. The truncation in the process of decomposition increases the diagonal elements of the triangular matrices by a random number and prevents the appearance of very small numbers during the Gauss decomposition and a negative radical expression in the Cholesky method. The number of additional operations required for obtaining an exact solution is estimated. The results of computational experiments are presented.

In this paper, we consider a mathematical model of a new-type micro- generator. The model is based on excitation by electrostatic forces of oscillations of a mobile electrode in a micro gap. The principle of operation of the micro-generator is similar to the well-known theory of clock with a striking escapement mechanism. The difference is that in the equation of motion, the form of the right-hand side takes into account the electrostatic nature of the pulsed action. The numerical analysis shows that with time the bounded oscillations in the phase plane tend to a stable limit cycle, and thus the emerging oscillations are stable towards external perturbations. Studying periodic oscillations in dependence on model parameters relies on the boundary value problem solution for an equation with a discontinuous right-hand side transformed so that to enable application of the method of solution continuation with respect to a parameter. In this way, the domain in the plane of model parameters in which stable limit cycles exist is defined.