Volume 34, Issue 6

A numerical model for the investigation of hydrodynamic stability of viscous incompressible shear flows in pipes of constant elliptic cross-section is proposed. The model is supplied with efficient algorithms for computing various hydrodynamic stability characteristics including the maximum possible amplification of the average kinetic energy density of disturbances and the so-called optimal disturbances. An efficient spectral reduction is also implemented, which allows one to eliminate spurious disturbances due to spatial approximation errors. The results of computing the maximum possible amplification of the average kinetic energy density of disturbances for pipes with different cross-sectional aspect ratios are presented.

We consider a method due to P. Vassilevski and Yu. A. Kuznetsov [4, 10] for solving linear systems with matrices of low Kronecker rank such that all factors in Kronecker products are banded. Most important examples of such matrices arise from discretized div K grad operator with diffusion term k 1 ( x ) k 2 ( y ) k 3 ( z ). Several practical issues are addressed: an MPI implementation with distribution of data along processor grid inheriting Cartesian 3D structure of discretized problem; implicit deflation of the known nullspace of the system matrix; links with two-grid framework of multigrid algorithm which allow one to remove the requirement of Kronecker structure in one or two of axes. Numerical experiments show the efficiency of 3D data distribution having the scalability analogous to (structured) HYPRE solvers yet the absolute timings being an order of magnitude lower, on the range from 10 to 10 4 cores.

The problem of constructing a numerically realizable model of a three-dimensional homogeneous random field in a layer 0 < z < H with given one-dimensional distribution and correlation function of the integral over coordinate z is solved. The gamma distribution with shape parameter ν and scale parameter θ is used in the work. An aggregate of n independent elementary horizontal layers of thickness h = H/n vertically shifted by a random value uniformly distributed in the interval (0, h ) is considered as a basic model. For each elementary random field, the normalized correlation function of the corresponding integral over z coincides with the given one, the gamma distribution with parameters depending on the number of horizontal layers is used as a one-dimensional distribution. It is proved that for the constructed model the normalized correlation function of the integral over z coincides with the given normalized ‘horizontal’ correlation function, and the parameters of the one-dimensional distribution asymptotically converge to given values for n → + ∞, but the corresponding mathematical expectation and variance coincide exactly with given values. To extend the class of possible models, an additional randomization of the basic model is considered. In the conclusion the results of computations for a realistic version of the problem are presented.

A new method is developed to calculate characteristics of contaminant transport (including non-classical regimes) in statistically homogeneous sharply contrasting media. A transport integro-differential equation in the space-time representation is formulated on the basis of the model earlier proposed by one of the authors (L. M.). Analytical expressions for transport characteristics in limiting time intervals in the one-dimensional case are derived. An interpolation form is proposed for the integral kernel of the transport equation. On a basis of this expression, an algorithm is developed for numerical modelling the contaminant transport in statistically homogeneous sharply contrasting media. Trial numerical 1D calculations are performed based on this algorithm. Good agreement was found between the numerical simulation results and the asymptotic analytical expressions.

We consider approaches to simulation of periodically correlated random processes based on the nonstandard spectral representation of the process with parameters periodically varying in time and also on spectral representations using the vector stationary Gaussian processes.

Human knee is one of the most complex joints. Different reasons may lead to knee instability. A personalized mathematical model of the knee may improve both diagnostic procedure and knee surgery outcomes. Such models require accurate geometric representation of bones and attachment sites of ligaments and tendons. This paper addresses automatic segmentation of knee bones and detection of origins and insertions for tendons and ligaments. The approach is based on anatomical features of bones and landmarks of tendons/ligaments attachments on the CT images. It provides a tool for the design of patient-specific geometrical knee models.