Том 9, № 5 (2017)

In the numerical simulation of gas-dynamic flows in domains with a complex geometry, it is necessary to use detailed unstructured grids and highly accurate numerical methods. The Galerkin method with discontinuous base functions (or the discontinuous Galerkin method) works well in dealing with such problems. This technique has several advantages inherent both in finite-element and in finite-difference approximations. At the same time, the discontinuous Galerkin method is computationally complex; therefore, the question arises about the most efficient use of the full potential of computers. In order to speed up the computations, we applied the operator programming method to develop the computational module. It allows presenting mathematical formulas in programs in compact form and helps to port programs to parallel architectures such as NVidia CUDA and Intel Xeon Phi. Earlier the operator programming method was implemented for regular three-dimensional Cartesian grids and three-dimensional locally adaptive grids. In this work, this method is applied to threedimensional tetrahedral grids. This example demonstrates that the method in question can be efficiently implemented on arbitrary three-dimensional grids. Besides, we demonstrate the use of the template metaprogramming methods of the C++ programming language in order to speed up computations.

Fermi-Dirac functions of integer indices are broadly used in problems of electronic transport in dense substances. Polynomial approximations are constructed for their fast computation. Such coefficients are found for functions of index 1, 2, and 3, which provide an error ratio of about 2 × 10^–16 with nine free parameters. In this work, we use the boost::multiprecision library of C++, which allows us to compute with any arbitrary number of digits. The precision of previously obtained relations is improved to ~5 × 10^–18 and the same relation is constructed for the index k = 4. Also, it is shown that simple global relation consisting of a few parameters reasonably describe the order of the value of the functions for all values of the independent variable and can be used for estimations.

A technique for locally rescaling (upscaling) the functions of the relative phase permeability (RPP) has been developed, which minimizes the error in the approximating the phase filtration rates for the superelement modeling of waterflooding a layered heterogeneous oil reservoir. The RPP is locally upscaled for each superelement based on the solution of two-dimensional two-phase filtration problems on a refined computational grid. Modified RPP functions (MFRPPs) are represented in the parametric form; i.e., the values of the parameters are sought when solving the problem of minimizing the deviations of the averaged and approximated phase velocities at the sites corresponding to the faces of the superelement. The efficiency of applying MFRPP to superelement modeling is illustrated by an example of a model reservoir region where oil is extracted using injection and production wells and by an example of waterflooding at a real oilfield.

The main steps of implementing the calculation technique on Chimera grids are considered. The attention is focused on the specifics of implementing the algorithm on grids that are composed of arbitrary polyhedrons. Methods of creating intersections of countable regions of arbitrary shape are proposed. The operability and efficiency of the technique are illustrated by examples with the available experimental data.

This paper compares the analytical model of the axisymmetric bending of a circular sandwich plate with the finite element method (FEM) based numerical model. The differential equations of the bending of circular symmetrical sandwich plates with isotropic face sheets and a nonlinear elastic core material are obtained. The perturbation method of a small parameter is used to represent the nonlinear differential equations as a sequence of linear equations specifying each other. The linear differential equations are solved by reducing them to the Bessel equation. The results of the calculations with the use of the analytical and FEM models are compared with the results obtained by other authors by the example of the following problems: (1) axisymmetric transverse bending of a circular sandwich plate; (2) axisymmetric transverse bending of an annular sandwich plate. The effect of the nonlinear elasticity of the core material on the strained state of the sandwich plate is described.

A computing technique for simulating the impact of a high-speed liquid jet on a wet wall is implemented. Such an impact generates shock waves in the jet, in the liquid layer on the wall, and in the gas surrounding the liquid. Also, the interphase boundary is strongly deformed by such an impact. The technique is based on the Constrained Interpolation Profile-Combined Unified Procedure (CIP-CUP) method combined with the dynamically adaptive Soroban grids. The gas-dynamic equations describing the liquid and gas flow are integrated without an explicit separation of the liquid-gas boundary. Such an approach is shown to be efficient for the considered problems. It allows us to obtain solutions without oscillations near the interfaces (including the case where they interact with the shock waves). For illustrative purposes, we provide the computational results for several one-dimensional and twodimensional problems with the typical features of the impact of a high-speed liquid jet on a wall, as well as a comparison with the known analytical and numerical solutions. The computational results for the problem of the impact of a high-speed liquid jet on a wall covered by a thin liquid layer are also presented.