and Computational Methods

Rubric: "01.02.00 Mechanics"

doi: 10.18698/2309-3684-2015-4-7591

This article deals with the finite-element RKDG method (Runge-Kutta Discontinuous Galerkin) and its application for numerical integration of three-dimensional system of equations of ideal gas on unstructured grids. By means of the described algorithm we solved two test tasks. For each task we conducted the analysis and compared the task solution with well-known analytical solutions or with tabular data. We also give error assessment in the solution.

Dimitrienko Y., Koryakov M., Zakharov A. Application of RKDG method for computational solution of three-dimensional gas-dynamic equations with non-structured grids. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 75-91

doi: 10.18698/2309-3684-2014-2-327

The paper considers two-dimensional boundary value problem of propagation of plane electromagnetic wave through a periodic stratified medium with one-dimensional photonic crystal structure. The structure contains a finite number of slabs. Each periodicity cell consists of two layers with different real values of constant dielectric permittivity and different thicknesses. We show that under certain additional condition, which connects the angle of incidence of the plane wave, thicknesses of the layers, frequencies and dielectric permittivity of the layers, we can solve the problem completely and explicitly, the solution leading to simple expressions for both the field reflected from the structure, and the field which has passed through it. Herewith in case of H-polarized field, unlike E-polarization, properties of this medium depend on the ratio of thickness of the layers multiplied by their dielectric permittivity (with E-polarization they depend on thickness ratio only). As a result, depending on the field frequency, photonic crystal can behave as perfectly reflecting structure, while with the same ratio of thicknesses of the layers in case of E-polarization, it becomes a wave guiding structure, and vice-versa. We have compared numerical computations with those for cases of E-polarization.

Apeltsin V., Mozzhorina T. Properties of one-dimensional photonic crystal as a reflective or wave guiding structure when excited by H-polarization. Маthematical Modeling and Coтputational Methods, 2014, №2 (2), pp. 3-27

doi: 10.18698/2309-3684-2015-2-2345

The article describes a class of promising anisogrid structures representing mesh shell of unidirectional carbon. A brief analysis of existing approaches to modeling deformation of grid structures is presented. New mathematical and numerical models are proposed for reliable description of complex behavior of anisogrid structures under different kinds of loads. A high degree of accuracy and stability of the numerical model based on the expansions of unknown functions in Chebyshev polynomials and Fourier series is caused by the lack of saturation of such approximations. Efficiency of the proposed models and techniques is demonstrated on the example of solving test boundary-value problems and a problem of axial compression of anisogrid cylindrical shell.

Golushko S., Semisalov B. Numerical modeling of anisogrid structures deformation using schemes of high accuracy without saturation. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 23-45

doi: 10.18698/2309-3684-2015-4-92109

The research explored questions of the convergence of iterative processes and correctness of the solutions on the example of the problem about a steady-state flat square lid-driven cavity flow of incompressible viscous liquid. The problem is solved for Reynolds numbers of 15000 < Re < 20000 and steps of grid 1/128 > h > 1/2048. The findings of the research illustrate that not for all relationships between Re and h the convergence of iterative processes is stable and the resulting steady-state solutions are qualitatively correct. We conducted a qualitative analysis of the solutions of the problem in the coordinate system (Re, 1/h) in terms of the convergence of iterative process, solution correctness and the required computing time. According to the literature and the results of systematic calculations we conclude that the stability of the convergence of iterative process on the coarse grid depends on the degree of influence of the artificial viscosity and/or the condition number of the matrix of difference elliptical linear algebraic equations, and on the detailed grid it depends on the grid Reynolds number. At high Reynolds numbers steady calculations can be carried out either on very coarse grids, or on very detailed ones. The width of the zone of instability in terms of parameter 1/h increases with increasing Reynolds number. Since the coarse grid solution is incorrect, and the use of detailed grid leads to very high costs of computer time, the further increase of the Reynolds number in the problem is associated with increasing the order of approximation of the differential equations.

Fomin A., Fomina L. On stationary solution of the problem of an incompressible viscous fluid at high Reynolds numbers. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 92-109

doi: 10.18698/2309-3684-2016-4-6783

The purpose of the work was to do mathematical modeling of axisymmetric body separation flow at subsonic velocities of incident flow. In our research we used the concept of viscous-inviscid interaction. We found velocities and pressures on the surface of the body under study according to the results of calculating of some equivalent body inviscid flow. The wake turbulence effect was simulated by the tailed section of the equivalent body. We examined the semi-infinite tailed sections of the equivalent body instead of the tailed sections of finite length. Moreover, we studied flow separation conditions in the base region. For the numerical simulation we used the discrete vortex method. The base pressure was found by Horner formula. We carried out mathematical modeling of the flow around cylindrical bodies with the head part of the ogival form.

Timofeev V. Construction of a semi-infinite equivalent body in mathematical modeling of subsonic separated axisymmetric flow. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 67-83

doi: 10.18698/2309-3684-2015-3-316

The article describes the method for calculating the stability of a rod under simultaneous action of axial force and torque, considering changing the torsion of the rod when it’s bent. The method is based on the use of the complete system of equations. The following cases are considered: end clamped rod, rod with a hinged support, the rod in the form of compressed and twisted console. Diagrams of dependence of the critical axial force versus the critical torque are obtained, i.e., the range of rod stability for the case of loading is determined.

Dubrovin V., Butina T. Modeling the stability of compressed and twisted rods in precise problem statement. Маthematical Modeling and Coтputational Methods, 2015, №3 (7), pp. 3-16

doi: 10.18698/2309-3684-2014-4-7487

The article describes performed simulation of force action on streamlined horizontal elements of engineering structures in the upper layer of sharply stratified flow associated with the generation of waves at the interface between the liquid layers. We obtained an integral representation of the wave drag and lift, made numerical calculations for a real marine environment. The conditions under which there is a significant increase in the hydrodynamic reactions on streamlined structural elements were revealed.

Vladimirov I., Korchagin N., Savin A. Simulation of wave action on horizontal structure elements in the upper layer of stratified flow. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 74-87

doi: 10.18698/2309-3684-2015-4-6674

Within solving the problems of active control of elastic and damping elements of multiwheeled vehicles (MWV) suspension, there arises an issue of utmost importance: that of studying the properties of suspension families, designed both for different strokes and different loads. We employed methods of experimental investigation and we conducted the verification of the mathematical model of the multi-wheeled vehicles movement with pliable on torsion by bearing system. We carried out calculation and experimental data analysis which indicates good results.

Zhileykin M., Sarach E. The verification of the mathematical model of the multiwheeled vehicles movement with pliable on torsion by bearing system. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 66-74

doi: 10.18698/2309-3684-2014-2-2848

We propose a method for calculating effective viscoelastic properties of composite materials under steady-state cyclical vibrations. The method is based on asymptotic averaging of periodic structures and finite-element solution of local problems of viscoelasticity in periodicity cells of composite materials. We provide examples of numerical simulation of viscoelastic properties for composites with unidirectional reinforcement, and of calculations of complex tensors of stress concentration in a periodicity cell. The paper presents a comparative analysis of dependencies of loss tangent of complex composite elasticity

modulus on vibration frequencies obtained through FEA calculations and rough mixed formulae. We show that rough mixed formulae, often used for calculating dissipative properties of composite materials, can yield appreciable calculation errors.

Dimitrienko Y., Gubareva E., Sborschikov S. Finite element modulation of effective viscoelastic properties of unilateral composite materials. Маthematical Modeling and Coтputational Methods, 2014, №2 (2), pp. 28-48