and Computational Methods

#### 536.2 Effective thermal conductivity of a composite in case of inclusions shape deviations from spherical ones

**Zarubin V. S. (Bauman Moscow State Technical University), Kuvyrkin G. N. (Bauman Moscow State Technical University), Savelyeva I. Y. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2014-4-317

On the basis of mathematical model of thermal interaction between inclusion and the matrix we estimated influence of inclusions deviations from spherical shape on the effective thermal conductivity coefficient of the composite and associated with such deviation a possible occurrence of the anisotropy of the composite with respect to the property of thermal conductivity. Using the dual variational formulation of the stationary problem of heat conduction in an inhomogeneous body we built bilateral estimates of effective thermal conductivity.

Zarubin V., Kuvyrkin G., Savelyeva I. Effective thermal conductivity of a composite in case of inclusions shape deviations from spherical ones. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 3-17

#### 519.612.2 Performance analysis of iterative methods of combined linear algebraic equations solution

**Marchevsky I. K. (Bauman Moscow State Technical University), Puzikova V. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2014-4-3752

When sampling partial differential equations one has to solve a system of linear algebraic equations. To select the optimal in the sense of the computational efficiency of iterative method for solving such equations, in addition to the rate of convergence we should take into account such characteristics of the system and method, as the condition number, the smoothing factor, the indicator "costs on." The last two characteristics are calculated by the coefficients of harmonics amplification that give evidence of the smoothing properties of the iterative method and its "costs on", i. e. how worse the method suppresses frequency components of the error as compared with the highfrequency ones. The suggested method of determining harmonic gain factors is based on of the discrete Fourier transform. As an example, an analysis of the effectiveness of the BiCGStab method with ILU and multigrid preconditioning when solving difference analogues of the Helmholtz and Poisson equations is described.

Marchevsky I., Puzikova V. Performance analysis of iterative methods of combined linear algebraic equations solution. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 37-52

#### 539.3 Asymptotic theory of thermocreep for multilayer thin plates

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2014-4-1836

The suggested thermocreep theory for thin multilayer plates is based on analysis of general three dimensional nonlinear theory of thermalcreep by constructing asymptotic expansions in terms of a small parameter being the ratio of a plate thickness and a characteristic length. Here we do not introduce any hypotheses on a distribution character for displacements and stresses through the thickness. Local problems were formulated for finding stresses in all structural elements of a plate. It was shown that the global (averaged by the certain rules) equations of the plate theory were similar to equations of the Kirchhoff–Love plate theory, but they differed by a presence of the three-order derivatives of longitudinal displacements. The method developed allows to calculate all six components of the stress tensor including transverse normal stresses and stresses of interlayer shear. For this purposes one needs to solve global equations of thermal creep theory for plates, and the rest calculations are reduced to analytical formulae use.

Dimitrienko Y., Gubareva E., Yurin Y. Asymptotic theory of thermocreep for multilayer thin plates. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 18-36

#### 517.9+532+536 Nonlinear delay reaction-diffusion equations of hyperbolic type: Exact solutions and global instability

**Polyanin A. D. (Bauman Moscow State Technical University/Ishlinsky Institute for Problems in Mechanics/MEPhI), Sorokin V. G. (Bauman Moscow State Technical University), Vyazmin A. V. (Moscow State University of Mechanical Engineering)**

doi: 10.18698/2309-3684-2014-4-5373

In the article we explored nonlinear hyperbolic delay reaction-diffusion equations with varying transfer coefficients. A number of generalized separable solutions were obtained. Most of the equations considered contain arbitrary functions. Global nonlinear instability conditions of solutions of hyperbolic delay reaction-diffusion systems were determined. The generalized Stokes problem for a linear delay diffusion equation with periodic boundary conditions was solved.

Polyanin A., Sorokin V., Vyazmin A. Nonlinear delay reaction-diffusion equations of hyperbolic type: Exact solutions and global instability. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 53-73

#### 532.58 Simulation of wave action on horizontal structure elements in the upper layer of stratified flow

**Vladimirov I. Y. (P.P. Shirshov Institute of Oceanology of the Russian Academy of Sciences), Korchagin N. N. (P.P. Shirshov Institute of Oceanology of the Russian Academy of Sciences), Savin A. S. (Bauman Moscow State Technical University)**

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

#### 533.16 Simulation of gas flow through the laminar boundary layer on the hemisphere surface in a supersonic air flow

**Gorskiy V. V. (Bauman Moscow State Technical University/JSC MIC NPO Mashinostroyenia), Sysenko V. A. (JSC MIC NPO Mashinostroyenia)**

doi: 10.18698/2309-3684-2014-4-8894

The article presents estimated accuracy of the engineering design procedure of the mass flow rate of gas through the laminar boundary layer on a hemisphere of [1]. A similar engineering method of extra accuracy is proposed.

Gorskiy V., Sysenko V. Simulation of gas flow through the laminar boundary layer on the hemisphere surface in a supersonic air flow. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 88-94

#### 519.63 Development and testing for methods of solving stiff ordinary differential equations

**Galanin M. P. (Bauman Moscow State Technical University/Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Khodzhaeva S. R. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2014-4-95119

The paper is aimed at research of the (m,k)-method, CROS, finite superelement method and 4-stage explicit Runge–Kutta method for solving stiff systems of ordinary differential equations. Analysis of tests results showed that the best choice for systems with high stiffness is CROS. The finite superelement method is the «precise» method for solving linear systems of ordinary differential equations, the best supporting method for its implementation is (4,2)-method. The variation of the finite superelement method has been built and tested for solving nonlinear problems, this method proved to be unsuitable for problems with high stiffness.

Galanin M., Khodzhaeva S. Development and testing for methods of solving stiff ordinary differential equations. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 95-119