and Computational Methods

#### 537.876.4:517.958 Properties of one-dimensional photonic crystal as a reflective or wave guiding structure when excited by H-polarization

**Apeltsin V. F. (Bauman Moscow State Technical University), Mozzhorina T. Y. (Bauman Moscow State Technical University)**

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

#### 539.3 Finite element modulation of effective viscoelastic properties of unilateral composite materials

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

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

#### 539.3 Theory of plates based on the method of asymptotic decompositions

**Sheshenin S. V. (Lomonosov Moscow State University), Skoptsov K. A. (Lomonosov Moscow State University)**

doi: 10.18698/2309-3684-2014-2-4961

The paper presents comparison of asymptotic analysis of a composite plate lateral bending under the impact of surface load against classical theories of thin and thick plates. Layers of the plate are assumed to be homogeneous linear elastic orthotropic materials.

Sheshenin S., Skoptsov K. Theory of plates based on the method of asymptotic decompositions. Маthematical Modeling and Coтputational Methods, 2014, №2 (2), pp. 49-61

#### 532.28 Simulation of wave influence of a stratified current on an underwater pipeline

**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: 10.18698/2309-3684-2014-2-6276

We have studied impact of near bottom flow ply boundaries waves’ forces on an underwater pipeline. We have performed numerical analysis of the obtained integral representation for the force of aqueous medium impact on the pipeline. We have revealed certain flow conditions with significant increase of hydrodynamic reactions.

Vladimirov I., Korchagin N., Savin A. Simulation of wave influence of a stratified current on an underwater pipeline. Маthematical Modeling and Coтputational Methods, 2014, №2 (2), pp. 62-76

#### 629.762:532.5.031 Boundary-element-method modelling of inside and outside non-stationary interaction of aircraft body and liquid

**Plyusnin A. V. (JSC MIC NPO Mashinostroyenia)**

doi: 10.18698/2309-3684-2014-2-77100

The article considers inner and outer problems of non-stationary interaction between aircraft body and incompressible ideal fluid and statement of the problems in the form of boundary integral equations. By inner problems we mean vibration of fuel in tanks and by outer problems we mean determination of additional masses and moments of inertia. We provide formula of efficient solutions for these problems by the boundary element method as applied to bodies of revolution and examples of calculations.

Plyusnin A. Boundary-element-method modelling of inside and outside non-stationary interaction of aircraft body and liquid. Маthematical Modeling and Coтputational Methods, 2014, №2 (2), pp. 77-100

#### 629.78 Mathematical modelling of deployment of large-area solar array

**Bushuev A. Y. (Bauman Moscow State Technical University), Farafanov B. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2014-2-101114

We have built a mathematical model for deployment of multibody solar array with a cable system of deployment. On the basis of analysis of the kinematic scheme of deployment system we have chosen the dimensions of the radii of the rollers and gear ratio of the two types of gear mechanisms which provide the preset sequence of fixation of sections. We used Lagrange equation of the second kind for studying deployment of the solar battery array. A distinctive feature of this approach is application of iterative method for taking into account deformation of the cables of synchronizing system. The mathematical model can be used to choose optimal design factors and deployment system performance requirements. It is also valuable for dealing with worst-case situations and verifying the reliability of deployment procedure.

Bushuev A., Farafanov B. Mathematical modelling of deployment of large-area solar array. Маthematical Modeling and Coтputational Methods, 2014, №2 (2), pp. 101-114

#### 551.5:517 Modelling global climate stabilisation by controlled emission of stratospheric aerosol

**Parkhomenko V. P. (Bauman Moscow State Technical University/Computing Centre of RAS)**

doi: 10.18698/2309-3684-2014-2-115126

During the last decades we are witnessing climate changes. Scientists assume global warming to be the result of man-generated increase of green house gases in the atmosphere, the most important one being СО2. The article deals with the problem and describes cutting-edge solutions for stabilising climate. The research makes use of a seasonal global combined threedimensional hydrodynamic model of climate. This model of climate includes model of the World Ocean with real depths and configuration of continents, model of evolution of sea ice and energy — moisture balance model of the atmosphere. The first stage covers estimation of climate change through 2100 following IPCC A2 СО2 increase scenario. The calculations yield rise of average annual surface temperature of the atmosphere by 3,5 С. A number of calculations have been made to estimate possibility of stabilising climate at the level of 2010 by means of controlled release of sulphate aerosol into stratosphere. The aerosol will reflect and disperse a part of the coming solar radiation. We have calculated concentration (albedo) of the aerosol from 2010 to 2100 which will enable us to stabilise the average annual temperature of the surface layer of atmosphere. We have shown that by this way it is impossible to achieve the seasonal uniform approximation to the existing climate, although it is possible to significantly reduce the greenhouse warming effect. Provided that the aerosol is distributed evenly in space in stratosphere, we can stabilize the average annual temperature of the atmosphere, herewith in middle and low latitudes the climate will be colder by 0,1…0,2 С and in high latitudes it will be warmer by 0,2…1,2 С. Besides, these differences are essentially seasonal in nature, they increase in winter. If we stop releasing the aerosol in 2080 the average annual global temperature of the atmosphere will rise, reaching the former value without the aerosol by the year 2100.

Parkhomenko V. Modelling global climate stabilisation by controlled emission of stratospheric aerosol. Маthematical Modeling and Coтputational Methods, 2014, №2 (2), pp. 115-126