and Computational Methods

#### 539.3 Modeling the elastic-plastic characteristics of monocrystalline intermetallic alloys based on microstructural numerical analysis

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University), Bazyleva O. A. (Federal State Unitary Enterprise “All-Russian Scientific Research Institute of Aviation Materials”), Lutsenko A. N. (Federal State Unitary Enterprise “All-Russian Scientific Research Institute of Aviation Materials”), Oreshko E. I. (Federal State Unitary Enterprise “All-Russian Scientific Research Institute of Aviation Materials”)**

doi: 10.18698/2309-3684-2015-2-322

The article presents a model of microstructure of two-phase monocrystalline intermetallic alloys in the form of a periodic structure of the hexagonal type, as well as a mathematical model of elastic-plastic deformation of monocrystalline alloy, based on the method of asymptotic smoothing periodic structures. Deformation plasticity theory under loading is used for the phases with due regard for the effect of their damage level during loading. For numerical calculations of the developed model the heat-resistant monocrystalline alloy of the type VKNA-1V was used. Finite element calculations of deformation and fracture micromechanical processes in monocrystalline alloy of the type VKNA-1V were carried out. It was found that under tension maximum values of phase damagability, which determine the beginning of the alloy micro-fracture zone, are achieved in the areas adjacent to the phase interface and in areas of maximum curvature of the geometric shape of the phases. Calculations of heat-resistant alloy strain diagrams in plastic range are proved to be consistent with experimental data.

Dimitrienko Y., Gubareva E., Sborschikov S., Bazyleva O., Lutsenko A., Oreshko E. Modeling the elastic-plastic characteristics of monocrystalline intermetallic alloys based on microstructural numerical analysis. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 3-22

#### 539.3:519.63 Numerical modeling of anisogrid structures deformation using schemes of high accuracy without saturation

**Golushko S. K. (Design Technological Institute of Digital Techniques/Institute of Computational Technologies), Semisalov B. V. (Institute of Computational Technologies)**

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

#### 539.3 Modeling of the dynamic stability of a cylindrical shell under the axial compressive load

**Dubrovin V. M. (Bauman Moscow State Technical University), Butina T. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2015-2-4657

The article describes a method for calculating the dynamic stability of cylindrical shell under axial compressive time-varying load. The case of linearly varying load was con-sidered as an example.

Dubrovin V., Butina T. Modeling of the dynamic stability of a cylindrical shell under the axial compressive load. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 46-57

#### 537.8+519.63 Modeling the electromagnetic effects in complex structures exposed to pulse radiation

**Berezin A. V. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Zhukov D. A. (JSC MIC NPO Mashinostroyenia), Zhukovskiy M. E. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Konukov V. V. (JSC MIC NPO Mashinostroyenia), Krainukov V. I. (JSC MIC NPO Mashinostroyenia), Markov M. B. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Pomazan Y. V. (Section of Applied Problems, RAS), Potapenko A. I. (The 12th Central Research Institute, RF Ministry of Defense)**

doi: 10.18698/2309-3684-2015-2-5872

The article describes a mathematical model of photon transport and generating the secondary electromagnetic fields in structures of complex geometry and package. A draft of the model design is given. The results of computing the photon flux in different elements of the model structure are demonstrated. It is shown that multiple-material stack-up of the enclosure can dramatically weaken the photon flux, scattering not only soft but hard photons as well. Intensity of absorption has pronounced maxima. There is space charge and the electrostatic field generated in the gas atmosphere inside the model. Electrostatic field can reach high amplitude in a small spatial domain inside the enclosure of the model.

Berezin A., Zhukov D., Zhukovskiy M., Konukov V., Krainukov V., Markov M., Pomazan Y., Potapenko A. Modeling the electromagnetic effects in complex structures exposed to pulse radiation. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 58-72

#### 681.5.015.23 Identification of parameters of regulator actuators for steam power turbines using matrix operator apparatus

**Kornyushin Y. P. (Bauman Moscow State Technical University), Egupov N. D. (Bauman Moscow State Technical University), Kornyushin P. Y. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2015-2-7386

In the paper we propose an algorithm of parameters (time constants of the turbine) iden-tification using the gradient method with an adaptive model. The adaptive mathematical model has the same structure as the identification object. The identification criterion is based on the loss function, which is the misalignment between the left and right sides of the equation, which describes the adaptive model. Thus it is avoided the need of finding the solution of a nonlinear equation for the adaptive model in an explicit form. In the model the signal observed at the output of the identified object is used instead of the output signal. Since mathematical models are nonlinear, the Newton – Kantorovich linearization and the matrix operator apparatus are applied to solve the problem. The features of gradient vector computation and features of the identification algorithm and its organization are considered. The results of the two time constants identification for the mathematical model of the turbine PT-12/15-35/10M are presented.

Kornyushin Y., Egupov N., Kornyushin P. Identification of parameters of regulator actuators for steam power turbines using matrix operator apparatus. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 73-86

#### 681.513.5 Stabilization of an unstable limit cycle of relay chaotic system

**Krasnoschechenko V. I. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2015-2-87104

The article presents an algorithm of synthesis for stabilization of an unstable limit cycle of relay chaotic system. One-dimensional discrete Poincare map is used in algorithm for finding fixed points of the period one (limit cycles of initial continuous system). It is shown, that classical OGY method of dead beat regulator synthesis does not solve the problem as it takes into account only speed of the target coordinate what is not sufficient for stabilizing. The proposed algorithm is based on search of the necessary regulator factor by solving an inverse problem: at first some factor is assigned and then two-step procedure of system transition to the following switching point (with correction) is carried out. The task of correction is performed in a complete neighborhood of target coordinate position and speed, and it provides stabilization of a limit cycle by adjusting small amplitude pulses in the chosen area of entry conditions (area of stabilization) as evidenced by the simulation results.

Krasnoschechenko V. Stabilization of an unstable limit cycle of relay chaotic system. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 87-104

#### 519.63 Parallel multigrid algorithms

**Martynenko S. I. (Baranov Central Institute of Aviation Motor Development)**

doi: 10.18698/2309-3684-2015-2-105120

The paper represents the main directions of development of the parallel classic multigrid algorithms and discusses their disadvantages. The possibility of efficient parallelization of smoothing iterations at the levels of coarse grids is shown using the Robust Multigrid Technique. Then multigrid structure is used for developing hybrid multigrid method. The paper describes estimations of speed-up and efficiency of different parallel multigrid algorithms as well as the results of numerical experiments.

Martynenko S. Parallel multigrid algorithms. Маthematical Modeling and Coтputational Methods, 2015, №2 (6), pp. 105-120