and Computational Methods

#### 539.3 Mathematical modeling of thermal stresses in a solid with an internal crack

**Valishin A. A. (Bauman Moscow State Technical University), Kartashov E. M. (MIREA - Russian Technological University/Moscow Technological University)**

doi: 10.18698/2309-3684-2018-3-321

The purpose of this study was to evaluate the influence of the inertial effects and their deviation from the same quasi-static results. The role of inertial effects in the problem of thermal shock is studied on the example of a massive body with an internal spherical crack. We study the thermal reaction of an elastic space with an internal spherical crack whose surface, initially stress-free and at a temperature of T0, is instantly heated to a temperature of TC > T0 and then maintained at that temperature. Thermal stress state occurs under different modes of heat exposure, creating heat stroke. The most common in practice, three cases: temperature heating, thermal heating and heating medium. The generalized dynamic thermoelasticity equation for all three cases in rectangular and curvilinear coordinates is obtained. Considered the thermal response of a massive rigid body with internal spiroborate crack. The exact analytical solution of the problem is obtained. Earlier in the works of one of the authors the solution of the dynamic problem in the form of bulky functional structures was obtained, which greatly complicated their practical use. In this paper, we propose a solution to the problem in new classes of functions, which makes the solution more convenient for numerical experiments. A generalized differential relation for dynamic thermoelasticity is proposed, which has an extensive field of practical applications in the study of thermal response to heat stroke of solids of different shapes. It is shown that the component of the radial stress is a spherical elastic wave propagating from the cavity surface into the material. Numerical calculations of dynamic effects are performed and it is shown that the quasi-static interpretation of time problems in the theory of heat stroke does not allow to take into account the basic laws of transient thermoelasticity and inertial effects.

Валишин А.А., Карташов Э.М. Математическое моделирование термических напряжений в твердом теле с внутренней трещиной. Математическое моделирование и численные методы, 2018, № 3, с. 3–21.

#### 519.6 Ψ-transformation optimization method in сomparison with canonical particle swarm optimization method

**Bushuev A. Y. (Bauman Moscow State Technical University), Maremshaova A. A. (Bauman Moscow State Technical University/JSC MIC NPO Mashinostroyenia)**

doi: 10.18698/2309-3684-2018-3-2237

When dealing with many applications there is a problem of finding the global extremum. Of particular relevance are the optimization methods that allow solving problems effectively when the objective function depends on a complex mathematical model that requires large computing resources for its solution. In this paper, a comparison is made between the Ψ-transformation optimization method and the canonical particle swarm optimization method. The flaws of some known algorithms of the Ψ-transformation optimization method are revealed and a modification based on the replacement of a random law with uniform distribution for generating statistical realizations on the second and subsequent iterations of the standard algorithm by the normal distribution law with parameters determined by the results of the previous iteration is proposed. On the basis of the extensive computational experiment, the advantage of the modified algorithm of the Ψ-transformation optimization method is shown in comparison with algorithm of the canonical particle swarm method.

Бушуев А.Ю., Маремшаова А.А. Сравнение модифицированного метода Ψ-преобразования и канонического метода роя частиц. Математическое моделирование и численные методы, 2018, № 3, с. 22–37.

#### 519.6 Modelling of quasi-static reliability of technical system design

**Dubrovin V. M. (Bauman Moscow State Technical University), Semyonov K. S. (Bauman Moscow State Technical University/RSC Energia)**

doi: 10.18698/2309-3684-2018-3-3848

We consider the technical system, comprising a plurality of structural elements operating under the influence of a complex external loads. For such a system, we proposed a method for calculating the reliability criterion for the occurrence of one or more of the limit states design elements.

Дубровин В.М., Семёнов К.С. Моделирование квазистатической надежности конструкции технической системы. Математическое моделирование и численные методы, 2018, № 3, с. 38–48.

#### 628.822 Analytical model of oscillations of the roller moving along a surface in a hydrodynamic lubrication regime

**Ivanov V. A. (Политехнический институт СФУ), Yerkayev N. V. (ICM SB RAS)**

doi: 10.18698/2309-3684-2018-3-4966

This article deals with the model of normal oscillations of the roller moving along the surface with a constant velocity in a presence of a liquid lubrication layer. Pressure distribution along the lubrication layer is obtained as a result of integration of the Reynolds equation taking into account both tangential and normal velocities of the roller with respect to the surface. A damping coefficient is determined as that of proportionality between the normal velocity and corresponding variation of the carrying capacity. After special normalizations, the problem is reduced to the stiff ordinary differential equation with small parameter multiplied on the highest order derivative term. For this equation, analytical solution is derived by method of asymptotic expansion on a singular small parameter. This solution contains regular terms of series expansion, as well as boundary layer functions decreasing rapidly with time. Characteristic decreasing time for these functions is proportional to the small parameter. The obtained analytical solutions is applied for the problem of roller relaxation to the new equilibrium state after sharp increase of the external loading. A peculiarity of this process is a rapid increase of the pressure peak just after the loading jump, which afterwards is gradually relaxing to the new stationary value corresponding to the increase external loading.

Иванов В.А., Еркаев Н.В. Аналитическая модель колебаний ролика, движущегося вдоль твердой поверхности в режиме гидродинамической. Математическое моделирование и численные методы, 2018, № 3, с. 49–66.

#### 532.5.013.2+534.113 Mathematical methods of identification of hydrodynamic loads at impact on water based on one-dimensional theories of elastic wave propagation in rods

**Yeroshin V. A. (Lomonosov Moscow State University), Plyusnin A. V. (Bauman Moscow State Technical University/JSC MIC NPO Mashinostroyenia)**

doi: 10.18698/2309-3684-2018-3-6794

The problem of longitudinal and transversal oscillations of elastic cylinder generated by high velocity impact of the forward end on the water surface is considered from the point of the identification of hydrodynamic forces by treating optical measuring data of the opposite end motions. The statements of the direct and reverse problems are derived, based on the one-dimensional theories of Saint-Venan and Timoshenko, which provides the hyperbolicity of the governing equations. The results of the direct problem calculations by the finite-difference method are compared with the available experimental traces and show rather accurate qualitative coincidence.

Ерошин В.А., Плюснин А.В. Математические методы идентификации гидродинамических нагрузок при ударе о воду, основанные на одномерных теориях распространения упругих волн в стержнях. Математическое моделирование и численные методы, 2018, № 3, с. 67–94.

#### 517:519.6 Computation of stress-strain condition of free body by finite element method

**Temis Y. M. (Центральный институт авиационного моторостроения им. П.И. Баранова), Azmetov K. K. (Baranov Central Institute of Aviation Motor Development/Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2018-3-95113

The technique of computation of stress-strain condition of free bodies by finite element method was offered. Implementation of suggested algorithm in two-dimensional formulation and calculation examples are given.

Темис Ю.М., Азметов Х.Х. Расчет напряженно-деформированного состояния свободных тел методом конечных элементов. Математическое моделирование и численные методы, 2018, № 3, с. 95–113.

#### 539.3 Modeling of the stresses in thin composite cylindrical shells based on the asymptotic theory

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

doi: 10.18698/2309-3684-2018-3-114132

The previously developed general asymptotic theory of thin multilayer shells is used for the case of cylindrical shells. The ratios are presented in explicit analytical form for all six components of the stress tensor in a thin multilayer elastic cylindrical shell, depending on the deformations, curvatures of the middle surface of the shell, as well as their derivatives along the longitudinal coordinates. The obtained formulas make it possible to calculate all the distributions of the components of the stress tensor over the thickness in a cylindrical shell after finding solutions to the two-dimensional problem of the theory of KirchhoffLyav shells. An example is given of the calculation of stresses in a cylindrical composite shell underaxisymmetric bending by pressure. To calculate stresses by these formulas, only a differentiation of displacements is required - a deflection and two displacements of the middle surface of the shell, for which an analytical solution is obtained.

Димитриенко Ю.И., Губарева Е.А., Пичугина А.Е. Моделирование напряжений в тонких композитных цилиндрических оболочках на основе асимптотической теории. Математическое моделирование и численные методы, 2018, № 3, с. 114–132.