State Heat Conduction Problems Research Articles

MODELING THE INFLUENCE OF THE SHAPE OF THE LOCAL HEAT FLOW INTENSITY DISTRIBUTION ON THE SURFACE OF A SEMI-INFINITE BODY ON THE STRESS STATE IN THE VICINITY OF A SUBSURFACE CRACK

Purpose. A mathematical model to determine the two-dimensional thermoelastic state in a semi-infinite solid weakened by an internal crack under conditions of local heating is examined. Heat flux due to frictional heating on the local area of the body causes changes in temperature and stresses in the body, which significantly affects its strength, as it can lead to crack growth and local destruction. Therefore, the study of the problem of frictional heat is of practical interest. This paper proposes to investigate the stress-deformed state in the vicinity of the crack tip, depending on the crack placement. Methodology. The methods for studying the two-dimensional thermoelastic state of a body with cracks as stress concentrators are based on the method of complex variable function. Reducing the problem of stationary heat conduction and thermoelasticity to singular integral equations (SIE) of the first kind, the numerical solution by the method of mechanical quadrature was obtained. Findings. In this paper, we present graphical dependencies of stress intensity factors (SIF) at the crack tip on the angle orientation of the crack as well as forms of the intensity distribution of the local heat flux. The obtained results will be used later to determine the critical value of the intensity of the local heat flux from equations of limit equilibrium at which crack growth and the local destruction of the body occur. Originality. The scientific novelty lies in the fact that the solutions to two-dimensional problems of heat conduction and thermoelasticity for a half-plane containing a crack due to local heating by a heat flux were obtained. This would make it possible to obtain a comparative analysis of the intensity of thermal stresses around the top of the crack, depending on the form of distribution of the intensity of the heat flow on the surface of the body. Practical value. The practical value is the ability to extend our knowledge of the real situation in the thermoelastic elements of engineering structures with the crack that operate under conditions of heat stress (frictional heat) in various industries, particularly in mechanical engineering. The results of specific values of SIF at the crack tip in graphs may be useful in the development of sustainable modes of structural elements in terms of preventing the growth of cracks.

Развитие и обобщение метода отражений в задачах электростатики и теплопроводности плоскослоистых сред

The method of mirror reflections of electrostatics for a point charge located next to a plane-layered medium consisting of a single film on a dielectric half-space is formulated. The method is generalized to the case of an arbitrary system of charges. The proposed approach is applied to mathematically similar problems of electrostatics and stationary heat conduction of plane-layered media. In particular, the problems of finding distributions of the electrostatic potential around a conducting sphere, an ellipsoid of revolution and a drop-shaped body located near the dielectric film on the dielectric half-space. It is shown how to apply the results obtained for electrostatic problems to similar problems of finding the temperature distribution of uniformly heated bodies of the same geometry located near a heat-conducting film in a heat-conducting half-space.

Development and generalization of the method of reflections in problems of electrostatics and thermal conductivity of plane-layered media

The method of mirror reflections of electrostatics for a point charge located next to a plane-layered medium consisting of a single film on a dielectric half-space is formulated. The method is generalized to the case of an arbitrary system of charges. The proposed approach is applied to mathematically similar problems of electrostatics and stationary heat conduction of plane-layered media. In particular, the problems of finding distributions of the electrostatic potential around a conducting sphere, an ellipsoid of revolution and a drop-shaped body located near the dielectric film on the dielectric half-space. It is shown how to apply the results obtained for electrostatic problems to similar problems of finding the temperature distribution of uniformly heated bodies of the same geometry located near a heat-conducting film in a heat-conducting half-space. Keywords: plane-layered medium, mirror reflection method, electrostatics, thermal conductivity.

Tolerance modelling of heat conduction in biperiodic fourth-component composite

The object of analysis is a heat conduction problem within the framework of tolerance modelling in fourth-component biperiodic composite. Two materials are isotropic and two are orthotropic, and additionally symmetry axes of orthotropic ones are rotated with respect to each other by an angle equal 90°. The results of some special boundary conditions for stationary problem of heat conduction were obtained from the local homogenization model (LHM). The model equations were derived by simplifying the equations of the standard tolerance model (STM), which were obtained based on two model assumptions: micro-macro decomposition of the temperature field and residual function averaging, after introducing the concept of “weakly slowly varying function” (WSV) and “slowly varying function” (SV) into the modelling process. The presented examples show the influence of the given boundary conditions on the macro-temperature distribution and on the distribution of the approximate temperature field θ(). The effect of thermal conductivity of the component materials and the number of periodicity cells on temperature distribution was also shown.

Вариационная формулировка математической модели процесса стационарной теплопроводности с учетом пространственной нелокальности

In creating and using new structurally sensitive materials, it is essential to construct mathematical models which make it possible to describe the materials’ behavior in widely changing external influences. Modeling of non-local media is a class of methods of generalized continuum mechanics. The capability of analyzing mathematical models of a continuous medium can be extended through variational methods. The paper describes the construction of a functional for the problem of stationary heat conduction in a homogeneous body, taking into account the effects of nonlocality and with a temperature-independent heat conduction coefficient. A variational formulation used in the case of a linear problem in combination with acceptable admissible functions allowed us to quantify this effect. The study gives an example of using the variational formulation, which takes into account the influence of the spatial nonlocality of the stationary heat conduction process on the parameters that determine the well-known phenomenon of thermal explosion with an exponential temperature dependence of the volumetric heat release rate in the plate. The quantitative analysis was carried out for two versions of the function that is admissible for the constructed integral functional and describes the possible temperature distribution over the thickness of the plate under consideration. Having compared the obtained results, we opted for the admissible function, as its use leads to a smaller difference between the maximum value of the parameter characterizing the intensity of heat release in the plate and the known result found with no account for the influence of spatial nonlocality

Influence of extended heat sources on the temperature distribution in profiled polar-orthotropic annular plates with heat-insulated bases

The solution of the stationary heat conduction problem for profiled polar-orthotropic annular plates with heat-insulated bases from N extended heat sources at their external border is presented. The temperature distribution in such plates will be non-axisymmetric. The solution of the stationary heat conduction problem for anisotropic annular plates of an random profile is resolved through the solution of the corresponding Volterra integral equation of the second kind. The formula of a temperature calculations in anisotropic annular plates of an random profile is given. The exact solution of stationary heat conduction problem for polar-orthotropic annular plate of an exponential profile is recorded. The temperature distribution in such anisotropic plate from N extended heat sources at its outer border is more complex than in the case of temperature distribution from N point heat sources at their external border.

A novel Boundary Element model for solving stationary inhomogeneous heat conduction problems

A new formulation of BEM applied to steady heat conduction problems, governed by the inhomogeneous Laplace Equation will be presented. The thermal property is isotropic, but varies according to a function known throughout the domain. DIBEM and DST are used together to generate the numerical model. Being used in the interpolation process, two distinct RBF will be used to evaluate the formulation performance. In order to show the generality and capacity of the proposed model, the heat flux variation in a machine tool temperature sensor is calculated. The comparison of the quality of the results achieved is made using FEM.

Heat Conduction Problems in a Homogeneous Pipe with Inner Nonhomogeneous Coating

The paper deals with the analysis of nonhomogeneous inner coatings for a homogeneous pipe with respect of heat loss from the outer pipe surface. Two kinds of the coatings in the form of ring layers are considered: (1º) with the thermal properties changing continuously along the coating thickness (called the coating A), (2º) multilayered coatings with piecewise continuous thermal properties (called the coatings B). The analysis is connected with the stationary heat conduction problems. Some special cases of the coatings A and B are investigated. The obtained analytical results and the comparison of the coatings are presented.

Two-Dimensional Problem of Thermoelasticity for a Half Space Whose Boundary is Either Free or Rigidly, Smoothly, or Flexibly Fastened with Heat Insulation in a Ribbon-Like Domain Parallel to the Boundary

We construct Green’s functions for the problems of stationary heat conduction and thermoelasticity posed for a semiinfinite body subjected to the action of a thermal dipole under plane-strain conditions with either free or rigidly, smoothly, or flexibly fastened boundary kept at a temperature equal to zero. To solve the problem of heat conduction, we use the logarithmic double-layer potential. In the problem of thermoelasticity, we use a thermoelastic potential of displacements in the infinite body with heat dipoles specularly located relative to the boundary of the half space. To satisfy the boundary conditions on the boundary of the body, we construct the Boussinesq functions. We present explicit expressions for temperature, displacements, and stresses that can be used to determine the thermoelastic state of the half space caused by the perturbations of a given heat flux by a heatproof ribbon-like domain parallel to the boundary.

The influence of the length of heat sources on the external border on the temperature distribution in profiled polar-orthotropic ring plates taking into account there heat exchange with the external environment

We study the influence of N extended heat sources at external boundaries on the nonaxisymmetric temperature distribution on profiled polar-orthotropic ring plates and take into account heat exchange with the external environment. The solution of the stationary heat conduction problem for anisotropic annular plates of a random profile is resolved through the solution of the corresponding Volterra integral equation of the second kind. The formula of a temperature calculations in anisotropic annular plates of an random profile is given. The exact solution of stationary heat conductivity problem for a reverse conical polar-orthotropic ring plate is recorded. The temperature distribution in such anisotropic plate from N extended heat sources at its outer border is more complex than in the case of temperature distribution from N point heat sources at their external border.

New Possibilities of Application of Artificial Intelligence Methods for High-Precision Solution of Boundary Value Problems

One of the main problems in modern mathematical modeling is to obtain high-precision solutions of boundary value problems. This study proposes a new approach that combines the methods of artificial intelligence and a classical analytical method. The use of the analytical method of fictitious canonic regions is proposed as the basis for obtaining reliable solutions of boundary value problems. The novelty of the approach is in the application of artificial intelligence methods, namely, genetic algorithms, to select the optimal location of fictitious canonic regions, ensuring maximum accuracy. A general genetic algorithm has been developed to solve the problem of determining the global minimum for the choice and location of fictitious canonic regions. For this genetic algorithm, several variants of the function of crossing individuals and mutations are proposed. The approach is applied to solve two test boundary value problems: the stationary heat conduction problem and the elasticity theory problem. The results of solving problems showed the effectiveness of the proposed approach. It took no more than a hundred generations to achieve high precision solutions in the work of the genetic algorithm. Moreover, the error in solving the stationary heat conduction problem was so insignificant that this solution can be considered as precise. Thus, the study showed that the proposed approach, combining the analytical method of fictitious canonic regions and the use of genetic optimization algorithms, allows solving complex boundary-value problems with high accuracy. This approach can be used in mathematical modeling of structures for responsible purposes, where the accuracy and reliability of the results is the main criterion for evaluating the solution. Further development of this approach will make it possible to solve with high accuracy of more complicated 3D problems, as well as problems of other types, for example, thermal elasticity, which are of great importance in the design of engineering structures.

A stationary heat conduction problem in low dimensional sets in {mathbb {R}}^N

We study a basic linear elliptic equation on a lower dimensional rectifiable set S in {mathbb {R}}^N with the Neumann boundary data. Set S is a support of a finite Borel measure mu . We will use the measure theoretic tools to interpret the equation and the Neumann boundary condition. For this purpose we recall the Sobolev-type space dependent on the measure mu . We establish existence and uniqueness of weak solutions provided that an appropriate source term is given.

The uniqueness of the solution of a nonlinear heat conduction problem under Hölder’s continuity condition

We investigate a stationary nonlinear heat conduction problem in which heat conductivities depend on temperature. It is known that such problem need not have a unique solution even when the conductivity coefficients are continuous. In this paper we prove that for 12-Hölder continuous coefficients the uniqueness of the weak solution is guaranteed.

Approximate analytical solution to the stationary two-dimensional heat conduction problem on infinite bar with the source of heat

An approximate analytical solution to the boundary-value heat conduction problem for an infinite bar with a heat source was obtained with the use of the integral method of heat balance, by introducing a complementary required function and complementary boundary conditions. The boundary - value problem for a partial differential equation is reduced to an ordinary differential equation with respect to this function due to the complementary required function that characterizes the change in temperature along the axis of symmetry in the cross-section of the bar. The complementary boundary conditions determined by the initial differential equation and the given boundary conditions are found so that their satisfaction is equivalent to the solution of the initial equation of the boundary value problem at the boundary points. The fulfilment of the equation at the boundary points as well as the heat balance integral results in the fulfilment of the initial equation inside the domain. The approximate analytical solution obtained can be used to identify the amount of internal heat generated by various production processes (vibration and deformation loads, electromagnetic fields effects, etc.) in thermal and nuclear power plants, in the rocket and space industry and other industrial facilities.

Axisymmetric stationary heat conduction problem for half-space with temperature-dependent properties

The study examines problems of heat conduction in a half-space with a thermal conductivity coefficient that is dependent on temperature. A boundary plane is heated locally in a circle zone at a given temperature as a function of radius. A solution is obtained for any function that describes temperature in the heating zone. Two special cases are investigated in detail, namely Case 1 with given constant temperature in the circle zone and Case 2 with temperature given as a function of radius, r. The temperature of the boundary on the exterior of the heating zone is assumed as zero. The Hankel transform method is applied to obtain a solution for the formulated problem. The effect of thermal properties on temperature distributions in the considered body is investigated. The obtained results were compared with finite element method model.

Mathematical Modelling of Stationary Thermoelastic State for a Plate with Periodic System of Inclusions and Cracks

Abstract Two-dimensional stationary problem of heat conduction and thermoelasticity for infinite elastic body containing periodic system of inclusions and cracks is considered. Solution of the problem is constructed using the method of singular integral equations (SIEs). The numerical solution of the system integral equations are obtained by the method of mechanical quadrature for a plate heated by a heat flow, containing periodic system elliptic inclusions and thermally insulated cracks. There are obtained graphic dependences of stress intensity factors (SIFs), which characterise the distribution of intensity of stresses at the tops of a crack, depending on the length of crack, elastic and thermoelastic characteristics inclusion, relative position of crack and inclusion.

Mathematical modeling of stationary thermoelastic state in a half plane containing a periodic system of cracks due to periodic local heating by a heat flux

Purpose. To determine the two-dimensional thermoelastic state in a semi-infinite solid (half-plane), weakened by a system of periodic internal cracks under conditions of local heating on the edge of the half plane. Heat flux due to frictional heating on the local area of the body, causes changes in temperature and stresses in the body, which significantly affects its strength, as it can lead to crack growth and local destruction. Therefore, the study of the problem of frictional heat is of a practical interest. This paper proposes to investigate the stress-deformed state in the vicinity of the crack tip, depending on the period of cracks placement. Methodology. The methods for studying two-dimensional thermoelastic state of a body with crack as stress concentrators are based on the method of complex variable function. Reducing the problem of stationary heat conduction and thermoelasticity to singular integral equations (SIE) of the first kind, the numerical solution by the method of mechanical quadrature was obtained. Findings. In this paper, we present graphical dependencies of stress intensity factors (SIF) at the crack tip on the angle of orientation and on the relative position of cracks. The obtained results will be used later to determine the critical value of the intensity of the local heat flux from equations of limit equilibrium at which crack growth and the local destruction of the body occur. Originality. The originality of our solution lies in the fact that the new two-dimensional problems of heat conduction and thermoelasticity for a half plane containing a periodic cracks due to local heating by a heat flux are obtained. Practical value. The practical value is the ability to extend our knowledge of the real situation in the thermoelastic elements of engineering structures with cracks that operate under conditions of heat stress (frictional heat) in various industries, particularly in mechanical engineering. The results of specific values of SIF at the crack tip in graphs may be useful in the development of sustainable modes of structural elements in terms of preventing the growth of cracks.

РАЗРАБОТКА АЛГОРИТМА ОТОБРАЖЕНИЯ РЕЗУЛЬТАТОВ ТОПОЛОГИЧЕСКОЙ ОПТИМИЗАЦИИ ПРИ РЕШЕНИИ ДВУМЕРНЫХ ЗАДАЧ СТАЦИОНАРНОЙ ТЕПЛОПРОВОДНОСТИ

РАЗРАБОТКА АЛГОРИТМА ОТОБРАЖЕНИЯ РЕЗУЛЬТАТОВ ТОПОЛОГИЧЕСКОЙ ОПТИМИЗАЦИИ ПРИ РЕШЕНИИ ДВУМЕРНЫХ ЗАДАЧ СТАЦИОНАРНОЙ ТЕПЛОПРОВОДНОСТИ

Stationary problem of heat conductivity for complex-shape multilayer plates

A method for calculating stationary temperature fields in a complex-shape multilayer plate when it is heated with film heat sources is proposed. Convective heat exchange takes place on the outer surfaces of the plate. The solution of the stationary heat conduction problem is obtained for a three-dimensional formulation based on the immersion method. A comparative analysis of the temperature distribution along the thickness of a square plate with the results obtained by finite element method has been carried out. As an example, the problem of stationary thermal conductivity for a five-layer glazing element of a vehicle when heated with a film heat source has been solved.

Визначення квазістатичного термопружного стану шаруватих термочутливих плит

The technique of determining the quasistatic thermoelastic state of the layered thermosensitive plates free of load is illustrated. Much attention is paid to finding analytical-numerical solutions of one-dimensional non-stationary heat conduction problems taking into account the temperature dependences of the thermal and temperature conductivity coefficients. Their finding involves use of the Kirchhoff transformation, generalized functions, Green's functions of the corresponding linear heat conduction problem, exact sums of the series, in particular those for which the Gibbs effect takes place, linear splines and solving the received recurrent systems of nonlinear algebraic equations relative to the values in the nodes of the spline of the Kirchhoff variable on the layer division surfaces and the derivative in time on inner flat-parallel surfaces of layers. The results of numerical calculations of temperature fields in two-layer plates with different thicknesses of layers and the external surface heated by a constant heat flux are presented. The accuracy of the found solution is investigated. The comparison of the temperature fields, which are determined assuming simple nonlinearity, stable thermophysical characteristics with the ones based on the exact solution of the corresponding nonlinear stationary heat conduction problem is fulfilled.