What is the difference between boundary and initial conditions? Boundary and initial conditions. See what “Initial and boundary conditions” are in other dictionaries

The initial conditions answer the question of what the temperature field was at the moment of time taken as the origin. They are described by the expression . Very often, the temperature of the components of technological subsystems at the initial moment of time can be taken equal to the ambient temperature, i.e. In this case, it is convenient, as noted above, to carry out calculations in so-called excess temperatures, conventionally assuming that , and then adding to the result at the end of the calculation. Boundary conditions are the conditions for the interaction of the surfaces of bodies with environment or other bodies. There are several types of boundary conditions. Under boundary conditions of the first kind (GU1), it is assumed that the law of temperature distribution on the boundary surfaces of the body is known . Let, for example, you want to determine the temperature field inside a part or tool. It is quite difficult to do this experimentally without destroying the measurement object, but measuring the temperature on the surface of a part, tool or other solid body experimentally is much simpler; this can be done without damaging the object. If we know GI1 in the form of the law of temperature distribution on the surfaces of the body, then, by solving the differential equation of thermal conductivity, we can calculate the temperature field inside the part, tool, etc. A special case of GI1 is the condition of isothermal surfaces of the body, i.e. .

Boundary conditions of the second kind (BC2) provide that the distribution law of heat flux density is known , passing through the boundary surfaces. In a special case. This means that the surface in question does not exchange heat with the environment, i.e. it is adiabatic. When performing thermal calculations related to technological subsystems, in many cases, with sufficient accuracy for practice, it is possible to neglect the heat exchange of a particular surface (or section of it) with the environment, i.e., accept , which simplifies the calculation.

Boundary conditions of the third kind (GBC) are used in the case when the heat exchange of the surface with the environment cannot be neglected. In this case, the temperature of the medium with which the given body is in contact and the so-called heat transfer coefficient, W/(m 2 × °C), characterizing the heat exchange between the medium and the surface, must be specified.

According to the Newton-Richmann law, the heat flux density is proportional to the temperature difference between the surface and its surrounding environment, i.e.

Formula (2.1) makes it possible to determine the amount of heat , W/m2, which per unit time from a unit of surface is released into the environment. As follows from Fourier’s law, a flow is supplied to the surface of the body

Hence,

or . (2.2)

Expression (2.2) is a mathematical description of boundary conditions of the third kind.

Boundary conditions of the fourth kind (BC4) arise when the solid body in question is in gap-free contact with another solid body and heat exchange occurs between them. This version of boundary conditions is very often encountered in the thermophysics of technological processes. For example, during pressure processing, the stamp parts are in almost gap-free contact with the workpiece being processed; When cutting metal, the surfaces of the tool in certain areas come into contact with the chips and the workpiece. Under boundary conditions of the fourth kind, when the contact between bodies is ideal, the temperature at any point of the contact surface on both the side of one and the other body is the same, i.e.

In order to simplify calculations, often instead of the equality of temperatures at each point of contact, the equality of average temperatures on the contact surface is taken as GI4, i.e., instead of formula (2.3) they assume

Boundary conditions of the fourth kind are used when solving balance problems, that is, when analyzing the distribution of heat between bodies in contact. Having distributed the heat generated on the contact surface between the contacting bodies and calculated the heat flux density in each of the bodies, then use the boundary conditions of the second kind.

Concluding our consideration of the issue of boundary conditions, we note that different boundary conditions may exist in different areas of real bodies. Consider, for example, the process of flat grinding a workpiece with the end of a cup wheel (see Fig. 2.5). If the problem of the distribution of grinding heat between the wheel and the workpiece is solved, then in relation to the workpiece we have the following boundary conditions: GU3 - on the surface of contact with the liquid; GU2 - on the contact surface with the circle, where the heat flux density is known, and at the end of the workpiece, which can be considered adiabatic if its heat transfer to the air is neglected; GU4 - on the surface where the workpiece comes into contact with the magnetic table of the machine.

Initial and boundary conditions. Integral and the most important element The formulation of any problem in continuum mechanics is the formulation of initial and boundary conditions. Their significance is determined by the fact that one or another system of resolving equations describes a whole class of motions of the corresponding deformable medium, and only setting the initial and boundary conditions corresponding to the process under study makes it possible to select from this class a special case of interest corresponding to the practical problem being solved.

Initial conditions are the conditions that set the values of the desired characteristic functions at the moment the consideration of the process under study begins. The number of specified initial conditions is determined by the number of main unknown functions included in the system of resolving equations, as well as the order of the higher time derivative included in this system. For example, the adiabatic motion of an ideal fluid or ideal gas is described by a system of six equations with six main unknowns - three components of the velocity vector, pressure, density and specific internal energy, and the order of the derivatives of these physical quantities does not exceed the first order in time. Accordingly, the initial fields of these six physical quantities must be specified as the initial conditions: at t =0,. In some cases (for example, in the dynamic theory of elasticity), not the components of the velocity vector, but the components of the displacement vector are used as the main unknowns in the system of resolving equations, and the equation of motion contains second-order derivatives of the displacement components, which requires setting two initial conditions for the desired function: at t = 0

When formulating problems in continuum mechanics, boundary conditions are specified in a more complex and varied manner. Boundary conditions are the conditions that specify the values of the sought functions (or their derivatives with respect to coordinates and time) on the surface S of the region occupied by the deformable medium. There are several types of boundary conditions: kinematic, dynamic, mixed and temperature.

Kinematic boundary conditions correspond to the case when displacements or velocities are specified on the surface S of a body (or part thereof) where are the coordinates of points on the surface S, which generally vary depending on time.

Dynamic boundary conditions (or stress boundary conditions) are specified when surface forces p act on the surface S. As follows from stress theory, in this case, on any elementary surface area with a unit normal vector n, the vector of specific surface forces pn forcibly sets the total stress vector?n = pn, acting in a continuous medium at a point on a given surface area, which leads to the relationship of the tensor stresses (?) at this point with the surface force and the orientation of the vector n of the corresponding surface area: (?) · n = pn or.

Mixed boundary conditions correspond to the case when the values of both kinematic and dynamic quantities are specified on the surface S or relationships between them are established.

Temperature boundary conditions are divided into several groups (genus). Boundary conditions of the first kind set certain values of temperature T on the surface S of the deformable medium. Boundary conditions of the second kind set the heat flow vector q at the boundary, which, taking into account Fourier’s law of thermal conductivity, q = -- ? grad T essentially imposes restrictions on the nature of the temperature distribution in the vicinity of the boundary point. Boundary conditions of the third kind establish a relationship between the heat flow vector q directed to a given medium from the environment and the temperature difference between these media, etc.

It should be noted that the formulation and solution of most problems in the physics of fast processes, as a rule, are carried out in the adiabatic approximation, therefore temperature boundary conditions are used quite rarely; mainly kinematic, dynamic and mixed boundary conditions are used in various combinations. Let's consider possible options for setting boundary conditions using a particular example.

In Fig. Figure 3 schematically shows the process of interaction when deformable body I penetrates deformable barrier II. Body I is limited by surfaces S1 and S5, and body II by surfaces S2, S3, S4, S5. Surface S5 is the interface between interacting deformable bodies. We will assume that the movement of body I before the start of interaction, as well as during its process, occurs in a liquid creating a certain hydrostatic pressure

Figure 3

and specifying the external surface forces рп = -- рп= -- рni ri, external to both bodies, acting on any of the elementary areas of the surfaces S1 of body I and S2 of barrier II, bordering the liquid. We will also assume that the surface Sz of the barrier is rigidly fixed, and the surface S4 is free from the action of surface forces (рп = 0).

For the given example, boundary conditions of all three main types must be specified on various surfaces bounding deformable media I and II. It is obvious that on the rigidly fixed surface Sз it is necessary to set kinematic boundary conditions? bodies: or The components of the stress tensor on the surface of the S4 barrier also cannot be arbitrary, but are interconnected with the orientation of its elementary areas as.

Boundary conditions at the interface (surface S5) of interacting deformable media are the most complex and relate to mixed-type conditions, which, in turn, include kinematic and dynamic parts (see Fig. 3). The kinematic part of the mixed boundary conditions imposes restrictions on the speed of movement of individual points of both media that are in contact at each spatial point of the surface S5. There are two possible options for setting these restrictions, illustrated in Fig. 4, a and b. According to the simplest first option, it is assumed that the speeds of movement of any two individual points in contact are the same (? = ?) - this is the so-called “sticking” condition, or “welding” condition (see Fig. 4, a). More complex and at the same time more adequate for the process under consideration is to set the condition of “impenetrability”, or the condition of “non-leakage” (? · n= ? · n; see Fig. 4, b), which corresponds to the experimentally verified fact: interacting deformable media cannot penetrate

Figure 4

into each other or lag behind each other, or can they slide relative to each other at speed? - ?, directed tangentially to the interface ((?I - ?II) · n = 0). The dynamic part of the mixed boundary conditions at the interface between two media is formulated on the basis of Newton's third law using the relations of stress theory (Fig. 4, c). Thus, in each of the two individual particles of deformable media I and II in contact, its own stress state is realized, characterized by the stress tensors (?) I and (?) II. Moreover, in medium I, at each elementary area of the interface with a unit normal vector nII , external to the given medium, the total voltage vector acts?nI = (?)·nI. In medium II, on the same area, but with a unit normal vector nII, external to this medium, the total stress vector?nII = (?)II · nII acts. Taking into account the reciprocity of action and reaction?nI = - ? n II, as well as the obvious condition nI = --nII = n, a relationship is established between the stress tensors in both interacting media at their interface: (?)I · p = (?) II · p or (?ijI - ?ijII) nj = 0. Possible options for specifying boundary conditions are not limited to the particular example considered. There are as many options for specifying initial and boundary conditions as there are many processes of interaction between deformable bodies or media in nature and technology. They are determined by the characteristics of the problem being solved practical problem and are set in accordance with the general principles given above.

Determines the temperature on the surface of the body at any time, that is

T s = T s (x, y, z, t) (2.15)

Rice. 2.4 – Isothermal boundary condition.

No matter how the temperature inside the body changes, the temperature of points on the surface obeys equation (2.15).

The temperature distribution curve in the body (Fig. 2.4) at the body boundary has a given ordinate T s , which may change over time. A special case of a boundary condition of the first kind is isothermal boundary condition under which the body surface temperature remains constant throughout the entire heat transfer process:

T s = const.

Rice. 2.5 – Condition of the first kind

To imagine such a state of the body, it is necessary to assume that symmetrically to the heat source acting in the body, there is another, fictitious heat source outside it with a negative sign (the so-called heat sink). Moreover, the properties of this heat sink exactly coincide with the properties of the actual heat source, and the temperature distribution is described by the same mathematical expression. The total effect of these sources will lead to a constant temperature being established on the surface of the body, in the particular case T = 0 8C , while within the body the temperature of the points continuously changes.

Boundary condition of the second kind

Determines the heat flux density at any point on the surface of the body at any time, i.e.

According to Fourier's law, the heat flux density is directly proportional to the temperature gradient. Therefore, the temperature field at the boundary has a given gradient (Fig. b), in the particular case constants, when

A special case of a boundary condition of the second kind is the adiabatic boundary condition, when the heat flow through the surface of the body is zero (Fig. 2.6), i.e.

Rice. 2.6 - Boundary condition of the second kind

In technical calculations, there are often cases when the heat flow from the surface of a body is small compared to the flows inside the body. Then we can accept this boundary as adiabatic. When welding, such a case can be represented by the following diagram (Fig. 2.7).

Rice. 2.7 – Condition of the second kind

At the point ABOUT heat source is active. To fulfill the condition that the boundary does not allow heat to pass through, it is necessary to place the same source outside the body, symmetrically to this source, at the point O 1 , and the heat flow from it is directed against the flow of the main source. They cancel each other, that is, the boundary does not allow heat to pass through. However, the temperature of the edge of the body will be twice as high if this body were infinite. This method of heat flow compensation is called the reflection method, since in this case the heat-impermeable boundary can be considered as a boundary reflecting the heat flow coming from the metal.

Boundary condition of the third kind.

Determines the ambient temperature and the law of heat exchange between the surface of the body and the environment. The simplest form of the boundary condition of the third kind is obtained if heat transfer at the boundary is specified by Newton’s equation, which expresses that the density of the heat flux of heat transfer through the boundary surface is directly proportional to the temperature difference between the boundary surface and the environment

The heat flux density flowing to the boundary surface from the side of the body is, according to Fourier’s law, directly proportional to the temperature gradient on the boundary surface:

Equating the heat flow coming from the body to the heat transfer flow, we obtain a boundary condition of the 3rd kind:

expressing that the temperature gradient on the boundary surface is directly proportional to the temperature difference between the surface of the body and the environment. This condition requires that the tangent to the temperature distribution curve at the boundary point passes through the guide point ABOUT with temperature located outside the body at a distance from the boundary surface (Fig. 2.8).

Figure 2.8 – Boundary condition of the 3rd kind

From the boundary condition of the 3rd kind, an isothermal boundary condition can be obtained as a special case. If, which occurs with a very large heat transfer coefficient or a very low thermal conductivity coefficient, then:

and , i.e. the body surface temperature is constant throughout the entire heat transfer process and is equal to the ambient temperature.

One equation of motion (1.116) is not enough for a mathematical description of a physical process. It is necessary to formulate conditions sufficient for an unambiguous definition of the process. When considering the problem of string vibration, additional conditions can be of two types: initial and boundary (edge).

Let us formulate additional conditions for a string with fixed ends. Since the ends of the string of length are fixed, their deviations at the points and must be equal to zero for any:

, .

(1.119)

Conditions (1.119) are called borderline conditions; they show what happens at the ends of the string during the vibration process.

Obviously, the process of oscillation will depend on how the string is brought out of equilibrium. It is more convenient to assume that the string began to vibrate at time . At the initial moment of time, all points of the string are given some displacements and velocities:

(1.120)

where and are given functions.

Conditions (1.120) are called initial conditions.

So, the physical problem of string oscillations has been reduced to the following mathematical problem: to find a solution to equation (1.116) (or (1.117) or (1.118)) that would satisfy the boundary conditions (1.119) and the initial conditions (1.120). This problem is called a mixed boundary value problem, since it includes both boundary and initial conditions. It is proven that under certain restrictions imposed on the functions and , the mixed problem has a unique solution.

It turns out that in addition to the problem of string oscillations, many other problems can be reduced to problem (1.116), (1.119), (1.120). physical tasks: longitudinal vibrations of an elastic rod, torsional vibrations of a shaft, vibrations of liquids and gases in a pipe, etc.

In addition to boundary conditions (1.119), boundary conditions of other types are possible. The most common are the following:

I. , ;

II. , ;

III. , ,

where , are known functions, and , are known constants.

The given boundary conditions are called boundary conditions of the first, second, third kind, respectively. Conditions I occur if the ends of the object (string, rod, etc.) move according to a given law; conditions II – in case specified forces are applied to the ends; Conditions III – in the case of elastic fastening of the ends.

If the functions specified on the right side of the equalities are equal to zero, then the boundary conditions are called homogeneous. Thus, the boundary conditions (1.119) are homogeneous.

Combining the various listed types of boundary conditions, we obtain six types of the simplest boundary value problems.

Another problem can be posed for equation (1.116). Let the string be long enough and we are interested in the vibrations of its points sufficiently distant from the ends, and over a short period of time. In this case, the mode at the ends will not have a significant effect and is therefore not taken into account; the string is considered infinite. Instead of a complete problem, a limit problem is set with initial conditions for an unlimited domain: find a solution to equation (1.116) for for , satisfying the initial conditions:

, .

the area under consideration, respectively.

Usually a differential equation has not one solution, but a whole family of them. Initial and boundary conditions allow you to select one from it that corresponds to a real physical process or phenomenon. In the theory of ordinary differential equations, a theorem on the existence and uniqueness of a solution to a problem with an initial condition (the so-called Cauchy problem) has been proven. For partial differential equations, some theorems on the existence and uniqueness of solutions for certain classes of initial and boundary value problems are obtained.

Terminology

Sometimes the initial conditions are also considered to be boundary conditions. stationary problems ah, such as solving hyperbolic or parabolic equations.

For stationary problems, there is a division of boundary conditions into main And natural.

The main conditions usually have the form where is the boundary of the region.

The natural conditions also contain the derivative of the solution along the normal to the boundary.

Example

The equation describes the motion of a body in the field of gravity. It is satisfied by any quadratic function of the form , where - arbitrary numbers. To identify a specific law of motion, it is necessary to indicate the initial coordinate of the body and its speed, that is, the initial conditions.

Correctness of setting boundary conditions

Problems of mathematical physics describe real physical processes, and therefore their formulation must satisfy the following natural requirements:

The solution must exist in some class of functions;
The solution must be the only one in some class of functions;
The solution must continuously dependent on data(initial and boundary conditions, free term, coefficients, etc.).

The requirement for a continuous dependence of the solution is determined by the fact that physical data, as a rule, are determined approximately from experiment, and therefore one must be sure that the solution to the problem within the framework of the chosen mathematical model will not significantly depend on measurement error. Mathematically, this requirement can be written, for example, like this (for independence from the free term):

Let two differential equations be given: with identical differential operators and identical boundary conditions, then their solutions will continuously depend on the free term if:

solving the corresponding equations.

The set of functions for which the listed requirements are met is called correctness class. The incorrect setting of boundary conditions is well illustrated by Hadamard's example.

Literature

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See what “Initial and boundary conditions” are in other dictionaries:

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Books

Numerical methods for solving inverse problems of mathematical physics, Samarsky A.A. In traditional courses on methods for solving problems of mathematical physics, direct problems are considered. In this case, the solution is determined from partial differential equations, which are complemented...