SU1374258A1 - Apparatus for solving inverted heat conductivity problem - Google Patents

Abstract

The invention relates to the field of computer technology and is intended to calculate the temperature dependence of the thermal conductivity of materials by solving an internal inverse problem. The purpose of the invention is to improve the accuracy of the solution. To achieve this goal, threshold units, OR elements, and Kirchhoff’s inverse transform units are introduced into the device, each of which includes a comparison unit, integrator, adders, and multiplication units. The implementation of the inverse Kirchhoff transformation is possible only under the condition of physical realizability of the transformed function D (T) 0. Therefore, in the complex with blocks of the Kirchhoff inverse transform, the blocks must be in operation, which ensure the tracking of the value of L (T) and the issuance of a signal of violation of this condition. In the proposed device, this is accomplished by threshold blocks and OR elements. 2 Il. g (L

Description

The invention relates to analog computing and is intended to calculate the temperature dependence of the thermal conductivity of materials by solving an internal inverse (inverse) problem.

When determining the dependence of thermal conductivity on temperature A (T) from the data on the stationary thermal process, it is advisable to base the calculations on the thermal conductivity model transformed by the Kirchhoff substitution

e (t) at. (one)

Then the simulation of thermal conductivity nonlinearity is carried out on a passive model (for example, a grid of resistors or a model made of electrically conductive paper), and the changes in analog signals caused by the mathematical model-1, 1, transformation are performed by function converters.

The purpose of the invention is to increase the accuracy of the solution.

On .fig, 1 shows the block diagram of the device; in fig. 2 is a block diagram of the inverse Kirchhoff transform.

The device for solving the inverse problem contains a passive model 1, four comparison blocks 2-5, a block 6 voltage dividers, five adders 7-11, three blocks 12 in the module section, two key elements -13, a former 14 pulses, a cyclic register 15 shift, two integrators 16 three blocks 17 of the inverse Kirchhoff transform, two elements OR 18 and 19, and two threshold blocks 20.

The Kirchhoff inverse transform unit 17 consists of a comparison unit 21, an integrator 22, three multiplication units 23-25 and two adders 26 and 27.

The device works as follows.

The value of the function 0 at a given temperature at the boundary is calculated by the adder 8. This is possible due to the representation of the desired function by a polynomial

) l (T) ao + a, H-a, jT2. (2)

Here, for simplicity, a second-degree polynomial is taken.

The device contains two key elements 13 and two integrators 16. In the general case, to obtain

five

0

five

0 0

five

dependencies, (t) in the form of a nth degree polynomial, the device must contain n key elements and n integrators. At. This block 17 of the inverse Kirchhoff transform will contain n + 1 multiplication blocks and n adders. The degree of the polynomial must also correspond to the number of inputs of adders 7, B, 10 and 11.

In accordance with (1), the function in at the boundary point takes the form

T2-GZ

,, 5 + ai5-,

where T is the temperature at the border of the floor.

In order to obtain at the output of the adder 8 a voltage proportional to 9, its input resistors are set in such a way that the transmission coefficients on the three inputs are proportional to T, and, respectively. In this case, potentials proportional to the values a, a, and a of the corresponding approximation are supplied to the inputs.

FIG. 1 shows one adder B for setting boundary conditions. In the general case, their number should correspond to the number of border sections with different values of T.

 From the output of the adder 8, the voltage proportional to bp is fed to the input of the passive model 1. To assess the degree of consistency of the results of the simulation with the initial data of the problem, the output potentials of the model 1j proportional to 0 must be changed accordingly. VII with Kirchhoff T (0) reverse transformation procedure. I

Since C is the interval of a strictly positive function; (T), the functions 0 (T) and T (b) are monotone, that is, each value 0 corresponds to one value T and vice versa. This means that T by the value of b can be transmitted by a change when T changes in the whole range, where L (T), until the minimum gap is reached.

T2 TK /

, T + a z

In the device, this procedure is carried out by a block 17 of the inverse Kirchhoff transform. Convertible value 0 is fed to the first input of the comparator unit 21, to the second input of which the function calculated according to the Horner scheme is applied

0 (T) ((| 2.T-b | i) T + ajT.

(3)

To calculate the polynomial (3), use multiplication unit 23, adder 26, multiplication unit 24, adder 27 and multiplication unit 25. The multiplication by constant coefficients (1/3, 1/2) is carried out by appropriate adjustment of the input circuits of multiplication unit 23 and adder 26. If the potentials at the inputs of the comparison unit 21 do not match, the input potential of the integrator 22 is different from zero. the integrator is an analogue of temperature T. A change in T occurs until the voltage at the output of multiplication unit 25 becomes equal to the voltage at the first input of unit 17. Kirchhoff's inverse transform. In this state, T will correspond to the value of the transformed function © with the a, a and a 2 operating at the given moment 2. For the stable operation of the circuit, it is necessary to match the time constants of the integrators 22 and 16 so that the rate of change of a, a and a was significantly lower than the rate of change of T at the outputs of blocks 17 of the reverse Kirchhoff transform.

From the outputs of the Kirchhoff inverse transform units 17, the T values obtained by simulation are fed to the inputs of the comparison units 3-5, the other inputs of which are supplied from the voltage divider 6 potentials corresponding to the experimentally known temperature values at the same points of the sample. Differential signals from the outputs of the 3-5 comparison modules, via the module 12 allocation module, arrive at the first three inputs of the adder 9, a quarter of which input at the moment of the violation of the positive condition I (T) receives a fine impulse. Otherwise, this input holds zero potential and the output voltage of the adder 9 is proportional to

.-

total error functions

3

.

..

where T- is the calculated temperature for the i-th point; T - experimentally measured

temperature.

The number of 3-5 comparison blocks and the allocation modules 12 connected to them must be equal to the number of thermometry points, i.e. the number of outputs of the passive model 1. Dunn

with

ju 15 20 25

about 35

40

45

50

55

the scheme was constructed under the assumption that there are three such points.

The values of the coefficients a, a. and a, determine the desired function when (4) the permissible value of ODV is reached. In order to detect such a situation, a device 2 compares the device, to one input of which a potential is supplied from the output of the voltage divider 6, proportional to E, and to the second input is a signal from the output of adder 9, proportional to E. If U and E are equal, the output signal of block 2 the comparison becomes zero. In this case, the pulse shaper 14 supplies a signal through the OR element 19 to the input of the cyclic shift register 15. Through the switched on key element, 13 a signal from the output of the comparator unit 2 is fed to the input of one of the integrators 16, which form the output signals of the device proportional to the polynomial coefficients a and a the desired dependency (2).

The coefficient a according to (2) is equal to

(, -. Tr, l (5) where is the known value of the thermal conductivity coefficient at temperature T.

To calculate a by formula (5), adder 7 is used, to the inputs of which, in addition to signals proportional to a. and aj, voltage is supplied from divider 6, corresponding to L.

From the output of the adder 7 and from the outputs of the integrators 16, potentials proportional to a, a and aj are also fed to the inputs of blocks 17 of the inverse Kirchhoff transform and adders 8, 10 and 11. Adders 10 and 11 serve to determine the values of the desired function D (T) in the extreme points of the temperature range acceptable for this task are T ;, „and T„ o. For this, the input resistors of the adders 10 and 11 are adjusted so that the transfer coefficients for the equal inputs are 1. P “n and 1. T„ „, s, T2,„ .. respectively. Then the adders 10 and 11, obtaining potentials proportional to a, a, and a to the inputs, calculate the values of l () and L (T = ax) in accordance with (2). If; () and) are positive, there is reason to believe that the condition of the physical realizability of the desired function A (T) is fulfilled. If the output voltage of the adder 10 or 11 decreases to zero, which

indicates that the device is in an unacceptable mode of operation, the potential of the outputs of one of the threshold blocks 20 abruptly changes, through the OR 18 element, to the adder 9 and through the OR 19 element to the cyclic shift register. Thereby, the minimized function is increased, i.e. it is fined, and, moreover, another integrator 16 is included in the integration mode. This leads to the fact that the parameter a., when varying which a violation of the positiveness of L (T) occurred, takes on a physically justified value and instead begins to tune another polynomial coefficient.

Claims (1)

The solution is found when function (4) reaches the value E, for each of the parameters a, and aj. The voltages of the outputs of the adder 7 and the integrator 16 correspond to the coefficients of the dependence L (T) and are the output signals of the device. Invention Formula A device for solving the inverse heat conduction problem, containing a passive model, comparison units, the first input of the first comparison unit is connected to the first output of the voltage divider unit, the second output of which is connected to the first input of the first adder, the second adder, the output of which is connected to the boundary node of the passive model, the outputs of the second, third and fourth comparison units are connected to the inputs of the respective allocation modules of the module, the outputs of which are connected to the first three inputs of the third adder, the output of which connected to the second input of the first comparison unit, the output of which is connected to the information inputs of the key elements and the input of the pulse former, the bit inputs of the cyclic shift register are connected to the control inputs of the key elements, the outputs of which are respectively connected to the inputs of the first and second integrators, the outputs which are the first and second outputs of the device, the third output of which is the output of the first adder, the outputs of the first and second integrators are connected respectively to the first and second th inputs of the second, fourth and fifth adders, as well as the second and third inputs of the first sum 0 5 five 0 five 0 five 0 torus, the output of which is connected to the third inputs of the second, fourth and fifth adders, characterized in that, in order to improve the accuracy of the solution, threshold blocks, OR elements and three Kirchhoff inverse transformation units are inserted, each Kirchhoff inverse transformation unit includes a comparison unit , integrator, adders and multiplication units; in each Kirchhoff’s inverse transform unit, the integrator output is connected to the first inputs of the first, second and third multiplication units; the output of the first multiplication unit is The key is connected to the first input of the first adder, the output of which is connected to the second input of the second multiplication unit, the output of which is connected to the first input of the second adder, the output of which is connected to the second input of the third multiplication unit, the output of which is connected to the first input of the comparator, the output of which is connected to the input the integrator of the Kirchhoff inverse transform unit; the outputs of the fourth and fifth adders, respectively, are connected to the inputs of the first and second threshold units, the outputs of which are respectively connected to the inputs the first OR element, the output of which is connected to the fourth input of the third adder and the first input of the second OR element, whose output is connected to the information input of the cyclic shift register, the output of the pulse former is connected to the second input of the second OR element, the outputs of the first adder, first and second integrators are respectively connected with the second input of the second adder, the second input of the first adder and the second input of the first multiplication unit, respectively, the Kirchhoff inverse transform units, The passive model nodes are respectively connected to the first input of the comparison block of the corresponding Kirchhoff inverse transformation blocks, the third, fourth and fifth outputs of the voltage divider block are connected respectively to the first inputs of the second, third and fourth comparison blocks, the second inputs of which are respectively connected to the outputs of the integrators corresponding Kirchhoff reverse transform units. FI.1 21 years FIG. 2.