SU1268554A1 - Device for solving heat conductivity inversion problems - Google Patents

Abstract

This invention relates to analog computing. The purpose of the invention is to improve the accuracy and the expansion of the class of tasks. The device contains a constant voltage source, a voltage divider, subtractors, integrators, an RC grid, a controlled resistor, multipliers, and zero-organs. The device allows solving unsteady heat conduction problems with a nonmonotonic change of the initial temperature data. 1 il. c: t (L s

Description

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| The invention relates to analog computing and can be used in solving inverse problems of unsteady heat conduction. The purpose of the invention is to improve the accuracy and the expansion of the class of automatically solved problems. The drawing shows the proposed my device. The device contains an RC grid. 1, consisting of capacitors 2 and controlled resistors 3, voltage divider 4, Goodman function generation units 5, subtractors 6, integrators 7, multipliers 8, zero organs 9, source 10 constant voltage wives The unsteady heat equation is:,, zty1, .aCs) (, iTi-g: PT (C) temperature, X, t are the spatial and temporal coordinates. Using the finite-difference approximation of the spatial coordinate, we replace equation (1) with the corresponding system grid equations, each of which for internal to sec sec body has the form (.) jLL (i) Il (i). - "(T;) x ,, Iii ± iiTii., (n ЗТ, Г) СЛТ;), g where 1 - discrete coordinate of space; h is a step in space. The first Kirchhoff law for a T-shaped RC chain is written as follows: (t) -U; . (t) .4, (t) 4jj (ij zi; (t) (3) where g, C is electrical conduction and capacitance; and is electric potential. Comparing formulas (2) and (3), we note that RC - the grid will be an electric model - an analogue of the thermal process subject to the following ratios between thermal and electric quantities: 4 where gpr is the scale factor.This approach to the construction of an electrical model implies the general principle of solving inverse problems of unsteady thermal conductivity with the help of an RC grid model consisting in setting the parameters of the node elements so that the potentials at the nodes of the model, taking into account the scale, coincide with the known, experimentally removed (reference) temperature values at the corresponding points. To solve this problem, the SI use the Goodman substitution of JH (T) –dT. ) to equation (1) reduces it to a form that allows it to be simulated on an RC grid with constant nodal capacitances. The solution process proceeds as follows, the potential of the node cf (t) is compared in the subtractor 6 with the reference voltage Ug ( t) proportional to body temperature according to boiling point. The received error signal (differential signal) is fed to the input of the forma- l control action (integrator). The generated control action changes the parameters of the node elements (conductivity or capacity) by a certain amount, which leads to a change in the nature of the flow of processes in the node cell of the grid model. To ensure the operability of the device, the control action should be automatically formed so that the variation of the parameters of the node elements leads. to reduce the error signal. At the same time, a change in the parameters of the node elements has a significant effect only on the distribution of currents in the branches x, converging in the node. The nodal potential is associated with the current flowing through the capacitor ij, (t), by the following relationship: (t) -f-i, (t) .dt. Thus, by changing the magnitude of the current ij, (t), it is possible to influence the rate of increase or decrease of the nodal potential and, thus, control its value in order to reduce the error signal. However, changing the parameters of node elements affects the error signal ambiguously. Suppose U (t) is greater than c | (t) and in the subtractor 6 the error signal is formed with the following: (t) U, (t) - Cf (t). (8) Then the signal (t) ,. entering the input of the integrator causes an increase in its output voltage, which leads to an increase in the value of the conductivity of the nodal resistors (decrease in resistance). If the ratio between the input voltages U, (t), and the node potential cp (t) is such that the capacitor is in a state of charge, an increase in the conductivity of the node resistors will lead to an increase in current through the capacitor, increasing the rate of increase of the node voltage. potential and, therefore, will reduce the error signal. In the case of a capacitor discharge, under the same conditions Ug (t) is greater than tfCt) and B (t) forming with the law (8), a decrease in the conductors of the gages will lead to an increase in the rate of discharge yes capacitor that increases the error ku The device is inoperable. Thus, if in the subtractor 6 the error signal (t) is generated according to the law (8), then the device is operational only for such initial data, when the capacitor is in the state of charge, i.e. automatic adjustment of the parameters of the node elements leads to a decrease in the error signal. Having taken the law of formation 5 (t) in the form of (t) (p (t) - U5 (t)

It is possible to ensure the operability of devices only in the state of capacitor discharge. Therefore, the law of formation (t) according to the law (8) or (9) imposes strict restrictions on the initial data and does not allow S.-i.ffc,

Thus, the increment of the capacitance value also has an ambiguous effect on the change in the magnitude of the error Xt) 544, which completely automates the solution process. The initial data must be such that in the process of solving the node capacitors are only in a state of charge or only a discharge, and this must be known in advance before the start of the solution and manually adjust each channel of the device parameters adjustment in accordance with the direction of current flow through the capacitor. the decision process. The dynamics of the RC grid node shown in the drawing, in accordance with the first Kirchhoff law, is described by an equation of the form Ч (0 ((t) - cp (t) gJ.) - (t) Ji (10) cdq (t) where i (t ) U (t), U2 (t) are the input voltages (potentials of neighboring nodes), g is the conductivity of the node resistors. Sensitivity to a change in the parameter of the node element k is described by the relation &; (11) From relation (11) , the conductivity increment S g will cause the current i to change by 8; (and + U ,, - 2tp); i., yo go °. Therefore, the sign and the magnitude of the increment ui is determined not only, but also by the magnitude and The direction of current flow ig .. Denoting node slew rate capacity dcp (t) / dt through tx:, determining the effect of its change in increments:

The known devices are an interconnected, multichannel system for automatically controlling the distribution of the nodal potentials of an RC grid. When controlling an inertial object, such as an RC grid, an overshoot and oscillation of the controlled variable can be observed. This means that in the process of solving a change may occur in the direction of current flow through the capacitors, due not only to the nature of the change) but due to the peculiarities of the flow of inertial object control processes. All this further narrows the range of applicability of the device for solving inverse problems of unsteady thermal conductivity, realizing their well-known method of automation of a solution,

In the proposed device, to convert the equation (1), the Goodman substitution (5) is used, the application of which allows to convert the equation (1) to the form (b), which allows to simulate the right-hand side of (6) using a constant capacity; and the left one with controlled resistive elements,

In the proposed device, signals proportional to changes in temperature at some points of the simulated body are obtained at the outputs of voltage dividers 4, which through blocks 5 arrive at the first inputs of subtractors 6 to the second inputs of which voltage is supplied from the corresponding node points of the grid model. In blocks 5, the conversion is carried out according to the formula (5). From the output of each subtractor 6, the error signal arrives at the second input of the multiplier 8 for the first time, the inputs of which post the signals from the outputs of the corresponding null distributors. The sign of the output voltage of the Zero Organ is determined by the in-flow of current through the nodal capacitor (the positive current is positive, the discharge is negative).

The null organ 9, in particular, can be performed on a negative feedback differential amplifier, the inputs of which are included in the interruption of the charge circuit of a capacitor so that a non-inverting input is connected

to the potential-zero bus, and inverting to the corresponding capacitor lining. This inclusion of a differential amplifier, without affecting the capacitor charging process, allows for the sign of the output voltage to determine the direction of the current in the capacitor.

Thus, the voltage oii (t) coming to the input of the integrator 7, is formed as

oi (t) U5 (t) -q) (t) signi (, (t). The signal from the output of the integrator 7 is fed to the control input of the controlled resistor. 3. Adjustment of the conductivities of the controlled resistors occurs as long as the signals the errors will not be equal to zero, i.e., until the voltage at the nodes corresponds to the temperatures at the corresponding points of the body under study. The conductivity of the controlled resistor measured during the adjustment process makes it possible to judge the dependence of the coefficients of equation (6) on the temperature;

Claims (1)

Invention Formula A device for solving inverse heat conduction problems containing a KS grid, a group of integrators, a constant voltage source, the output of which is connected to the input of a voltage divider, a group of outputs of which is connected to the first inputs of the subtractors through the corresponding units of forming the Goodman function. groups; the second inputs of which are: are connected to the corresponding central nodes of the KS grid, which means that, in order to increase the accuracy and expansion of the class of tasks to be solved, the group of multipliers and the zero group organs whose outputs are connected to the first inputs of the corresponding multipliers of the group, the outputs of which through the corresponding integrators of the group are connected to the corresponding inputs of the assignment of the conductivity of the CS grid, the groups of boundary nodes of which are connected to the inputs of the corresponding zero-organs of the group teley group are connected to second inputs of corresponding multipliers group.