Solving Partial Differential Equations on Active Grid Models

I. G. Vintizenko, O. V. Smirensky

(Presented by the scientific seminar of the computing laboratory)

Second-Order Equation

\[ a\nabla^{2}\varphi-b\varphi+f(\varphi)=c\frac{\partial\varphi}{\partial t} \tag{1} \]

describes the behavior of the function \(\varphi(\vec r,t)\) in a system with distributed sources and sinks. The function \(f(\varphi)\) is positive definite; \(a, b, c\) are positive coefficients, which may be functions of the coordinates. Such equations occur in problems connected with heat conduction, diffusion, chemical processes, etc. When the domain in which the solution is sought has a complicated shape, the equation can be solved only by numerical methods.

The solution of equation (1) is easily obtained on an analog computer by approximating it with a system of finite-difference equations (with respect to the coordinates). But if the solution domain is sufficiently large and the finite-difference grid sufficiently fine, then modeling requires too large a number of operational amplifiers.

An equation of the indicated type can be solved on a model that is a combination of an electrical grid with analog functional elements. Indications of such a possibility are found in the literature [1]. The present paper describes the principle of constructing an active grid model for solving nonstationary second-order partial differential equations.

The active grid model is an electrical network consisting of resistors, capacitors, and nonlinear negative resistances [2]. The circuit of one cell of the model for solving a two-dimensional problem is shown in Fig. 1. A grid of resistors \(R_1\) implements the Laplace operator \(\nabla^{2}\varphi\); the resistors \(R_3\) are associated with the magnitude of the sinks \(b\varphi\); the value of the capacitance \(C_3\) is determined by the right-hand side of equation (1). The distributed sources \(f(\varphi)\) are modeled by nonlinear resistances, which may include active analog elements of the type of operational amplifiers, multipliers, and function devices. Negative resistances are connected to the grid nodes corresponding only to those points of the modeled system at which sources are present.

Let us write the original equation (1) in finite-difference form:

\[ \frac{\varphi_{i-1,k}+\varphi_{i+1,k}+\varphi_{i,k-1}+\varphi_{i,k+1}-4\varphi_{i,k}}{\dfrac{1}{a}} -bh^{2}\varphi_{i,k}+h^{2}f(\varphi_{i,k})-ch^{2}s\,\varphi_{i,k}=0, \tag{2} \]

where \(h\) is the spacing of the spatial grid, identical in both coordinates \(x\) and \(y\).

The current-balance equation for node \((i,k)\) of the model has the following form:

Fig. 1. Cell of an active grid model

\[ i_{1}^{\mathrm{I}}+i_{1}^{\mathrm{II}}+i_{1}^{\mathrm{III}}+i_{1}^{\mathrm{IV}}-i_{3}+i_{2}=0, \tag{3} \]

where the current \(i_{4}\) is neglected.

If we take into account that the potential at the output of the nonlinear resistance (the functional converter and operational amplifier) is determined by the expression

\[ V_{1}=m_{f}\frac{R_{5}}{R_{4}}f(V_{i,k}), \]

where \(m_{f}\) is the conversion scale in the functional converter, then equation (3) can be written as follows:

\[ \frac{V_{i-1,k}+V_{i+1,k}+V_{i,k-1}+V_{i,k+1}-4V_{i,k}}{R_{1}} -\frac{V_{i,k}}{Z_{3}} +\frac{1}{R_{2}}\left[m_{f}\frac{R_{5}}{R_{4}}f(V_{i,k})-V_{i,k}\right]=0. \tag{4} \]

Dividing all terms of equation (2) by the base value \(a^{*}\) and multiplying all terms of equation (3) by the base value \(R^{*}\), we put the coefficients in the equations into dimensionless form; here the quantities \(a^{*}\) and \(R^{*}\) are chosen arbitrarily. In addition, we introduce the modeling scale \(m\) of the desired function, i.e., we set

\[ V=m\varphi \tag{5} \]

and substitute expression (5) into equation (4). After these transformations, equations (2) and (4) take the following form:

\[ \frac{\varphi_{i-1,k}+\varphi_{i+1,k}+\varphi_{i,k-1}+\varphi_{i,k+1}-4\varphi_{i,k}}{\dfrac{a^{*}}{a}} -\frac{bh^{2}}{a^{*}}\varphi_{i,k} +\frac{h^{2}}{a^{*}}f(\varphi_{i,k}) -\frac{ch^{2}s}{a^{*}}\varphi_{i,k}=0, \tag{6} \]

\[ \frac{\varphi_{i-1,k}+\varphi_{i+1,k}+\varphi_{i,k-1}+\varphi_{i,k+1}-4\varphi_{i,k}} {\dfrac{R_1}{mR^*}} -\frac{mR^*}{Z_3}\varphi_{i,k} +\frac{R^*}{R_2}\left[m_f\frac{R_5}{R_4}f(V_{i,k})-V_{i,k}\right]=0. \tag{7} \]

Comparing equations (6) and (7), we obtain the modeling conditions, i.e., the conditions under which the distribution of potentials at the nodal points of the grid will correspond to the space-time distribution of the function \(\varphi\) in the system being modeled.

\[ R_1=\frac{ma^*}{a}R^*, \tag{8} \]

\[ \frac{h^2 f(\varphi_{i,k})}{a^*} = \frac{R^*}{R_2}\left[m_f\frac{R_5}{R_4}f(V_{i,k})-V_{i,k}\right], \tag{9} \]

\[ \frac{mR^*}{Z_3}=\frac{bh^2}{a^*}+\frac{ch^2s}{a^*}. \tag{10} \]

The complex \(Z_3\) is a parallel connection of the resistor \(R_3\) and the capacitor \(C_3\). From condition (10) we obtain

\[ R_3=\frac{ma^*}{bh^2}R^*, \tag{11} \]

\[ C_3=\frac{ch^2m_\tau}{ma^*R^*}, \tag{12} \]

where \(m_\tau=\dfrac{s}{p}\) is the time scale of modeling,

\(s\) and \(p\) are the Laplace-transform operators in the physical system and in the model. Finally, condition (9) can be written in a more convenient form:

\[ \frac{R_5}{R_4}f(V_{i,k}) = \frac{h^2}{m_fa^*}f(\varphi_{i,k})+\frac{1}{m_f}V_{i,k}. \tag{13} \]

When solving problems by means of the model considered, a significantly smaller number of operational amplifiers is required than when solving them on analog computers. Thus, the use of active grid models for solving problems of mathematical physics makes it possible to construct specialized models of systems with distributed parameters more simply and economically.

REFERENCES

1. I. M. Tetelbaum. Electrical Modeling. GIFML, Moscow, 1959.

2. I. G. Vintizenko. Construction of negative resistances based on an operational differential amplifier and their application to solving problems of field theory. Second All-Union Conference on Analog Means and Methods for Solving Boundary-Value Problems. Kiev, 1965.