THE APPLICATION OF FINITE FOURIER TRANSFORMS TO ANALOG COMPUTER SIMULATIONS. Eric Liban. Grumman Aircraft Engineering Corp. Bethpage, New York

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1 55 THE APPLICATION OF FINITE FOURIER TRANSFORMS TO ANALOG COMPUTER SIMULATIONS Eri Liban Grumman Airraft Engineering Corp. Bethpage, New York SUIIl!Ilary An Analog Computer tehnique for the solution of ertain lasses of boundary value problems of partial differential equation based on Finite Fourier Transforms is presented, whih requires onsiderably less omputer omponents than onventional finite differene methods. The derivation of the Finite Fourier Transform method is briefly stated and then applied to analog omputer simulations of heat transfer equations with linear and nonlinear boundary onditions. Introdution A well known method for the solution of ordinary linear differential equations with onstant oeffiients is the Laplae transform method whih redues the differential equation to an algebrai equation. The solution of the latter is a funtion of a parameter "s" and given initial onditions, and its inverse Laplae transform is the solution of the differential equation. This method is easily extended to partial differential equations in two independent variables. A Laplae transform with respet to one of the independent variables gives an ordinary differential equation, whose solution is the Laplae transform of the funtion satisfying the partial differential equation and its initial and boundary onditions. The inappliability of this method for analog omputations arises from the fat that the inverse Laplae transform, a neessary step to extrat the solution, is an integration over an infinite path in the omplex domain. To solve partial differential equationson an Analog Computer by a transform method the inverse transform must be obtainable by operations in the real domain only. The partiular transform disussed in this paper is the Finite Fourier Transform, whih is appliable to equations in whih only the even order derivatives (of the funtion) with respet to the transformed variable appear. The heat equation, wave equation and bending beam equation are of suh nature. Many other transforms exist whih may be used for other types of equations. (Refs. 1,, and 3). A feature, whih makes the Finite Transform a very eonomial method for analog omputers, is that the inverse transform may be solved only for regions of interest. The trunation error in the analog simulation may be made smaller than the error in the amplifiers themselves, and therefore the solution obtained by the Finite Fourier Transform method is the exat solution within the auray of the Analog Computer; and the bounds on the deviation from the true solution an be stated preisely in terms of the usually small amplifier noise and integrator drifts. Finite Fourier Transforms The transform exists for all bounded, pieewise ontinuous funtions over a finite interval. The extension of a ontinuous funtion F(x) defined for o < x <~, into an odd periodi funtion F(x) ;ith period ~ may be expressed by the Fourier series ""' F(x) o f (n)sin nx. s The oeffiients are given by I ~ f (n) s F(x) sin nx dx, o,,.... (1) () From the olletion of the Computer History Museum (

2 56 ANALOG APPLICATIONS AND TECHNIQUES This set of oeffiients, defined by Eq. (), is the Finite Fourier Sine Transform of t{x), and the inverse Sine Transform is F{~) as defined by Eq. (l). Note ~hat F(x) = F(x) for 0 < x <~. But F(O) and F(~) will always equal zero, whatever F{O) and F{~) may be. This follows from the extension of F(x) into an odd funtion and the property of Fourier series to onverge at finite disontinuities to the average value of the limits the funtion approahes from the right and left respetively. The following expression is obtained by two suessive integrations by parts and a substitution using Eq. (): r - n f (n) s o (3) + n[f{o) - (-l)~{~)]. E~panding F{x) tion F{x) of period Fourier series into an even fun- ~ leads to the - n f (n) - F' (0) + (-l) ~' (71") (6) by integrating the left side by parts twie and substituting Eq. (5), F'{O) and F' (71") are the values of the derivative of F(x) at x = 0 and 71", respetively. Two other useful Finite Fourier Transforms are stated below: Finite A-Transform n-l F{x) sin -- x dx,,,.... Inverse A-Transform ()() (7) F(x) 1. f (O) + ~ 71" C 71" f (n)os nx (4) where the set of oeffiients f(n) is the Finite Fourier Cosine Transform (FPC~ given by f (n) 71" J F(x)os nx dx, o n=o, 1,,.... (5) The inverse FFCT is the funtion F(x) as given by Eq. (4). Here F{x) = F{x) fqr the losed interval 0 < x < 71"; while df/dx = df/dx holds generally only fqr o < x < 71", sine the derivative of F may be disontinuous at x = n71". The following relations are obtained for the FFCT: and F{x) S1.n n-l fa (n) sin -- x,~_~ n-l + (n -l)f(0) - (-l)~' (71"). Finite ~-Transform o...,.,_, \ ).LA \11) 71" n-l f (n) F(x)os -- x dx, ~ J,,.... (8) (9) (10) From the olletion of the Computer History Museum (

3 THE APPLICATION OF FINITE FOURIER TRANSFORMS TO ANALOG COMPUTER SIMULATIONS 57 and Inverse I-L-Transform L '=I n-1 F(x) 7T" fl-l (n) os -- x (11} r7t"..lm 1'\ 1 dx (n-1) fl-l(n) dx os -- x J o U l' Ln-.l - F' (0) + (-1)~(7T") (1) Only the relations expressing the transforms of the seond derivative of a funtion, in terms of the transform of the funtion and its boundary values, were shown. However, relations may be obtained for all even order derivatives. For an elementary disussion of the Finite Fourier Transform see Churhill, Modern Operational Mathematis in Engineering.~ For a more advaned treatise on the Finite Fourier Transform and other Integral Transforms see bibliography. Analog Simulations of the Heat Equation The Re-entry Problem The Finite Fourier Transform method is applied first to the one-dimensional heat equation with boundary onditions enountered in the "re-entry problem". Let ~ be the spae oordinate of a onedimensional slab of length L, whih is the idealization of a longitudinal setion of the nose one of a missile, T the time and T(T,~) the temperature. At ~ = 0 a heat input or heat flux is given, while at ~ = L the slab is assumed to be insulated, that is, the temperature gradient is zero at ~ = L. In this problem the main interest is in the temperature of the fae ~ = 0, to determine whether it rises above or remains below the melting point. Also of onern is the temperature at ~ = L, whih must remain below the melting temperature of the bak-up struture. The mathematial statement of this problem for homogeneous slabs with temperature independent thermal onstants and uniform initial temperature TO dt d T is given by dt = a d~ ' o ~ ~ ~L T > 0 (13) Initial Condition: Boundary Conditions: T(O, ~) TO (14) dt k d~ = - q(t) at ~ 0 dt d~ = 0 at L. The symbols used in the above equations are defined in the following table: T _ temperature - or T == time - se ~ _ spae oordinate - in a == k == thermal diffusivity - in/se (15) thermal ondutivity - BTU/se-in-oR q == heat input rate per unit area - BTU/se-in The linear heat equation may be solved for the temperature deviation from the initial temperature, expressed by U(T, ~) = T(T, ~) - T 0 (16) Equation (16) and the substitutions x = E ~ t 7T" L, = a T, L transform Eqs. (13), (14), and (15) to the equation du d U (17) dt = ' o < x < 7T", t > 0 (18) dx From the olletion of the Computer History Museum (

4 58 ANALOG APPLICATIONS AND TECHNIQUES Initial Condition: Boundary Conditions: u(o, x) o. (19) eu L ex = - 11"k q(t) - - Q(t) at x = 0 o at x = 11" (0) Equation (18) and boundary onditions as shown in Eq. (0) indiate the use of the Finite Fourier Cosine Transform with respet to x. The FFCT of u(t, x) is given by f (t, n) I~ u(t, x) os nx dx, o (1) n=o, 1, under disussion, equals zero. If the problem were solved for the temperature T with initial onditions T(O, x) = TO, then the initial onditions for Eqs. (3) would still all be zero with the exeption of f(t, 0), sine the integral of a onstant times os nx between 0 and 11" always vanishes when n j O. When the initial temperature distribution is a funtion of x, then the initial onditions for f(t, n) may be either readily omputable by hand or, easily evaluated on the analog omputer. The set of solutions f(t, n) of Eqs. (3) form the FFCT of u(t, x); i.e., they are the Fourier oeffiients as a funtion of time t of the temperature u(t, x). The inverse FFCT, as given by Eq. (4), for x = 0 and x = 11" respetively, beomes: u (t, 0) 1 f (t, 0) + 1 \ f (t, n) (5) 11" 11" ~ C 00 It is easily seen that eu et os nx dx n=o, 1,.... f (t, n), () u(t, 11") 1 f (t 11" C ' +1 0) 00 (_l)n f (t, n) L 11" (6) By multiplying both sides of Eq. (18) with os nx dx and integrating from 0 to 11" using the notation defined hy Eqs. (1) and (), the differential equations for the terms of the FFCT are formed. Equation (6) with the given Boundary Conditions as shown in Eq. (0) gives: f (t, n) n f (t, n) + Q(t) n=o, 1,, with initial onditions: f (0, n) = 0, (3) n=o, 1,,... (4) The initial ondition of f(t, n) equals the FFCT of the initial ondition of u(t, x), whih for the ase Clearly, it is not possible to solve Eqs. (3) for all n from 0 to 00. However, the inverse Ftnite Fouri.er Cosine Transform is a fairly rapidly onvergent series, beause it is a ontinuous funtion for all x. The oeffiients behave like 1/n for inreasing n. When Q(t) > 0 for all t, it an easily be seen from Eqs. (3) that f(t, n) ~ 0 for all nand t. Hene,there exists an N for any E, no matter how small, suh that maxiu(t, 0) - 1 f 11" (t, 0) (7) N _ 1 f (t, n) 1 < E 11" C L From the olletion of the Computer History Museum (

5 THE APPLICATION OF FINITE FOURIER TRANSFORMS TO ANALOG COMPUTER SIMULATIONS 59 That is, there exists an N the trunation error 00 EO max L 'IT n=n+l suh that f (t, n) (8) will be smaller than the auray of the omputer or some other desired auray ompatible with analog omputer harateristis. The maximum absolute errors in the omputer solutions of u(t, x) 1 f (t, 0) 'IT L N + 'IT C f (t, n)os nx (9) for x ~ 0, will all be smaller than EO' sine for positive f(t, n) max 'IT 00 n=n+l < max l. 'IT f (t, n) os nx 00 n=n+l f (t, n). (30) The number of Fourier oeffiients to be generated on the analog omputer may therefore be obtained by an a priori estimate or by inreasing N until f(t, N) ~ 0 for all t. Figure 1 shows the analog omputer setup for the Finite Fourier method solution of Eqs. (18), (19), and (0) with N = 7. For test purposes the heat input Q(t) was a steep saw tooth and it was found that the temperature at x = 0 for all time t differed always by less than 1% from the analytially obtained exat solution. The slab assumes a uniform steady state temperature when the heat input is of finite duration. The Fourier Transform method of simulation will always give the orret steady state temperature regardless of the number of Fourier oeffiients used. This an easily be seen from the fat that the output voltages of all the integrators will eventually be zero, when Q(t) = 0, exept the output of the integrator generating f(t, 0). When the slab is at uniform onstant temperature, then the Fourier series of the temperature distribution is simply the onstant (l/'it)f(t, 0). Convetive Boundary Condition The boundary with onvetive heat transfer may be either ~ = 0 or ~ = L, or both. In addition a heat input q(t) may be given. In this setion the analog simulation of the heat equation is shown for the following boundary onditions: k dt d~ - q(t) at ~ 0 (31) k dt h(t(t, L) - T.) at ~ L, (3) d~ 1 where h - onvetive heat transfer oeffiient - BTU/inse of. The boundary onditions, stated in words, mean that the bar is heated at the rate q(t) at the fae ~ = 0 while the fae ~ = L is in ontat with a substane at temperature Ti' whih may be a funtion of time, and heat is transferred from the bar to this substane aording to Eq. (3). Let the bar be initially at a uniform onstant temperature T(O, ~) = TO. With the substitutions Eqs. (16) and (17), ui = Ti - TO, and (L/'lTk)q(t) = Q(t) the problem is stated: Initial Condition: Boundary Conditions: t > 0. (33) u(o, x) o. (34) - Q(t) at x = 0 (35) From the olletion of the Computer History Museum (

6 60 ANALOG APPLICATIONS AND TECHNIQUES du L [ ] -- = - h u(t ~) - u. dx ~, 1 at x = ~. (36) Sine the boundary onditions presribe the derivatives at x ~ 0 and x ~ ~, the use of the Finite Fourier Cosine Transform is indiated; and from Eq. (6) and the above boundary onditions is obtained the set of differential equations f (t, n) = - n f (t, n) + Q(t) The Flat Plate Radiating into Vauum The term IIf1at p1ate" is taken here to mean a solid slab bounded by a pair of parallel planes ; = - Land ; = L. The initial temperature TO is uniform and Stefan-Boltzmann radiation into vauum takes plae at the two faes ~ = ± L. Clearly, the temperature distribution will be symmetrial about ~ = 0 and therefore it is more onvenient to state the problem as follows: + (-1) n b h f u (t. ~) - u. 1 ~ L' 1J (37) dt d T h = a d~ 0 < ~ L T ~ > 0 (39) n=o, 1,,.... The essential differene between Eqs. (37) and the equations for the Fourier Transform of the heat equation with insulated boundary Eqs. (3) is that now the boundary ondition is a funtion of the temperature at x =~. But u(t,~) is given by Eq. (6), as in the previous example and may be generated ontinuously. Hene the last term of Eq. (37) is easily formed and fed bak to the appropriate integrators in the simulation. The analog iruit for the heat equation with a heat input on one fae and a onvetive heat transfer at the other fae is shown in Fig.. The FFCT was solved for n = 0 to n = 7. Radiation Heat Transfer 5 Examples hosen for this type of boundary onditions are a flat plate and a sphere radiating in a vauum. The boundary ondition for radiation heat transfer at the fae ~ = L is given by T_.!.....: _1 Condition:.1.U.1.L..1.Cl.1. Boundary Conditions: T(O, ~) TO (40) dt d~ 0 at ~ 0 (41) - oe[t(t, L)]4 at ~ L (4) Equation (41) follows from the symmetry ondition, and Eq. (4) is obtained from Eq. (38) by letting Ta = 0 for radiation into vauum. Sine Eq. (4) is nonlinear it an no longer be solved for the temperature deviation from the initial temperature TO' It is probably just as simple to simulate the original equations, and ertainly the results would be more easily interpreted; however, for simpliity of presentation let where o _ Stefan-Bo1tzmann's onstant - BTU/se in R4 E _ emissivity - dimensionless (38) and set x = 1L ~ and t = a L T L L g 3 b EO U o 'Tf 1< (43) (44) Ta - temperature to whih surfae radiates - or. and u (45) From the olletion of the Computer History Museum (

7 THE APPLICATION OF FINITE FOURIER TRANSFORMS TO ANALOG COMPUTER SIMULATIONS 61 The problem may now be stated here Radiating Into Vauum Initial Condition: t > 0 (46) The heat equation for a sphere with temperature distribution a funtion of time T and distane p from the enter of the sphere, expressed in polar oordinates is given by Boundary Conditions: u(o, x) 1 (47) au 0 at 0 ax x = (48) au 4 ax - gu (t, 7r) at x = 7r (49) The differential equations for the FFCT are derived in the same way as in the previous examples and are given by The initial the FFCT's u(o, x) = 1 equals zero term in Eq. 1y from u(t, 7r) - n f (t, n) n 4( - (-1) gu t, If) n=o, 1,,.... (50) onditions of f(t, n) are of u(o, x), whih for equals 7r when n = 0, and for all n ~ O. The last (50) is generated ontinuous f (t 7r ' 7r 0) N (-1) n f (t, n) I (51) raised to the 4th power, whih in the simulation indiated in Fig. 3 was performed by two suessive servo-multipliations. The simulation proved to be stable and in agreement with the solution obtained by Abarbane1. 5 If boundary Eq. (4) were replaed by Eq. (38), the simulation would differ only slightly. at a a ( at) JT = 0 op p ap By the hange of variables,- (5) r = 7r 7r R t..i.. p = a T, u = r (53) R TO where R is the radius of the sphere, Eq. (5) beomes for 0 < r < 7r and t > O. Initial Condition from Eq. (53): (54) u(o, r) = r (55) The radiating boundary ondition as given in Eq. (38) beomes au 4 or = u(t, 7r) - gu (t, If) at r = 7r (56) where Boundary Condition: g (57) u(t, 0) = 0 at r = 0 (58) Equation (58) insures that the temperature T(T, 0) remains finite. In this ase the temperature is presribed on the boundary r = 0 while at the boundary r = 7r the temperature gradient au/ar is a given funtion of the temperature. The Finite A-Transform, as given by Eq. (7), is to be used for these boundary From the olletion of the Computer History Museum (

8 6 ANALOG APPLICATIONS AND TECHNIQUES onditions. From Eqs. (9), (56), and (58) one obtains the following set of differential equations Sine initially u(t, If) = If from Eq. (53), one an perform a steady state hek on the omputer setup and on the onvergene of the trunated series; namely, from Eqs. (61) and (64) - (-1) n [ u ( t, If) - gu 4 ( t, If)] (59) n 1,,.... N 1 I 1 If (n - ~) ~ Ji (65) From Eq. (8) u(t, r) and ()() ()() f,(t, n)sin(n - ~)r 1\ (60) u(t, If) ~\ "L -( -1) n fa (t, n). (61) Equation (61) is used in the simulation to form the last term in Eq. (59). From Eq. (53) the temperature at p = R is given by T(t, R) TO -:;- u(t, If). (6) The ini.tial onditions f~\(o, n) are obtained from Eqs. (55) and (7) by On the analog omputer (as well as numerially) % auray T~y be ahieved with eight oeffiients. The analog simulation is shown in Fig. 4. Conluding Remarks Problems with onvetive as well as radiating boundary onditions presribed may be solved by this method by the obvious extension of ombining the onvetive term h(u - ui) and the radiation term g (u4 - ua4) in the differential equations generating the Finite Fourier Transforms. The temperature at points other than the boundary, if they should be needed, may be readily obtained by summing the Fourier oeffiients weighted by their proper trigonometri terms. This simply requires a potentiometer for eah term and an inverter when neessary. For onvetion and for radiation the temperature at the boundary should be as aurate as possible, beause the boundary onditions themselves are given by them. The Finite Fourier Transform method whih gives the exat boundary temperature ~AJithin the omputer auray"" an be very suessfully applied. lf f o r sin(n - ~)r dr,,,.., N whih upon integration beomes 1 -( ) (_l)n n - ",,..., N (63) (64) The most widely used methods for the solution of partial differential equations are based on finite differenes. These methods require ertain assumptions about where the finite differene equals the derivative whih by neessity have to be most loosely made on the boundaries; the very points where the greatest auray is required. One must form many grid points, at least, near the boundary in order for the solution of the differene equation to ome lose to the solution of the differential equations. For substanes with low ondutivity, as used for nose ones in missiles, the spaing of the grid points beome even more From the olletion of the Computer History Museum (

9 THE APPLICATION OF FINITE FOURIER TRANSFORMS TO ANALOG COMPUTER SIMULATIONS 63 ritial. The Finite Fourier Transform method may also be used for equations with more than one spae variable. Stephens and Karplus6 have employed this transform for analog simulations of two-dimensional diffusion equations. Two approahes are given in their paper. In the first only one spae variable is transformed, whih redues the given problem to a partial differential equation with one independent spae variable. This they solve then by a finite differene method. In the seond approah the dependent funtion is transformed with respet to both spae variables, thus produing a system of ordinary differential equations. In priniple, this method may be extended to analog simulations of heat equations in three spae variables. The number of ordinary differential equations, however, beomes quite large when Fourier transformations are performed with respet to three variables. Their analog simulation would be pratial only if the analog omputer is apable of time sharing or repetitive modes of operation, and possesses dynami storage elements. The Finite Fourier Transform method may also be a very effiient tehnique for the solution of multidimensional heat equations on a linked Analog-Digital omputing system Churhill, IlModern Operational Mathematis in Engineering,1I MGraw-Hill, New York, Abarbanel, P.S., "Time Dependent Temperature Distribution in Radiating Solids,1I Journal of Mathematis and Physis, Vol. 30, Stephens, P.A., Jr. and Karplus W.J., IIAppliation of Finite Integral Transforms to Analog Simulations,1I AlEE Transations, Vol. 78, Part I, Burns, A.J. and Kopp, R.E., IICombined Analog-Digital Simulation,1I Proeedings of the Eastern Joint Computer Conferene, Deember The number of amplifiers required for the solution of the heat equation by the Finite Fourier Transform method is about one-half to one-fifth of the number of amplifiers needed for a finite differene simulation of equal auray. The ratio depends on the shape of the heat input urve, the thermal onstants, and the physial dimensions. Referenes Sneddon. I.N. "Fourier Transforms. II MGraw-Hill, New York, ' Tranter, C.J., "Integral Transforms in Mathematial Physis,1I Wiley & Sons, In., New York, Leonard, J.L., flintegral Transforms for Appliation to Partial Differential Equations,1I Grumman Airraft Engineering Corporation, Researh Department Report RE-I03, September From the olletion of the Computer History Museum (

10 64 ANALOG APPLICATIONS AND TECHNIQUES -Q (t) J _ 7 f,. (t,7) ~~ I 'lj I 7r'( I~~I u(t,o) ~ Fig. 1 - Analog Simulation Of The Re-entry Problem Fig. - Heat Equation With Convetive Boundary Conditions From the olletion of the Computer History Museum (

11 THE APPLICATION OF FINITE FOURIER TRANSFORMS TO ANALOG COMPUTER SIMULATIONS _gu 4 (t,7t) 65 Fig. 3 - Flat Plate Radiating Into Vauum I.e.: 40.0 Fig. 4 - Sphere Radiating Into Vauum From the olletion of the Computer History Museum (

12 From the olletion of the Computer History Museum (