A SHORT ACCOUNT OF RELAXATION METHODS

Transcription

1 A SHORT ACCOUNT OF RELAXATION METHODS By L. FOX (Natinal Physical Labratry, Teddingtn, Middlesex) [Received 4 Nvember 947] SUMMARY The relaxatin methd is a prcess f steadily imprved apprximatin fr the slutin f simultaneus equatins, and any prblem that can be frmulated in terms f simultaneus equatins can, theretically, be slved by this methd. After the first sectin f this accunt, in which the physical basis leading t the title and vcabulary f relaxatin is discussed, the methd is presented as a simple mathematical technique. The slutin f rdinary simultaneus equatins is illustrated by an example, and devices are suggested fr increased cnvergence. Applicatin t the prblem f vibrating structures is next described, in which Rayleigh's principle and relaxatin are used t btain the frequencies and mdes f vibratin. An example is included. The use f finite differences is then demnstrated, whereby differential equatins are replaced by finite difference equivalents, and the slutin f a differential equatin is effected by slving a set f simultaneus equatins, the unknwns being the values f the required functin at pivtal pints f a range r ranges f integratin. Attentin is fcused n partial differential equatins f the secnd rder in tw variables, and practical details are illustrated by an example f the slutin f Laplace's equatin, functinal values being specified n a clsed bundary. The applicatin t ther partial differential equatins, including the biharmnic equatin and the equatins f vibratin f membranes and f flat plates, is briefly cnsidered, and the accunt ends with a shrt summary f prblems already successfully slved by relaxatin methds.. Intrductin OF the varius indirect methds fr slving algebraic linear simultaneus equatins, the methd f relaxatin is the mst recent, the mst interesting cmputatinally, and prbably the mst pwerful. Thugh similar in many respects t the iterative methds invented by Gauss and Seidel, and refined by Mrris and thers, it is very much mre flexible. A measure f skill in securing rapid cnvergence can be develped by practice and intelligence t a much greater degree than in the standard methds, and this psychlgical aspect is very imprtant t the cmputer, engaged fr the mst part in rutine calculatins. Any prblem which can be reduced t the slutin f a set f simultaneus equatins can in thery be slved by relaxatin. Apart frm the rdinary simultaneus equatins which arise in everyday wrk, there aretw ther imprtant prblems which can be reduced immediately t this frm. The first invlves the determinatin f the frequencies and mdes f vibratin f engineering structures, and the secnd invlves the slutin g

2 254 L. FOX f a certain imprtant class f differential equatins. Examples f these applicatins are given in 3 and 4. Mst f the available literature is due t R. V. Suthwell, wh, with a small and variable team f research wrkers, has been engaged since abut 936 with the develpment f the methd and its applicatin t a large number f imprtant prblems in engineering and mathematical physics. These researches have been cllected in tw bks (refs. and 2), and a third embdying the latest and mst difficult applicatins is in curse f preparatin. Suthwell first used the methd in the prblem f the determinatin f stresses in laded framewrks, and all his subsequent wrk keeps the vcabulary and ntatin apprpriate t this prblem. Every set f equatins is cnsidered frm this angle. If the equatins arise frm a differential equatin in tw variables the framewrk becmes a tensined net, and the 'equilibrium f the jints f the laded framewrk' becmes the 'equilibrium f the ndes f the tensined net'. In this accunt the methd f relaxatin will be presented as a simple mathematical technique, but fr a prper appreciatin f Suthwell's bks it is desirable t have an understanding f the physical picture which led him t use the methd fr framewrks, and frm which the name 'relaxatin' is derived. 2. Algebraic linear simultaneus equatins A. The framewrk analgy Suppse a framewrk, initially unladed, has cnstraints f sme srt applied t its jints, preventing any displacement.f If the external lads are nw applied the framewrk will remain unstressed and all the lad is brne by the cnstraints, sme particular cnstraint taking the largest share f the lad. Nw let this cnstraint be 'relaxed', that is, allw the jint it cntrls t displace s that it will relieve the cnstraint f its lad. Sme f the latter will be transferred t the framewrk, and sme t ther cnstraints. If at every stage that cnstraint taking the largest lad is relaxed, it is physically evident that mre and mre f the external lad will be transferred t the framewrk, and that ultimately the residual frce still brne by the cnstraints will be less than the uncertainty f the applied lads. The prcess can nw cease, the prblem being slved with t In this sectin 'displacement' is understd t include 'rtatin', and 'frce' includes 'cuple'. The fllwing accunt suggests that each jint has nly ne degree f freedm, that is, its mvement is cmpletely specified by ne crdinate x. In general, f curse, each jint has six degrees f freedm, three f displacement and three f rtatin, and six crdinates are necessary fr its cmplete definitin. Hwever, this restrictin t ne crdinate in n way affects the ideas underlying this physical cncept f relaxatin.

3 A SHORT ACCOUNT OF RELAXATION METHODS 255 all the accuracy attainable, having regard t the engineer's uncertainty as t the exact magnitude f the applied lads. This idea f the 'systematic relaxatin f cnstraints', r 'relaxatin methds' fr shrt, can be carried ut in cmputatin. A typical framewrk equatin has the frm a z +a 2 a; a ln a; n = b t +R v () Here x r is the displacement f the rth jint, b t is the external lad applied at this jint, and a r3 is an 'influence cefficient', being the frce exerted at the rth jint due t a unit displacement f the sth jint. The left-hand side f equatin () thus represents the frce exerted by the framewrk n jint, due t jint displacements x v x 2,..., x n. The residual R v the difference between the frce exerted by the framewrk and the frce exerted n the framewrk, represents the lad taken by the cnstraint applied t jint. The prblem is t reduce the residuals systematically t negligible amunts by making apprpriate changes in the x's. In the fllwing table the cmputatinal steps n the right crrespnd t. the physical prcesses n the left. TABLE. Apply cnstraints and external lads. 2. Take cnstraint supprting largest lad and relax it, relieving it cmpletely f its lad. 3. Sme f lad is transferred t adjacent cnstraints. 4. Relax cnstraints systematically, always taking that supprting largest lad. 5. When all cnstraints are relieved f lad, measure the final displacements, thus btaining the equilibrium psitin.. Take all x =. Initial residuals are R T = -6 r. 2. Find largest R T. Make a change f R r /a rr in x T. R r becmes zer. 3. Other residuals R, are altered by amunts a r,x r. 4. Cntinue making changes in the x's, s that at each stage the largest residual is brught t zer. 5. When all the residuals are negligibly small, accumulate the cntributins t the x's, thus btaining the slutin t the equatins. B. Example f slutin f simultaneus equatins As a simple illustratin cnsider the fllwing three simultaneus equatins: -3^-6*2+2*3-93 = R 2, (2) +^-2*2-5x3-49 = R 3. It is required t find values f x lt x 2, and x 3 fr which R v R 2, and R 3 are negligibly small. It is helpful t cnstruct an 'peratins table', shwing at a glance the effect n each residual f a unit change in each variable. This is shwn

4 256 L. FOX in Table 2. The 'matrix' f cefficients is clearly the transpse f the matrix f the riginal equatins. Sx, Sx, «* = i = i = Si?, TABLE Si?, Table 3 gives details f the relaxatin prcess. 8i? 3 + i 2-5 TABLE 3 Rw n. i *i *, I Accurate slutin *i = + I 3"95 x, = *s= +75 Rw gives the residuals crrespnding t the initial guess x = x 2 = x 3 =. R x is the biggest residual, and is almst cmpletely liquidated by a change f +8 in x x. Rw 2 recrds this change and the new residuals f which R 2 is nw the largest. When this is liquidated (rw 3) by means f a change f 24 in x 2, B t again becmes the largest f the three residuals, and is in turn dispsed f by a change f 4 in x x (rw 4). At this stage R 3 becmes fr the first time the largest residual, and is remved by a change f +2 in x 3 (rw 5). Finally, in rw 6, a change f +2 in x 2 reduces the residuals t +, +, and. In rw 7 the cntributins t the a;'s are accumulated, and the residuals are checked by substitutin in equatins (2). The residuals are nw very small, and there is n pint in further relaxatin unless a mre accurate slutin is required. Devices fr the acceleratin f cnvergence f the relaxatin prcess cme by experience and practice. Fr example, the largest residual is nt the all-imprtant quantity. Quicker cnvergence is secured if at each stage that residual is liquidated which requires the largest' displacement'f fr its liquidatin. This change is the largest f the quantities Brla^, and may nt crrespnd t the largest residual, particularly if the magnitude f the diagnal cefficients f the equatins varies at all cnsiderably. t The wrd 'displacement', shrt fr 'jint displacement' f the framewrk analgy, is ften used in Suthwell's publicatins t mean the change in the value f a variable and is a very cnvenient shrthand.

5 A SHORT ACCOUNT OF RELAXATION METHODS 257 'Under-relaxatin', in which nly a prtin f a residual is remved, and 'ver-relaxatin', in which the sign f a residual is deliberately changed, are tw useful devices. If nt immediately apparent frm the peratins tame, it is ften easy t decide, as relaxatin prceeds, whether t under-relax r ver-relax. In the present example, fr instance, successive changes in bth x x and x 2 are f ppsite sign s that, if further relaxatin were perfrmed, it wuld pay t under-relax when liquidating residuals JR X and R 2. Again it may happen, after sme relaxatin, that the remaining residuals are rughly prprtinal t their initial values, when a simple multiplying factr n the values s far btained can be used with great effect. One* f the mst imprtant devices is that f 'grup-relaxatin', in which tw r mre variables are altered at the same time by different amunts (if the change in each variable is the same, the device is ften called 'blck-relaxatin'). The idea is t btain cmbinatins fr which ne residual is changed by a large amunt, the thers by relatively small amunts. Referring t the peratins table, Table 2, tw gd gruprelaxatin peratrs wuld be as shwn in Table 4. TABLE 4 i = ' i -7 Evidently these peratins fr the liquidatin f residuals R x and R 3 respectively, wuld be far mre pwerful than thse f the riginal table. In practice subsidiary peratins f this srt wuld be included as additinal rws in the peratins table. Further examples f these cmputatinal tricks are given in 3 and 4. Meanwhile the fllwing pints may be nted.. It is generally useless t make any residual exactly zer, by an exact change RJa^ in the crrespnding variable. This residual will almst certainly change again subsequently, and much unnecessary arithmetic has been dne. One f the advantages f the methd lies in the fact that 'easy' peratins can be used, and much f the wrk can be dne mentally withut undue strain. The calculating machine need be used nly fr the calculatin f initial residuals (if sme guess ther than x r = were taken t start with) and the checking f the final nes. 2. In practical examples the cefficients are nt usually simple integers, but this entails little extra effrt. Having calculated the residuals using the exact cefficients, the latter may be runded ff t cnvenient numbers fr

6 258 L. FOX use in the peratins table. When residuals are subsequently checked (using the exact cefficients) they will nt agree exactly with thse btained frm the relaxatin prcess, but this merely requires a little extra relaxatin. 3. In cmmn with all methds f iteratin, mistakes are nt serius. If sme residual is incrrect, the mistake will be fund in the final check and further relaxatin will be necessary. This final calculatin f residuals is f curse very imprtant, and it als pays t check them fairly frequently in the curse f the wrk, say whenever they have been reduced t netenth f their previusly calculated values. 4. Again like ther iterative methds, it is nt necessary t start with a large number f figures t secure a result f given precisin. Figures can be added as the wrk prceeds, and accuracy builds up, as it were, frm belw. C. Limitatins f the methd The prcess f liquidating at every stage the largest residual des nt always cnverge: the residuals may increase indefinitely. Temple has prved (ref. 3) that there will always be cnvergence if the simultaneus equatins are derived frm the cnditins that sme psitive definite quadratic functin f the variables is a minimum. Framewrk equatins are always f this kind, ne characteristic being that the equatins are symmetric, that is, a rs = a^. Any set can be reduced t this frm by nrmalizatin, that is, by btaining a new set f equatins frm that typified by () in the frm A 2 ^ = (5= t n). (3) X * r=l This cnditin is nt essential fr cnvergence, either f relaxatin r f standard iterative methds; in the example f 2B, which was highly cnvergent, the cnditin was nt satisfied. Mrever, the relaxatin methd, by virtue f its greater flexibility, can be applied by a skilled cmputer t equatins fr which nrmal iterative methds wuld fail. The questin f theretical cnvergence is nt indeed very imprtant. Much mre significant is the questin f practical cnvergence, which may be s slw as t render the methd impracticable. Equatins f this srt are called 'ill-cnditined'. Cnsider, fr example, the fllwing set f equatins, devised by T. S. Wilsn: 5^+ x 2 + 6x a + 5x 4 23 =, 8x 3 + 7z 4-32 =, ( - 9z 4 33 =, ' 52;!+ 7x 2 + 9a; 3 -)-a; 4 3 =±, the slutin f which is x x = x 2 = x 3 = x t =.

7 A SHORT ACCOUNT OF RELAXATION METHODS 259 These equatins are in the nrmal frm sufficient fr cnvergence, but any indirect methd f slutin is quite impractical. Table 5 shws three sets f residuals crrespnding t three distinct apprximate slutins. TABLE 5 *. *4 Ri R 2 R* R, I-2I + I-O2I + + -i -i O-OOI -i -i O-OOI + O-OI + - The residuals given in the first rw are quite small cmpared with the initial cnstants ccurring in equatins (4), whereas the crrespnding apprximate slutin bears n resemblance t the crrect ne. Even the third slutin, whse residuals are less than / 5 f the initial cnstants, has errrs f per cent. Thugh relaxatin is ften impracticable in cases f this srt, especially with a large number f unknwns, it is hardly likely t lead t incrrect slutins. It is fund in the curse f the wrk that small residuals require very large displacements fr their liquidatin, and the whle prcess is tedius and slw, a sure sign f lack f cnditin f the equatins. Relaxatin, in fact, is mst useful when the set f equatins has large cefficients dwn the principal diagnal, r when sufficient effective gruprelaxatin peratrs can be wrked ut t prduce a similar effect. Here again the skill f the cmputer plays a big part, and the glden rule shuld always be brne in mind that any device, hwever revlutinary, can be used, subject t the final check that the slutin btained satisfies the riginal equatins. The basic principles and ideas f relaxatin methds have nw been described, and it remains t demnstrate their applicatin t tw imprtant prblems. 3. Vibratin f engineering structures A. Relaxatin and Rayleigh's principle The first imprtant applicatin cncerns the determinatin f the frequencies and mdes f vibrating systems. In terms f simultaneus equatins it is required t find the n nn-trivial slutins f equatins f the frm (a X)x +a 2 x a ln x n = a li x +(a 2Z -\)x a 2n x n = a ln x +a 2n x (a nn -X)x n = (5)

8 26 L. FOX and the n values f A fr which slutins exist. These values f A are the rts f the detenninantal equatin i n A a l2.. a ln =. (6) a 2 a 22 A.. a 2n I «ln a 2n a nn ' In mst cases suitable fr relaxatin the n rts are real and psitive. The methd f slutin is a cmbinatin f relaxatin methds and Rayleigh's principle. Writing equatins (5) becme X 2 Xx 2 =, Rayleigh's principle states that the 'best' value f the frequency crrespnding t a guess x v x 2,..., x n at the mde f vibratin, is given by n A - -h-i (9) etc. and that this value is statinary with respect t small changes in the mdal cmpnents. The mdes and frequencies are btained ne at a time, the prcedure fr sme particular mde being as fllws: (i) Make a guess at the mde, and calculate the crrespnding value f A using equatin (9). (ii) Insert this value f A in equatins (8), and calculate the residuals R r = X r -Xx r, () X r having been already btained in the calculatin f A. (iii) Try t liquidate the residuals by relaxatin f equatins (5) (frm which the peratins table, with the calculated value f A inserted in the diagnal elements, is btained in the usual way). It is nt pssible t btain zer residuals unless the calculated value f A is a rt f equatin (6), but they can be reduced r 'smthed' t smaller values than the riginal set. (iv) When further relaxatin appears t effect n imprvement in the residuals, take the new values x r, calculate a new value f A and hence new residuals, make the necessary crrectins in the diagnal elements f the peratins table, and cntinue the relaxatin prcess.

9 A SHORT ACCOUNT OF RELAXATION METHODS 26 (v) Cntinue until n further change in A takes place, and the residuals are effectively zer, t the accuracy required. T btain a secnd mde and frequency it is necessary merely t make a different guess z z n at a mde and repeat the prcess. The nly danger is that in the curse f relaxatin this new guess degenerates int the ld, and n new mde is btained. This can be prevented by using the rthgnal prperty f nrmal mdes, which states that if x v x z,..., x n and y x, y 2,..., y n are distinct slutins f equatins (5), then 2*r</r=. () Thus the new guess z prbably cntains a cmpnent f the first mde x, which can be remved by taking, instead f z, y = z cx. Then if y is t be rthgnal t x, f x r {z T -ax r ) =, (2) I r «= (z r x r )l (**). (3) I ' i The mde y then cntains n cmpnent f x (thugh it des cntain cmpnents f nrmal mdes nt yet btained). If it is used in place f z, there is n initial danger f 'slipping back' int the ld mde x, and subsequent danger can be prevented by repeating the eliminatin prcess at any stage. In practice the initial eliminatin is ften sufficient. An alternative and smetimes mre useful methd fr btaining a secnd mde and frequency is as fllws. Suppse a mde x, with cmpnents x lt x 2 x n and frequency X lt has been btained. A new determinantal equatin replacing (6) is then frmed, the element a rg in the rth rw and sth clumn being replaced by b ra, where One f the rts f this determinantal equatin, crrespnding t the knwn mde x, is zer. The remaining rts and mdes are the same as thse f the riginal prblem, and can be fund in a similar manner. The writer has fund this methd t wrk well in cases which were prving difficult by the first methd. B. Numerical example As a simple example cnsider the fllwing prblem with three degrees f freedm:,.,., A (5 A)^ 3* 2 +4a;3 = -3a: +(2-A)a ;2-2 : c3 =. (4) 4x -2a; 2 +(-A)z3 = j

10 262 L. FOX Equatins (7) in this case becme X x = +5x 3z X 2 = -3x +2x 2-2x 3. X 3 = +4x -2x 2 +x 3 ) The first stage in the slutin is shwn in Table 6. (5) TABLE 6 I 2 3 ' 4 X X R + I i X 5 3 8x,= 8x = S/f, SR t i? The initial guess (rw ) is taken, fr want f a better, t be Values f.x\, X 2, and X 3, calculated frm equatins (5), are given in rw 2. The value f A, calculated frm equatin (9), is fund t be 5-. Initial residuals (X r Xx r ) are then btained (rw 3). The peratins table, btained frm equatins (4) with the calculated value f A inserted, is given n the right f the table. Cmparisn f the residuals and the peratins table shws that the mst efficacius change fr the reductin f residuals is ne f 2 in x x, resulting in new residuals as shwn in rw 4. At this stage it is decided fr several reasns that further relaxatin is likely t be valueless. First, the diagnal elements f the peratins table depend n A, which, in view f the change in the whle character f the mde due t the single peratin perfrmed, must nw be expected t change cnsiderably frm its initial value. Secnd, the residuals are nw all f ppsite sign t the crrespnding mdal cmpnents, and it is clear frm equatin () that this is a satisfactry state f affairs, in that a decrease in A will imprve all the residuals simultaneusly. The new apprximate mde is x =, x 2 = -j-, x 3 = +, and the prcess is repeated as shwn in Table 7. The new value f A is fund t be 3-67, causing a substantial change in the diagnal elements f the peratins table. The last rw in the peratins table is a grup-relaxatin peratr, cnstructed t give a mre effective liquidatin f R 2. This peratr is used twice, f in rws 4 and 5, and effects a substantial reductin in all the residuals. The last peratin (rw 6) is rather cunning! Nt nly des it have the effect f making the residuals all f the same sign as the crrespnding mdal cmpnents, t A single peratin, cmbining rws 4 and 5, wuld have meant heavier arithmetic (cf. 2B, pint ). All the present cmputatins, except the final calculatin f A, were perfrmed mentally.

11 A SHORT ACCOUNT OF RELAXATION METHODS 263 TABLE 7 I X X R X t = -2 \ X, = - / Xg = -6 \ x 8 = -3 / x, -i -I +i +i xj = 8x, = 8x, = 8x, = 2, 8x3 = 3 8/?, + ' SR, \34 SR t X X -i A _ 8544 _ R I s that an increase in A will imprve them all, but als the magnitude f each residual is nw rughly prprtinal t the mdal cmpnent, giving cnsiderable all-rund benefit (if they were exactly prprtinal, a suitable change in A wuld yield an exact slutin). The last part f the table shws the imprved mde (rw 7), resulting in a A f 3-78, and the final residuals (rw 9) are very small. This value f A is in fact crrect t three decimal places, and the mdal cmpnents t tw. It is clear frm this example that the general remarks f 2B apply equally here. The main difference is that the mde is indeterminate t the extent f a cnstant multiplier, s that the residuals culd be made negligible by taking this multiplier sufficiently small, leading ultimately t the trival and useless slutin x x = x % =... = x n =. T prevent this it is advisable t maintain at least ne f the mdal cmpnents at a reasnable size, using a multiplying factr if necessary. It is nt always pssible t decide which particular mde has been btained, thugh in practice the general shape f the fundamental, the mst imprtant mde, is ften knwn frm physical cnsideratins. If the rts are well separated, the identity sum f rts = sum f elements f principal diagnal gives valuable infrmatin. Ill-cnditining manifests itself when tw r mre rts lie fairly clse tgether. This des cause truble, but in many practical prblems nly a few f the n rts and mdes are required, and it appears generally (and frtunately) that these are the nes mst easily btainable by relaxatin methds. Determinants f rders up t the twelfth have been successfully slved, including the mre difficult case in which the unknwn A appears in every element, and nt just in the diagnal terms f the equatins.

12 264 L. FOX 4. Differential equatins A. Derivatin f the simultaneus equatins The secnd prblem, the slutin f differential equatins,, prvides the mst interesting and the mst imprtant applicatin f relaxatin methds. As in the well-knwn step-by-step methds f integratin, numerical values f the required functin are btained at pivtal pints f a range r ranges f integratin, the distance r interval between pivtal pints being at the discretin f the cmputer. The tw methds f slutin are, hwever, quite distinct, and there is a mst imprtant difference in the types f differential equatin fr which the tw methds are respectively suitable. Relaxatin favurs thse differential equatins fr which the bundary cnditins are specified n a clsed bundary, step-by-step methds prefer the pen bundary type. Fr example, a secnd-rder differential equatin in ne independent variable needs tw bundary cnditins fr its slutin. Cmmn bundary cnditins are (i) the functin r its first derivative is specified at the end-pints f the range f integratin, r (ii) bth the functin and its first derivative are specified at ne end f the range. The first case wuld be slved by relaxatin, the secnd by step-by-step methds. Similarly, fr partial differential equatins in tw independent variables, relaxatin favurs the elliptic type, fr example, Pissn's equatin v 2 / = sy/a^+a 2 //^2 = g% y), ( 6 ) fr which cmmn bundary cnditins are that either / r its nrmal derivative df/dv is specified at every pint f a clsed bundary. With ther types f differential equatins, fr example, the parablic equatin d ,.,,., = (heat cnductin, etc.), (7) x* t and the hyperblic equatin -~ = (vibratin f strings, etc.), (8) 3a; 2 at* the bundary cnditins are seldm specified n a clsed bundary, and relaxatin methds are generally useless. There are, f curse, satisfactry step-by-step methds, bth by hand and by machines such as the differential analyser, fr the slutin f such equatins. The simultaneus equatins, whse unknwns are the values f the required functin at pivtal pints f the range r ranges f integratin,

13 A SHORT ACCOUNT OF RELAXATION METHODS 265 cme frm the replacement f derivatives by finite-difference expressins. Fr first and secnd derivatives, the equatins are respectively as fllws: (9) where h is the interval f tabulatin f the functin/(a;), and 8J is the rth central difference f the functin at the pivtal pint 5, arranged accrding t the fllwing scheme: a; /(*) 2h /_, S 9 8 a 8, i -h / - i Six 8 l -i g4 etc. (8J = i(8l } -M)\), etc.) h " 8 8 +h A 8? 8 8J 8} + 2h ft 8? 8J Any f the differences < ccurring; in equatins (9) canbe expressed in terms ( >f functinal values. Treating the first and secnd differences nly in this way, the equatins becme 2 Many imprtant prblems invlve the Laplacian peratr V 2 = --\ 5 in plane Cartesian crdinates x,y. dx* cy 2 Fr prblems f rtatinal symmetry, using cylindrical plar crdinates (r, 9, z), we have g2 x Q Q2 V 2 ^. 8r 2^r dr^dz 2 With the aid f equatins (2), and with reference t Fig., these tw peratrs can be expressed in finite-difference frm as where A = $ and -...) (2I) where.(22)

14 266 L. FOX In these equatins the subscripts x, r, etc., mean 'in the a;-directin', 'in the r-directin', etc., and A, a functin f differences higher than the secnd in the tw directins, will be called 'the difference crrectin'. Fr cnvenience the interval f tabulatin h is taken t be the same in bth directins. u r r O t h *~ h -> -> x r 2 - FIG.. Nw imagine the regin ver which the slutin is required, and n the bundary f which sme cnditin is specified, t be divided int a square mesh by lines parallel t the axes. Ignring fr the mment the differencecrrectin A, equatins like (2) and (22) can be used t prvide, at each mesh pint, an equatin cnnecting the values f the required functin at this and surrunding pints. If there are n mesh pints there will be n simultaneus equatins, and the slutin f these by relaxatin will prvide an apprximate slutin t the differential equatin. The number f equatins may be large, but the number f unknwns present in each equatin is small, a situatin ideal fr relaxatin methds. B. Numerical example Cnsider, fr example, the slutin f Laplace's equatin J \ J (\ fcyx\ ver a rectangular area, n the bundary f which values f/ are given. The relevant finite-difference equatin, btained frm equatin (2), is (/i+/-i)*+(/i+/-i) v -4/ =. (24) Nw cnsider Fig. 2, in which equatin (24) is t be satisfied at every internal mesh pint, bundary pints having an assigned value. It is clear frm the diagram that the value at every internal pint is an unknwn, but nly the pints nt adjacent t the bundary, that is, thse marked with a crss, are surrunded by fur unknwn values. Fr pints marked with an pen circle, ne f the terms in equatin (24) is a knwn bundary value, and fr the pints marked with a clsed circle, tw f the / 's have knwn values. Thus fr pints marked with a crss there are five

15 A SHORT ACCOUNT OF RELAXATION METHODS 267 unknwns in each equatin, fr pints marked with an pen circle there are fur, and fr pints marked with a clsed circle nly three. i ( V f 6 ) / ) K / V N * \ c ) \ f > ^ J g f ( ( ) ( J / / ^ s \ \ y N J ^ ( "S J i \ r ) C \ ( ) \, g FIG. 2. The methd f slutin fllws familiar lines. Plausible values are attached t each mesh pint, and initial residuals calculated frm the equatins ^ = (A+/_ I)X+(/I+/ _ I)J/ _ 4/O. (25) The peratins table is simple: clearly a change f + in/ alters B by 4, and changes the residuals at adjacent internal pints in the (x, y)- *! 4ls +, - (a) FIG. 3. directins by -j- (since the cefficients f / in the expressins fr the. residuals at these pints are always +!) It is fund cnvenient t recrd values f the functin t the left f the ndal pints, residuals t the right. Figs. 3 (a), (6), and (c) shw the effect f a unit change in the functinal

16 268 L. FOX value at each f the three types f pints f Fig. 2. The bundary is shwn with bld lines, the mesh with fine lines. As an illustratin f the methd f setting ut the wrk, Fig. 4 gives full details f the relaxatin prcess fr the slutin f equatins f type (24), fr a square bundary n which functinal values are given. Only fur internal pints are taken, each being f the type f Fig. 3(c). IOO 'O O UO Z 6 z - 2O IOO - s -IO -2O bo UO -u -y O s -lo 7O SO IO 25 s io O? a -a 6 IOO -IOO 3OO ZOO Fi. 4. The initial value zer is attached t each internal pint, and the initial residuals, calculated by equatin (25), are recrded n the right f the ndal pints. The relaxatin prcess then prceeds, just as fr any set f simultaneus equatins, t impse displacements, liquidating at every stage the largest residual. These displacements are recrded t the left f the ndal pints, the residuals still unliquidated t the right. At the stage finally reached in Fig. 4, the remaining residuals shw that n further imprvement is pssible, t this rder f precisin. The displacements are then accumulated, and the final residuals checked by equatin (25). The result is shwn in Fig. 5. Evidently n errrs were made in the relaxatin! Many f the remarks made in 2B apply equally here. Fr example, nly ccasinally in Fig. 4 is a residual liquidated cmpletely and exactly, multiples f being always used until the residuals are sufi&ciently small t warrant single-figure displacements.

17 A SHORT ACCOUNT OF RELAXATION METHODS 269 The devices f under- and ver-relaxatin and blck and grup displacements require special cmment. It is nt always easy, in the slutin f a general set f simultaneus equatins, t decide when t ver-relax r hw t cnstruct effective grup peratins. Fr difference equatins f the type (24), the prblem is quite easy, wing t the simplicity f the peratins table. Clearly ver-relaxatin is necessary when residuals f z O s s -s 8 25 O -IOO 3 FIG. 5. 2OO O the same sign are clse tgether, fr liquidatin f, say, a psitive residual, requires a psitive displacement, which impses a psitive additin t residuals at surrunding pints. Similarly when residuals f different sign are clse tgether, it pays t under-relax. In the wrk shwn in Fig. 4, tw ver-relaxatins were perfrmed, in each case a change f + 2 being made at a pint at which the residual was -f 6. It is clear frm Fig. 3 that the algebraic sum f the residuals changes if, and nly if, sme pint adjacent t the bundary is relaxed:f relaxatin at any ther internal pint (f type 3 (a)) has n effect n the algebraic sum. Cnsequently residuals predminantly f ne sign can nly be made t vanish by 'pushing' them twards the bundary, and this is ften dne mst effectively by blck-relaxatin. (It is imprtant t ensure the f It will be nticed that the wrd 'relax' is lsely used in many cntexts. Thus we 'relax', amngst ther things, a prblem, an equatin, a residual, a cnstraint, and nw a pint! 592.3

18 27 L. FOX effective vanishing f the algebraic sum f residuals, since small residuals f ne sign may in the aggregate require quite large displacements fr their liquidatin.) A typical blck-relaxatin is shwn in Fig ~\ -2 - O FIG. 6. A unit psitive displacement applied at every internal pint (the bundary is shwn in bld lines) prduces changes in the residuals as shwn in the figure, giving a ttal change f 2 in the algebraic sum. Nw cnsider the prblem shwn in Fig. 7, where the bundary values are specified as shwn, and the initial value zer attached t each mesh pint results in residuals all f the same sign, with an algebraic sum f 24. Applicatin f the blck-relaxatin peratr f Fig. 6, with 2 as unit instead f, will clearly reduce the algebraic sum t zer, and leave the psitin shwn in Fig. 8. The residuals can nw be liquidated quickly by pint relaxatin, and in fact displacements f +5 at the pints with residuals +2 in Fig. 8, and 5 at the pints with residuals 2, reduces all the residuals t zer and cmpletes the slutin. Anther useful device is a grup relaxatin f a particular kind, f great value in the slutin f difference equatins f the type (/i+/-i)»+(/i+/-i)»- 4 / = cnstant, K say, (26) where / is zer n the bundary f the regin cnsidered. At sme stage

19 A SHORT ACCOUNT OF RELAXATION METHODS 27,OO. 3 O ZOO IOO O 3OO t 2 2O IOO ZOO 2 2OO ZOO 2 2 fcoo 3O O 2 FIG. 7. 3OO U. l+oo OO IOO 2 2 z 2 O 2 IOO IOO z FIG <i.

20 272 L. FOX in tb,e relaxatin it may arise that the residuals are predminantly f ne sign, thugh n ne is very large. The device cnsists in calculating the average value f the left-hand side f equatin (26), and then multiplying the slutin s far btained by a cnstant such that this average value is equal t the required value K. Clearly the algebraic sum f the remaining residuals will be small, psitive and negative signs mingling freely, and pint relaxatin shuld then be effective. It is nt pssible in this accunt t list all the tricks which a cmputer acquires fr the acceleratin f cnvergence f the relaxatin prcess. As in all ther branches f numerical mathematics, facility in the relaxatin technique cmes with practice, and the need fr such devices becmes apparent nly when ne tries t slve a differential equatin withut using them. C. Special prblems in the slutin f secnd-rder differential equatins The example given in the last sectin related t the slutin f the easiest type f partial differential equatin. The peratins table was simple, and was effectively the same at all pints. A mre general case f a linear secnd-rder partial differential equatin might lead t a finitedifference equatin f the frm (A/ +^f. ) x +(Cf +I)f-i)v-W = F, (27) where the cefficients A F are either cnstants r functins f the independent variables x and y. There is clearly n new principle invlved in slving by relaxatin this mre general prblem, and n further cmment will be made. Similarly the applicatin f the methd t differential equatins in ne independent variable is straightfrward and bvius. There is, hwever, a prblem in the slutin f differential equatins additinal t that f relaxing simultaneus equatins. The latter can be perfrmed t any required degree f accuracy, but the questin remains t what extent the simultaneus equatins are an adequate representatin f the differential equatin. The chief surces f errr lie in (i) the neglect f the difference crrectin A, and (ii) the derivatin f difference equatins fr use near a curved bundary. These prblems are discussed nly briefly here; a fuller treatment, with examples, is cntained in ref. (9). In mst published wrk the difference crrectin is frankly ignred, and the mesh made sufficiently fine t ensure that the neglect is nt serius. The writer prefers t use as carse a mesh as pssible, n which a first apprximate slutin is btained by neglecting the difference crrectin. The latter is then calculated at each mesh pint frm the apprximate slutin, and is inserted as a new residual, t be relaxed in turn. This is repeated until the full finite-difference equatin is satisfied, the number

21 A SHORT ACCOUNT OF RELAXATION METHODS 273 f repetitins necessary rarely exceeding tw. By this means the large number f equatins required by the first methd is avided, and dubts as t the accuracy f the slutin are t a large extent dispelled. The methd is particularly advantageus whenever the differential equatin cntains a first derivative, fr example, in equatin (22). The prblem presented by a curved bundary is illustrated in Fig. 9. FIG. 9. A finite-difference equatin f type (24) cannt be used withut mdificatin at a pint such as, since it invlves a knwledge f the external pint. One methd f dealing with this prblem is t express the value at in terms f the knwn bundary value B and the internal pints, 2, 3,... The expressin cmes frm the Gregry-Newtn interplatin frmula.., > f B =/ + za+qa +QAg+..., (28) Where x = OB/h, A=A-/> A 2 =/2-2A+/, etc. Frm this equatin f can be btained in terms f as many internal pints n the line, 2, 3,... as desired. Fr simplicity Suthwell retains nly the first frward difference A in equatin (28), that is, he assumes the functin t be linear at the bundary. The writer preserves the accuracy btainable

22 274 L. FOX by the difference crrectin methd by retaining as many differences as are significant in equatin (28). The resulting frmulae fr pints adjacent t the bundary are then smewhat cmplicated, and may cntain mre than the five unknwns present in the equatin fr an rdinary internal pint, but again there is n inherent difficulty in the relaxatin. A prblem f a similar kind is invlved when the nrmal derivative f the functin, instead f the functin itself, is specified at the bundary. If the bundary is curved, the setting up f accurate finite difference equatins is particularly difficult. One methd is given in Suthwell's bk (ref. 2), a secnd in ref. (4). 5. Further prblems in differential equatins A. The Inharmnic and allied equatins The applicatin f relaxatin methds t the slutin f equatins cntaining the biharmnic peratr ii+$) a (29 > is described in the paper given as ref. (5). The principle is the same, t b 2 5 II 3 O I q 7 U- 8 2 FIG.. thugh the technique is mre difficult. The finite-difference equatin, fr example, here cnnects 3 pints, and is given, with the ntatin f Fig. = 2/ -8 2/ r +2 i/ r + f/r- (3) 5 9

23 A SHORT ACCOUNT OF RELAXATION METHODS 275 Tw bundary cnditins are necessary fr a unique slutin, usually the functin and its nrmal derivative being specified. The chief feature f relaxatin with this peratr is the slwness f cnvergence, making very necessary the use f grup- and blck-relaxatins. The prblem in finitedifferences f accurately satisfying f the secnd bundary cnditin is als trublesme. Hwever, several imprtant prblems in elasticity have been slved with all the accuracy necessary fr engineering applicatins. Occasinally (in the thery f flat elastic plates) the biharmnic equatin can be replaced by tw secnd-rder equatins f the frm 8x\8z = (3) where A is cnstant, and u and v take specified values n the bundary. The apprximate finite-difference equatins, with the ntatin f Fig., are given by (l+a)(u +u 3 )+A(u 2 +u i )-2(l+2A)u +^(v 5 -v 6 +v,-v s ) = (32) A(v +v 3 )+(l+a)(v 2 +v i )-2(l+2A)v +i(u 5 -u 6 +u 7 -u 8 ) = O 7 d. 8 FIG.. Here there are tw functins, u and v, t recrd at every ndal pint, and als the residuals R u, R v crrespnding t the tw equatins. When a ^-displacement is made, five residuals R u are changed, and als fur residuals B v (by a much smaller amunt). This wuld seem t be cmplicated, but in fact the relaxatin is easier and much mre cnvergent than in the biharmnic equatin. Examples are given in the paper f ref. (5).

24 276 L. FOX B. V.ibratin prblems It was seen in 3 that the mdes and frequencies f vibratin f engineering structures can be btained by relaxatin methds. Similarly the equatin f membrane vibratin, given by V 2 /+A/=, (33) and the general equatin f buckling f flat plates, dx\ x dz dy)^dy\ v By 8x)\ (in which P x, P y, and 8 are knwn functins f x and y) can be slved by these methds. The technique is identical with that f 3. A guess is made at the required mde f vibratin r buckling, a first apprximatin t A btained by Rayleigh's principle, and the mde imprved by relaxatin f the gverning differential equatin, expressed in finite-difference frm. Fr equatin (33), Rayleigh's principle gives jjp Pdxdy the finite-difference frm f the differential equatin is (35) =, (36) and the rthgnal prperty f nrmal mdes is expressed by the equatin jjf dxdy = O (r^s). (37) r f 8 The difference crrectin in equatin (36) is the same as that f equatin (2), and it is an interesting characteristic f prblems f this srt that neglect f A in the relaxatin prcess leads t an incrrect value f A, but a substantially accurate mde. The A can then be imprved by including the difference crrectin in the calculatin f V 2 /, fr insertin in equatin (35). Examples are given in ref. (9). Similar treatment f equatin (34) enables the mde f buckling and the critical applied edge frce t be determined. Examples f the vibratin f membranes (which has imprtant electr-magnetic applicatins) and f flat plates are given in refs. (6) and (7) respectively. G. Miscellaneus prblems and cnclusin The fllwing is a shrt list f prblems invlving the slutin f partial differential equatins, already successfully treated by relaxatin methds, (a) Secnd-rder equatins (i) Cnfrmal transfrmatin: V 2 / =, / r 8fj8v given n bundary.

25 A SHORT ACCOUNT OF RELAXATION METHODS 277 (ii) Trsin f bars, unifrm sectin: V 2 / = cnst.,/ = n bundary, (iii) Trsin f circular shafts f varying sectin: f given n bundary. (iv) Incmpressible fluid flw: V 2 / =,/ = cnstant n fixed bundary, (v) Flw f cmpressible fluid thrugh a nzzle: v 2 (x<a)-«av 2 x =, -mm if/ = n the nzzle wall, = n the centre line, and the frm f the functin / is knwn. (vi) Membrane vibratin: V 2^+A^ =, tf> = n bundary. (6) Furth-rder equatins (i) Bending f flat elastic plates: \ V*w = Z, Slw mtin f viscus fluid: / w, dwjdv knwn n bundary, (ii) Stretching f plates: 8x\dx dy u and v given n bundary. (iii) Centrifugal stresses in rtrs. This is a very difficult prblem t relax, the differential equatins being f the frm ( _H \ \ 2 rdr + 8z*) X ~ dr 2 with bundary cnditins dz 2 ' v) = ^ r as where A, and B are cnstants, and (r, v) is the angle between the directin

26 278 L. FOX f r and the nrmal v t the bundary. Details f the methd, and ther prblems in cylindrical crdinates, are given in ref. (8). (c) Prblems in which ne bundary is unspecified in advance In these prblems sme extra cnditin is necessary, the satisfactin f which enables the psitin f the free bundary t be fixed. A D E ~B FIG. 2. (i) Seepage f fluid thrugh a prus wall. The differential equatin fr the pressure is V*p =. The bundary cnditins are (Fig. 2): n AB, p = y, n BC, = cnstant, By n CD, p = y y D, n DE, p = ; and n the free surface AE, tw cnditins have t be satisfied: _, dp dx p =, and - - = -;- (ii) Oil pressure in jurnal bearing. The differential equatin is 8v as c 5»!jP(l+cc8)«= -6csin, where B 2 and c are cnstants. The bundary cnditins specify that p is zer n a clsed bundary in the (z, )-plane, but the psitin f the tp bundary, 6 = w+a, is

27 A SHORT ACCOUNT OF RELAXATION METHODS 279 unknwn. The value f a is btained by satisfying an extra cnditin n this line, namely, at the pint, =. 86 Many ther imprtant prblems f this type are dealt with in ref. (2). The list given is sufficient t shw the pwer f relaxatin methds. D z = _ J_ B C FIG. 3. Finally, it may be remarked that the relaxatin methd is nw being used in the imprtant wrk f tabulatin f mathematical functins, particularly in the case in which a limited degree f accuracy will suffice. In this cnnexin it may be nted that any regular functin f a cmplex variable has real and imaginary parts, each f which satisfies Laplace's equatin, s that tabulatin f duble entry functins f this kind becmes a practicable pssibility by relaxatin. With regard t the pssibility f treating a given prblem by relaxatin, certain criteria must be brne in mind. Success can almst always be guaranteed if (i) the differential equatins are linear, (ii) the bundary cnditins are given n a clsed bundary and are als linear, (iii) the slutin t the prblem is unique and free frm singularity, and (iv) the prblem invlves at the mst tw independent variables. On the ther hand, this methd is s flexible that ccasinally nn-linear prblems, either in differential equatin r in bundary cnditin, can be slved. Further, it is ften pssible t recgnize the type f any singularity

28 28 A SHORT ACCOUNT OF RELAXATION METHODS present, and t remve this first analytically, leaving a nn-singular prblem fr relaxatin. Finally, thugh the labur f relaxatin in three dimensins is prhibitively great, the future use f the new electrnic calculating machines in this cnnexin is a distinct pssibility. REFERENCES. R. V. SOUTHWELL, Relaxatin Methds in Engineering Science (Oxfrd, 94). 2. Relaxatin Methds in Theretical Physics (Oxfrd, 946). 3. G. TEMPLE, ' The general thery f relaxatin methds applied t linear systems', Prc. Ry. Sc. A, 69 (939), L. Fx, 'Slutin by relaxatin methds f plane ptential prblems with mixed bundary cnditins', Quarterly f Applied Mathematics, 2, N. 3 (Octber 944). 5. Fx and SOUTHWELL (94), 'Biharmnic analysis as applied t theflexure and extensin f flat elastic plates', Phil. Trans. Ry. Sc. C,, 5-56; A, 239 (945), ALLEN, FOX, MOTZ, and SOUTHWELL (94), 'Free transverse vibratin f membranes, with an applicatin (by analgy) t tw-dimensinal scillatins in an electr-magnetic system', ibid. C,, 85-97; A, 239 (945), CHKISTOPHERSON, FOX, GREEN, SHAW, and SOUTHWELL (94): 'The elastic stability f framewrks and f flat plating', ibid. C,, 57-83; A, 239 (945), ALLEN, FOX, and SOUTHWELL (942), ' Stress distributins in elastic slids f revlutin', ibid. C,, 99-35; A, 239 (945), L. Fx, 'Sme imprvements in the use f relaxatin methds fr the slutin f rdinary and partial differential equatins', Prc. Ry. Sc. A, 9 (947), References 5, 6, 7, and 8 are respectively parts VII A, VII C, VII B, and VII D f the series 'Relaxatin Methds applied t Engineering Prblems'. Fuller lists f references are cntained in Suthwell's bks.