THE AUTO-ADJUSTABLE DAMPING METHOD FOR. (Received June 20, 199-5; Revised June 10, 1996; Communicated by Zhong Wanxic) Abstract

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1 Applied Mamatics and Mechanics (English Edition, Vol. 19, No. 2, Feb. 1998) Published by SU, Shanghai, China THE AUTO-ADJUSTABLE DAMPING METHOD FOR SOLVING NONLINEAR EQUATIONS Chang Haiping (~)~ Huang Taiping (~:~-~)' (Received June 20, 199-5; Revised June 10, 1996; Communicated by Zhong Wanxic) The general approach.for solving tiw nonlinear equations is /ineori:ihg llw equatiolls and.formhtg various iterative procedures, l~ e.vecutiltg munerical shnulation. For strottg O' nonlinear prob/et~ts, solution ohlaiued ih ilerative process is always dijfieuh, evelt di)'ergent due to nutnerical hlstabi/it)'. It calt not an analytic solution only when polytropic index of detonation products equals to three. In general, a fulfill enghteerbtg analysis is requirements. required. In Newton's this paper, method however, and its by variants utilizing cart not "weak" settle shock behavior of this problem. reflection As shock a resuh, in explosive application products, of and shmdation applying for small strongly parameter purterbation method, nonlinear an problems analytic, is first-order limited. An approximate auto-adjustable solution damphzg is obtained method has for been problem presented of flying plate driven in by this various paper. high This explosives is a furr with improvement polytropic of Newton's indices or method than with but thtmping.[aetoi'. nearly equal to three. Final velocities A set of o./" flying vector plate of damphtg obtained./2wtor agree is very httro~hwed. well with This set oj veetor results cajs by he computers. ad.ju.vted Thus continuously durhtg iterative process ht aet'or(htnce with judgement and adjustment. An e.f.fectire convergence coej.ficient and quiehening eo~:['['ieient are emplo),ed to rehtx restricted requiremettts for initial vahtes ttud to shorten iterative process. Then, 1. Introduction stabuit.r will he ensured.for sohttiotl qf eomplieated strong D' nonlhtear equations. Ushtg this otethod, some complicated strongly nonlhtear heat transfer probh, ms ht aiu~htnes am/ aeroenghws have been ly simuhtted successfully. It caji be used for numeriea/ shmdation of of common strongly interest. nonlhtear problems h7 enghwerhtg such as nonlinear li.rth'od.rnamics and Under aerodynamics, assumptions heat of transfer one-dimensional and structural plane dynamic detonation re.wojlse and etc. rigid flying plate, normal approach of solving problem of motion of flyor is to solve following system of equations Key words nonlinear equation, stability, Newton's method, auto-adjustable governing flow field of detonation products behind flyor (Fig. I): damping method, vector of damping factors I. Introduction ap +u_~_xp + au There are a lot of nonlinear problems au au in engineering. 1 For examples, heat transfer with nonlinear radiation-convection boundary and y limited =0, heat capacity, nonlinear hydrodynamics and aerodynamics, as nonlinear as structural dynanaic responses and l i ke. It is almost impossible to solve nonlinear equations by direct method except fo," some special nonlinear equations. The general approach is linearizing eqt'ations and forming various iteration procedures. For strongly nonlinear problems, solution obtained in itcrative where process p, p, is always S, u are uneasy, pressure, even density, divergence specific due entropy to and particle velocity instability, of detonation especially products for much respectively, more complicated with problems trajectory such R of as reflected shock of calculation detonation of wave D tempe,'atu,'c as a boundary distribution and trajectory F of flyor as anor boundary. Both are unknown; position of R and state parameters i Department on it are governed of Power by Engineering, flow field Nanjing I of central University rarefaction of Aeronautics wave behind and Astronautics, detonation Nanjing wave D and by initial stage of motion of flyor also; position of F and state parameters of products , P. R. China 163

2 164 Chang Haiping and Huang Taiping of wall of flame tube. The wall thickness is very thin. Heat is transferred to wall by flow convection and radiation of gas. When cooling film is used, heat flux by radiation heat transfer to wall is ag'.ainst that by convection. In this situation, divergence of solution in iterative process Occurs easily. Anor example is heat transfer problem in afterburning jet pipe of turboengine and rear fuselage fl'ame of engine cabin. Newton's method and its variants (with relaxation factor and or damping factor) can not settle this problem due to restricted requirement to initial vatucs and very slow convergence. Because re was not any effective way to solve this problem, simulation for predicting strongly nonlinear problems such as heat transfer with nonlinear radiation-convection boundary, is limited. A new; method, auto-adjustable damping method, has been presented in this paper. It can solve problem with relaxing restricted requirement to initial value, shortening iterative process and ensuring The one-dimensional stability tbr problem strongly of nonlinear motion system of a tu' rigid 121. Pl. flying plate under explosive attack has an analytic In order solution to relax only when restricted polytropic requirements index of to detonation initial products values equals in to three. In general, calculation a of nonlinear analysis is problems, required. In to this avoid paper, appearance however, by of utilizing ill-condition "weak" of shock behavior matrices of due to reflection nonlinear shock in parameters explosive during products, iterative and applying process, small a set parameter of vector pur- of terbation method, an analytic, first-order approximate solution is obtained for problem of flying damping factors has been introduced. These damping t:actors can be autonaaticatly adjusted plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final continuously' velocities during of flying plate iterative obtained process agree to very ensure well with calct, results lation by does computers. not diverge Thus an but analytic converge formula quickly) with Therefore. two parameters it is called of high as auto-adjustable explosive (i.e. detonation damping velocity method(ad;m). and polytropic There index) arc a for judgement estimation and of an adjustment velocity of processes flying plate in is ADM. established. A conception of effective convergence and quickening coefficient are employed. The effective convergence coefficient and quickening coefficient are determined 1. in accordance Introduction with properties of problem. The convergence rate is controlled by effective convergence coel;ficient and quickening coefficient Explosive in driven adjustrnent flying-plate process. technique ffmds its important use in study of behavior of II. General Methods for Solving Nonlinear Equations of common The general interest. lbrm of nonlinear equations is expressed as folloxvs. Under assumptions of one-dimensional plane detonation and rigid flying plate, normal approach of solving A(xl,x2,'",*.) problem of motion = of 0 flyor is to solve following system of equations governing flow field of k(=t,x2,'",x,,) detonation products = 0 behind flyor (Fig. I):. ~ f.(xt,x2,'",x,,) = 0 ap +u_~_xp + Rewritten as following shortened au form: au 1 au (i = 1,2,'",n)} (2.1) F(x) = 0 (2.2) as as The general iterative methods are as followsl4j: (1) Newton's method for solving p =p(p, nonlinear s), equation The main idea of Newton"s method is Successively linearizing nonlinear equations and where fornling p, p, an S, itcrative u are pressure, p,'ocedure. density, If tile specific vector entropy function and F(X), particle in a velocity neighborhood of detonation #cdincluding products respectively, Xk E D. can with be approximately trajectory replaced R of reflected by shock linear of l'unction detonation wave D as a boundary and trajectory F of flyor as anor boundary. Both are unknown; position of R and state parameters on it are governed by flow field LI,(X) I of = central AkX.+ rarefaction bl, wave behind detonation (2.3) wave D and by initial stage of motion of flyor also; position of F and state parameters of products n. nonlinear equation (2.1) will be app,'oximatcly replaced by linear equation Lk(X) = At.X+ bk = 0 (2.4)

3 Damping Method for Nonlinear Equations 165 The solution of equation (2.4) will be approximate solution of equation (2.1). This is so called linearizing method, which turns nonlinear problem.to be linear problem. Equation (2.4) is linearizing equation of equation (5.1). Then. F(Xk) = Lk(Xk) = A~Xk + bk (2.5) DF(Xk) = DLk(Xk) = Ak (2.6) Solving DF(Xk) and substituting it into equation (2.4), we obtain q(x) = DF(Xk)(X- Xk) + V(Xk) = 0 (2.7) If DF(Xt) is nonsingular, n Xk+1 = Xk - [DF(Xk)]-tF(Xk) (k = 0,1,2,'") (2.8) Equation (2.8) is called Newton's iterative formula for nonlinear equation (2.1). an analytic (2) Variants solution of only Newton's when method polytropic index of detonation products equals to three. In general, (a) a Newton's procedure analysis is with required. relaxation In this factor paper, however, by utilizing "weak" shock behavior In of order to reflection relax shock restricted in requirements explosive products, to initial and applying values, a relaxation small parameter factor wk pur- is terbation introduced method, into Newtofl's an analytic, procedure. first-order approximate solution is obtained for problem of flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities of flying Xk+l plate = Xt obtained - wt[df(xt)]-tf(xt) agree very well with (k = 0,1,'-') results by computers. (2.9)" Thus (b) Newton's procedure with damping factor In order to overcome singularity and or ill-condition of matrix DF(Xt).. a damping factor /zk is applied, n matrix becomes nonsingular and or recovers from illcondition 1. Introduction Explosive driven Xk+, flying-plate = Xk - [DF(Xt) technique + ffmds,ukl]-'f(xt) its important (k use = in 0, i,'-') study of behavior (2.10) of (c) Newton's method with relaxation and damping factors of common interest. Xt+l = Xt - wt[df(xt) + tzkl]-'f(xk) (2.11) Under assumptions of one-dimensional plane detonation and rigid flying plate, normal approach Although of solving methods problem mentioned of motion above of flyor may is relax to solve restricted following requirements system of equations to governing i,fitiat values, flow or field max of overcome detonation products Singularity behind and or flyor recover (Fig. from I): ill-condition of matrix DF(Xt). it is very difficult to ensure that all Xi(i-1,'",n) do not diverge but would quickly converge with selected value of cot and /-*k. for a complicated problem. Gene,'allv. it would appear that singularity ap of +u_~_xp some X + i is au overcome while convergence rate of or Xi decreases ve,-v much. It can au not fulfill au 1 eneineerin~ requirements. III. The Auto-Adjustable Damping Method as as In order to overcome shortages of above methods. Newton's method with relaxation and damping factors is fu,-r improved as Xk+, = Xt - oj[of(xt) +,u]-lf(xt) (k = 0,1,.'-) (3.1) where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, where with trajectory R of reflected shock of detonation wave D as a boundary and trajectory F of flyor as anor cot boundary. = diag(wkl, Both wtz,"', are unknown; oj~) position of R and state parameters on it are governed by flow field I of central rarefaction wave behind detonation wave,ut = diag(t, kt,,ut=,"',,u~) D and by initial stage of motion of flyor also; position of F and state parameters of products e < o2ti < (2- r -fl < /2ti < 7/,8 = mini{ I a.i 12/(2Re2i) I Re,l,i > 0}

4 166 Chang Haiping and Huang Taiping r/ = mini{ I Ai 12/- (2Re2~) I ReA~ < 0} where 13= + oo or 'r/= + no if re is no eigenvalue with ReAi> 0 or Re2~ < 0 respectively. The values of damping factors oju and /zta are different for each point X~. Furrmore, values are varied during iterative process. Because number of nodes of system is very large, it is necessary to have an effective algorithm for adjusting damping factors during iterative process automatically. Orwise, calculation procedure will be unable to execute. Equation (3.1) could be rewritten as Xk+~ = Xk + AXk AXk = ojl, E DF( X~) +/tt,]-lf(xt) (k = 0,1,-") (3.2) Let an analytic solution only when polytropic index of detonation products equals to three. In (~( Xk) = ajk[ DF( Xt) general, a analysis is required. In this paper, however, by utilizing "weak" shock (3.3) behavior of reflection shock r in = explosive AXu products, and applying small parameter purterbation Define method, convergence an analytic, rate as first-order approximate solution is obtained for problem of flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities of flying plate 2~ = obtained ~(Xk,i)/~(Xk_l.i) agree very well with (i = 1,2,'",n) results by computers. (3:4) Thus The convergence rate will be in following conditions: [ tou/v, and /zu invar: Ai ~<- a adjust~ 1. gu" Introduction v (if I-tu = O, npu =c), and tota invar; "wu/vand ] /z u 9 v;(if /-tta = 0, n /.tta = c) materials under - a < intense Ai < 0 impulsive do loading, not adjust shock synsis of diamonds, and explosive welding and cladding of metals. The method of estimation f C.Oki of II ) flyor and velocity ~k/ invar; and way of raising it are questions of common interest. 0 < Ai < b adjust~ I.tki/v, and r invar; Under assumptions of one-dimensional plane detonation and rigid flying plate, normal 1. approach of solving problem of motion tota9 of v, flyor and is l.zm/v to solve following system of equations governing ~.i >1 flow b field of detonation do not products adjust behind flyor (Fig. I): where a is effective convergence coefficient; b is quickening coefficient: a. b. c and v is real number respectively, where a ~< I, b ap < +u_~_xp 1,c ~< + 1 and au v >'1 One of calculation procedure au is as au lbllows. 1 (I) Giving an initial approximated valuc X0 and initial damping factor vector ato and /.to. At beginning, wo and /z o can as be selectcd as us ~ool and /zol. In order to adjust wu~ and /zki, effective convergence coefficient a and quickening coefficient b should be choosen in accordance with requirements or properties of calculation problem. For instance, a=0.95, 0.8, 0.7; b=0.1, 0.25; v=2. 5 or 10 even I00 or more can be choosen. where (2) p, p, Calculate S, u are F(Xo). pressure, and density, DF(Xo) specific. Solve entropy equatio~a and particle velocity of detonation products respectively, with trajectory R of reflected shock of detonation wave D as a boundary and trajectory F of flyor as anor ~(Xo) boundary. = ~o[ Both DF( are Xo) unknown; +/z0]-tf(xo) position of R and state parameters on it are governed by flow field I of central rarefaction wave behind detonation wave n Xl is obtained. D and by initial stage of motion of flyor also; position of F and state parameters of products (3) Calculate F(XI) and DF(Xl). Solve cqtmtion ~( XI) = ojo[ OF( Xt) + po]-i F( Xt)

5 Damping Method for Nonlinear Equations 167 n X2 is obtained. (4) Calculate Ai. If 21< -a, n adjust (oli/vl and,ul~ invar or adjust /tti'v (if /zll = 0 n /tti = e) and ~ll invar, or adjust r and /qi'v (if/~11 =0 n/~tl = e). If O< A~ < b,n adjust ojti'v and /zu invar or adjust ]lsti/v and call invar, or adjust o~ll. v and /tll/v. The new values tnl and/at replace tn0 and/z0 n do 3. (5) If kth approximation X~ has been obtained, calculate F(XI,) and DF(XI,). Solve equation #(Xt,) = coj,[ OF(Xk) +,uk]-tf(xk) n Xt, 1 is obtained. (6) Calculate 2,1. If ).i<- a, n adjust to~/v and,u~ invar or adjust,uta'v (if /ttu =0 n /zta = e) and tota invar, or adjust a~ta/v and,uu'v. If 0< Ai < b~ n adjust tota "v and,utl invar, or adjust /t~//v and to/a invar, or adjust wt~'v and,u~i/v. Then an do analytic 5. solution only when polytropic index of detonation products equals to three. In general, (7) a If I[ ~ (X t,) analysis H ~< e. is n required. Xk+l In is this approximation paper, however, which by utilizing has fulfilled "weak" accuracy shock behavior requirement. of Orwise.Xk reflection shock o~k in and explosive /zk replace products, Xk, and tok and applying,uk respectively, small parameter n do pur- 5. terbation Repeat method, above an analytic, steps until first-order requirement approximate is solution fulfilled. is obtained for problem of flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final IV.. velocities Calculation of flying Example plate obtained agree very well with results by computers. Thus an analytic As an formula example, with two parameters analysis of high of explosive temperature (i.e. detonation field of velocity rear and fuselage polytropic frame of an airplane has been done by using this method. The inner edge of frame is situated at position with high temperature radiation fi'om afterburning jet pipe. The secondary flow passes through passage between 1. Introduction afterburning jet pipe frame for cooling frame. The outer edge of frame is surrounded by high velocity air flow. The temperature field of frame is calculated under flight condition of Mach number M=2.0 at altitude H = 13000m. Without using ADM, solution still can not converge to required accuracy of after common more interest. than 400 iterations. When ADM is applied, effective convergence coefficient a=0.7, Under quickening assumptions coefficient of one-dimensional b=0.1 and v=2. plane 5. detonation 10 is selected and respectively, rigid flying plate, calculation normal approach resulis quickly of solving converge problem to required of motion accuracy of flyor by is only to solve 16 iterations. following Some system of of calculation equations governing results are listed flow field and compared of detonation products measuring behind data as flyor follows, (Fig. I): measuring data ap +u_~_xp + au point at upper part of outer edge au au ~ point at lower part of outer edge as as average temperature of inner edge 125 ~ calculation results 116 ~ ~ 139 "C (2 V. Conclusions where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products The iteration for solving tile complicated strongly nonlinear problems is always unable to respectively, with trajectory R of reflected shock of detonation wave D as a boundary and trajectory be convergent, F of flyor even as to anor be divergent boundary. due Both to are unknown; instability. position of It R can and not state fulfill para meters engineering on it are requirements. governed by The flow traditional field I of Newton's central rarefaction method and wave its behind variants detonation could not wave solve D this and problem. by initial stage Thus of motion application of flyor of also; position simulation of F and fo," state parameters complicated of products strongly nonlinear problems is limited. The new method, auto-adjustable damping method, presented in this paper, is a furr improvement o'" Newton's method with damping factor. The

6 168 Chang Haiping and Huang Taiping vector of damping factors is introduced and can be automatically adjusted corttinuously during iterative process in accordance with judgement and adjustment. The effective convergence and quickening coefficients have been.applied for relaxing l~he requirement to initial values and shortening iterative process, n stability can be eusured for complicated.strongly nonlinear problems. Some of complicated strongly nonlinear heat transfer problems in airplanes and aeroengines have been ly simulated successfully through this method. This method can be widely used for simulation in strongly nonlinear problems in engineering such as nonlinear flowing, heat transfer and structural dynamic responses and so on. References [!] Chang Haiping, T.he analysis for temperature distribution of frame of aircraft by FEM, CSAA-PC88273 (1988). an [2] analytic Chang solution Haiping, only The when polytropic calculation index for of detonation wall temperature products of equals flame to three. tube In of general, combustion a chamber, analysis CSAA-PC88-74 is required. In (1988). this paper, however, by utilizing "weak" shock behavior [3] Chang of Haiping( reflection Huang shock in Taiping explosive and Chen products, Wanbin, and applying Numerical small simulation parameter of pur- 3-D terbation temperature method, an distribution analytic, first-order of flame approximate tube of solution combustion is obtained for chamber problem with of air flying film plate driven cooling, by various Jottrnal high of Thermal explosives Science, with polytropic bltel"nati~ltal indices Joltrnal or of than Thermal but nearly and Fluid equal Sciences, to three. Final velocities of flying plate obtained agree very well with results by computers. Thus 5, I (I 996), index) [4] for Wang estimation Deren, of The Methodolog.r velocity of flying for Soh'ing plate is established. Nonlhrear Eqt~ations and Optimization, The People's Education Press, Beijing (1979), (in Chinese) 1. Introduction of common interest. Under assumptions of one-dimensional plane detonation and rigid flying plate, normal approach of solving problem of motion of flyor is to solve following system of equations governing flow field of detonation products behind flyor (Fig. I): ap +u_~_xp + au au au 1 as as where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products respectively, with trajectory R of reflected shock of detonation wave D as a boundary and trajectory F of flyor as anor boundary. Both are unknown; position of R and state parameters on it are governed by flow field I of central rarefaction wave behind detonation wave D and by initial stage of motion of flyor also; position of F and state parameters of products