The method  presented  in  the paper ensures consistent  convergence  acceleration  for  the so-
lution  of the diffusion  problem  in  electromagnetic field  by finite  difference  and  the pseudo-
transient  methods  of integration  of the partial  differential  equation  system,  providing it is 
a  priori  known that the solution is  periodical.  The effectiveness  of the method is  illustrated 
in  the  numerical  results  of an  actual  problem

Füzi, János

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODICA POLYTECHNICA SER. EL. ENG. VOL. 39, NO. 1, PP. 53-62 (1995) A METHOD FOR CONVERGENCE ACCELERATION IN PSEUDO-TRANSIENT INTEGRATION OF DIFFUSION EQUATION SYSTEMS Department of Electrical Engineering 'Transilvania' University of Brasov 2200 Brasov, Romania Str. Politehnicii Nr.1-3 Fax & Phone: (+40 68) 152362 Received: March 5, 1995 Abstract The method presented in the paper ensures consistent convergence acceleration for the so-lution of the diffusion problem in electromagnetic field by finite difference and the pseudo-transient methods of integration of the partial differential equation system, providing it is a priori known that the solution is periodical. The effectiveness of the method is illustrated in the numerical results of an actual problem. Keywords: finite difference method, problem of diffusion in electromagnetic field, pseudo-transient method. 1. Description of the Method The method of finite difference is relatively often used in the solution of the electromagnetic field diffusion problem in conductive media [1], especially in cases of non-linear (e.g. ferromagnetic) media, where analytical solutions are not at hand. A comfortable way to find the permanent solution of the problem is the pseudo-transient method applying to a domain with zero initial field values the source term at its borders and carry out the computations until the stabilised solution occurs. The main inconvenience consists of the low rate of convergence leading to large values of computing time. In case of the expression of the source term and the field equations guarantee a periodical solution, with zero mean value in each point of the considered domain, the following strategy leads to important gain in convergence speed without loss of stability or accuracy: - carry out computations for several periods of the source term, \vith zero initial values (pseudo-transient method). 1 The paper has been accom plished during a post-grad uate scholarship at the Faculty of Electrical Engineering of the Technical University of Budapest. 54 1. FUZE symmetrize the obtained solutions. This means the computation of the mean value for the solution functions in each spatial grid point and its subtraction from the actual value at the end of the last period determined. The values thus obtained are the initial values for further corn pu tations. continue computations for several other periods. repeat symmetrization if needed. The effects of this strategy are illustrated by an actual problem. z B [T] o 30 60 90 H [A/cm] Fig. 1. Geometrical configuration and magnetisation characteristic of a cylinder 2. The Symmetrizing Technique ~20 Let f(t) be a periodical function of time, with the follo·wing feature: f (to + ~) = - f(to) , (1) where T stands for the period and a(t) a combination of it ·with a linear function: a(t) = mt + n + f(t) . (2) The mean value of a(t) computed for a period is: (3) CONVERGENCE ACCELERATION 55 while the mean value of f(t) is zero. This case approximates quite well the structure of the functions the intermediate solution of most eddy current problems stand of. In fact the real structure contains an exponential instead of the linear function, nevertheless, due to the lmv rate of decrease of the exponential, it can be replaced by a linear function during a period of the source term, providing to be far enough from the starting moment of computing process (i.e. we have computed several periods). Because of the low value of m (the decrease factor), we can approximate the mean value of a(t): (4) thus simplifying the algorithm, only the extremes of a(t) having to be determined, to have the approximation of the mean value. An even better result can be obtained by the following correction term instead of the mean value: .6. = amax + amin _ a(to) - a(to + T) . 2 2' (5) including the effect of the decreasing trend in the mean value. Applying ao = a(to +T) -.6. (6) as initial value for the next period, the permanent solution of the problem is much closer to, as it is illustrated in the following example. 3. An Actual Problem Let us consider a cylinder made of ferromagnetic material, plotted in Fig. 1, 'with electrical conductivity: ()" = 5 . 106 S/m. The magnetic permeability varies with respect to the magnetic field intensity according to: (7) where !lo = 471 .10-7 H/m is the permeability of free space and the coef-ficients ml = 0.85 T; kl = 0.00328 m/A; m2 = 0.4 T; k2 = 0.00024 m/A have been determined based on the minimising of error square by means of a non-linear optimisation algorithm [5] in a manner that the model thus obtained would be fitted to the measured data of the anhysteretic mag-netisation curve of a certain material (steel). In Fig. 1 the circles stand for measured points of the magnetisation curve and the plotted curve was ob-tained by Eq. (7). 56 1. FUZI 1.0 , .......................... , .......................... c·················· .... • .. J· ...... •· .. ·••••• ...... ···T ........ ············· .... "· .. ······················c O.S 1 ................................................. 1 ........ ~~~~ 0.6 ; .. ---.----;.--.----,.'iW-0.0 r---!lir----j----j-------j"'----i---;,ffIL-'-----j -0.2 ;.-".---... -+-.--.-.-.. -.---'-... -.-... -.. ---+ -0.4 ; ............. "'l~ =............. ......................... ··I·~r.:· .. ·· .. ·······l·········· .. ·········;··· .... ······ .... · ......... !, -0.6'·· .. ··· ........ ···"···· ...... '- .......... J, ...................... L .............. , ........ 0 -150 -100 -50 o 50 100 150 H [A/cm] Fig. 2. Pseudo-transient magnetisation loop as computed without convergence acceleration A magnetic field of maximum intensity HOrn = 105 . Vi A/m and frequency f = 500 Hz is applied at the surface of the cylinder with radius ro = 20 mm. In the eddy current simulation; the governing equations of the elec-tromagnetic field are [2]: rot H = J, aB rotE=--. at ' div B = 0, J = O"E , (8) B = JL(H)H, with the non-linear magnetic permeability JL(H) given in Eq. (7) . Applying a z- directed external magnetic field at the surface a magnetic field results parallel to the axis and azimuthal oriented current density in the material: H = kH(r, t); J li<pJ(r, t) . (9) CONVERGENCE ACCELERATION 57 1.0 . r--'I ~ 0.8 S O.S '---' 'S. 0.4 0.2 0.0 -0.2 -0.4 ..•. " ...• " •..•.. ~ .. " .••. " ................... j -O.S ......... _ ... _--_.-.--.. -_ .. __ .--0.8 H [A/c~] -150 -100 -50 0 50 100 150 Fig. 3. Pseudo-transient magnetisation loop as computed with the simple convergence acceleration scheme The equations of the electromagnetic field yield the following differential equation, written in cylindrical co-ordinates [3]: oH ot with the following boundary conditions: oH or (0, t) = 0, H(ro, t) = HOrn sin 271 ft. ; r E [0, ro], t ~ 0 , (10) t ~ O. (11) According to the solution in time domain obtained by the pseudo-transient method the initial condition is selected as: H(r,O) = 0; r E [0, ro] (12) 58 J. FCJZI 1.0 r-"1 ~ 0.8 S 0.6 L.....J 'S. 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 H [A/cm] -150 -100 -50 0 50 100 150 Fig. 4. Pseudo-transient magnetisation loop as computed with the refined convergence acceleration scheme Formulating the solution by finite difference method, discretisation is per-formed over a grid selected as: T = j;, t = IT j = O,n l '2. 0 n = 200 T = .) . 10-7 S (13) A three time-level discretisation scheme had been chosen, with the follo\,,ring approximations of the partial differentials: HI+ 1 HI+l HI HI H I - 1 H I - I aH 1 j+l - j-I + HI - j-I + j+1 - j-I -a-r (r) t) ~ 3' . --"-~--"--=-----''-'---:h'----=-------'.-'-''---'--=- (14) 2 H I+ I _ H I+ I 2H I+ I + HI, ..L HI _ 2HI ..L H I - I ..L H I - I _ 2H I- I a H (ro t) ~ ~ . --=.j..:.+.:::.l_---=-j_-.:::.I __ .=...l __ -=-l-_, _1_'--,..:;;1-_1 __ 10-'_,,-1-"-.:.., =-1 _'_"-1-....:1=--_-'1'---ar2' 3 h2 where Hj = H(jh, IT) h = TO n (15) CONVERGENCE ACCELERATION 59 1.50 90 H !:A/mm] 60 -60 I -90 , -l.2O -1.50 0.0 0.5 1..0 1..5 2.0 t [ms1 Fig. 5. Magnetic field intensity \vith respect to time and radius The reason for selection of a three-level scheme is the presence of the vari-able coefficient in the denominator of the right side of Eq. (10). Approxi-mating it with a central value in both spatial and time direction, the fol-lowing solution can be applied: (16) The system resulted can easily be solved by Gausslan elimination [4] at every time level. Due to the cylindrical symmetrv. at the axis ofthe cylinder the condition . (82 H 1 8 H) 82 H lim -8 2 + --8 = 2-8 2 (O,t) r-O r r r r (17) has to be followed. The condition of convergence, the criteria for stopping computations, IS: H(r, t + T) = H(t); r E [0, ro] . (18) 60 B [ T ] J. FlJZI 2.4 /,,/'j/Ni/'~l : _~ -~H-H--f-' i lr-'~~: .;; i i i ' I \~~ ...... 1 ... ~ ..... ~ ..... L ......... ' ....... ' ....... ·····f\::::;···i"~····'·······i ...... i ....... : : ! i !bin V : I i"-~: ······:······i ..... , ····1-······f "7 :::~:,···] .. ·····l ~ ! -- . _)c.-._ ·1J. :.15~5 ~', : vt--H 2.0 1.6 .1.2 O.B 0.4 0.0 : ~,{t-!---il--"':r'- --r -0.4 4-!\I--+--;~-J...i-..lJ~l/-I+ -O.B I -.1.6~: \ ~\ ....-2.0;-' -: +! +-~-H-t"~\~\~\~'~: ~. : i.. I': -2.4 , .......•....... i ... _ ... L .....•....... , ............ _, ....... i ....... i ....... i ..... ' ....... , .....•...... J ....... i ..... ' .. _ ..•............... .! ..... . 0.0 0.5 1.0 1.5 2.0 t [ms] Fig. 6. Magnetic flux density with respect to time and radius \Vith the calculated values of the magnetic field intensity the corresponding flux densities can be computed and integrated over the cross-section of the cylinder, to find the total magnetic flux at each temporal grid level (19) Providing the flux density functions are periodical with zero mean value, the total flux is expected to be a similar function of time as well. Plotting the total flux with respect to the field intensity at the surface of the cylinder, information can be obtained regarding the level of convergence achieved by the level of symmetry of the resulting loop with respect to the co-ordinate axes. CONVERGENCE ACCELERATION 61 V 2.0 E-! • ~ ti~ (U a:J 60 1..5 r [mm] Fig. 7. Effective values of magnetic field intensity and flux density with respect to radius 4. Numerical Results Computations ,"",ere carried out for the example given above in three cases. First, without any convergence-accelerating procedure, ten periods of the source field lead to the result shown on Fig. 2 where the extremely low rate of convergence can be observed. Then the simple accelerating procedure was applied after the third and following periods, just by subtracting the mean values of the field intensity functions computed for each period from the actual values at the end of the respective periods. In Fig. 3 an oscillating process can be observed around the symmetrical position of the loop. The permanent solution could be reached after six seven periods, a serious improvement regarding the situation presented in Fig. 2. Further improvement can be seen in Fig. 4 obtained by the refined accelerating process, taking into account the decreasing tendency of the mean values as well. \Vith this strategy the permanent solution can be reached as soon as the fourth period of the source field, while the track of 62 the operating point follows in the fifth period the loop plotted during the fourth period. The resulting wave-forms of the magnetic field intensity and flux den-sity functions with respect to time obtained by means of the refined accel-eration scheme can be seen on Figs. 5 and 6, for different depths in the cylinder. The symmetry of these wave-forms stated in Eq. (1) can be no-ticed as well as the fact that their mean value is zero. Thus one can con-clude that a fairly sufficient convergence level has indeed been achieved. Although it is not represented in Figs. 5 and 6, due to scale incompatibil-ity, the wave-forms at much smaller radius, closer to the axis, maintain the symmetry mentioned above. Computing the effective values of field inten-sity and flux density at different depths, and plotting the graph given in Fig. 7, the skin-effect occurring under the conditions given at the begin-ning of point 3 can be observed. 5. Conclusions The method described in point 1 leads to substantial gain in convergence speed applying the pseudo-transient algorithms for solution of diffusion problems in time domain. Implementation of the method into the existing program source does not require much supplementary programming effort and does not affect the accuracy of the obtained results. Combined with an appropriate finite difference algorithm it provides a comfortable and fast code for simulation of electromagnetic field in both linear and non-linear media. References 1. IV.4.NYI, A. (1994): Numerical Methods in Computer Aided Field Simulations, (in Hungarian), Technical University of Budapest. 2. SZABO, W. (1977): Fundamentals of Electrotechnics, (in Romanian), University of Brasov. 3. FUZI, J. (1994): Eddy Currents in a Ferromagnetic Cylinder Induced by an External Magnetic Field, Proceedings of the OPTIM'94 Conference Brasov, Vol I, pp. 63-66. 4. DAUTRAY, R. - LIONS, J-L. (1993): Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5-6: Evolution Problems, Springer-Verlag, Berlin-Heidelberg. 5. Fuzy, J. (1993): Non-linear Optimisation. Collection of Problems, (in Romanian), University of Brasov. 

A METHOD FOR CONVERGENCE ACCELERATION IN PSEUDO-TRANSIENT INTEGRATION OF DIFFUSION EQUATION SYSTEMS

https://pp.bme.hu/ee/article/download/4436/3541

A METHOD FOR CONVERGENCE ACCELERATION IN PSEUDO-TRANSIENT INTEGRATION OF DIFFUSION EQUATION SYSTEMS

Abstract

Similar works

Full text

Available Versions

Periodica Polytechnica (Budapest University of Technology and Economics)