Scientific Research of the Institute of Mathematics and Computer Science DIFFERENT VARIANTS OF THE BOUNDARY ELEMENT METHOD FOR PARABOLIC EQUATIONS

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1 Scieniic Research o he Insiue o Mahemaics and Compuer Science DIERENT VARIANTS O THE BOUNDARY ELEMENT METHOD OR PARABOLIC EQUATIONS Ewa Majchrzak,, Ewa Ładyga Jerzy Mendakiewicz, Alicja Piasecka Belkhaya Deparmen or Srengh o Maerials and Compuaional Mechanics Silesian Universiy o Technology, Gliwice Insiue o Mahemaics and Compuer Science Czesochowa Universiy o Technology, Czesochowa Absrac. In he paper he dieren algorihms o boundary elemen mehod or parabolic equaion are presened, his means he s and nd scheme o he BEM, he BEM using discreizaion in ime and he BEM using Laplace ransorm.. Inroducion The ransien emperaure ield in D domain oriened in Caresian co-ordinae sysem is described by he equaion T ( x, ) x : = a T ( x, ) () where a = λ/c (c is he volumeric speciic hea, λ is he hermal conduciviy), x = {x, x } denoes spaial co-ordinaes, is he ime. The equaion () is supplemened by ollowing boundary-iniial condiions: ( ) x : T x, = T ( ) ( ) (, ) T x x : q( x, ) = λ = q n = : T x, = T x b b () where T b is he known boundary emperaure, T ( x, ) / n is he normal derivaive a he boundary poin x, q b is he given boundary hea lux, T is he iniial emperaure. In order o solve he problem discussed, he ollowing varians o he boundary elemen mehod can be used: s scheme o he BEM [-3], nd scheme o he BEM [,, 4],

2 8 E. Majchrzak, E. Ładyga, J. Mendakiewicz, A. Piasecka Belkhaya he BEM using discreizaion in ime [,, 5, 6], he BEM using Laplace ransorm [].. s and nd schemes o he BEM Iniially, we ormulae he weighed residual crierion or he problem analyzed T ( x, ) a T ( x, ) T (ξ, x,, ) d d = (3) where [, ] is he ime inerval, T * is a undamenal soluion and i is a uncion o he orm [-4] r (ξ,,, ) = exp T x 4 π a ( ) 4 a ( ) (4) where ξ denoes a poin in which he concenraed hea source is applied, while r is he disance rom he considered poin x o he poin ξ. The normal hea lux o undamenal soluion should be ound in analyic way where * q ( ξ, x, T, ) = λ * ( ξ, x, n, ) = 8πa λd ( ) r exp 4a( ) d= ( x ξ )cos α + ( x ξ )cos α (6) (5) while cosα, cosα are he direcional cosines o he normal boundary vecor. In order o consruc he inegral equaion corresponding o he problem considered, he nd Green s ormula is applied or he irs componen o equaion (), while he second componen o he equaion () is inegraed by pars. Nex, using he properies o undamenal soluion [-4] one obains ollowing inegral equaion B(ξ ) T (ξ, ) + T (ξ, x,, ) q( x, ) d d = = q (ξ, x,, ) T ( x, ) d d + T (ξ, x,, ) T ( x, ) d (7) where B(ξ ) (, ]. The numerical approximaion o inegral equaion (6) consiss in he discreizaion o considered inerval o ime [, ], namely

3 Dieren Varians o he Boundary Elemen Mehod or Parabolic Equaions 9 + = < < K < < < < K < < (8). wih consan sep = In his place wo approaches can be aken ino accoun, i.e. he s or he nd scheme o he BEM. The idea o he s scheme consiss in he reamen o ransiion rom o as a cerain separae problem wih adequae pseudo-iniial condiion. In he case o he nd scheme he BEM he inegraion process sars rom = and hen he knowledge o successive pseudo-iniial condiions is needless, bu emporary values o boundary emperaures and hea luxes or =, =,..., = mus be regisered. So, i he s scheme o he BEM is used, hen he boundary inegral equaion (7) akes a orm [-3] B(ξ ) T (ξ, ) + T (ξ, x,, ) q( x, ) d d = = q (ξ, x,, ) T ( x, ) d d + T (ξ, x,, ) T ( x, ) d (9) while in he case o he nd scheme o he BEM one has [,, 4] s B(ξ ) T (ξ, ) + T (ξ, x,, ) q( x, ) d d = s s= s = q (ξ, x,, ) T ( x, ) d d + T (ξ, x,, ) T ( x, ) d s= s () 3. BEM using discreizaion in ime The boundary elemen mehod using discreizaion in ime consiss in he subsiuion o ime derivaive T/ or [, ] by he adequae dierenial quoien [,, 5, 6] and hen he equaion () can be wrien in he orm or (, ) (, ) T x T x [, ]: = a T ( x, ) T ( x, ) T ( x, ) + T ( x, ) = a a () ()

4 3 E. Majchrzak, E. Ładyga, J. Mendakiewicz, A. Piasecka Belkhaya Using he weighed residual crierion one has T ( x, ) T ( x, ) + T ( x, ) T ( ξ, x) d = a a (3) where T (ξ, x) is he undamenal soluion and or domain oriened in Caresian co-ordinae sysem i is a uncion o he orm [, 5, 6] r T ( ξ, x) = K π a (4) where K is he modiied Bessel uncion o zero order. The hea lux resuling rom he undamenal soluion is he ollowing T ( ξ, x) λ d r q ( ξ, x) = λ = K n π r a a (5) where K is he modiied Bessel uncion o irs order, d is deined in ormula (5). The boundary inegral equaion resuling rom ransormaion o equaion () can be expressed as ollows [, 5, 6] B(ξ ) T (ξ, ) + T (ξ, x) q( x, ) d = λ = q (ξ, x) T ( x, ) d T ( x, ) T (ξ, x) d λ + a (6) 4. BEM using Laplace ransorm We inroduce he Laplace ransorm [] [ ] L T ( x, ) = U ( x, s) = T ( x, ) e d (7) where s (real number) is he ransormed parameer. Because s T ( x, ) L = su ( x, s) T ( x, ) so he equaion () can be ransormed as ollows (8) su ( x, s) T ( x, ) = a U ( x, s) (9)

5 Dieren Varians o he Boundary Elemen Mehod or Parabolic Equaions 3 while he boundary condiions ake a orm: ( ) ( ) x : U x, s = U = T / s b b x : Q x, s = Q = q / s b b () Using he weighed residual one has s U ( x, s) U ( x, s) + T ( x, ) U ( ξ, x, s) d = a a () where U ( ξ, x, s) is he undamenal soluion and or D domain oriened in Caresian co-ordinae sysem i is a uncion o he orm [] s U ( ξ, x, s) = K r π a () The hea lux resuling rom he undamenal soluion is he ollowing U ( ξ, x, s) λ d a s Q ( ξ, x, s) = λ = K r n π r s a (3) Aer a cerain mahemaical manipulaions we obain he boundary inegral equaion B(ξ ) U (ξ, s) + U (ξ, x, s) Q( x, s) d = λ = Q (ξ, x, s) U ( x, s) d T ( x, ) U (ξ, x, s) d λ + a (4) 5. Resuls o compuaions The square domain o dimensions. m. m has been considered. The ollowing hermophysical parameers have been assumed: hermal conduciviy λ = 33 W/mK, volumeric speciic hea c = J/m 3 K, iniial emperaure T = C. On he righ and upper suraces he Dirichle condiion T b = 5 C has been acceped, on he le surace he Robin condiion q(x, ) = α(t T ), where α = 5 W/m K, T = 3 o C has been aken ino accoun. On he lower surace he no-hea lux condiion has been assumed. The boundary o he domain considered has been divided ino 4 consan boundary elemens, while he inerior has been divided ino consan inernal

6 3 E. Majchrzak, E. Ładyga, J. Mendakiewicz, A. Piasecka Belkhaya cells. Time sep: = s. The deails concerning numerical realizaion o dieren varians o he BEM can be ound in [-5]. In igures and he emperaure ield or imes 4 and 8 s is presened. The soluions have been obained boh applying he s scheme o he BEM as well as he BEM using discreizaion in ime. The dierences beween hese soluions are very small. ig.. Temperaure ield or = 4 s ig.. Temperaure ield or = 8 s Reerences [] Brebbia C.A., Telles J.C.., Wrobel L.C., Boundary elemen echniques, Springer-Verlag, Berlin, New York 984. [] Majchrzak E., Meoda elemenów brzegowych w przepływie ciepła, Wyd. Poliechniki Częsochowskiej, Częsochowa. [3] Piasecka A., Modelowanie procesu krzepnięcia meali i sopów za pomocą meody elemenów brzegowych, Poliechnika Śląska, Gliwice 996. [4] Mendakiewicz J., Symulacja krzepnięcia żeliwa jako sposób oceny jego skłonności do zabieleń, Poliechnika Śląska, Gliwice 994. [5] Ładyga E., Zasosowanie meody elemenów brzegowych z dyskreyzacją czasu do modelowania nieusalonej dyuzji, Poliechnika Częsochowska, Częsochowa 997. [6] Szopa R., Modelowanie krzepnięcia i krysalizacji z wykorzysaniem kombinowanej meody elemenów brzegowych, Hunicwo 54, Wyd. Pol. Śląskiej, Gliwice 999.