Heat Conduction Basic Research Part 15 ppt

Meshless Heat Conduction Analysis by Triple-Reciprocity Boundary Element Method 339 Numerical solutions are obtained using the interpolation functions for time and space. If a constant time interpolation and time step 1 () kk tt   are used, the time integral can be treated analytically. The time integrals for * (,,,) f T pq t  are given as follows: * 11 1 (,,,) ( ) 4 F f t f t Tpqt d Ea     , (71) * 1 (,,,) 1 exp( ) 2 F f t f t Tpqt r da nrn         , (72) where 2 4( ) f F f r a tt    . (73) Assuming that functions (,)TQ  and (,)TQ n    remain constant over time in each time step, Eq. (65) can be written in matrix form. Replacing (,)TQ  and (,)TQ n    with vectors Tｆ and Qｆ, respectively, and discretizing Eq. (65), we obtain (Brebbia ,1984) 11 FF ff   fF f fF f 0 HT GQ B , (74) where B 0 represents the effect of the pseudo-initial temperature. Adopting a constant time step throughout the analysis, the coefficients of the matrix at several time steps need to be computed and stored only once. If there is heat generation, the following time integrals are used (Ochiai, 2001). 2 * 211 1 (,,,) { ( ) [ ( ) 16 F f t ff t f r T pq t d Ea Ea a     ln( ) 1 exp( )]} ff aC a    (75) * 2 1 1exp( ) (,,,) [()] 8 F f t f f t f a Tpqt rr dEa nna          (76) 4 * 311 1 (,,,) {() [4() 256 F f t ff t f r T pq t d Ea Ea a     2 1 4ln( ) 4 1 exp( )] [2 ( ) fff f aC a Ea a  2ln( ) 2 2 3 3exp( ) 5 ]} ff ff aCa aa     (77) * 3 3 1 2 1exp( ) (,,,) {() 64 F f t ff f t f aa Tpqt rr dEa nn a           1 1 [2 ( ) 2ln( ) 2 1 exp( )]} ff f f Ea a C a a  (78) Heat Conduction – Basic Research 340 Additionally, the temperature gradient is given by differentiating Equation (65), and expressed as: 2* 1 0 (,) (, ,,) [( ,) t ii Tpt T pQt TQ xxn        * 1 (, ,,) (,) ] i TpQt TQ dd nx         * 2 1 0 1 (, ,,) ( ,) (1) [ t ff f i f TpQt WQ xn            2* 1 (, ,,) (,)] f f i TpQt WQ dd xn         * 3 3 0 1 (, ,,) (,) M t P m m i m Tpq t Wq d x          *0 2 1 1 (, ,,0) ( ,0) (1) [ ff f i f TpQt TQ xn         2* 1 0 (, ,,0) (,0)] f f i TpQt TQ dd xn      * 0 3 3 1 (, ,,0) (,0) M P m m i m Tpq t Tq d x       (79) The derivative of the polyharmonic function * (,,,) f TP q t  and the normal derivative with respect to i x in Eq.(79) are explicitly given by * 1 22 (,,,) , exp( ) 8( ) i i Tpqt rr a x t          , (80) 2* 1 22 (,,,) 1 [exp( )2, exp( )] 8( ) ii i Tpqt r naar a xn n t            , (81) * 2 (,,,) i Tpqt x      )exp(1 2 , a r r i   , (82) 2* 2 (,,,) i Tpqt xn    2 1 {[1exp( )]2, [1exp( ) exp( )]} 2 ii r nar aaa n r     , (83) * 3 1 (,,,) , 1exp( ) () ln() 1 8 i i Tpqt rr a Ea a C xa            , (84) 2* 3 1 (,,,) 1 1 exp( ) 1 exp( ) { [ () ln() 1 ] 2, [1 ]} 8 ii i Tpqt ar a nE a a C r xn a n a          , (85) where i i xrr  /, . The time integrals for * / f i Tx   and 2* (,,,)/ fi TP q txn    in Eq. (79) are given as follows: * 1 (,,,) , exp( ) 2 F f t i f t i Tpqt r da xr         , (86) Meshless Heat Conduction Analysis by Triple-Reciprocity Boundary Element Method 341 2* 1 (,,,) F f t t i Tpqt d xn      2 1 [2,(1)]exp() 2 ii ff r nr a a n r     , (87) * 2 1 1exp( ) (,,,) [()] 8 F f t f f t iif a Tpqt rr dEa xxa          , (88) 2* 2 (,,,) F f t t i Tpqt d xn      1 11 1 { ( ) [1 exp( )]} 2 , [1 exp( )] 8 if f i f ff r nEa a r a ana     , (89) * 3 3 2 11 1exp( ) (,,,) , { 64 1 ( ) [2 ( ) 2ln( ) 2 1 exp( )]} F f t ff i t i f fff f f aa Tpqt rr d x a Ea Ea a C a a          , (90) 2* 2 3 11 1 2 2 (,,,) 2 {() [()ln() ] 64 1 [1 exp( ) exp( )]} 2 , { ( ) 21 [1 exp( )] [1 exp( )]} . F f t if f f t if ff f i f f ffff f f Tpqt r d nEa Ea a C xn a r aa a r Ea n a aaaa a a                (91) 3. Numerical examples To verify the accuracy of the present method, unsteady heat conduction in a circular region with radius a, as shown in Fig. 6, is treated with a boundary temperature given by [1 cos( )] H TT t    . (92) We assume an initial temperature T ０=0 C  , and R denotes the distance from the center of the circular region. A two-dimensional state, in which there is no heat flow in the direction perpendicular to the plane of the domain, is assumed. Figure 6 also shows the internal points used for interpolation. A thermal diffusivity of   16 mm２/s and a radius of a=10 mm are assumed. H T =10 C  in Eq. (92) and a frequency of /2    rad/s are also assumed. The BEM results at R =0 and R=8 mm and the exact values are compared in Fig. 7. The exact solution for the temperature distribution is given by (Carslaw, 1938) 22 (,) [1 cos H ber Rber a bei Rbei a TRt T t ber a bei a           Heat Conduction – Basic Research 342 Fig. 6. Circular region with temperature change at the boundary. Fig. 7. Temperature history in circular region. 22 sin ber Rbei a ber abei R t ber a bei a           3 2 2 0 '242 1 0 () 2 exp( ) ] ()( ) ss s s ss JR t a Ja          (93) where ber( ) and bei( ) are Kelvin functions, and s  ( s=1, 2, ) are the roots of 0 ()0Ja   . Constant elements are used for boundary and time interpolation. Meshless Heat Conduction Analysis by Triple-Reciprocity Boundary Element Method 343 Appendix A (3D) The higher-order functions for 3D unsteady heat conduction are  * 2 ,,,Tpqt  1/2 3/2 1 {(1.5,) [1exp()]} 2 aa a r    3/2 1 (0.5, ) 2 a r     (A-1)   * 2 3/2 2 ,,, 1 (1.5, ) 2 Tpqt r a nn r         (A-2) )]}exp(1[3 1 ),5.1(3),5.2(3),2(6),5.1(3{ 12 2/12/112/1 2/3 * 3 aa a aaaaaaa r T     )]}exp(1[2 1 ),5.1(2),5.0({ 4 2/1 2/3 aa a aa r     (A-3) . n r a a a n T      )],5.1( 1 ),5.0([ 4 1 2/3 * 3   . (A-4) where (,)  is an incomplete gamma function of the first kind (Abramowitz, 1970) and ,/ ii rrx  . Using Eqs. (44) and (A-3), the polyharmonic function with a surface distribution is obtained as follows:  3/2 *3/23/2 32211212211 1/2 2( ) {2 (1.5,)2 (1.5,)2(3,)23, 6 (2,)6 (2,) 3 B Akt Tuuuuuuuuuu r           1/2 1/2 2211 62.5,62.5,uuuu   22 21 11 22 uu      1/2 1/2 221121 61.5,61.5,6(2,)6(2,)uuuuuu    21 2 1 3 3 3exp( ) 3exp( )}uu u u     , (A-5) where 2 1 () 4( ) rA u t      (A-6) 2 2 () 4( ) rA u t      . (A-7) The time integral of Eq. (62) can be obtained as follows: * 1 3/2 1 (,,,) (0.5, ) 4 F f t f t T pq td a r     (A-8) * 1 3/2 2 (,,,) 1 (1.5, ) 2 F f t f t Tpqt r da nn r         (A-9) * 2 (,,,) F f t t T pq td    3/2 1/2 12 [ (1.5, ) (0.5, ) ( 0.5, )] 8 ff f f f r aa a a a     Heat Conduction – Basic Research 344 3/2 1 [ (0.5, ) ( 0.5, )] 8 ff f r aa a    (A-10) * 2 3/2 (,,,) 11 [ (1.5, ) (0.5, )] 8 F f t ff t f Tpqt r daa nna         (A-11) * 3 (,,,) F f t t Tpqt d    3 3/2 1 [ 6 (1.5, ) (0.5, ) 96 ff f r aa a     3/2 1 8(2, ) f f a a   2 1 3(2.5, ) f f a a   1/2 4 f a  2 1 3(1.5, ) 3(0.5, ) ff f aa a   3/2 4 6(1.5, )] f f a a  (A-12) * 3 (,,,) F f t t Tpqt d n      2 3/2 1 [6 (1.5, ) 3(0.5, ) 96 ff f rr aa na       2 1 3(2.5, ) f f a a   1/2 8 f a  2 3 (1.5, ) 3 ( 0.5, )] ff f aa a   , (A-13) where 2 4( ) f F f r a tt    (A-14) and (,) is an incomplete gamma function of the second kind (Abramowitz, 1970). The time integral of Eq. (A-5) can be obtained as follows: * 3 (,,,) F f t B t T pq td    5 11 1/2 () 11 {2 (1.5, ) (0.5, ) 5 48 ff f ArA aa a r      1 5/2 1 41 (3, ) 5 f f a a   1 3/2 1 1 4(2, ) f f a a   1 2 1 1 3(2.5, ) f f a a   1/2 1 1 f a  11 2 1 13 3 (1.5, ) ( 0.5, ) 5 ff f aa a   1 5/2 1 12 1 (2, ) 5 f f a a   3/2 1 2 f a  1 3(2.5, )} f a   , (A-15) where 2 1 () 4( ) f F f rA a tt     . (A-16) For the sake of conciseness, the terms involving 2 u in Eq. (A-5) are omitted. The derivative of the polyharmonic function * (,,,) f TP q t  and the normal derivative with respect to i x are explicitly given by * 1 35 22 exp( ) 16 [ ( )] ii Tr r a xx kt        (A-17) Meshless Heat Conduction Analysis by Triple-Reciprocity Boundary Element Method 345 * 1 i T nx     35 22 1 2, ,exp() 16 [ ( )] ijji nurnr a kt     (A-18) * 2 3/2 2 13 , 2 2 ii Tr a xx r           (A-19) * 2 i T nx     3/2 3 13 5 ,2,,, 22 2 ii jj na arnr r                (A-20) * 3 3/2 1 11 3 ,, 22 8 ii T r aa xux                     (A-21) 2* 3 3 2 11 3 { [ (0.5, ) (1.5, )] , , [ (0.5, ) (1.5, )]} 8 iijj i T na arrna a nx u u r         (A-22)  * 3/2 3/2 3 11 1 1/2 2( ) 1 [ {2 (1.5, ) 2 3, 3 B ii dT akt r uu u dx x r r       11 6(2,)uu     1/2 11 62.5,uu   2 1 1 2 u   1/2 111 61.5,6(2,)uuu   11 33exp()}uu   1/2 11 1 2 {3 (1.5 , ) uu r   1 6(2, )u     3/2 11 32.5,uu    1    3/2 11 31.5,uu    1 1 3u   1 11 3 exp( )}]uu   (A-23) The time integrals of Eqs. (A-18), (A-20) and (A-22) can be obtained as follows: * 1 F f t t i T d nx     3 3 2 153 2, , , , 22 2 ijj f i f rnr a n a kr           (A-24) * 2 F f t t i T d nx     3 2 113 11 (3,,) , ( ,,)[ ,] 222 8 iijj f iijj f f nrnr a nrnr a a kr                    (A-25) 2* 21 1/2 3 3/2 21 (,,,) 311 3,6,3,16 222 192 311 ,, 9 , 6 , 3 , 222 F f t if f f f f f t i ijj f f f f f Tpqt r dnaaaaaa nx rrn a a a a a                                                    (A-26) * 3 F f t B t i T d x      4 1/2 211 {2 1.5, (0.5, ) 32 5 3 ff if arr aa xk a       5/2 41 3, 5 f f a a    3/2 1 42, f f a a    2 1 32.5, f f a a   1/2 1 f a  Heat Conduction – Basic Research 346   2 13 31.5, 0.5, 5 ff f aa a    5/2 12 1 2, 5 f f a a   2/3 2 f a  3(2.5,)} f a   (A-27) Appendix B (1D) The functions for 1D unsteady heat conduction are   * 2 ,,,Tpqt  12 12 (,0.5) exp( ) 2 r aa a         (B-1)   * 2 1/2 ,,, 1 (0.5, ) 2 Tpqt r a nn         (B-2)  3 *11/23/23/2 3 12 , , , {( 1.5) (0.5, ) exp( ) 2 exp( )} 12 r T pq taaaaaaaa       (B-3)  * 2 3 11 12 ,,, [(1 ) (0.5, ) (1.5, )] 4 Tpqt rr aaaa nn         , (B-4) where (,)  is an incomplete gamma function of the first kind (Abramowitz, 1970). The time integral of Eqs. (49) and (B-1)-(B-4) can be obtained as follows: 1 12 0.5 1exp( ) (,,,) [(0.5, ) ] 2 F f t f f t f a r Tpqt d a a       (B-5) * 1 (,,,) F f t t Tpqt d n      12 1 (0.5, ) 2 f r a n     (B-6) * 2 (,,,) F f t t Tpqt d    12 32 3 12 2 () 122 (0.5, ) (0.5, ) exp( ) 33 8 ff ff f f f aa r aa a a a              (B-7) * 2 (,,,) F f t t Tpqt d n      2 12 12 11 { (0.5, ) 2[ (0.5, ) exp( )]} 8 ff f f f rr aa a na a       (B-8) 5 * 3 12 2 1/2 3/2 5/2 5/2 34 ( , , , ) {15( ) (0.5, ) 12 (0.5, ) 2880 294 48 12 (0.5, ) 6( )exp( ) } F f t ff t f f ff f fff r Tpqt d a a a a aa aaa a        (B-9) * 4 3 12 2 12 32 exp( ) 2exp( ) (,,,) 11 1 {( ) (0.5, ) (0.5, ) } 3 2 16 3 3 F f t ff ff t f f ff aa Tpqt rr daa nna a aa          , (B-10) Meshless Heat Conduction Analysis by Triple-Reciprocity Boundary Element Method 347 where 2 4( ) f F f r a tt    . (B-11) Appendix C (Linear time interpolation) The time integrals of Eq. (62) using linear time interpolation in the two-dimensional case can be obtained as follows:  1 22 1 * 1 11 1 1 exp( ) exp( ) 1 () ( ) ()() 44 4 f f t ff fFfff t ff aa rr tTd tt Ea Ea aa                              (C-1)  1 22 1 * 11 1 1 1 1 1 exp( ) exp( ) 1 () ( ) ()() 44 4 f f t ff fFfff t ff aa rr tTd tt Ea Ea aa                             (C-2) 1 * 1 () f f t f t T td n        1111 2 11 1 ()exp()exp() ()() 24 Ff f f f f r t t a a Ea Ea n R                   (C-3) 1 * 1 1 () f f t f t T td n         11 111 2 11 1 ()exp()exp() ()() 24 Ff f f f f r tt a a Ea Ea n R                     (C-4) 1 * 2 ()(,,,) f f t f t tTpqtd      2 1 1 ()ln() 1exp( ) 16( ) ( ) 256 ff f Ff f f Ea a C a r ttEa a        2 1 1 2 1exp( ) 2()2ln( )2 1exp( ) () fff ff f f f aEa aC aa r Ea a a            2 111 1 1 1 11 2 1 1 1exp( ) 2( )2ln( )2 1exp( ) () fff ff f f f aEa aC aa r Ea a a                       (C-5) 1 * 12 ()(,,,) f f t f t tTpqtd       2 1 11 ()ln() 1exp( ) 16( ) ( ) 256 ff f Ff f f Ea a C a r tt Ea a             2 1 1 2 1exp( ) 2()2ln( )2 1exp( ) () fff ff f f f aEa aC aa r Ea a a            Heat Conduction – Basic Research 348 2 111 1 1 1 11 2 1 1 1exp( ) 2( )2ln( )2 1exp( ) () fff ff f f f aEa aC aa r Ea a a                       (C-6) 1 * 1 2 11 1 1 1exp( ) 1exp( ) () ()() ()() 8 f f t ff fffff t ff aa Trr t d t t Ea tt Ea nn a a                            2 1 2 11 11 1 2 2 2 2 2( ) {1 exp( ) exp( ) exp( )} 2( ) {1 exp( ) exp( ) exp( )} f ff ff f f ff ff f tt aa aa a r tt aa aa a r                  (C-7)   1 4 * 311 1 ()(,,,) {36( ) 4()4ln()4exp()1 9216 f f t fFfffff t f r tTpqtd ttEa Ea aC a a          1 2 2()2ln()2 3exp( )35 ff ff f Ea a C a a a         () 11 11 1 1 1 1 () 4()4ln 41exp fff f f Ea Ea a C a a           1 2 1 2()2ln()2 3exp( )35 ff ff f Ea a C a a a        2 1 11 1exp( ) () f f f a r Ea a                11 1 1 1 2 1 18 ( ) 18ln( ) 18 1 exp( ) ff f f f Ea a C a a a       2 11 1 1 1 1 3 1 12 ( ) 12ln( ) 12 16 16exp( ) 28 11 ff fff f Ea a C a a a a           1 9exp( ) () f f f a Ea a         1 2 18 ( ) 18ln( ) 18 9 exp( ) 27 ff ff f Ea a C a a a   1 3 12 ( ) 12ln( ) 12 16 16exp( ) } ff f f Ea a C a a            (C-8) [...]... 6, pp 603-620 Ochiai, Y., (1995b), Axisymmetric Heat Conduction Analysis by Improved MultipleReciprocity Boundary Element Method, Heat Transfer-Japanese Research, Vol 23, No 6, pp 498-512 Ochiai, Y and Sekiya, T., (1995c), Generation of Free-Form Surface in CAD for Dies, Advances in Engineering Software, Vol 22, pp 113-118 350 Heat Conduction – Basic Research Ochiai, Y., (1996a), Generation Method... Using Integral Equations, Computer Aided Geometric Design, Vol 17, No 3, pp 233-245 Ochiai, Y., (2001), Two-Dimensional Unsteady Heat Conduction Analysis with Heat Generation by Triple-Reciprocity BEM, International Journal of Numerical Methods in Engineering, Vol 51, No 2, pp 143 -157 Ochiai, Y., (2003a), Multidimensional Numerical Integration for Meshless BEM, Engineering Analysis with Boundary Elements,... Transient Heat Conduction Analysis by TripleReciprocity Boundary Element Method, Engineering Analysis with Boundary Elements, Vol 30, No 3, pp 194-204 Ochiai, Y and Takeda, S., (2009a), Meshless Convection-Diffusion Analysis by TripleReciprocity Boundary Element Method, Engineering Analysis with Boundary Elements, Vol.33, No.2, pp.168-175 Ochiai, Y and Kitayama, Y., (2009b),Three-dimensional Unsteady Heat Conduction. ..349 Meshless Heat Conduction Analysis by Triple-Reciprocity Boundary Element Method tf t f 1 (t f   ) exp(  a f  1 ) E1 ( a f  1 )  ln( a f  1 )  C T3* r5 r    d  6   E1 ( a f  1 )  2 n a f 1 153 6  n   a2 1 f    1  exp(  a f  1 ) a2 1 f 4 1  exp(  a f  1 )  a f  1   a3... Element Method, Engineering Analysis with Boundary Elements, Vol 33, No 6, pp 789-795 Partridge, P W., Brebbia, C A and Wrobel, L C., (1992), The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, pp 223-253 Sladek, J and Sladek, V., (2003), Local Boundary Integral Equation Method for Heat Conduction Problem in an Anisotropic Medium, Proceedings of ICCES2003, Chap 5 Tanaka,... Academic Press, London Micchelli, C A., (1986), Interpolation of Scattered Data, Constructive Approximation, Vol 2, pp 12-22 Nowak, A J., (1989), The Multiple Reciprocity Method of Solving Transient Heat Conduction Problems, Advances in Boundary Elements, Vol 2, pp 81-93, Eds C A Brebbia and J J Connors, Computational Mechanics Publication, Southampton, Springer-Verlag, Berlin Nowak, A J and Neves,... (1984), Boundary Element Techniques-Theory and Applications in Engineering, pp 47-107, Springer-Verlag, Berlin Carslaw, H S and Jaeger, J C., (1938), Some Problems in the Mathematical Theory of the Conduction of Heat, Phil Mag., Vol 26, pp 473-495 Dyn, N., (1987), Interpolation of Scattered Data by Radial Functions, in Topics in Multivariate Approximation, Eds C K Chui, L L Schumaker and F I Utreras, pp... Conduction – Basic Research Ochiai, Y., (1996a), Generation Method of Distributed Data for FEM Analysis, JSME International Journal, Vol 39, No 1, pp 93-98 Ochiai, Y and Sekiya, T., (1996b), Steady Heat Conduction Analysis by Improved MultipleReciprocity Boundary Element Method, Engineering Analysis with Boundary Elements, Vol 18, pp 111-117 Ochiai, Y and Kobayashi, T., (1999), Initial Stress Formulation... Problem in an Anisotropic Medium, Proceedings of ICCES2003, Chap 5 Tanaka, M., Matsumoto, T and Takakuwa, S., (2003), Dual Reciprocity BEM Based on TimeStepping Scheme for the Solution of Transient Heat Conduction Problems, Boundary Elements XXV, WIT Press, pp 299-308 Wrobel, L C., (2002), The Boundary Element Method, Vol 1, John Wiley & Sons, West Sussex, pp 97-117 . ) (,,,) {() 64 F f t ff f t f aa Tpqt rr dEa nn a           1 1 [2 ( ) 2ln( ) 2 1 exp( )]} ff f f Ea a C a a  (78) Heat Conduction – Basic Research 340 Additionally, the temperature gradient is given by differentiating. 22 (,) [1 cos H ber Rber a bei Rbei a TRt T t ber a bei a           Heat Conduction – Basic Research 342 Fig. 6. Circular region with temperature change at the boundary interpolation. Meshless Heat Conduction Analysis by Triple-Reciprocity Boundary Element Method 343 Appendix A (3D) The higher-order functions for 3D unsteady heat conduction are  * 2 ,,,Tpqt  1/2 3/2 1 {(1.5,)