24 1 2007 1 CHINESE JOURNAL OF COMPUTATIONAL PHYSICS Vol 24,No 1 Jan, 2007 [ ] 10012246X(2007) 0120019210,,, (, 411105) [ ], Hessian,,, [ ] ; ; ; Hessian [ ] O357153 ; O24211 [ ] A 0 (inertial confinement fusion,icf), ( ) 3, ( ) [13 80 %, ],,,, [ ] 2005-08 - 30 ; [ ] 2005-11 - 28 c e c i [ ] (10376031) (2005CB321701,2005CB321702) 9 T e 9 t 9 T i 9 t 4 c r T 3 r - - 1 ( K e 1 ( K i 9 T r 9 t - 1 ( K r T e ) = ei ( T i - T e ) + er ( T r - T e ), T i ) = ei ( T e - T i ), T r ) = er ( T e - T r ), T e, T i, T r ;,, ; K e, K i, K r, K= AT (= 5Π2 e,i), K r = A r T 3 + r ; ei, er, ei = A ei T - 2Π3 e, er = A er T - 1Π2 e ; c, A(= e,i,r),, A ei, A er,, ICF,,,, 3 2,r2 p2 2,, r2, [4,5 ] p2 [6,7 ] 2 [8 ]UG [ ] (1970 - ),,,, 1994-2007 Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet (1)
20 24, Hessian, (1),, 1 111 Hessian,[9 ], Hessian, R n, u C 2 ( g) u H2, ( T c 0 2 u) ( x) c 0 T H( x), R n, x R, (2) H2 u Hessian ( 2 u) ( x) H2 H p = (det H) - 1 2 p+ n H, p 1 (3) T N R n, N T N 0 1, HH 0 1 0 T H T H( x) 1 T H, R n, (4) i gd2, i g 2Πn max T g 0, ΠT N ; min T g 1, (5) g H p,gd, i i [9, ] 1 u C 2 ( g), T N, u I T N u, u - C = C( n, p, c 0, 0, 1, 0, 1 ) u I p L ( ) CN - 2Πn n det ( H) pn, L 2 p+ n () ( ),, Hessian ( ( 2 u) ( x) ),, Hessian Hessian,,, ; ( u) ( u), ( 2 u) ( x) = 9 2 u 9 x 2 1 9 2 u 9 x 1 9 x 2 9 2 u 9 x 2 9 x 1 9 2 u 9 x 2 2, (6) recover patc Zikienwicz2Zu recover [10,11], Bank Xu [12 ] L 2 recover,, Carstensen Bartels [13 ],, 1994-2007 Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet
1 21 ( u N ) ( x i ) = ( i i u N ), L 2 recover 112, (7), (1) T 0 k T k ( k = 0,1,2,, T 0 = T 0 ), T k + 1 S ( B) 1 step 1 T k T 0 ; step 2 T 0 S ; step 3 T 0 gd ; step 4 T 0 gd,, ; step 5 T 0, T k + 1, T 0 T k + 1 1, AB, A ( x 1, y 1 ), B ( x 2, y 2 ), S ( A) gs = S ( A) + S ( B) 2, X = ( x 2 - x 1, y 2 - y 1 ) T,AB AB S = X T gs X (8) T N N,gd, i gd = T N 3 N 3 gd, i i = 1 (9) 1, (1), 3 S, 3 ( Grad ), S S = 9 T r 9 x 0 0 9 T r 9 y (10) Hessian ( Hess ), S = H p (11) ( KGrad ), S S = K r 9 T r 9 x 0 0 K r 9 T r 9 y (12) 2 3 (1) { ( x, y, t) ( x, y) xy,0 t 100}, xy = 3 i = 1 i (1), 3 1994-2007 Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet
22 24, 1 90, 2 90,95, 3 95,132, K T n 2 = 0,= (e,i, r), n ; K Tn 1 = 0, = e,i, T r = T r ( x, y, t) 1 = 210 ; T( x, y,0) = T 0 ( x, y) = 310 10-4, = e,i,r = = A ei = 0109, in 1, 215, in 2, c i 111, in 3, 110, in 1, 214, in 2, 310, in 3, 2 000, c e c r A e = = 115 e, = 115 i, = 110 r, 200, 60, 81 e = A i = 4 000, A er = 140, in 2, 7 000, 10, in 1, 79, in 3 1 Fig 1 Solution domains 35, 35, in 1, 40, i = 40, in 2, r = 01007 568, 45 5, 01000 17, 0102, A r = 70, in 3, 118 10 7 Π, 910 10 2 Π 115, 211 10 3 Π 2,, 2 356 mes0 ;mes0, 4 9 424, mes1, 2 2 mes0 mes1 Fig 2 mes0 and mes1 [14 ] (1),, AMG CG, l 2 5 l 2 6, [8 ], E flux, E own, E 0, Err Err = E flux - ( E own - E 0 ) E flux (13) 1,u, 0 u, k u k + 1 (= e,i,r) T 0, T k T k + 1 3, T 0 < T k 1994-2007 Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet
1 23 1) T k T 0, 2) T 0 T k + 1 u 0 ( x) = u k ( x), Π x T 0 (14) e = c e T e, i = c i T i, r = c r T4 r, x T k + 1, 2 x 1, x 2 T 0, 0 r ( x 1 ) = c r ( u0 r ( x 1 ) ) 4, 0 r ( x 2 ) = c r ( u0 r ( x 2 ) ) 4, k+1 r u k+1 e ( x) = u0 e ( x 1 ) + u 0 e ( x 2 ) 2 u k+1 i ( x) = u0 i ( x 1 ) + u 0 i ( x 2 ) 2 r ( x) = 4 k+1 r ( x)πc r u k+1,, ( x) = 0 r ( x 1 ) + 0 r ( x 2 ) 2, 1, 2 step1 T k T 0 ; step2 T 0 S ; step3 T 0 gd ; step4 T 0 gd, ( 113), ; step5 T 0, 1, step6 T 0, T k + 1, T 0 T k + 1 211 3, 2, 1 mes0 2 mes1 3 mes0 T 0, Hess 4 mes0 T 0, Grad 5 mes0 T 0, KGrad (1),, 3 4 5 3 600 3 3 t = 3108 t = 11185, 3, 4 5 3, 3 4 5, 4 5 6 1) 3, 4 (a) ; 4 (b), 2, 1 3, Hess,,, 1994-2007 Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet, (15)
24 24 3 3 Fig 3 Poton temperature contour in Exp 3 2) 4, 5 3) 5, 3 4,Hess Grad KGrad 4 3 Fig 4 Grid in Exp 3 5 4 Fig 5 Grid in Exp 4 6 5 Fig 6 Grid in Exp 5 1 1 5 110, 510, 1010, 2010,10010 ( ) 1 1) 2 1 52 % 2 1, 1 Table 1 Energe conservation errors 1Π% 2Π% 3Π% 4Π% 5Π% 1 0 8 69 4 16 3 19 4 87 4 12 5 0 6 50 3 14 2 52 3 96 3 19 10 0 5 68 2 74 1 72 3 38 2 78, 2) 3 1 63 %, 2 24 % 20 0 5 16 2 48 1 56 3 41 2 52 100 0 4 32 2 08 1 59 3 16 2 10 3) 4 1 27 %, 2 52 % 1994-2007 Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet
1 25 4) 5 2,7 1) 7 7, 4 1 2, Grad,, 3 2, Hess 2) 3 4, 7 1,2,3,4,5, Fig 7 Energy conservation errors in experiments 3) 2 5, 2 1 5 100 CPU Intel 214GHz,110G PC 2 2 CPU, Hessian Table 2 CPU times in experiments 1Πs 2Πs 3Πs, 4Πs 5Πs 6 662 44 59 364 56 12 338 06 17 112 53 13 145 34 3 4 7,,, T k + 1 T k 8,, T k + 1 T k,, 8 8,, Grad 212 8 3 4 Fig 8 Zoom of energy conservation error in Exp 4 and Exp 3, 2 step6,, p 1994-2007 Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet
26 24 T k + 1 p T k,u ( p) T k + 1 = u ( p) T k, p, 2,mes0 T 0, 6 7, 6 Hess, 7 Grad 3 6 7 CPU 3 6 7 5 Table 3 Energy conservation errors and CPU time in Exp 6 and Exp 7 ( ) 100 CPU 9 9 1) 6 2,3,4,6,7 Π% CPU Πs t = 110 t = 5 0 t = 10 0 t = 20 0 t = 100 0 6 3 39 2 67 2 38 2 17 1 85 15 701 95 3 7, ; 7 4 70 3 61 3 13 2 82 2 36 15 801 15 2) 6 3 14 %,CPU 27 % ; 3), 7 4 25 %,CPU 8 %,,,,Hess,,,, 213, 1 310 10-4 210,, mes0 1, 2 854, T 0, 8 9,, 8 Hess, 9 Grad, 4 750 10 2,3,4,8,9, 4 8 9 5 CPU 9 2,3,4,6,7 10 2,3,4,8,9 Fig 9 Energy conservation errors in Exps 2,3,4,6,7 4 10 Fig 10 Energy conservation errors in Exps 2,3,4,8,9 4 8 9 CPU Table 4 Energy conservation errors and CPU time in Exps 8, 9 1) 8 2 Π% 2 ), t = 1 0 t = 5 0 t = 10 0 t = 20 0 t = 100 0 CPU Πs ( 2 77 %, 1 89 % 8, 8 1 24 1 02 0 30 0 18 0 48 19 666 64 9 2 61 2 33 2 12 2 48 2 43 45 436 13 1994-2007 Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet
1 27 113 %,, 0148 %, 2) 9 4 23 %, 2 8 3,,,,,,Grad Hess 3, 1) Hessian,,, 2),, Hess,,,,,, 2 3) 5, Hessian,,,, 4), 5) (1),,,,,,,, ;, [ ] [ 1 ] Fu Sangwu, Sen Longjun, Huang Suke 2D numerical simulation metod and code for laser2driven implosion[j ] Hig Tecnolgy Letters 1998,5 51-54 [ 2 ] Fu Sangwu, Fu Hanqing, Sen Longjun, Huang Suke, Cen Guangnan A nine point difference sceme and iteration solving metod for two dimensional energy equations wit tree temperature[j ] Cinese J Comput Pys, 1998,15 (4) 489-497 [ 3 ] Fu Sangwu, Huang Suke, Li Yunseng Numerical simulation of indirectly driven ig2convergence implosions [J ] Cinese J Comput Pys,1999,16 (2) 162-166 [ 4 ] Gottlieb J J, Hawken D F, Hansen J S Review of some adaptive node2movement tecniques in finite2element and finite2difference solutions of partial differential equations[j ] J Comp Pys,1991,95 (2) 254-302 [ 5 ] Huang W, Sun W Variational mes adaptation Error estimates and monitor functions[j ]J Comput Pys,2003,184 619-648 [ 6 ] Gui W, Babugka Te, p and - p version of te finite element metod in one dimension part Te adaptive - p version [J ] Numerisce Matematic,1986,48 [ 7 ] Oden J T, Racowicz W, Demkowica L Toward a universal - p adaptive finite element strategy, part 3 Design of - p meses [J ] Comp Met in Appl Mec and Engng, 1989,77 181-212 [ 8 ] Mo Zeyao, Sen Longjun, Gabriel Wittum Parallel adaptive multigrid algoritm for 22D 32T diffusion equations[j ] Int J of Computer 1994-2007 Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet
28 24 Mat, 2004,81 (3) 361-374 [ 9 ] Cen Long, Sun Pengtao, Xu Jincao Optimal anisotropic simplicial meses for minimizing interpolation errors in L p 2norm [J ] Submitted to Mat Comp,2003 [10 ] Zienkiewicz O C, Zu J Z Te superconvergence patc recovery and a posteriori error estimates part 1 Te recovery tecniques[j ] Int J Number Metods Engrg,1992,33 1331-1364 [11 ] Zienkiewicz O C, Zu J Z Te superconvergence patc recovery and a posteriori error estimates part 2 Error estimates and adaptivity [J ] Int J Number Metods Engrg,1992,33 1365-1382 [12 ] Bank R E, Xu J Asymptotically exact a posteriori error estimators, part General unstructured grids[j ] SIAM J on Numerical Analysis,2003,41 (6) 2313-2332 [13 ] Carstensen C, Bartels S Eac averaging tecnique yields reliable a posteriori error control in FEM on unstructured grids I low order conforming, nonconforming, and mixed FEM[J ] Mat Comp, 2002,71 (239) 945-969 [14 ] Ma X, Su S, Zou A Symmetric finite volume discretization for parabolic problems[j ] Comput Metods Appl Mec Engrg,2003, 192 4467-4485 A Mes Adaptive Metod for Two2Dimensional Tree2Tempeature Heat Conduction Equations J IANGJun, SHU Si, HUANG Yunqing, CHEN Long ( Scool of Mat and Computational Science, Xiangtan University, Xiangtan 411105, Cina) Abstract A mes adaptation approac based on Hessian matrix is proposed to solve two2dimensional eat conductione equations wit coupled electron, iron and poton temperatures Tree kinds of adaptive mes and two adaptation metods based on gradient and flux of te poton finite element solution are used It is sown tat te energy conservation error and computation efficiency of te approac are improved Key words laser2driven inertial confinement fusion ; two2dimensional tree2temperature eat conduction equation ; mes adaptation ; Hessian matrix Received date 2005-08 - 30 ; Revised date 2005-11 - 28 1994-2007 Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet