CHINESE JOURNAL OF COMPUTATIONAL PHYSICS ] A. 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 ),

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1 CHINESE JOURNAL OF COMPUTATIONAL PHYSICS Vol 24,No 1 Jan, 2007 [ ] X(2007) ,,, (, ) [ ], Hessian,,, [ ] ; ; ; Hessian [ ] O ; O24211 [ ] A 0 (inertial confinement fusion,icf), ( ) 3, ( ) [13 80 %, ],,,, [ ] ; [ ] c e c i [ ] ( ) (2005CB321701,2005CB321702) 9 T e 9 t 9 T i 9 t 4 c r T 3 r ( 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 [ ] ( ),,,, Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet (1)

2 20 24, Hessian, (1),, 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) p+ n H, p 1 (3) T N R n, N T N 0 1, HH 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 u 9 x 1 9 x u 9 x 2 9 x u 9 x 2 2, (6) recover patc Zikienwicz2Zu recover [10,11], Bank Xu [12 ] L 2 recover,, Carstensen Bartels [13 ],, Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet

3 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 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), Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet

4 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) = , = 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 = , 45 5, , 0102, A r = 70, in 3, Π, Π 115, Π 2,, mes0 ;mes0, , 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 Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet

5 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 , 2, 1 mes0 2 mes1 3 mes0 T 0, Hess 4 mes0 T 0, Grad 5 mes0 T 0, KGrad (1),, t = 3108 t = 11185, 3, 4 5 3, 3 4 5, ) 3, 4 (a) ; 4 (b), 2, 1 3, Hess,,, Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet, (15)

6 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 Fig 5 Grid in Exp Fig 6 Grid in Exp , 510, 1010, 2010,10010 ( ) 1 1) % 2 1, 1 Table 1 Energe conservation errors 1Π% 2Π% 3Π% 4Π% 5Π% , 2) %, 2 24 % ) %, 2 52 % Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet

7 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, CPU Intel 214GHz,110G PC 2 2 CPU, Hessian Table 2 CPU times in experiments 1Πs 2Πs 3Πs, 4Πs 5Πs ,,, T k + 1 T k 8,, T k + 1 T k,, 8 8,, Grad Fig 8 Zoom of energy conservation error in Exp 4 and Exp 3, 2 step6,, p Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet

8 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 CPU 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 = , ; ) %,CPU 27 % ; 3), %,CPU 8 %,,,,Hess,,,, 213, ,, mes0 1, 2 854, T 0, 8 9,, 8 Hess, 9 Grad, ,3,4,8,9, CPU 9 2,3,4,6,7 10 2,3,4,8,9 Fig 9 Energy conservation errors in Exps 2,3,4,6, Fig 10 Energy conservation errors in Exps 2,3,4,8, 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 = CPU Πs ( 2 77 %, 1 89 % 8, Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet

9 %,, 0148 %, 2) %, 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, [ 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) [ 3 ] Fu Sangwu, Huang Suke, Li Yunseng Numerical simulation of indirectly driven ig2convergence implosions [J ] Cinese J Comput Pys,1999,16 (2) [ 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) [ 5 ] Huang W, Sun W Variational mes adaptation Error estimates and monitor functions[j ]J Comput Pys,2003, [ 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, [ 8 ] Mo Zeyao, Sen Longjun, Gabriel Wittum Parallel adaptive multigrid algoritm for 22D 32T diffusion equations[j ] Int J of Computer Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet

10 28 24 Mat, 2004,81 (3) [ 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, [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, [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) [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) [14 ] Ma X, Su S, Zou A Symmetric finite volume discretization for parabolic problems[j ] Comput Metods Appl Mec Engrg,2003, 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 , 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 ; Revised date Cina Academic Journal Electronic Publising House All rigts reserved ttp//wwwcnkinet