AP DUNG PHU'aNG PHAP PHAN TU' HU'U HAN KET HQP PHU'QNG PHAP RUNGE-KUTTA GlAl HE PHU'QNG"
TRiNH SONG NU'CVC NONG
NCS. Nguyin Hodng Minh GS.TS. Trdn Dinh Hffi
Tom tat:
Bai viet trinh bay mdt thuat toan giai he phuong trinh sdng nudc ndng (sdng ddng luc 2 ehidu ngang) dua tren co sd phuang phap phan tu- hu-u han Galarkin bien doi phuang trinh vi phan dao ham rieng v i dang he cac phuo'ng trinh vi phan thudng va giai he phugng trinh vi phan thudng bang phuong phap Runge-Kutta.
Thuat toan nay dugc ap dung cho diin toan ngap Iut tai khu Ha Ddng, huyen Thanh Ha, tinh Hai Dugng, kit qua tinh toan cho sai sd cd t h i chap nhan dugc.
Tif khda: Sai phan, phan tu hiiu han.
Summary:
APPLYING FINITE ELEMENT METHOD IN INTEGRATION WITH RUNGE- KUTTA METHOD TO SOLVE SURFACE-WATER EQUATION The water flow in nature is a complicate phenomenon. For describing it can be used only physical or mathematical modeling. This paper is dealing with a study on mathematical approach to describe this water flow characteristics, using the finite element method in combination with Runge-Kutta method to solve surface-water differential equation.
Keywords: Difference, finite element.
I. DAT V A N 0^
Ddng chay trong thien nhien thudng la ddng chay rdi khdng dirng, ba chiiu, nhiiu pha; viec md phdng vimg nghien cihi chi cd the bing cac md hinh gin dung nhu md hinh vat ly hoac md hinh toan.
Trong nhiiu trudng hgp ddng chay trong thien nhien nhu ddng chay lu viing ddng bing, ddng chay viing cua sdng, ddng chay tran trong cac vung phan cham lu, trong cac viing bi vd de... thanh phin van tic theo phirong thang diing cd thi bd qua, viing nghien ciiu hoan toan cd thi dugc md phdng bang bai toan 2 chiiu ngang.
Hien nay do dien biln bit thudng ciia thai tilt, hang nam viing ddng bing ven biln miin Trung hay cac viing ven de d miin Bic... thudng cd nguy co bi ngap, Iut do mua lu cd cudng do Idn, keo dai va nhiiu khi diin ra lien tiep, vi vay viec nghien ciru
280 TUYEN TAP KHOA HOC CONG NGHE 50 NAM... TAP II
md phdng dien bien ddng chay 2 chieu theo ca khdng gian va thdi gian la rat can thiet, giiip cho cac nha quan ly cd the dy bao dugc quy md va pham vi anh hudng cua cac tran mua lii nay.
II. PHU'aNG PHAP NGHIEN ClJU
Trong sd cac thuat toan hien dang dugc sir dyng md phdng bai toan 2 chieu, phuang phap phan tir hiru han da dugc quan tam nghien cim d trong va ngoai nudc do phuong phap cd kha nang md phdng khdng gian vdi dp chinh xac cao. Vdi myc dich tiep can, nam bat va ty chii ve cdng nghe, bai viet trinh bay mpt thuat toan chi tiet giai bai toan 2 chieu hay la giai he phuang trinh sdng nudc ndng dya tren co sd xap xi khdng gian nghien ciru bang cac phan tir hiru han, sir dyng ham ndi suy khdng gian tuyen tinh de dua he phuang trinh dao ham rieng ve dang he phuong trinh vi phan thudng theo thdi gian va giai he bang phuang phap Runge-Kutta bac 3 [1], [2], [3], [4], [5].
He phuang trinh sdng nudc ndng xay dyng bang each tich phan theo chieu sau ddng chay he phuang trinh Navier-Stoke ciia ddng chay khdng nen dugc va phuong trinh lien tuc, ta dugc he phuang trinh nhu sau:
(1)
Trong do:
U, V - van tdc dugc trung binh hda theo do sau, ling vdi tryc ox, oy;
h - dp sau ldp ddng chay;
Sox, Soy - dp doc day theo true ox, oy tuong iing;
•"^ox- •'^oy" 'i'^g suat tiep theo hudng ox va oy;
Dai lugng: S,^=^^ va 5 , - = ^ ^ pgh ' pgh
la dp dde thiiy lyc (dp dde can) theo hudng ox va oy tuong iing, trong trudng hgp chay rdi dugc xac dinh qua cdng thirc Chezy nhu sau:
„
u^u-
+v-
, ^v^u-
+v-
•^A = ^ v a S. = -^
^ C-h * C-h Trong dd: C - he sl Sedi
Theo phuang phap phan tir hun han, khu vye tinh toan dugc chia thanh cac phin tu. Cac phan tir c6 the la hinh tam giae, tir giae diu hoac khdng diu cd kich thudc khac nhau va sd lugng niit khac nhau. Trong trudng hgp tdng quat, cac phin tir tam giae vdi 3 diem nut thudng dugc lya chpn (hinh 1).
dh dan,) d(m dh ,,dh , dU „dh , dV -^ + — ; — + = — + ( ' — + /; + V + /l =
dl d.x dy dl d.x dx fy' fy' dU ,,<?(' ,,dU dh (^ r "l
dl d.x fy ''d.x ^1, °' pgh) dy ,,dy ,dv dh (^ T ]
cl CX fy re { ' pgh)
= 0
Hinh 1. Phdn tir tam gidc
Cac an ham U(x, y, t), V(x, y, t), h(x, y, t) trong moi phin tir dugc x i p xi nhu sau:
U{x,y,t)^Yu,{t)F,{x,y)
1=1
V{x,y,t)^Y.^.{t)F,{x,y)
;•=]
N
h{x,y,t)^Y.^,{t)P'M,y)
Trong dd:
Fj - ham ndi suy trong trudng hgp la quan he tuyin tinh cd dang:
1
2A {a. + b-x + c.y)
(2) Phuang phap sd dy trgng sd Galerkin the hien nhu sau:
f FJAD = 0
Trong dd:
D - khdi chira cac phan tir;
R - sd du khi xap xi cac bien sd ddng thdi phu thudc khdng gian-thdi gian bang tdng cac ham sd thdi gian va khdng gian rieng re.
Nhu vay, phuang phap Garlekin cho rang sd du xuat hien khi md phdng khdng gian bang cac phan tir hiru tryc giao vdi cac ham trpng sd npi suy. Hay ndi mpt each khac ban chat cua phuang phap Garlekin la vdi ham trpng sd dugc lya chpn, tdng sai sd md phdng theo khdng gian tren toan mien bang khdng.
Ap dung phuang phap Galerkin cho he phuang trinh (1) ddi vdi phan tir 1 thu dugc:
rAdU ,,eu ,,dU ch ,„
J/ [ dl B.x dy d.x cAdP' ,,dV ,dV dh , „
-5„)i/;(/n = 0
(3)
rr\dh ,,dh ,dh ,dh , 5K1 , ^ ^ 'J[dt d.x fy' fy cfyj Trong dd: Q - mien gidi han tinh toan. •
He phuang trinh (3) sau khi dugc tich phan sd. dirge viet nhu sau:
282 TUYEN TAP KHOA HOC CONG NGHE 50 NAM... TAP II
Ne f JTT "I
i^{ dV 1
^' { Hh 1
Trong dd: Ne - sd cac phan tir cua ludi tinh.
De nhan thay rang tich phan Galerkin (2) dua he phuang trinh dao ham rieng (1) ve dang he cac phuang trinh vi phan thudng. He phuang trinh (4) sau khi tdng hgp cho tat ca cac phan tir thupc viing nghien ciiu cd dang phuang trinh ma tran:
c/{W}
dt :{r}-[C]{W}
Trong dd:
[C], [A], [B] - cac ma tran vudng;
[C] = [A]-' * [B]
'A,. 0 0 0 A: 0
0 0 /4,,
B. 0 Bf
0 £)/
B,. Df B' 5 , N - sd an can tim n = Ne x 3;
{W} - vec to an sd can tim: U, V, h;
{T} - vec ta ve phai.
Phuong trinh (5) vdi diiu kien ban diu {W},=o va dieu kien bien {Wl^^^ dugc giai theo phuong phap Runge-Kutta bac 3 [6, 7] va kilm tra lai bien theo phuang phap ndi suy tuyen tinh ndi tiep (successive linear interpolation):
Budc 1: Giai he phuang trinh (5) theo Runge-Kutta AW'" = {TY'''At-[Cp{W{t)}M AW'" ={TY"M-[Cr {W {I) + AW^ -}AI
AW'''={TY''At-[Cr>{W{l)+^^^}At n An can tim sau khoang thdi gian Ar thu dugc cd dang:
W.{t + Al) = W,{t) + AlV.
DWi la nghiem ngoai suy: DWj = 0,25 DW'"+ 0,75 DW^^'
Budc 2: Giai bai toan bien phuang phap ndi suy tuyin tinh nli tiip (successive linear interpolation) [6].
Tai bien, vdi diiu kien bien W^.^^ = W{t) cho trudc, dat:
'^r(' + ^0 = W^i){l + At) = W^{t + Al)
Xet ham sd tai bien: f,^{W) = (w,,., - W^)'
(i) Niu: fi^{W)>E udc tinh:
Gan: W^{t + Al) = W^^^ trd vl budc 1 tinh lai xac dinh dugc ff;^^,,.
Xet ham sai sd tai bien:
^.,)(w,,„,)=(w„,„-w,)'
(ii)Niu: ^ , „ ( W , ^ , „ ) > f ude tinh:
W -W fik*i)(^ii,*,))
(^)(f^*,)-^.„(»;*.„))/(^^„-(^^*„,))
Gan: W^{t + At) = W^^^^^ trd vl budc 1 tinh lai.
(iii) New. f^|„,^{W^„^,^) <£,ta cd:
Wp {I + At) = W(/ + At) qua trinh tinh dugc thyc hien timg budc cho khoang thdi gian tinh tiep theo.
Thuat toan giai khdng gian bang phuang phap phan tir hihi han, giai thdi gian bang phuang phap Runge-Kutta dugc lap trinh tinh toan va vilt bing ngdn ngir Fortran 95 tren nen Digital Fortran 6, ket qua cho dudi dang file text, phan dd hpa sir dung phan mem tecplot.
Phan mem xay dyng da tinh toan thiiy lyc cho mdt doan kenh ehir nhat dai 580 m, rdng 58 m, day phang va dugc kiem chung ciing vdi phan mem MIKE21, ket qua tinh toan cho thay sai sd khoang 5% va cd the chap nhan dugc.
III. K^T QUA VA THAO LUAN
Ap dung phan mem nay cho viec md phdng lai qua trinh ngap lut do va de ta song Thai Binh tai khu Ha Ddng, huyen Thanh Ha, tinh Hai Duong nam 1996.
Khu Ha Ddng dugc bao bpc bdi 4 con sdng la: Thai Binh, Giia, Van Uc va Mia, de bao ve ddi sdng dan sinh, kinh te, he thdng de dieu dugc dap bao quanh khu nay, de cd dp cao trung binh 3,8 m, mat de rdng 3 m. Dudng true 190A chay xuyen sudt ca viing cd chieu cao trung binh la 1,80 m.
284 TUYEN TAP KHOA HOC CONG NGHE 50 NAM... TAP II
Sy cd vd de nhu sau:
{+) Thdi gian vd va ngap lyt tir 24/8 den 11/9/1996.
(+) s l vi tri vd la 2 vi tri tai Km54 va Km56 tren de ta sdng Thai Binh.
(+) Trong khoang thdi gian 8-^ lOh dau cao dp myc nudc dat khoang +1,8 m, nudc chua tran qua dudng 190A. Sau khoang 3 ngay myc nudc dat cao nhat +2,3 m, dien tich ngap khoang 2.700 ha, chiiu sau ngap trung binh tir 0,5 -=-1,1 m.
Md hinh thiiy lyc 2 chiiu dugc sir dung de md phdng lai qua trinh ngap lyt tai khu Ha Ddng.
Cac so lieu dugc sir dyng tir cac ngudn Cue Thuy lgi, Vien Quy hoach Thiiy Igi, Cue Quan ly De diiu va PCLB - Bp Ndng nghiep va PTNT va Vien co hpc.
(+) Ludi tinh toan: Tir cac ban do (1/5.000, 1/25.000) thu thap duge qua cac co quan neu tren, sir dyng phin m i m chia ludi ty ddng cua phan mem Telemac de tao ludi. He tpa dp XOY duge gin vdi he qudc gia, ludi tinh toan khdng cau triie cua khu Ha Ddng gIm 10845 phin tir tam giae vdi 5576 mit ludi. Tai khu vyc gan cac vi tri xay ra vd de va tren cac dudng giao thdng cac tam giae cd kich thudc nhd. O cac vimg cdn lai cac tam giae cd kich thudc ldn hon.
(+) Diiu kien bien: Bien ddng, cac mit ven de: U = 0, V = 0. Bien hd, ddng chay vao, quan he luu lugng Q ~ t (U-t, V-t). Lii tran qua vit va tai 2 doan, mdi doan qua 3 dilm nut. Doan 1 qua cac niit: 188, 211, 151; doan 2 qua cac mit: 247, 278, 44. Luu lugng phan bd deu qua cac nut.
(+) Diiu kien ban diu: Thdi diem sau gid thu 2 {liic da co cd 2 vet va) ke tir khi nudc tran vao khu Ddng, do sau ngap nudc trong viing tir 0,1 -^0,4 m, cao dp mat nudc trong vimg +1,3 m {loan vimg da co nuac dim).
Chuong trinh se thyc hien viec md phdng lai mdt qua trinh truyen lii, xac dinh cao dp myc nudc, van tdc U, V tai tat ca eae diem trong mien tinh toan tir gid thir 3 den gid thir 8 ke tir de ta sdng Thai Binh hi vd.
Ket qua tinh toan: Sau 8 gid, nudc lii da gay ra ngap toan bd cac xa ben hilu dudng 109A tuy nhien ehua tran qua dudng tryc nay. Dp sau ngap nudc tir 0,6 + 0,9 m.
Cao dp myc nudc khoang +1,8 m.
Cac ket qua tinh toan da duge kiem tra vdi mdt vai sd lieu thyc te thu dugc tai vimg ngap khu Ha Ddng, huyen Thanh Ha, tinh Hai Duong nam 1996 {so lieu lay lic nguon Xi nghiep khai thdc cong trinh thuy lai huyen Thanh Hd vd Trgm qudn ly de dieu Thanh Hd, tinh Hdi Duang) [8] (bang 1).
Bdng 1. So sdnh kit qud linh loan vd so lieu quan sdl thuc te TT
1 2 3 4
Dia diem VTnh Lap Thanh Hong 2 Trudng Thanh Hgp Difc (ben huu)
So lieu quan sat thuc tg 1,80
1,80 1,70 1,70
Ket qua tinh 1,850 1,863 1,865 1,864
Chenh lech 0,050 0,063 0,165 0,164
Cac ket qua ti'nh toan d tren da dugc so vdi sd lieu thyc te, cao hon tir 0,05 H- 0,16 m, tuy nhien cd the chap nhan dugc do trong qua trinh md phdng da bd qua mpt sd dieu kien bien ben trong {cdc cong trinh lieu, cong qua duang...), he sd nham dupc lay trung binh bang 0,03.
IV. K^T LUAN
Md hinh sd tri dugc thiet lap tu he phuong trinh sdng nudc ndng dugc giai bang phuong phap phan tu hiru han Galerkin ket hgp vdi phuang phap Range-Kutta md phdng dugc cac dac trung thuy ddng lyc hgc cua ddng chay 2 chieu ngang trong khdng gian.
Ve ca ban thuat toan la don gian, cd do dn dinh cao, khdng khd lap trinh so vdi cac sa do sai phan khac da cd do vay cd the ung dyng de giai quyet cac bai toan ngap lut, tran de hoac cac vung phan cham lii.
Chuong trinh khi dugc nghien cuu hoan thien cd the dimg de ve ban dd ngap lyt khi xay ra hien tugng vd dap hoac khi phai xa lii Idn, dieu nay rat phii hgp vdi cac dieu khoan da dugc quy dinh trong Nghi dinh 72/ND-CP ban hanh thang 5/2007 cua Chinh phil ve viec Quan ly an toan dap trong ca nudc.
TAI LIEU THAM KHAO
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[2]. Forsythe G.E., Malcolm M.A.., Moler C. B. (1977): Computer Method for Mathematical Computations. Prentice-Hall (Russian translation from English, 1980).
[3]. G.I. Marchuc, V.V. Saidurov (1979): Nang cao dp chinh xac giai cae sa dd sai phan. NXB Nauka, Matxcova (Tilng Nga).
[4]. J.F. Bellemare, G. Dumas. G. Dhatt (1990): A Finite Element Method of Estuarian and River Flows with Moving Boundaries. Adv. Water Resources Joumal, Vol.
13, No. 4.
[5]. Luang Tuin Anh, Trin Thuc (2003): Mpt phuang an nang cao dp In dinh ciia so dd phin tii hiru han sdng ddng lyc 2 chiiu ngang. Tuyen tap bao cao Hdi thao Khoa hpc-Vien Khi tugng Thiiy van lin thir 8, Ha Ndi-12/2003, Trang 1-5.
[6]. Nguyin Van Diep (2001), Bao eao dy an "Xay dyng edng nghe md phdng sd phyc vu viec dl xuit, danh gia va diiu hanh cac bien phap phdng chdng lu tren ddng bing sdng Hdng - Thai Binh", Vien Co hpc.
[7]. V. Aizinger, C. Dawson (2002): A Discontinuous Galerkin method for two-dimensional flow and transport in shallow water. Advances in Water Resources 25, 67-84.
[8]. Ventechow, David R. Maidment, Lary W.Mays (1988): Applied Hydrology - Mc Graw - Hill Book Co (Thiiy van img dung, 1994).