Luận án tiến sỹ Áp dụng phương pháp phần tử hữu hạn giải một số bài toán tĩnh và động của vật rắn có biến dạng phức tạp

DA.IHOC Quac GIA TP.HCM
TRUONG DA.IHOC KHOA HOC nJ NHIEN
NGUYEN PHU VINH
, Ap DUNG PHu'(iNG PHAp PHAN TV HUU HAN GIAI MOT
A' , - 'A ? A ~ , A"
SO BAI TOAN TINH VA DONG CUA V~ T RAN CO BIEN
D~NG PHUC T~P
Chuyen nganh: ca HOC V~T THE RAN BIEN DA.NG
Ma S6:1.02.21
TOMTAT LU~N ANTIEN si ToAN HOC
, ' ;~
\:}".~H. nr Nt-HEN
THI1\lIEN
THANH PHa HO CHi MINH-2005
C6ng trlnh du<;choan thanh t<:tiKhoa Tmln Tin, Truong
B<:tiH9C Khoa H9C Tif Nhien, B<:tiH9C Quac Gia Thanh Pha H6 Chi Minh
Nguoi huang din khoa h9C:
PGS.TS Ng6 Thanh Phong
Truong B<:tiH9C Khoa H9C Tif NhienTp.HCM
Phan bi~n 1: GS.TSKH. Ng6 Van Lu<;c.
Lien hi~p HQiKhoa H9C Ky Thu~t. Ba Ria- Vung Tau.
Phan bi~n 2: PGS.TS. Nguy€n Luong Dung.
Truong B<:tiH9CBach Khoa. BHQG.Tp.HCM.
Phiin bi~n 3: PGS.TS. Nguy€n Dung.
Vi~n Co H9C Ung D\lng. Tp.HCM.
Lu~n an sf: du<;cbaa v~ truac hQi d6ng chilli lu~n an cip nha nuac h9P t<:ti
Truong B<:tiH9C Khoa H9C Tif Nhien Tp.HCM.
Vao h6i gio ngay thang nam 2005
C6 th€ tlm hi€u 1u~nan t<:ti:
Thu Vi~n Truong B<:tiH9C Khoa H9C Tif Nhien Tp.HCM.
Thu Vi~n Khoa H9c T6ng H<;pTp.HCM.

1
, ? '
PHANMODAU
1. Tinh dIp thie't cua d~ tai
L9 thuyet bien d'.lng dan h6i da d'.lt duQc noting thanh tt;!'uto lOn,
noting v§:n clitia Gap ling du'Qc nhu cftu ngay cang cao v~ v~t li~u mdi
coo cong ngh~ mdi hi~n nay. Co noting v~t li~u khong mo ta trong ph'.lm
vi 19 thuyet dan h6i du'Qc,vi dlJ linn khang thu~n nghich khi bien d'.lng,
cac v~t lit%:uco linn bien d'.lng phuc t'.lp. C~n nghien cuu cac lo'.li v~t lit%:u
da d'.lng lam vit%:ctrong moi tru'ang phti'c t'.lP, c1€chili du'QcIt;!'cngoai laC
dlJng cling vdi cac ann hu'ang khac cua mai tru'ang nhu' nhi~t dQ cao, ap
sua't lOn, st;!'thay d6i thai tiet Bai roan khao sat ling xiI cua v~t lit%:u
trong tq.ng thai dan-deo, dan-nhdt-deo v§:ncon mang linh thai st;!',bai Ie
linn da d'.lng va v~ m~t 19 thuyet. Cac Wi giai coo cac bai roan nay v§:n
con g~p kho khan v~ m~t roan hQc, nha't la ph'.lm vi hai chi~u, ba chi~u,
th~m chi cii lai giai g~n dung v§:n con dt h'.ln che. V~y vit%:ckhao sat,
coo cac lai giai sa g~n dung v§:n la nhu cftu dp thier, mang linn thai st;!'
d~t ra coo noting nha roan hQc ding nhu' cd hQc.
2. M1}cdich nghien CUll
MlJc lieu cua lu~n an la ket hQp ba lInh vt;!'cCd hQc-Toan hQc-Tin hQc
d~ giai quyet mQt sa ma hinh cac bai roan cd hQc moi tru'ang rcln bien
d'.lng phuc t'.lp. Nghien cuu cac giiii thu~t qui h6i coo cac bai roan 1
chi~u va 2 chi~u. SiI dlJng chu d'.lo PPPTHH ket hQp vdi sai phan htiu
h'.ln, giai rich ham, cac ph~n m~m h6 trQnhu'Matlab, Maple.
3. D6i ttiqng va ph~m vi nghien CUll
Dai tu'Qngnghien cau cua lu~n an la: PPPTHH coo noting bai roan bien,
co ling xiI cua v~t lit%:uphi tuyen va co linn bien d'.lng phuc t'.lp. Nen ht%:
phu'dng trlnh thiet l~p la phu'dng trlnh vi roan ba't kha rich. Khi v~t Mu a
tr'.lng thai dan-deo till t6n t'.li song song mi~n dan h6i du'Qc ma ta bai
phu'dng trlnh elliptic va mi~n deo thoa phu'dng trlnh hyperbolic. Bien

roan c.ach gitia mi~n dan h6i va mi~n deo clitia bier tru'dc va du'QCxac
Ginn trong qua trlnh giai sa bai roan dan-deo, tren bien roan cach cua 2
mi~n, cac thanh ph~n tenxd ling sua't, bien d'.lng, chuy~n vi phai thoa
man di~u kit%:nlien tlJc, do la nQi dung cat 16i cua thu~t giiii ann X'.lqui
h6i. TIm nghit%:mgiai rich cac phu'dng trlnh nay, con g~p noting kho khan
khong th~ khclc phlJc du'Qcv~ m~t roan hQc. Dai vdi v~t lit%:uma tr'.lng
thai ling sua't phlJ thuQc cii bien d'.lng va tac dQbien d'.lng, ta co ma hlnh
moi tru'ang dan-nodi va ht%:phu'dng trlnh se la phu'dng trlnh rich roan
2
Voltera. VI the' cac bai roan bien phi tuye'n a trong lu~n an cling chua
giiii ho~c chi giiii duQc mQt phgn tuong ling vdi s6 h,~tngphi tuye'n Clfth~.
4. T6ng quaD v~ dng d1.J.ngPPPTHH trong v~t dn hie'n d~ng
Bai roan dan-deo, dan-nhOt-deo dii khai thac tri~t d~ v~ m~t ly thuye't
tren the' gidi nhu: Bedukh6p.N.I, Khachan6p.L.M va A.Ilyushin,
Timasenka, Zeinkiewicz, George E.Mase, Hill.R, Bao Huy Bich, B~c
bi~t cac cang trlnh cua Sima.J.C, Hughes va Owen.J, Hinton.E dii duc
ke't thu~t giiii cho cac bai roan nay c6 ten la cac thu~t giai return-
mapping ma lu~n an t~m dich la "anh x~ qui h8i". Nhung cae vi dlf s6 Clf
th~ hgu nhu hie'm c6, va l<,tithu~t giiii nay tuong d6i phlic t<,tpd ph<,tmvi
mQt ehi~u va hai chi~u. Lu~n an sU'dlfng PPPTHH cho thu~t giai anh x<,t
qui h8i a chuang 2 va chuang 3. Xet cae bai roan Clfth~, c6 loi giai s6
d~ so sanh vdi nghi~m chinh xae, vi dlf a chuang 3, trong truong hQp
deo ly tuang a Khachan6p.L.M dii c6 Wi giai giai rich. Cling trong ph~m
vi PPPTHH eho cae bai roan c6 bie'n d~ng phUc t~p, lu~n an dii xet them
hai bai roan ti:nhva dQng a chuang 4 va chuang 5. Lien quail tdi llnh v\fe
nay nhu: Lions.J.L, Johnson.Claes, NT.Long. Cac tac gia nay khao sat
s\f t8n t~i va duy nha't nghi~m mQt caeh tri~t d~. d day lu~n an ma
phOng hai bai roan Clfth~, dung PPPTHH, ke't hQp vdi phuong phap giai
rich ham, sai phan hil'u h<,tn.xa'p xi nghi~m trong khang gian hil'u h~n
ehi~u, chung minh s\f t8n t<,tiva duy nha't nghi~m cua bai rain xa'p xi v~

nghi~m cua bai roan xua't phat. So sanh nghi~m xa'p xi vdi nghi~m chinh
xae. Chung minh 6n dinh cua luQe d6 sai rhino PPPTHH xuyen su6t d
lu~n an, nh~m M giiii s6 ggn dung la phuong phap v<,tnnang, hi~u qua
nha't, cho phep ta chuang trlnh h6a, t\f dQng h6a, d~ thay d6i dil' li~u ban
dgu cua bai roan.
5. B6 C1.J.c va nQidung ciia lu~n an
Lu~n an c6 163 trang g8m ph~n ma d~u, nam chuang, ke't lu~n
va cae ke't qua eua lu~n an dii duQe eang b6, tai li~u, phgn phlfllfe.
Chu'dng 1: CO Sa ToAN HQC
1.1. Quy lu~t dng xU'phi tuye'n trong v~t riin hie'n d~ng
C6 3 lo<,tiphuong trlnh: phuong trlnh can b~ng Kd=f, phuong trlnh ling
xU'(dinh lu~t Hooke) (J=DE, phuong trlnh hinh hQc E=Bd, trong d6 K: la
matr~n dQ cling, D: ma tr~n v~t li~u, B ma tr~n hinh hQc (d<,toham ham
d~ng), f: l\fc ngoai, d: ehuy~n vi, (J :ung sua't, E: bie'n d~ng. Cae phuong
3
phap giai t6ng quat nhu la: phuong phap d(j cling, phuong phap ling sua't
ban diiu, phuong phap bie'n d,.lllgban diiu.
1.2 Tien d~ cd ban ly thuye't d~lllh6i phi tuye'n:
0 tr~ng thai ling sua't-bie'n d~ng phuc t~p, duong cong ling xU'v<i.tli~u
cling c6 d~ng gi6ng nhu keo lien don gian, hie d6 quail M giQ'aling sua't
va bie'n d~ng c6 d~ng a =E (I-CD)E, trong d6 E mOduli dan h6i cua v<i.t
li~u, con CDla ham giai rich cua d(j dan dai tuong d6i tuc la CD=CD(E).
Tien d~ duong cong duy nha't phat bi~u: au = E(I-CD)Eu,au cuong d(j ling
sua't tie'p, Eucuong d(j bie'n d~ng truQt. Ne'u bi~u deSkeo lien cua v<i.tli~u
c6 th~ thay the' giin dung b~ng hai do~n thing nghieng thl
£e Ell £e Ell
() u = E [1- (1- -)(1- -)] Eu' (0 =(1- -)(1 - -) ,
£u E £u E
e Ell
Ne'u vie't l~i: (0 = A (1- ~) thl A =1 - -

£u E
1.3. M(it vai khai ni~m va ky hi~u cong thU'c hie'n phan daD cleo.
D~ c<i.pde'n cac cong CtJ, ky hi~u cua giai rich ham, cong thuc bie'n
phan, d~o ham suy r(jng, d~o ham Frechet, Gateaux. D~ ti~n vi~c trlnh
bay, ta dung ky hi~u tenxo h~ t<?ad(j thing tn;1'cgiao va qui uac Einstein.
.Phuong trlnh bie'n phan Euler-Lagrange: Tli vi~c c\;1'cti~u phie'm ham
nang luQng rc(ui ) = fH {W(Eij)- hi ui }dx, Uj=O V xi E 3B , ta thu
B
dt(Qcphuong trlnh can b~ng trong v<i.tth~ ran bie'ndang () U,j + hi =O.
1.4. Xay dt!ng d~.lllghie'n phan hai tmin daD h6i t6ng quat trong R3
Chuy~n vi u=(Uj), trong mi~n Q, truong ling sua't d6i xung a=(aij),
aij=aji tenxo bie'n d<:tng E=(Ejj)i,j=I,2,3. Tli phuong trlnh can b~ng:
()(j,j + ii =0 trong Q, vai di~u ki~n bien Uj=0 tren f2 va (Jij nj = gi
tren fl, i=1,2,3. Ta c6 th~ phat bi~u d<:tngbie'n phan nhu sau: TIm UiE V
sao cho : a(uj,vi)=L(v;) '\I Vi E V, trong d6
a(u, v) = Hf(Adiv(u)div(v) + IlEij (U)EU(v),ktx
Q
L(Vi) = ffffiVidx + ffgiVids, V={V=(Vi) E [HI(Q)]3: Vi=0 tren f2}.
Q rl
Ta c6 th~ ki~m chung r~ng V la khong gian Hilbert vai rich vo huang
4
(e, e) tuong ling vai chuifn II-IIv , va aCe, e)y 1a d?ng song tuye'n tinh
tren VxV, thoa di~u ki~n V-elliptic a(vi, Vi) ~ a Ihll~ \iVi EV, trong do
Ilvill~ =11vII~'(Q)=lhll~'(Q)' Ihll~'(Q) = gs(vl +(~::)2)dX.
Chu'dng 2: MO HINH PHI TUYEN M(n cHrEu
2.1. Dflll deo m(}t chi~u
e Cac phuong trlnh ling xU',dinh 1li(~tchay deo cua mo hlnh deo 1:9tuCing
duQc mo ta nhu bang 2.1, mo hlnh cho v~t 1i~u tai b~n d~ng huang nhu
bang 2.2.
Bang2.]

il Quan M ling suilt -bie'n d?ng dan h6i: a = E ( £- £P)
iil Dinh 1u~tchay: iP =y sign(a)
iiil Di~u ki~n chay: f(a)= Icrl- cry = 0
ivl Di~u ki~n Kuhn-Tucker: y ~ 0, f(a) S;0, Yf(a) =o
v/Di~u ki~n trung boa: rj(a-) = 0 ne'uf(a)=o
Bang2.2
il Quan M ling suilt -bie'n d?ng dan h6i: a = E ( £- £P)
iil Dinh Lu~t chay: iP =y sign(a), eX = r
iiil Di~u ki~n chay: f(a,a) = Icrl-(cry + Ka) =0
ivl Di~u ki~n Kuhn-Tucker: y ~ 0, f(a,a) s;0, y f(a,a) =o
vi Di~u ki~n trung boa: rj(a-,a) = 0 ne'uf(a,a) = o
e Bai roan bien tq ban dftu (bai roan BTBD): D?ng vi phan cua bai roan
BTBD mQtchi~u: pv; - cr~- Pb = 0 trong Bx]0,T[, di~u ki~n bien:
u = Ii tren
auB x ]0,T[, a =cr tren acrBx ]0,T[, v = itt
b :B x[0,T]~ IR, b = hex,t) 1tfckhoi. V~t th~ dan h6i tuye'ntinh thl
a(x,t) = EE(X,t). Ne'u coi a-(x,t) 1a khong tuye'n tinh, mo hlnh dan
d
' . E(K+H).
d
O"
k
o~
h
'
d
'
f
o
j

.
o B
[0 T]
eo: a =
E
H)
E, leu len c ay eo: =, = trong x , .
+(K+ .
cr = EE cac traCinghQp con l?i.
5
.D,~tngy€u (bi€n phan) eua bai loan BTBD mQt ehi~u
Dinh nghla tru'ong ehuy€n vi dQng hQe kha dI:
St ={u(8,t):B~IR, u(8,t)lauB =~(8,t)}, co St CHI(B)
vai m6i t e6dinh,vai H I(B) 1akhong gian Sobo1evcap 1tren B.
ta dinh nghla: V ={17Eel (B) : 17(0)= a} la khong gian cae ham co
d'.loham tren B va tri~t lieu khi x
=O.Do 1akhonggiancae hamthii',
hay bi€n phan dQng hQe khii dI. Khi do 11= M E HI (B) :11(0)=a}
Vai cae dinh nghla tren, phat bi€u d'.lng y€u eiia bai tOaDBTBD nhu' sau:
TIm ham U(8,t) E St sao eho:
fpv17dx+G(o-,17) =0, V17EV, VtE[O,T],
B
f
/
f
- / a17
trangdo G(o-,17)= 0-17dx- pb17dx-o-17la B va 17 =
B B ()" ax
Chu yding u(x,t) la ham a"n trang &= Ux va v=Utt.
Chung minh du'Qeslf tu'ong du'ong eua hai d'.lng vi phan va bi€n phan. Slf

duy nhat eua nghi~m eua bai loan BTBD.
.Thu~t loan Anh X'.lqui h6i eho bai loan tai b~n d~ng hu'ang: MQt so d6
xuyen su6t eho thu~t loan anh X'.lqui h6i la bi€n d'.lng deo pht,l thuQe
vao loan bQ lieh sii' d~t tai eua v~t li~u. Trong 1y thuy€t deo ngu'oi ta
phat bi€u cae M thue giila ung soar va bi€n d'.lng thong qua dQ gia tang
bi€n d'.lng llEn(x)qu,a hlnh sau
DQ gia tang bi€n d'.lng llEn(x)
~n (X),E~ (x),exn (x) }=>IAnhx~~uih6i I=>~n+l (X),E~+l (x),exn+l (x)}
Tu qui t:le sai phan trung tam ta du'a ra each giai ggn dung tung bu'ae
b~ng bang 2.3:
Bang 2.3
1.1Dilki~n (E~' exn ' En' llE J
2./ &n+1= &n + ll&n
3./ Tinh ung soar thii' daD h6i va ki€m tra di~u ki~n deo
trial -
E(
P
)
{'trial _
I
trial
!
[
K
]
o-n+1 - &n+I-&n ' In+1 - o-n+1 o-y + an
6
IF f~~fl ::;0 then (tinh cae bu'de d~lnh6i)
( ) ( )
trial, .

set. n+1=set. n+! va exIt
ELSE (tinh deo)
I"trial
f:,.y= ~ > 0
E+K
[
1 f:,.y.E
1
trial
ern+l = -Ier~-~~ll ern+l
p - P A .
(
trial
)
Gn+l - Gn + uy sign (J n+l
an+! =an +f:,.y
END
a
a
an
ay
trial
CJ"n :-1
an+!
En~l ~n
I 1
~En <:0
1 I
: :
1

En+1 :
~
1
1
trial - - - - - - - - - - - - -
!
CJ"n+1 r "
,
,
,
, ,-
an+l ~
ay
an
, ,0;
, ,I
, "
- - - -I - - - - -"' " 1
'I ' 1
, ' 1
, " 1
, " I
,':,' I
" ,1 :
, ,': :
1 i Ei, Ent1
E'I : :: ~L'1En>O
p 1 1 .l1li 1
G 1 1 1
n .: 1

p 1
Gn+1 :
I
I
En+! .
l1li
f
trial <
0
Hlnh2.1a. n+! -
.Bid toaD qui hOl)ch H~irOi rl)c
Ta xet phiern ham hai bien X(a-, a) :
1 .
I 1
.
I
1
X
(er a)=-
(
ertna -er
)
E-
(ertna -er)+-(a -a)K
(
a -a
)
.
, 2 n+l n+l 2 n n
Hlnh 2.1

b
f
trial >
0
. n+l -
7
phie'm ham x(o-,a) la nang lu<;ingbu b6 sung khi co dQ tang giua 2
tr~ng thai ungsuilt (0-,a).
Ta se ct!c ti6u hoa phie'm hamx(o-,a) tren mi~n dan h6i EO"
EO"={(o-,a)EIRxIR+ :f(o-,a)::;O}.
Ta gia thie't mi~n EO" la mi~n 16i nghla la ham f: IR x IR ~ IR la
ham 16i.Bay gio ta phat bi6u bai loan qI'c tq:
TIm (o-n+l,an+l) E EO"sao cho X(o-n+l,an+l) = Min x(o-,a),
(O",a)EEa-
trang do E > 0 va K > O. Bai loan la tlm ct!c ti6u vai (0-, a) thoa man
rang buQc f(o-,a)::; 0, ta co th6 dung phuong phap nhan tU'Lagrange,
va hon the' nua trang ly thuye't qui ho~ch t6i u'u, Bertseka da chung minh
du<;ictint duy nhilt nghi~m. Ham Lagrange:
L(a,a,t'J.y) = x(a,a) + t'J.yf(a,a)
va bai loan tim ct!c tri co di~u ki~n trd v~ bai loan ct!c tri dia phuong
cua ham Lagrange, Sau khi ct!c ti6u hoa ta thu du<;iccac phuong trlnh
ling xU' trang ant x~ qui h6i d bang 2.3. d day co th6 md rQng cho ma
hint tal b~n ding huang dQng hQc.
2 2 Giiii thu~t s6 imh x~ qui h6i cho mo hlnh daD cleo hii b~n d£ng
hu'ong: Ap dlfng ly thuye't thu~t giai anh x~ qui h6i cho vi~c tinh roan
thie't ke' vai ma hlnh dan-deo tal b~n ding huang mQtchi~u. Ma hlnh va
giai thu~t du<;icap dlfng M giai bai roan dan ba thanh d6ng qui chili It!c
keo, nen. Ma hint nay cling du<;icmd rQng cho bai roan tal b~n ding
huang dQng hQc. Trang ma hlnh dan-deo tal b~n ding huang, mi~n dan
h6i trong khang gian ling suilt se du<;icmd rQng rhea ca hai phia keo, nen

mQt khi ling xU'cua v~t Mu niim trang giai do~n tal b~n ding huang ma
trang [61] da ma ta. Cac ke't qua so' cho thily ma hint phu h<;ipvai tht!c
nghi~m. Ta co th6 ling dlfng trang cang ngh~ san xuilt v~t Mu d6 tang
module dan h6i biing cach lam cho v~t li~u chay deo truac de'n mQt ling
suilt mong mu6n. Khi cho cac tham so' tal b~n tri~t tieu ta thu du<;icke't
qua bai roan ba thanh deo ly tudng nhu LM. Khachanc/p [19].
Ma hlnh mQt chi~u la mQt co sd t6t d6 phat tri6n cac ma hlnh 2
chi~u, 3 chi~u. So d6 kh6i:
I B5't I
8
Nh~p dli li~u xae dinh cae kieh thu'de hlnh hQe, tal trQng tae
d\lllg, cae di6u ki~n bien,d~e tinh v~t Mu,
T'.lOcae array ban diiu zero
~
.
I
_.A " , pint
Dti'hyubandaut'.llv~tnxEB: n' an,dn,Fn ,Cn+l
Tinh roan ma tr~n de)eti'ng, veeW tal phiin tti'
Yang l~p gia tang tai
Up rap ma tr~n phiin eti'ng phgn tU'va veetd tal phgn tU'
d~ tinh ma tr~n de)eti'ng t6ng th~ va veeW t6ng th~.
Tinh roan veetd ehuy~n vi tri nut gia tang
l'.dn+! ti'ng vdi gia tal (F;:tl - FneXI)
Yang l~p
nghi~m
ehu'a hQi t\l
Tinh ehuy~n vi tri nut t6ng: dn+l= dn+ l'.dn+l
Tinh roan tru'Cing bie'n d'.lng t6ng t'.li di~m XEB: En+l=Bedn+l
Trong do Be: Veetd ehua cae; d'.lo ham eua ham d'.lng.

Tinh ti'ng suat thU'dan h6i va ki~m tra di6u ki~n ehay deo:
atrial = E
(
£ - £p
) f
trial =
l
atriall-
[
a + Ka
]
n+l n+l n' n+l n+ll Y n
2
I'lrwl < 0
J n+l -
Dung
Giai do'.ln dan h6i
trial
a n+l = a n+l
p p
£n+l = £n ,an+\ = an
Sat
~ 9.
Giai do£!.ncleo
Giai thu<\itanh x£!.qui h6i
{
trial
~ = . n+\ > 0
Y (E + K)
I trial A

E
.
(
trial
)J
a n+\ =Lan+\ - DoY sIgn a n+\
P P A .
(
trial
)
tn+\ =tn + DoYSIgnan+\
an+l=an+~y
I
Tinh lOan vectd 11,I'cnQi fjnt (a n+l) cua tUng ph~n tt'!'va Hip rap tinh lOan
. n. n
1
A.':;
F
illt
A
f.
illt
( )
T d
'
A T
' ?
I
"
h

'
vectd 1,I'cnQl tong n+\ = e a n+\ . rang 0 : oan tU' ap g ep
e=l e=l
Trong do dn : vectd chuy~n vi nut ling
vai tai trQng laC dl,lllg F:xt
F
int
1
A":;
C d 1 d
'
d
?
n : vectd 1,I'cnQl tong, n+l: mo u e an- eo.
.
f
trial
0
EK
kh
. trial
0
Cn+1= E khl
+1
~ , Cn+1= - 1 f
+\
~
n E+K n
.Ap dl;lllgso cho bai loan 3 thanh hlnh 2.2 khi st'!'dlfng giai thu<\itanh x£!.
qui h6i hoan lOan phil h<;lpvai 1ythuye't dan-cleo tai b~n ding huang

hlnh (2.3, 2.4, 2.5). Khi cho cac thalli so tai b~n trit;t lieu ta thu dU<;lCke't
qua bai loan ba thanh cleo 1y tudng nhu LM. Khachan//p [19].
I-fmh 2.3. Quan ht; p-8
Sai
Xac dinh ~dn+1ling vai gia tai (F::~ (O"n+l)- Fne:;)
L
K
J
-I
[
in!
(
exl
]
~dn+1 = - n+l Fn+1 0"n+l) - Fn+'
vai Kn+' = A keln+' : Ma tr<\indQ cling t6ng th~
e=l
keln+' = fB:Cn+,Bedx:Matr<\indQclingph~ntt'!'
Nghit;mcua bai loan
p p
a=an+\,t =tn+l,
a = Cln+l
[ Ke't thuc
p
T
Hlnh 2.4. Quan M crl- El
2 ~"-~-~Ir~ cr,-'t
i . cr
cr,
1.

0'-
-L5
-1
-115
0
05
10
r
L
L
p
Hinh22: H~ dim 3 thanh
Hlnh 2.5. Quan h~ cr2- E2
2.
l
x1d_T~nh~_!
2
.
t ! -
.
crT+
.
k
.
",,(I',
.
,)1~
i cr,+K,a2(P,)
1. r- . cr(PI)='
1 t 1- !

0
0, .'0
cr, OJ
.1
.2
.1
.O.fi
0
O.fi 1.fi
x103
.1fi
,.~
'"
E, x11f3
2.3. Bai tmin dan- nhot- cleo mQt chi~u
ta xet mG hlnh d~ln-nhot-deo mQtchi~u, trong d6 c6 mG hlnh do Prezyna
d~ nghi nghla 1a ti'ng swlt phat sinh cua mG hlnh con phI;!thuQc VaGt6c
dQ chay deo. Ung suilt phI;!thuQc VaGt6c dQbie'n d<;lngva san khi d<;lt
nguong deo thl thai gian 1a dQc l~p, ling suilt khOng d6i nhung bie'n d<;lng
v~n tang. £>~tie'p c~n bai to~n nay, chung ta gioi thi~u mQt khai ni~m
hfc gia, duQc xU'dl;!ngtrong nhling buoc tinh toan khi hi~n tuQng dan-
nhot-deo xay fa. Xem hlnh 2.6. Ma sat khO khGng hO<;ltdQng khi
CTp<Y. Bie'n d<;lngt6ng cQng tren mG hlnh 1a: E =Ee + Eyp, CTe= E Ee, va
CT =CTd +CTp, Evp =y(cr-Y),trongd6:CTp =CT ne'u CTp<Y,CTp=Y,
ne'u CT=CTp;::Y. Ung suilt d bQph~n giiim chiln nhot 1a CTdlien h~ voi bie'n
dE
d h
'
d
?

b
? vp.
1
,
h
A K
h
'
<;lngn at- eo a1: crd = 11dt = IlEvp' fJ- a y so n at,
r . 1
I 8 H . 1.0-1.Hi Ni-Iie:
\TrUJviEN'l
11 r-' =-1
y = 1-, H 11aM s6 goc tai b~n cua du'dng chay cleo tai b~n'
L
I
11 1
Tru'dc khi d":ltdu'<;jctr":lngthai nhdt-deo ta co : Evp= 0 di~u nay d~n de'n
<Td= O. Thie't 1~p quail h~ U'ngxli' cho mo hlnh d hai tr":lngthai: tr":lng
thai daD h6i va tr":lngthai nhdt-deo daD h6i (dan-nhdt-deo):
Evp = Y(o"- (O"y+ H'Evp)) = Y(o"-y),
E(t)= O"A + O"A-o"y (l- e-YH't
)
E H' \
co d6 thihlnh2.10.
lOp <Tp
'
l
<Tp
Ee

' 4>2
HI ~O ~ 0"A + (0"A - 0"Y ht
E
. Vi dl;! biing s6: Bai toan 1: v~ stf bie'n d":lngdan-
nhdt-deo cua mQt thanh don gian (Hlnh 2.9a) chiu tai
tn;mg keG khong d6i, chi€u dai thanh L = 10 don vi,
co thallis6 gidi h(;lnchay cleo1a:O"y=10,tai trQng
P = 10 don vi, E = 10.000, tie't di~n thanh A=l don vi
di~n rich, h~ s6 nhdt y =0.001 va thong s6 tai b~n
bie'n d":lng HI=5000, Ke't qua chuy€n vi Hlnh 2.7 phu
h<;1pvdi hlnh 2.10.
Hlnh 2.7
]
Hlnh 2.8
Q.Jan ~ chu~n vi, lhOigian
~ ~
cj>,
l'
QlliI ~ ~ \oj,dit .,;on
'4> - T i-iT i"""T I
~, j ~.H~-1
.=
, z
-e., ~
I
+~ +-
;:
,~
"

I
.
I I'
+-1 j
N
i ,,-
"
~.
~.
;:-
~
+ 1
ThOigian
t
Hlnh 2,10: Bie'n d(;lngtuye'n tinh v~t 1i~u tai b~n
+-
" ~ ~ ~ u -"-" r
1Jij Wan
Bai roan 2: Hai thanh don gian n6i song song (Hlnh 2.29b), chiu tai
trQng keG khong d6i, chi~u dai hai thanh biing nhau, L = 10don vi, tai
trQng P =15 don vi, trong do hai thanh co tinh chat d~c tru'ng v~t Mu:
12
hai thanh co cling thalli so module dim h6i E =10.000, tie't di~n hai
thanh b~ng nhau A=l, nhung giai h':ln U'ngsullt chay deo cho phep cua
hai thanh khac nhau 19nluQt la cry=20, cry=10, lien quail tai h~ so nhat
y =0.001 (thong so d~c trung cho tr':lng thai long ), h~ so giai h':ln slf gia
tang bie'n d':lng nhat-deo r =0.1. CM Y thong so tai b~n bie'nd':lngtrong
mo hlnh nay HI=0. Ke't qua slf bie'n d':lng dan-nhat-deo th6 hi~n tren
hlnh 2.8, la chuy6n v~cua nut cuoi vai thai gian cho phgn tU' mQt thanh
co xet de'n lam vi~c dan-nhat- deo, ch~utai tr9ng keG thong d6i. Ke't

qua la hlnh 2.8 v~n phil hQp vai hlnh 2.10 nhutrong ly thuye't bie'n d':lng
tuye'n tinh v~t Mu teEb~n, duang cong chuy6n v~rhea thdi gian m6i 19n
gia tai, bi6u di6n dung cong thU'cbie'n d':lng, U'ngsullt thong d6i nhung
bie'n d':lng v~n tang, do hi~n tuQng chay cMm nhat-deo.
Chu'dng3: MO HINH DAN-DEO HAl CHn~:U
3.1. Ly thuye't dan-deo hai chi~u : Nghien cU'umo hlnh dan-deo hai va
ba chi~u. Trinh bay ly thuye't chay deo, thong gian bie'n d':lng va thong
gian U'ngsullt, cac phuong trinh U'ngxU' dan-deo thong thu~n ngh~ch,
cac di~u ki~n chay deo, module tie'p tuye'n dan-deo, thong gian bie'n
d':lng vai cac di~u ki~n d~t tai va cilt tai, ke' do trinh bay v~ ly thuye't
bie'n d':lng phiing van Mises, deo ly tudng, deo tai b~n ding huang dQng
h9C, tai b~n chay ke't hQp trong thong gian U'ngsullt. Trinh bay nguyen
ly qtc d':li haG tan nang luQng deo la di~u ki~n dn cho d~nh lu~t chay
ke't hQp trong thOng gian U'ngsullt, va tinh 16i cua mi~n dan h6i. Minh
h9a d~nhlu~t chay deo ke't hQp nhula mQt hilt ding thU'cbie'n phiin.
3.2. Mo phOng so'bai toaD daD deo hai chi~u: Ung d1?-ngthu~t roan anh
X':lqui h6i cho mo hlnh bai roan dan-deo hai chi~u: Ong trlJ dai, co ban
kinh trong r=lOOmm, ban kinh ngoai R =200mm, ch~u ap Ilfc d~u
xuyen tam tU bell trong hlnh trlJ, xet bai roan trong tr<;tngthai bie'n d':lng
phing, rhea quail di6m chay deo van Mises. Ke't qua so phan anh dung
slf lam vi~c cua v~t li~u d tr':lng thai dan-deo tai b~n. E= 2.1 xl04
dN/mm2, M so Poissions v =0.3, nguong U'ng sullt deo cry =24.0
dN/mm2, thong so tai b~n ding huang HI=0.00. Mo hlnh doi xU'ngnen
ta chi xet mQt phgn tumo hlnh nhu hlnh ve tren, ta phan rich luai phgn
tU'huu h':lnnhu hlnh 3.1(576 phgn tU')va hlnh 3.2(1080 phgn tU').
.Ke'tlu~nv~ cac ke'tqua so
Ke't qua so phil hQp vai slf lam vi~c cua v~t Mu v~ m~t d~nhtinh. Do la
vi dlJ kinh di6n v~ bai roan bie'n d':lng phing thea tieu chuin chay deo
13
eua yon Mises. Truy~n ling suitt tie"p tuye"n thay d6i rhea rIa (jr tang nSi

giam diin rhea r =100, 200. Cae d5 tht sau bi€u dih quail h~ giua R ya
(j tu'cingling yoi cae ap ltfe P liin lu'<;itbing 8dN/mm2, 12dN/mm2,
14dN/mm2, 18dN/mm2. Hlnh 3.3 :lam yi~e trang tr<.tngthai dan h5i. Hlnh
3.4 :b~t diiu xuat hi~n tr<.tngthai dea. Hlnh 3.5, hint 3.6 :ban kinh m~t
ehay dea IOndiin. Vi dlf hint 3.6 : co ban kinh m~t ehay dea r=165.
Tru'ong h<;ipdea 1.1tu'dng (cae M s6 tai b~n biing khOng ), cae ke"tqua
trung yoi LM. Khachan6p [19]
576 phin III tam giac
CI
,-;0
~
;!'
,:3
-!
lim kinhr
Hlnh 3.3, P=8dN/mm2
IVOVpIlaIllU lam glac
80
60
20
0
0 ~ ~ w 00 ~ ~ ~ ~ 00 ~
CI
':0;,
~
","
=
,';:)
-,
r

i
\00 "" ,. ,. '"
rnn kinhr
Hlnh 3.4, P= 12dN/mm2
"~li
.
~:~
p
'~~
H
-T '~L
,
~I
H
-"
,
1
'" "
I
.
~t-
'
'
O
! , +- ', -
-t'+
, ," I
I
" 'I ",1 I I
cr' -

III-~ + ;
I
"<:T

++ r-~
~ I ," I , I 1
~ i
IT
I"'.I
tt
i ' I
,
, -( "'- 1
~ i I I I "'" I
~'-+-+-
11tt
! ' ! I'_+
,
'~ ~
I I I! i t I I I
Li- '-IT
,
+
1 +-+
1
-H
, 1 ' I
+-+
I ,
~-'

l
-ll: -, ,
, i ILi i 1
rn '" w m ,.
Iimkinhr
2
Hinh 3,5, P=14dN/mm
Om;r!r~[ ,
,
i
'~=+
,
I ;-~
'<I-' ' I -'" '-'
t
'Y
, -~ +-~ + l
I ,/ I I ,
:.' -+ t£- +~ l
l
[
+7~
ffi

,
1 = j
'<:: - t ~r I I '
" L- L +~~
-fI-~ =f~+ +-
,

- I
f
'-~ 1
;4 r-t-+ ,
i I, i .1
" " . , .
ClJuy€n vi ~t troog
14
,-
<~'
,J,
'00 n, '" m '~ ,. ,m- '00
bankinbr
Hinh 3.6, P=18dN/mm2
Hinh 3.7 quan M P va ehuy6n vi
m?t trang
Chtidng 4: MO PHONG BA.I ToAN VON THANH DA.N Hal PHI
TVYEN NHUNG TRONG M(n CHAT LONG N4,NG
4.1. Md dftu: Xet mQt thanh dan h6i phi tuyen
co chi~u dai L voi kh6i lu<;ingrieng 10 bi gill
ch?t mQt d~u va du<;icnhung chlm rhea phuong
thing dung trang mQt ch!lt long co kh6i lu<;ing
rieng 11' Do stf khae nhau v~ cac kh6i IU<;ing
rieng eua thanh va eh!lt long, ltfe My
Arehimede va cae ngo?i ltfc khac tic dl,lng tren
thanh lam cho thanh bi u6n congo GQi u(x) la
goc giua tiep tuyen voi thanh d tr?ng thai bi
u6n t?i di6m eua thanh co hoanh dQ cong x va
trl,lcthing dung Oy. Tucsnak trdng [63], da
y Hinh 4.1

15
thie't l~p bai roan bien phi tuye'n cho u(x) bi€u di~n goc giil'a tie'p tuye'n
va du'ong dan h6i vdi tn:lc oy nhu' hlnh 4.1, tren cd sa mQt s6 ke't qua cua
V.Valcovici du'a ra v~ du'ong dan h6i cua thanh bie'n d~ng trong moi
tru'ong chao khong vao nam 1971 nhu'sau:
-d I
-M(x,u (x)) + g(x)sinu(x) =0, 0 < x < L,
dx
u(O) = 0, (4.2)
M(L,ul (L))+YIG(L)sinu(L) = 0, (4.3)
trang do A. la mQt hiing s6 du'dng bi€u th~ h~ s6 lIfc nen d9C tn,lc cua
thanh, g(x) = -A, + (YO - Yl)F(x) + GI (L) la ham cho tru'dc co y nghia
cd hQc nao do, G(x)la moment u6n t~i di€m x, M(x,ul (x)) la d6i ng~u
lIfc ma sat anh hu'ang tren do~n thanh cong tu d~u thanh de'n v~tri x Trang
cong tdnh cua mlnh [63]. Tucsnak xet slf phan nhanh cua phu'dng tdnh vi
phan ma ta goc l~ch t~i m6i v~tri cua thanh, nhu'ng ang cling chi dung l~i
trang mQt s6 tru'ong hQp rieng cua ma hlnh. Vao nam 1992 mQt s6 tac gia
trong [30] dii tie'p tl;!Ckhao sat vdi tru'ong hQp moment u6n cua thanh chi
phl;! thuQc vao to~ dQ cua di€m dang xet tren thanh. Trang lu~n an nay,
khao sat t6ng quat hdn, do la ngoai slf phl;! thuQc vao v~ tri tQa dQ,
moment u6n con phl;! thuQc van goc giil'a tie'p tuye'n cua du'ong dan h6i
vdi trl;!cc6 d~nh, va thanh ph~n d~o ham rhea x du'Qcthem mQt Ilfc dQc
trl;!c N(x,u(x)), d€ lam t6ng quat hoa bai roan hdn nhu' sail:
- d [M(x, ul (x)) + N(x, u(x))] + g(x) sin u(x) = f(x), 0 < x < L ,(4.4)
dx
va di€u ki~n bien t~i hai d~u thanh la: u(O) = 0, (4.5),
M(L, ul (L)) + N(L,u(L)) + bl sin u(L) = b2, (4.6)
trong do L > 0 bl, b2 la cac hiing s6 cho tru'dc, cac ham s6
M,N:[O,L]xR~R, j,g:(O,L)~R la cho tru'dc thoa cac di~u
ki~n ma ta se d~t sail. Nhu' v~y bai roan (4.4) -(4.6) la ma hlnh bai roan

u6n mQt thanh dan h6i phi tuye'n nhung trang mQt chat long t6ng quat
hdn so vdi Tucsnak [63] dii thie't l~p tu'dng ling vdi tru'ong h<;ip:
g(x)=-A.+(Yo -YI)F(x)+G'(L), bl =YIG(L),b2 =0, j(x)=O,
N(x,u) = O. (j day phu'dng phap ph~n ta hil'u h~n du'<;icsa dl;!ngd€ giai
bai roan bien phi tuye'n (4.4) -(4.6). Trang chUng minh co sa dl;!ngb6 d~
(4.1)
16
Brouwer ke't hQp vdi mQt so b1ltding thuc danh gia nQi suy da thUc. Thu
duQc cac ke't qua v~ danh gia sai so giil'a nghi~m x1lpXlva nghi~m chinh
xac. Ke't qua thu duQc d day Ia mQt t6ng quat hoa tuong doi cua cac ke't
qua trong [40], [54], [45].
4.2. Phtidng phap phiin to' hUll h~n Xay dl!ng khong gian ham va hQ
cac khong gian hil'u h<;lnchi~u d~ Kay dl!ng thu~t giai x1lp Xl b~ng
phuong phap ph~n t11hau h<;ln.Ta thanh l~p cac gia thie't sail:
(HI) M,N: [O,L]x IR ~ IR Ia cac ham thoa di~u ki~n Caratheodory,
Wc Ia:
(i) M(e,y),N(e,y) do duQctren [O,L] vdi mQi Y E IR,
(ii) M(x,e),N(x,e) lien t\Ic tren IR vdi h~u he't x E [O,L].
(H2) (M(x'YI)-M(x'Y2» (YI-Y2»0, \iYI,Y2 EIR'YI :;tY2.
(H3) T6n t<;licac h~ng so duong CI,C2,C3 vdi CI > C3(~)p-l va
VP
/ I 1
cac ham ql E LI, q2, q3 E LP ,vdi - + / =I sao cho:
p p
(i) yM(x,y)ZCIlyIP-lql(X)I, \iYEIR,vdih~uhe'txE[O,L].
(ii) IM(x,y)1 ~ C21y1P-l +lq2(X)I, \iy E IR, vdi h~u he't x E [O,L].
(iii) IN(x,y)I~C3IyIP-l+lq3(X)I, \iYEIR,vdih~uhe'txE[O,L].
f
I I
(H4) E V , gEL.

(H5) T6n t<;lih~ng so C4 > 0 sao cho: \iYI, Y2 E IR vdi h~u he't
x E [O,L], ta co: (M(x'YI) - M(x'Y2» (YI - Y2) Z C41YI - Y212
(H6) :JKI >O:IN(x'YI)-N(x'Y2)I~KIIYI-Y21 \iYI'Y2 EIR, vdi
h~u he't x E [0, L]. Nghi~m ye'u cua bai roan (4.4)-(4.6) duQc thanh
l~p tu bai roan bie'n phan: Bai roan (P): TIm UE V sao cho:
<M(x,u/) + N(x,u), vi) + <g(x)sinu, v) + (bi sinu(L) - b2)v(L) = <f, v)
, \iv E V(4.9). Ta x1lpXl bai roan (P) bdi hQ cac bai roan hil'u h<;ln
chi~u (Pm) nhu sail: Bai roan (Pm) : TIm um E Vm sao cho
<M(x,u~) + N(x, um)' W) + <g(x)sinum' IV;)+
17
+(blSinum(L)-b2)Wj(L)=<f,wj> Vi, l~j~m.
Dinh 1:91:Cho bl,b2 E IR. Cia sa (H1)-(H4) dung. Khi do:
i) Bili loan (Pm) co nghifm Um E Vm.
ii) Bili loan (P) co nghifm UE V.
Hun nrla nlu ta thay cae giG thilt (H1)-(H4) Mi (H1), (H3) -(H6)
vil(H7)nhusau: (H7) (Kl +lbll+llgIILdL <C4. Khido:
(4,10)
iii) Bili loan (Pm) co duy nhtft m(jt nghifm Um E Vm.
4i) Bili loan (P) co duy nhtft m(jt nghifm UE V.
5i) um ~ u hQi tl,ltrang CO(0) (h(ji tl,uliu tren dO{;m[0, L]),
D€ chung minh co xiI dl,lllgblSd~ Brouwer ke't hcjp vdi mQt s6 ba't dhg
thuc danh gia nQi suy da thuc.
4.3. Thui;Hgiai s6: Vdi L = 1, Nghi~m bai loan (4.4)-(4.6) du'cjcxa'p xi
m
bdi mQtday hQitl,l {um}: um = I Cm)Wj, Khi do Vm la mQtkhong
)=1
gian can huu h~n chi~u cua V sinh bdi m ham cd sd Wj(x) ham d~ng
hlnh rang cu'a,1~ j ~ m nhu'sau:
{
(x-xj)/h, ne'u Xj-l ~X~Xj'

-vdi l~j~m-l: Wj(x)= (X)+l-x)/h,nA~u x) ~X~Xj+l'
0, lieu x~[Xj-l,xj+d
_
{
(X-Xm-l)/h, ne'u xm-l ~x~L,
wm(x)- ,
0, lieU 0 ~ x ~ Xm-\'
Hdn nua cac ham Wj con thoa tinh cha't W/Xi) =/5ij'
1
< '<
0
< '< s::
I
'
k
'
h
'A
Kr k
- } - m, - 1- m, vi; a y HfU onec er.
- d / /3 1 1
-[(2 + sin x)(u (x) + U (x» + -cos(xu(x»] + -sin u(x) =l(x),
~ 4 4
0 < x < 1 (4.41), u(O)= 0 (4.42),
. / /3 cosu(1)+sinu(1)
(2 + Sill l)(u (1)+ u (1» + =b2 (4.43), trong d6
4
- vdi j = m :
18
1 h 77:213 377:4 .

b2 =-(1 + -v3)+ -[1 + z](2 + SIn 1) (4.44),
8 36 (36)
lex) = ~sin(2asin ax) - 2a2[1 + 4a4 cos2 (ax)] cos(ax) cos x
4
+ 2a3[1 + I2a4 cos2 (ax)]sin(ax)(2 + sin x)
+ ~a[sin(ax) + ax.cos(ax)]sin(2ax.sin ax), a =77:/ 6 (4.45).
2
Nghi~m chinh xac cua bib loan (4.41)-(4.45) Ia Vex (x) =~sin(~).
4.4. Philo tich cae ke't qua s6: EHnh gia sai s6 giG'a nghi~m xap Xl
Urnva nghi~m chinh xac: Ilurn- ullHI(O,L) ::;; ~. Dung thu~t giai
Newton-Raphson cho bai loan (4.41)-(4.45). Chung toi thu duQc cac ke"t
qua tinh loan va so sanh voi nghi~m chinh xac Vex (x) =~sin(~)
tticlng ling voi: m =20 xem hlnh 4.2, m =30 xem hlnh 4.3. Tinh loan
lftn lUQtvoi m = 5,10,15,20,30,50, cho ta thay sai s6
E
(k) _
II
(k)
U
II
I
(k)
U
(
)1
.?
d
"
kh
' -

d
"
m = cm - ex = max cmi - ex xi glam an 1 m tang an.
. ct:) l:::;/:::;m
'*' Ngj1iemPPPIHH, '-'Nghiemchinh xac
- -
'*' Nghiem PPPTHH, '-Nghiem chinh xac
ooc r r !
00__+-+__1
I
!
:~ H~-
oj- _o
f-
+-
.

l
!
"
L OIia ra :30 doan
LOJiara:20cban
mnh 4.2, L chia 20 do,!-n mnh 4.3, L chia 30 do,!-n
Cac rich phan s6 trong bai loan duQC tinh biing cong Cl;lrich phan s6
Gauss. DQ phUc t'!-Ptrong cac rich phan s6 la dang k~ VIdn chinh xac
rat cao, VIphai lay rich phan s6 tren cac phftn tii' chi~u dai kha be khi
chia nho thanh, ne"ukhOng, sai s6 rich lGy se lam anh huCingde"nnghi~m
cua bai loan, nhat la cac phftn tii' ci dftu mui thanh. Phftn l~p trlnh duQc
19
h6 tn;1b~ng Maple6 ke't h<;5pvdi l~p trlnh tinh roan b~ng MatlabR 12 d~

cho ra ke't qua s6 va bi~u d6 so sanh. Ynghia thlfc ti~n cua mo hlnh co
th~ ap dl:Ing cho cac bai to<lnthlfc te' nhu' xac dinh dQ sai l~ch cua mlii
clan khoan d~u tren m~t bi~n, hay lien quan de'n va'n d~ xac dinh vi tri
cua mQt thanh dai du'a vito moi tru'ong cha't long, ching h,!-nnhu' bai roan
nhling 6ng dftn vito moi tru'ong bi~n khi Hip d~t du'ong 6ng.
Chu'dng 5: !"fO PHONG SO BAI TOAN D,AO DONG CUA THANH
DAN HOI YOI RANG BUOC DAN HOI NHOT (j MA T BEN
Khao sat bai toan mo ta slf va ch,!-mnhu' la dao dQng, CI;1th~:
5.1. Md dftu: Ta xet mQt v~t dn kh6i lu'<;5ngM chuy~n dQng vdi v~n
t6c ban d~u, khi va ch,!-m vao thanh co chi~u dai L qua bQ ph~n giam
cha'n g.ln d d~u thanh co dQ cling k . D~u Kia cua thanh ti,tatren mQt n~n
cling. Gia thie't d m~t ben, thanh chili mQt IlfCma sat dan h6i nhdt va
mQt ngo,!-iIlfc phl;1j(x,t). Llfc ma sat dan h6i nhdt du'<;5cxa'p Xlnhu' mQt
llfc kh6i. Khi do dQ dich chuy~n dQc u(x,t) thOa phu'dng trlnh song:
2
utt-a uxx+Ku+AUt=f(x,t), O<x<L,t>O, (5.1.1)
trong do a = .J(A + 2G)/p la v~n t6c truy~n song dan h6i dQc cua
thanh; K = Klr /F, A = Air/F, vdi r va F l~n lu'<;5tla chu vi va di~n
rich cua thie'tdi~n ngang; KI, Al l~n lu'<;5tla h~ s61lfCcan dan h6i va M
s6 nhdt (im~t hen. Ngoai ra con co cac di~u ki~n bien
crx
=Eux(0,t) = -pet), t,!-id~uthanhx = 0, (5.1.2)
u(L, t) = 0, t~icu6i thanh x = L, (5.1.3)
va cac di~u ki~n d~u: u(x,O) = uO(x), Ut(x,O)= ul (x). (5.1.4)
Llfc dan h6i P(t) lac dl;1ngten d~u thanh x = 0 thoa man bai roan
Cauchy cho phu'dngtrlnh vi phan thu'ong[30]
II k k
P (t)+-P(t)= Utt(O,t), t>O,
M F
~ I ~

P(O) = PO' P (0) = Pl' (5.1.6)
Y nghia thlfc ti~n cua bai roan la mo hlnh mQt mQt boa may dong CQCbe
tong, tren d~u CQCco bQ pMn giam cha'n, chung quanh CQCchili ma sat
dan h6i nhdt, t6ng quat hdn no chili them mQt llfc phl;1.Trang Wc dong
CQCd~u CQCKiagi,ipphai n~n da cling ching h,!-n.
(5.1.5)
20
Lu~n an trlnh bily mQt thu~t giai so cho bili roan (5.1.1) -(5.1.6). f)gu
lien dung mQt so d6 sai phan theo bien thai gian dua bili roan v~ mQt h~
phuong trlnh elliptic, sau do ta giai xap xi h~ do bJng phuong phap phgn
tti' huu h<).ncap mQt, chung minh st! t6n t<).ivil duy nhat nghi~m trong
tung bude thai gian, st! hQi tl;!nghi~m xap xi v~ nghi~m eua bili roan
xuat phat, danh gia sai so eua chung.
5.2. Sai pMn theo bie'n thoi gian: Xap xi sai phan cho M (5.1.1)-
(5.1.6) thanh h~ (5.2.6)- (5.2.8): ti = iM, i = O,l, ,N, f t= T/ N.
II
-ui (x)+aui(x) = Fi(x), 0 < x < L,
u{(O)-~Ui(O)=Gi' ui(L)=O,2:::;i:::;N,
1
[
AI
)
k 1
a=- K+-+- A=
2 2 ' jJ ,
a f t (f t) EF 1+ ~(M)2
M
~ =P(ti)' ui(x)=u(x,tJ, Gi =G(ui-i(O),ui-2(O),Pi-j,Pi-2)'
(2 + AM)ui-l - ui-2 1
Fi(X) = 2 +2fi(X), Pi =-E(~Ui(O)+Gi)

(af t) a
5.3. Khao s:lt bai tmin (5.2.6)-(5.2.8): Nghi~m yell cua bili roan (5.2.6)-
(5.2.8) dU<;icthilnh l~p tU bili roan bien phan sau:
Bilitocln (~):TIm ui EV={VEHl(O,L):v(L)=O} saoeho:
a(ui,v)=(Li'v), VVEV, vdi (5.3.1)
I
f
I I
a(ui'v)= [ui(x)v (x)+aui(x)v(x)]dx+~ui(O)v(O) (5.3.2)
0
I
(Li, v) = fFi(X)V(x)dx - Giv(O).
0
Nha dinh 1;' Lax-Milgram ta co:
Dinh ly 1: Bili roan (Pi) t6n t<).ivil duy nhat mQt nghi~m ui E V.
5.4. Xftp XIbili toaD (Pi) b~ng phu'dngphap phiin to' hU'uh:;tn:
Bay gia ta xap xi bili roan (Pi) bdi hQ cac bili roan huu h<).nchi~u
(p/m) nhu sau: Bili tocin (p/m): TIm u;m) E Vm sao cho
(5.2.6)
(5.2.7)
(5.2.8)
(5.3.3)
21
a(ujm) ,Wj) = (Li' Wj), Vi, 0::; j ::;m -I. (5.4.3)
Nha dinh ly Lax-Milgram ap dl,lng eho d'.lng song tuye'n tinh a(.,.) va
d'.lng tuye'n tinh Li tren khong gian huu h'.lnehi~u Vm, ta co:
fJjnh ly 2: Bai roan (p/m» t6n t'.li va duy nha't mQt nghi~m u}m) E Vm.
fJjnh ly 3: (i) m~~IHm)- Ui!lv = O.
(ii) Ne'u nghi~m ui E V n H2, thl ta co daub gia sai s6:
max

ll
u;m)-u,
11
=O(~2).
05,,5,N v
m-I
5.5. Giiii hili toaD (p/m»: Di;it ujm) = I cj~)w j' Xet bai roan
.
0
1 .
1=
(5.1.1) -(5.1.6) vdi :L=I,a=l, K=I,A=I, E=I,M=I, T=l.
Cho tru'de N ;:::3. Di;it: ti =i!1t, i = 0,1, ,N, !1t= T / N = ]/ N ,
Xj =jh=j/m, j=O,I, ,m, h=L/m=l/m.
2
f(x,t) = [(l +~)cas(JT x) + JTsin(JTX)]ex-t, uo(x) = ex cas( 7tx),
4 2 2 2
7tX
ul (x) =-ex cas( 2)' Nghi~m ehinh xae eua bai roan (5.6.1)-(5.6.6)
la: ueX(x,t)=ex-tcas(7tx), pex(t)=-e-t,
2
m-I
u(m)
(
x
)
= " c~m)w.
(
x
)

p(m) =_~u(m)
(
O
)
-G(m)
1 L !J 1'1 """I I'
j=O
G~m) = G
(
(m)
(
0
)
(m)
(
0
)
p(m) p(m»
I UI-I' UI-2 ' I-I' 1-2 .
D~ daub gia sai s6 ta quail sat cae sai s6 sau day khi eho N (bu'de thai
gian i = O,I, ,N,), m (bu'dekhong gianj = O,I, ,m,) tang d~n. Chli y
khi tang mIen, d6ng thai v§:n eho N tang len ( tu'dng ling !1t = ]/ N
be) "khong h<;1ply" thl ke't qua tho du'<;1ekhong t6t.
E(N,m) = m.ax max
I
c~m) -uex(Xj,ti)
l
,
2'5,I'5,NO'5,I'5,m-) .
E(N,m) = max

I
p(m) - pex (ti )
1
' trong do:
2'5,i'5,N I
22
Hlnh 5.11a sai s6 cua E(N,m), Hlnh5.21a sai s6 cua E(N,m) ling vdi
m = 25,36,64,81,100,121,144, N=5,6,7,8,10
Cac duong, giam, n?imngang, cho tha'yslf hQitl;}theo buoc khong gian
tuong ling voi m6i buac chia thai gian la 6n dinhtheo luQcd6 sai phan.
m=25,36,64,81, 100, 121,144
Hlnh 5.1 Sai s6 E(N,m)
m=25.36,64,81,loo,121,144
Hlnh 5.2 sai s6 E(N,m)
KET LUA.N vA KIEN NGHJ
.Ke't qua tho duQC trong lu~n an
1/ SO'dl;}ngphuong phap rhein to' huu h~n (PPPTHH) voi trlnh dQ chuyen
sau trong ta't d cac chuang cua lu~n an d~ mo phong giai s6 cac bai
roan bien t'inh va dQng cua moi truong r~n bie'n d~ng VOlqui lu~t ling xO'
phi tuye'n va bie'n d~ng phuc t~p. 0 day ke't hQp co hi~u qua giua
PPPTHH VOlcac phuong phap cua giai rich ham d~ phat bi~u bai roan
duoi d~ng ye'u, xa'p Xl bai roan d~ng ye'u trong khong gian huu h~n
chi~u, Kay dlfng cac khong gian ham xa'p Xl, ham d~ng co sa sinh ra
khong gian ham huu h~n chi~u. Danh gia sai s6, chung minh slf hQi tl;}
cua thu~t giai, trong mQt s6 truong hQp, chung minh duQc slf t6n t~i va
duy nha't nghi~m va so sanh nghi~m xa'p Xl voi nghi~m chinh xac.
2/ Ap dl;}ngthu~t roan anh X~qui h6i trong ba chuang (1,2, 3), t~p trung
vao mo hlnh ling xO'dan-cleo, dan-nhot-deo mQt chi~u va hai chi~u, d~
xac dinh duong chaydeo b?ing cach dua cac ling sua't thO'khi gia tai nam
ngoai duong chay cleo v~ dung duong chay cleo cua bai roan mQt chi~u.

Tuong tlf cho mi:j.tchay cleo cua bai roan hai chi~u, thu~t roan dua ling
sua't thO' n?im ngoai mi:j.tchay cleo v~ dung mi:j.tchay cleo b?ing each
EJ;1u,;;;
-

n__ _
>

___n______-
w
_n
__n_ ___-
EJ;6.oV
>
_nn
EJ;5.nV
_n-__On
_n-

.m

+-n______-'_n__""' n
"
£"0-
£(8,m)
_n
- n_- __n
"
£(l

;'u
!! 0 " '
23
chie'u xu6ng di€m g<innhat la di€m tren tie'p tuye'n cua mi.itchay. Neu
len cac di~u ki~n Kuhn-Tucker, di~u ki~n chay deo cho tung tr<;tngthai
lam vi~c cua v~t li~u. Danh gia nang hiQng hao tan ti~n dinh, do la h~
qua cua Hnh chat: co slf lieu hao nang hiQng d bell trong cac M co hQc.
Chung minh clfc ti€u boa phie'm ham b6 sung nang luQng, thu duQc ke't
qua tuong duong la cac phuong trlnh ling xli' trong thu~t roan anh X<;tqui
h6i trong bai roan 1chi~u. Chung minh slf tuong duong cua vi~c clfc d<;ti
hoa nang luQng hao tan deo va bat ding thuc bie'n philo trong bai roan
dan-deo trong bai roan 2 chi~u. Lu~n an di:'isli' dl,lng cong Cl,lgiai rich l6i
mQt cach d~y thuye't phl,lc VI giai rich l6i lam thay d6i bQ mi.itcac bai
roan dan-deo nhu cac cong trlnh cua Moreau (1976-1977), Johnson
(1976-1978), lL.Lions[68]. vv
3/ Khao sat bai roan Tucsnak md rQng: u6n thanh dan h6i phi tuye'n
nhl1ng trong mQt chat long nii-ng. Chung minh slf t6n t~i va duy nhat
nghi~m cua bai roan d d~ng ye'u. Tu do cling chung minh duQc slf t6n t<;ti
va duy nhat nghi~m cua bai roan d d~ng vi philo. Chung minh duQc slf
hQi tl,lcua Wi giai xap Xlv~ lai giai cua bai roan xuat phat. Thu duQc cac
ke't qua v~ danh gia sai so' giua nghi~m xap Xlva nghi~m chinh xac. Sli'
dl,lng thu~t roan Newton-Raphson d€ giai M phuong trlnh phi tuye'n
trong R" trong chuang b6n. Th€ hi~n dQphuc t~p cua thu~t roan Newton-
Raphson trong R" so vdi trong R.
4/ Khao sat slf va ch~m sinh ra Ian truy~n song: mo phong bai roan dao
dQng cua thanh dan h6i vdi rang buQc dan h6i nhdt d mii-tbell. Thie't l~p
so d6 sai philo theo bie'n thai gian M dua bai roan v~ mQt M phuong
trlnh elliptic, sau do xap XlMd6 b~ng phuong phap ph~n tli'huu h~n cap
mQt. Sli' dl,lng dinh 19Lax-Milgram d€ chung minh slf t6n t~i va duy nhat
nghi~m cua bai roan d d~ng ye'u. Tu do cling chung minh duQc slf t6n t~i

va duy nhat nghi~m cua bai roan d d~ng vi philo. Lu~n an di:'ichung
minh d<iydu cac chi tie't, k€ d b6 d~ 3 la ti~n d~ cho slf danh gia sai so'
cap mQt theo budc chia khong gian dQc theo chi~u delicua thanh. Chung
minh thu~t roan hQi tl,ltheo luQc d6 sai philo. Cu6i cling minh hQa thu~t
giai tren mQt bai roan Cl,lth€. f)<lnh gia sai so' hQi tl,lcua thu~t roan, cac
ke't qua thu duQc d dily di:'ilam sang to hon cac cong trlnh truoc do.
5/ D€ thu duQc cac Wi giai so', lu~n an di:'i t~n dl,lng slf h6 trQ cua
Maple6 d€ l~p trlnh cac mo mnh tinh roan rat thu~n ti~n, ke't hQp voi
l~p trlnh tinh roan b~ng MatlabR12 cho ra cac d6 thi bi€u di~n ke't qua