13.1
.
13.2
1.
1
2
3
4
5
6
7
8
13.3
.
sz=tzx=tzy=0, gyz= gxz=0, ez . ez=gxz=gyz=0, tzy=tzx=0, sz . , .
13.1.
Sp ( ), Sf , Su . d w V. - {e} {s} c
,
sx,sy X Y, txy ; ex,ey X Y, gxy .
. (13.1)
, [D] . [D]
, (13.2)
E , m - .
, .
[D]
. (13.3)
. (13.4)
(13.4) , Sp , Sf , Su ( ).
, , , , .
, . 7.2.
|
|
.13.2
I,J,K ,
U2i-1,U2j-1,U2k-1 ,
U2i,U2j,U2k - .
c
(13.5)
e ;j(e) - ;a1,a2,a3 ; X,Y .
(13.5)
, (13.6)
i,j,k ; X,Y ; Fi,Fj,Fk ; a1,a2,a3 .
(13.6)
A .
, (13.7)
Ni, Nj, Nk .
j(e) = NiFi + NjFj + NkFj.
13.1
(13.8)
(13.9)
[N(e)] ;
{U(e)} ;
{u(e)} .
Q .
(), (13.1), (13.8)
(13.7) ,
. (13.10)
(13.10) (13.1).
.
(13.11),
.
(13.10)
(13.12).
[B(e)] , {e(e)} {s(e)}.
(13.4) . , (13.4) . , (13.1),
(13.13).
(13.12) (13.4)
(13.14)
(13.14) , .
. , , () , .
, . c c = L - Y, (13.15)
L - , Y .
(7.15) (13.16).
|
|
(13.17),
v(e) . (13.17) (13.10) (13.11),
(13.18).
, , Yf () Yp.
Yf = {u}T{P} (13.19),
{u} , {P} - . , Px Py , ..
.
Yp Sq, .
Sq .
,
qx,qy , S(e) . C (13.9), Yp(e)
(13.20).
(13.18), (13.19), (13.20) (13.16)
(13.21).
{u}
(13.22).
(13.23),
[k(e)] , [f(e)] .
, (13.22)
(13.24),
[K] , {F} .
(13.24) , {u}.
[B(e)] [D] , (7.23)
,
t , A(e) .
, . ( ), .
, , . . . , .
, . . . . X,Y .
x,h. x,h . x h -1 1. .
.
(13.25)
Xi, Yi - ; Ni(x,h) - ; X,Y - , ; 8 - .
. Ni x h. (13.25) , - . .
(13.25)
N1=-0.25(1-x)(1-h)(x+h+1); N3= 0.25(1+x)(1-h)(x-h-1); N5= 0.25(1+x)(1+h)(x+h-1); N7=-0.25(1-x)(1+h)(x-h+1); | N2 = 0.5(1-x2)(1-h); N4 = 0.5(1-h2)(1+x); (13.26) N6 = 0.5(1-x2)(1+h); N8 = 0.5(1-h2)(1-x). |
.13.3.
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13.3
X,Y x,,h . 1 8. 1 x=h=-1. . .
2,4,6,8 . .
-1/2 < x < 1/2, -1/2 < h < 1/2. (13.27)
, . . . . 4 ( ). , , , 0. , . . 13.3. . , :
1. . . . , X Y .
2. . . . 13.3.
3. x h. x , h - . , 1.
4. . . (13.25) ( ). , x = -1, h = +1 ( x -1 +1) ( h +1 -1).
5. . , , .
13.4
|
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ProFEM, Windows 98 . ProFEM .
1. .
2. .
13.4. . .
|
. 13.4
3. , k=1..Mf, Mf .
4. ( - ). j=1..Mn, Mn .
5. X Y . xk,hk. j=1..Mn 13.1.
(Mf=2, Mn=13)
13.1
j | j | ||||
Xj | Yj | Xj | Yj | ||
4.2 | 4.2 | ||||
1.86 | 0.78 | ||||
5.5 | 2.25 | ||||
0.78 | 1.84 | ||||
2.36 | 5.47 | ||||
1.41 | 1.4 | ||||
2.77 | 2.76 |
6. .
13.2
7. ( ) . - .
13.3
. | x | h | ||||||||
8. ProFEM - (. 13.5).
.13.5.
ProFEM
, (. 13.6).
. 13.6.
. 1-4 .13.6 13.2, 5,6 7-14 13.3. .13.6 13.1. .
. .
9. , , (.13.7).
. 13.7.
, . (.13.6) . .
10. . . 13.8.
. 13.8.
. , . 2 13.9.
|
|
. 13.9.
, , . . , .
11. - () ( ). . . . . , , . 1 , 2 . , .
.13.10
,
12 ProFEM , , , 13.11.
.13.11
(
) ( ) .
.13.12
. , .
13.5
:
- , .
- .
- , , , .
- , .
13.6 .
1. , ?
2. ?
3. ?
4. ?
5. ?
6. ?
7. ?
8. ?
9. ?
10. ?
11. ?
12. ?
13. - ?
14. ?
13.7 .
1. . .- ., , 1979.
2. ., . .- .,,1986.
3. .., . .
. - ., , 1980.
4. .. . .: , 1987.