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XFem. , . . : 3 , . .

. . , , , ..

x,y,z ,, u,v,w x,y,z . u , , Gi . . v Pi i dV F [D] E Poison [B] [J] [] [Ke] Hi . Gauss f R view . c xv,yv,zv xw,yw,zw [Dep] [D] [Dp] [D], [Dep] J 2 I1 Lode ,c ,

1. 1996 [1], . : . , XFem. , . C/C++ UNIX. . Fortran . (Object Oriented Programming), C++, . (C) class (C++). , . , .

2

,k m,s C,T

Drucker-Prager Hoek-Brown ,

2. , 2.1. . , 501-2. . . , ( ) ( 16-32 Kb, MFLOP- ). () .. ( 128 Mb, 200 MFLOP ). , . , 70 90-120 30-40 . 50-60x103 8-10x103 . , : 2 , ( 3 ). : (), , , .. . , .

2.2. , . ( ) , . [10] [12] (2.1),(2.2), (2.3) (2.4). , ( ) Dirichlet (2.5) Neumann (2.6) (, ). : :

x =

u v w , y = , z = x y z u v v w w u , zx = + , yz = + + y x z y x z

(2.1)

xy =

(2.2)

:2 2 2 i j ij + 2 = 2 j i ij

(2.3)

i, j x,y y,z z,x

ix iy iz + + + Fi = 0 x y z :

:

(2.4)

i = x,y,z.. Fi .

ij v j = Gi

(2.5)

v Gi . : (2.6) ui=FI . [3] [7]. ij ij . Pi i ,

Argyris, J.H. and Kelsey, S. (1960) Energy Theorems and Structural Analysis, Plenum Press Clough R.W. (1960), The Finite Element Method in Plane Stress Analysis, Proc. 2nd Conf. Electronic Computation, ASCE, Pittsburg, Pa., Sept. 8-9, 19602

1

3

:

1. / 2.2.1.

P = i i V

ij ij

dV

(2.7)

, , , . ( 8 ) (3 x,y,z ). . ( ). 2,3 . To [3] [5] [7]:[K].u=F

(2.8)

u n , (ui=) Dirichlet. F n , ( ). [K] nxn . . . [K].u=F u. u . 1 . , , (/) .

[K] [Ki] . [K] Gauss. ( - shape functions), / . , . , 8 . , 20 . , , . (,,) . -1 1. ,, [Bi] .

2. Gauss

2 Gauss [5]. 3 . Gauss . 27 3 , 3 3 3 3/5. [Ke] :

K e = [ B ]T [ D][ B ]dVV

(2.9)

:

4

K e = H i [ B( i , i , i )]T [ D][ B ( i , i , i )] J ( i , i , i )i =1

n

(2.10) Hi Gauss n ( i ,i , i ) , J X,Y,Z ,,. [D] E,.2.2.2.

[K].u=F u, ( x,y,z ). . , . , , . , (.. ) . , . 2.3. - ( / ) [K].u=F, [K] F / u. - , /. [K].u=F [K].du=dF [K] (-) (.. - ). [K].u=F (.. ). , . , . F, u [K].u=F. (u, F) , u .

[K] u F f, R=F-f. R. R R (). 3 Newton-Raphson [6] [13] . NewtonRaphson [K] (. 4). [K] , .

3. Newton - Raphson

4. Newton Raphson

- . - . , - - ( Hook). - [K] .

5

2.4. 5.

6. Mohr Coulomb 1, 2, 3

5. .

, . . . (flow rules) Hook . , . 1,2,3. , , . ( - ), ( ) ( -). , . Von Mises 1,2,3, 45 . 6 MohrCoulomb.

, [10]. , . ( . , , ). , Von Mises, (dilatancy) Mohr-Coulomb. ( ) , . . =30 c=0. , , . . , , , . . [K].u=F . , . .. [K].u=F, [K].u=F [K] .

3. .. [1] . (visualization)

6

.. . 3.1. . 1995. PC 486/40z, MS-DOS Borland C++ 3.1. T 8 . - 1995, , UNIX. . UNIX . ... ( IRIX Silicon Graphics). UNIX DOS/WINDOWS , . UNIX / Internet. C/C++ Standard UNIX UNIX. UNIX : Hermes () (Silicon Graphics IRIX 5.3), Indigo 2 Silicon Graphics IRIX 6 ... / , Apollo 9 Hewlett Packard . UNIX. Linux. UNIX , (http://www.linux.org), / . 486/40 MHz Pentium 350 MHz Workstation. 1996 . / X-Forms Library Dr. T.C.Zhao Mark Overmars. Internet (http://bragg.phys.uwm.edu/xforms). , . :

/ 3 , : ()

7. ..

3.2. / . , [], [u] [F] ---, . (.. ), . , . , , , . [K] . [K] [K] Gauss - . Gauss [] w Gauss. : , [J-1] ( ,, X,Y,Z), [] ( ), [D] ( . ). [D] Gauss [D] [Dep]=[D]-[Dp] [Dp] . ,

7

. . Skyline. : HBW (Half Bandwidth/ ).

HBW.

, , . . , ( 4 bytes ) ( 4 bytes ). , . : , , , . , , (.. 8 20 ), , . , . , ( ). Gauss, 27 , /. . : , Gauss, . , (classes) C++, , . 3.4. . .

3.3. , , , . , . , . , , . , , . 8: 1 ... n

1

...

8.

. , . . , , . (, , ) 3 , ( ).

8

3 :

, ( ) .. 3.5. , . : . . . . . : , : . , [K] . . [] . . [K]. :

9.

: . ( ) . . . ( ), ( ) [14]. :[xv,yv,zv,1]=[xw,yw,zw,1]Tview

(3.1)

, 4x4 Tview X,Y,Z . [c.xv,c.yv] [c.xv/zv,c.yv/zv] c . . , , , , Gauss, .. , ,

9

. .

.

,

4. . . , (, , , ), .

, DXF (.. AutoCAD). . .

9a. XFem

4.1. , : ( ) :

11.

.

4.2. . . :

10.

:

10

14.

: . , , , .

12. 1-6

. .

13. 8-13

, , .. 8 ( ) . . ( 1), ( 2) ( ). : ( ) . . , :

. 4 , , .

15.

11

, , : . 4 . . , . . . . .

(Associative non Associative MohrCoulomb). ( ) . :' J2 I1 ' sin + J 2 cos sin sin c cos = 0 3 3

(4.2) Drucker-Prager [4] [6], Mohr - Coulomb Von Mises. c & k, associative non associative . :' aI 1 + 3 (J 2 ) 2 k ' = 0 1

(4.3)

4.3. . . KPa, (KN/m3). ( ).

, Von Mises, (T) (c). : ' (4.4) 3 J 2 + ( c T ) I 1 c T = 0 Hoek-Brown[15]. . m s c. . :m c I1 sin ' ' ) s c2 = 0 + 4 J 2 cos2 + m c J 2 (cos 3 3 (4.5)

,