PSI - Issue 6

Pavel A. Akimov et al. / Procedia Structural Integrity 6 (2017) 182–189 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

184

3

Let physical and geometrical parameters within other group of subdomains from (2) is arbitrary varying. FEM can be effectively used for approximation here. Thus, combined application of DCFEM and FEM is advisable. Operational formulation of two-dimensional problem of elasticity is given at [1, 2]. So-called approximation parameter k  can be introduced in accordance with the following rule : 1  k  for approximation with the use of FEM ; 2  k  for approximation with the use of DCFEM . Various approaches can be used for numbering of subdomains. The first approach provides separate numbering of subdomains with different types of approximation:      k s s k k k 1 1 1 | 2 | ( )  ;      k s s k k k 1 2 2 | 1| ( )  , (3)

where k is initial number of subdomain approximation with the use of FEM ;

k  ; ( ) 1 1 k k k  is the corresponding number of subdomain with ( ) 2 2 k k k  is the corresponding number of subdomain with approximation

with the use of DCFEM. We can certainly construct inverse relationships by tabulating the results of calculations using formulas (3).

( ) 2 k k k  ,

( ) 1 k k k  ;

(4)

Thus we can rewrite (2) in the following form:

N

  dc fe N k k fe k 1 1 1    

.

(5)





k 

dc

2

1

2

Besides, the following relations are valid:

   1 1 b n s

   1 1 b n s

1    b dc fe N N n ,

fe N

dc N

;

;

(6)

|

2 |

|

1 |



s 





s 



fe k , 1, 2, ..., 1 1   k

N

dc k , 1, 2, ..., 2 2   k

N

are subdomain with approximation with the use of FEM ;

are

where

fe

dc

subdomain with approximation with the use of DCFEM. The second approach, on the contrary, is based on a linked numbering of subdomains with different types of approximation. Formula (2) can be used, where

  

, if , if

1

k   k

k   k

 

fe

(7)

k 



2

dc

while formulas (3) and (4) are not required. The second approach is used in this paper. Let’s consider arbitrary subdomain fe k  . We can introduce notation

.

2, 1    b k fe k l x x 2,

if

1

k 



(8)

b

k

2,

are coordinates coordinates (along 1 x and 2 x ) of nodes of

fe k i , 1, , 1, 2, ...,  i x

N

x

j

N

fe k j , 2, ,

1, 2, ..., 

Let’s

and

fe k

fe

k

1,

2,

finite elements, which are used for approximation of domain fe k  ; ( elements along coordinates 1 x and 2 x , which are used for approximation of fe numbering of finite elements, which are used for approximation of fe 1) 1,  fe k N

k N

and (

1)

2,  fe

are numbers of finite

k  . Three-index system is used for

k  . Typical number of has the form ( , , ) k i j ,