PSI - Issue 6

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

185

4

where k is the number of subdomain with approximation with the use of FEM, i and j are numbers of elements (along 1 x and 2 x ) ; fe k i j , ,  is corresponding finite element. Let’s consider arbitrary subdomain dc k  . We can introduce notation

.

l

2, 1    b k x

x

if

2

k 



(9)

dc

b

k

k

2,

2,

are coordinates (along 1 x ) of nodes (nodal lines) of discrete-continual finite elements,

dc k i , 1, , 1, 2, ...,  i x

N

Let’s

dc k

1,

k N

which are used for approximation of domain dc k  ; (

1)

1,  dc

is the number of discrete-continual finite elements, which

dc k  . Two-index notation system is used for numbering of discrete-continual finite

are used for approximation of

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

elements, which are used for approximation of dc subdomain, i is the number of element (along 1 x ). We can introduce notation

N N

  

if if

1

k   k

 

fe k

2, k N N 

if

1

k 



N



;

(10)

fe

1,

2;

k

dc k

2,

k

1,

1,

x x

l l

  

  

if if

1

if if

1

k   k

k   k

 

 

, dc k i fe k i ,

fe

x

x

if

1



k 



x

l





;

(11)

, fe k j

k

1,

2,

2;

2.

k j

dc

2, ,

2,

k i

k

1, ,

2,

k

1,

2,

It should be noted that in the simples cases (such case in considered in the distinctive paper) discretization of structure is constant along 1 x throughout the domain (otherwise the mathematical constructions given below are substantially more complicated). We have

1 1, 2, ..., N i  .

;

,

(12)

N N k

n

, x x k 1, ,  i k i 1,

n

, 1, 2, ...,

1

1, 2, ...,

1

 







k

b

b

1,

1

In should be noted that notation from [2] is also used in this paper.

2. Approximation models for subdomains and domain

Let’s consider arbitrary subdomain fe k  . Discrete (finite element) approximation model for the considering two dimensional problems presupposes finite element approximation along 1 x and 2 x . Thus extended subdomain fe k  is divided into finite elements,

  1 1 1 1 2 ,    N i N j k 

1

, x x x x x x { ( , ) : 1 2 1, 1  i 1 1,   i

x x

}





 

(13)

,

.

, , fe k i j

k 





fe

, , fe k i j

k j

2, , 1  k j

2, ,

2

Lame constants for finite element are defined by formulas:

  

1,

;



   

, , fe k i j

;





     

(14)





, where

k

0,

;



k i j , ,

k i j , ,

k i j , ,

k i j , ,

, , fe k i j

, , k i j

k

k i j , ,  is the characteristic function of element fe k i j , ,  .