PSI - Issue 52

J.C. Wen et al. / Procedia Structural Integrity 52 (2024) 625–646

630

6

Author name / Structural Integrity Procedia 00 (2019) 000 – 000

(

) 2 / ,

 and T are two free parameters. The solution of the

where Laplace parameter

0,1,..., , K

k s

ik T k

  = +

=

static case can be obtained simply by letting mass density

0  = in the procedure of analysis directly.

2.2. Mapping technique Firstly, we divide the physical domain Ω into several blocks Ω j ( j=I , II …) in the numerical process FBM. For the sake of convenience of analysis with the block characteristics, we define ( , ) x y and ( , )   as variables in the coordinate system in the physical domain and the transformation domain, respectively, as shown in Fig. 1(a) and Figure 1(b). In the case of two-dimension, the coordinate transformation with 8 seed quadrilaterals is represented as

8

8





( ) , M x y   ,

( ) , M y  

(10)

,

x

=

=

i

i

i

i

1

1

i

i

=

=

where ( , ) i i x y indicates the coordinate of node in physical domain. Then, the shape functions (same as the finite element method) are defined as follows ( )( )( ) ( ) ( ) ( ) ( ) 2 2 1 1 1 1 for 1,2,3,4, 4 1 1 1 for 5,7, 2 1 1 1 for 6,8, i i i i i i i i i M i M i M i         = + + + − = = − + = = − + = (11) in which ( , ) i i   presents the coordinate of node i in the mapping space. Then, the first partial differentials of the shape function with normalized axes  and  are ( )( ) ( )( ) 1 2 , 1 2 , 1,2,3,4 i i i i M M i           = + + = + + = 2

i

i

i

i

i

i

4

4









M

M





i 

(

)

(

)

2 1 , 

1  

,

5,7

i

(12)

=− +

= −

=

i

i

i

2









M

M



i 



(

)

(

)

2 1 , 

1  

,

6,8

i

= −

=− +

=

i

i

i

2









a

b

Fig. 1. One block in: (a) physical domain; (b) normalized domain.

In addition, the first order partial derivatives with respect to Cartesian coordinates are expressed in terms of partial derivatives with respect to normalized coordinates as