PSI - Issue 52

J.C. Wen et al. / Procedia Structural Integrity 52 (2024) 625–646 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

635 11

The finite block method mentioned above is a strong form method. In the next section, we extend the finite block method to the weak form by introducing the concept of Galerkin method. 3. Galerkin finite block method 3.1. Weak form governing equations By utilizing Galerkin method with following weight functions ( ) ( ) ( ) ' ' ' ' , , ' 0,1,2, , , ' 0,1,2, , . m n m n w T T m M n N     = = = (41)

Then, the governing equations in the mapping domain yield

1 1

+ +

  

  

2

2

T

T

T T

T

T





   





M N

u    mn 

( ) ( ) ET T T d d     n

1 AT

1 BT

C

1 DT

+

+

+

+

m

n

m n      

m

1

1

'

'

n

m

n

m

m

n

2

2

T 















0 0

m n

= =

1 1

− −

1 1

+ +

  

 

2

2

T

T

T T

T









M N

v     mn

( ) ( ) J T T T d d      n m m n 1 ' '

1 FT

1 GT

H

1 I T

+

+

+

+

+

m

n

m n  

m

1

n

m

n

2

2

















0 0

m n

= =

1 1

− −

1 1

+ +  

( ) ( ) m n T T d d       , ' ' ( )

0,

X

(42)

−

=

− 1 1

−

1 1

+ +

  

  

2

2

T

T

T T

T

T





   





M N

u    mn 

( ) ( ) E T T T d d     n

2 AT

2 BT

C

2 DT

+

+

+

+

m

n

m n      

m

2

2

'

'

n

m

n

m

m

n

2

2





T 

T 









0 0

m n

= =

1 1

− −

1 1

+ +

  

 

2

2

T

T

T T









M N

v     mn

( ) ( ) J T T T d d      n m m n 2 ' '

2 FT

2 GT

H

2 I T

+

+

+

+

+

m

n

m n  

m

2

n

m

n

2

2  − m















0 0

m n

= =

1 1

− −

(       = = = , ( ) ( ) ' ' ) m n T T d d ' 0,1,2, , , M n Y

0,

' 0,1,2, , . N

(43)

Therefore, a set of linear algebraic equations of the Galerkin finite block method (GFBM) can be obtained

0 0 0 1 , M M M M   = K f  0

(44)

2 ( 1) ( 1) M M N =  +  + is the number of linear algebraic equations formulated from

where the number in subscript

0

the governing equation Eqs. (21) and (22). 3.2. Weak form boundary conditions For the Chebyshev polynomial interpolation, the weak form boundary conditions can be obtained by integrating along the edges with the displacement boundary conditions as below

1

1

+

+

M N

( ) 1    u mn m T 

( ) ( ) T T d u   

(

) ( ) n T d    ' ,



0 = 

1,

(45)

'

n

n

x

0 0

m n

= =

1

1

−

−

1

1

+

+

M N

( ) 1    v mn m T 

( ) ( ) T T d u   

(

) ( ) n T d n    '



0 y

1,

, ' 0,1,2, , , N =

(46)

= 

'

n

n

0 0

m n

= =

1

1

−

−