Modelos Não Lineares do Método dos Elementos de Contorno para ...

Ponto Fonte não alinhado com o elemento
Matriz S
{
Figura B.1 Ponto fonte não alinhado com o elemento. WUTZOW (2003).
1 C6
3
S11 = ⋅ ⎡(
2⋅ C4 + 4⋅υ ) ⋅( sen( θ2 ) ⋅ cos ( θ2 ) + θ2 ) − 2⋅ sen(
θ2 ) ⋅ cos(
θ2
) +
a ⎣
( ) ( ) ) ( ) ( ) ( ) 2
a
+ 3⋅ sen θ2 ⋅ cos θ2 + 3⋅ θ2 + 2⋅ C4 − C5 ⋅θ ⎤ 2 φ ⎡
⎦
⋅ 1 − ⋅ − 2⋅ C4
+ 4⋅υ ⋅ cos θ2
+
L ⎣
4 C6
+ 2⋅ cos( θ2 ) − ( 2⋅ C4 −C5 ) ⋅ Ln( cos( θ2 ) ) ⎤} − ⋅ ⎡(
2⋅ C4 + 4⋅υ ) ⋅( sen(
θ1 ) ⋅ cos(
θ1) + θ1)
+
⎦ a ⎣
3
( 2 sen( ) cos( ) 3 sen( ) cos( ) 3 ) ( 2 C C )
− ⋅ θ1 ⋅ θ1 + ⋅ θ1 ⋅ θ1 + ⋅ θ1 + ⋅ 4 − 5 ⋅θ ⎤
1 ⋅ φ1
+
⎦
a
2 4
⎫
− ⋅ ⎡− ( 2⋅ C4 + 4⋅υ ) ⋅ cos( θ1 ) + 2⋅ cos( θ1) − ( 2⋅ C4 − C5 ) ⋅ Ln(
cos(
θ1
) ) ⎤⎬
L ⎣ ⎦⎭ (B.9)
Anexo B – Integrais Hiper-Singulares __________________________________________
{
(
1 C6 ⎧ 2 4 a
⎫
S12 = ⋅⎨⎡− ( C4 + 2⋅υ ) ⋅ cos( θ2 ) + 2⋅ cos( θ2 ) ⎤ ⋅φ1 − ⎡( C4 + 2⋅υ ) ⋅( −sen ( θ2 ) ⋅ cos(
θ2
) +
) ⎤⎬
a ⎩⎣ ⎦ L ⎣ ⎦
⎭
( ) }
3
) 2 sen( ) cos ( ) sen(
) cos(
)
θ2 θ2 θ2 θ2 θ2 θ2 ⎤
+ − − ⋅ ⋅ + ⋅ +
⎦
+
C6 ⎧ 2 4 a
− ⋅⎨⎡− ( C4 + 2⋅υ ) ⋅ cos( θ1 ) + 2⋅ cos( θ1 ) ⎤ ⋅φ1 − ⎡(
C4 + 2⋅υ ) ⋅( −sen ( θ1) ⋅ cos ( θ1
) +
a ⎩⎣
⎦ L ⎣
1 1
21 12
( ) }
3
) 2 sen ( ) cos ( ) sen(
) cos(
)
360
θ1 θ1 θ1 θ1 θ1 θ1 ⎤
+ − − ⋅ ⋅ + ⋅ +
⎦ (B.10)
S = S
(B.11)