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

2 ⎧ 2 a
⎫
D12 = −C3 ⋅⎨⎡C 4 ⋅ Ln( cos( θ2 ) ) − cos( θ2 ) ⎤ ⋅ φ2 + ⋅ ⎡−C4 ⋅( tan ( θ2 ) −θ 2 ) − sen ( θ2 ) ⋅ cos ( θ2 ) + θ ⎤ 2 ⎬ +
⎩⎣ ⎦ L ⎣ ⎦
⎭
⎧ 2 a
⎫
+ C3 ⋅⎨⎡C 4 ⋅ Ln( cos( θ1 ) ) − cos( θ1) ⎤ ⋅ φ2 + ⋅ ⎡−C4 ⋅( tan ( θ1 ) −θ1 ) − sen(
θ1) ⋅ cos(
θ1 ) + θ ⎤ 1 ⎬
⎩⎣ ⎦ L ⎣ ⎦
⎭ (B.28)
2 ⎧ 2 a
⎫
D21 = −C3 ⋅ ⎨⎡−C4 ⋅ Ln( cos( θ2 ) ) − cos( θ2 ) ⎤ ⋅ φ2 + ⋅ ⎡C4 ⋅( tan ( θ2 ) −θ 2 ) − sen(
θ2 ) ⋅ cos(
θ2 ) + θ ⎤ 2 ⎬ +
⎩⎣ ⎦ L ⎣ ⎦
⎭
⎧ 2 a
⎫
+ C3 ⋅⎨⎡−C 4 ⋅ Ln( cos( θ1 ) ) − cos( θ1) ⎤ ⋅ φ2 + ⋅ ⎡C4 ⋅( tan ( θ1 ) −θ1 ) − sen(
θ1) ⋅ cos(
θ1 ) + θ ⎤ 1 ⎬
⎩⎣ ⎦ L ⎣ ⎦
⎭ (B.29)
( ) ( ) ( ) ( ) ( ( ) ) ( ) 2
2 ⎧ a
⎫
D22 = −C3 ⋅ ⎨⎡ C4 + 1 ⋅θ2 − sen θ2 ⋅cos θ2 ⋅ φ2 + ⋅ ⎡ −C4 − 2 ⋅ Ln cos θ2 − sen θ ⎤
⎣ ⎤⎦
2 ⎬ +
⎩ L ⎣ ⎦⎭
( ) ( ) ( ) ( ) ( ( ) ) ( ) 2
⎧ a
⎫
+ C3 ⋅ ⎨⎡ C4 + 1 ⋅θ1 − sen θ1 ⋅cos θ1 ⋅ φ2 + ⋅ ⎡ −C4 − 2 ⋅ Ln cos θ1 − sen θ ⎤
⎣ ⎤⎦
1 ⎬
⎩ L ⎣ ⎦⎭
(B.30)
( ) ( ) ( ) ( ) ( ( ) ) ( ) 2
2 ⎧ a
⎫
D31 = −C3 ⋅⎨⎡ − C4 + 1 ⋅θ2 − sen θ2 ⋅cos θ2 ⋅ φ ⎡
2 + ⋅ C4 − 2 ⋅ Ln cos θ2 − sen θ ⎤
⎣ ⎤⎦
2 ⎬ +
⎩ L ⎣ ⎦⎭
( ) ( ) ( ) ( ) ( ( ) ) ( ) 2
⎧ a
⎫
+ C3 ⋅⎨⎡ − C4 + 1 ⋅θ1 − sen θ1 ⋅cos θ1 ⋅ φ2 + ⋅ ⎡ C4 − 2 ⋅ Ln cos θ1 − sen θ ⎤
⎣ ⎤⎦
1 ⎬
⎩ L ⎣ ⎦⎭
(B.31)
( θ )
( θ )
5
⎧
2 ⎪ 2 a ⎡ 2⋅
sen 2
D32 = −C3 ⋅ ⎨⎡( −C4 − 2) ⋅ Ln( cos( θ2 ) ) − sen( θ2 ) ⎤ ⋅ φ2 + ⋅ C4
⋅( tan ( θ2 ) − θ2
) + +
⎣ ⎦
⎢
⎪ L ⎢⎣
cos 2
⎩
3
( θ2 ) ( θ2 ) ( θ2 ) ( θ2 ) θ2 } 3 ( 4 ) ( ( θ1
) )
Anexo B – Integrais Hiper-Singulares __________________________________________
364
{ }
+ 2⋅ sen + cos + 3⋅ cos ⋅ sen − 3⋅ ⎤ + C ⋅ ⎡
⎣
−C − 2 ⋅ Ln cos +
⎤
⎦
⎦
( θ1
)
( θ )
5
2 a ⎡ 2⋅
sen
⎤⎫
3
⎪
− sen( θ1 ) ⎤ ⋅ φ2 + ⋅ C4 ⋅( tan ( θ1 ) − θ1 ) + + 2⋅ sen(
θ1) + cos(
θ1
) +
⎦
⎢ ⎥⎬
L ⎢⎣ cos 1
⎥⎪ ⎦⎭
( θ1) sen(
θ1 ) θ1
}
+ 3⋅ cos ⋅ − 3 ⎤⎦ (B.32)
Ponto Fonte alinhado com o elemento posicionado atrás do mesmo
Figura B.2 Ponto fonte alinhado com o elemento posicionado atrás do mesmo. WUTZOW (2003).