BEM October 07th 2015
Name Phung Duc Thuan
Student ID P46047158
Content Derive the boundary Integral Equation
HOMEWORK
Problem
Let U be a function in the region D with the boundary B. U satisfies the following
equation:
Or
ΔU x, y = f x, y
ΔU r⃗ = f r⃗
U is an unknown function need to be determined.
Green’s function G associated with Laplace’s operator has the form as follow:
ΔG r⃗, ⃗⃗⃗⃗
r =
In the polar coordinates, we have:
ΔG r⃗, r⃗⃗⃗⃗ =
∂
r ∂r
r
∂G
∂r
⇒r
⇒
∂G
∂r
∂G
∂r
r⃗ − ⃗⃗⃗⃗
r
= 0 where r = |r⃗ − ⃗⃗⃗⃗|
r
=C
=
r
⇒ G = C lnr + D
Choose C=1 and D = 0. The Green’s function becomes:
Green’s Second Identity
Where
G(r⃗, r⃗⃗⃗⃗) = ln r
∫ (U
∂V
∂U
− V ) dS = ∫
∂n
∂n
UΔV − VΔU dV
B is a piecewise smooth contour enclosing domain D
�⃗⃗ is a normal unit vector
U, V are functions which must have continuous second order partial derivatives
Boundary Integral Equation for two domain problems
Let U be an unknown function satisfying the equation below:
ΔU r⃗ = f r⃗
G be Green’s function associated with Laplace’s operator:
ΔG r⃗, ⃗⃗⃗⃗
r =
r⃗ − ⃗⃗⃗⃗
r
Assume that U exists on the domain D with the boundary B and r⃗⃗⃗⃗ ∈ D
Green’s function is infinite at r⃗ = ⃗⃗⃗⃗
r so we enclose r⃗⃗⃗⃗ by a very small circle D�.
In the region D-D Green’s second identity becomes
∫
∫
+��
+��
(G
(G
∫ (�
∂G
∂U
− U ) dS = ∫
GΔU − UΔG dV
∂n
∂n
−��
∂U
∂G
− U ) dS = ∫
Gf dV
∂n
∂n
−��
∂U
∂G
∂U
∂G
Gf dV
− U ) dS + ∫ (G
− U ) dS = ∫
∂n
∂n
∂n
∂n
−��
��
Set � → 0, we obtain:
lim[∫
�
�→
∂
∂n
−U
∂G
∂n
dS + ∫�
�
G
∂
∂n
−U
∂G
∂n
We have three terms which need to be evaluate:
lim ∫
�→
−��
lim ∫� �
�→
�
�→
�
lim ∫� �
Gf dV − lim ∫
Gf dV = ∫
��
��
��
�→
�
−��
dS] = lim ∫
�→
f ln dV
=∫
Gf dV − lim ∫ f ln � dθ
=∫
Gf dV − lim � ��� ∫ fdθ
�→
�� = lim ∫�
�→
�
�→
�
��
��
�� = lim ∫� �
��
= lim ∫
�→
�
�
�→
��� �� = lim ∫
�
��
−��
�→
� ��
��� �� = lim ∫
�→
−��� = −2�� ⃗⃗⃗⃗
�
Plug back to the Green’s second identity, we obtain:
��
�
���� �� = 0
�
����
−��
���
Gf dV
∫ (�
∂U
∂G
− U ) dS + 2πU = ∫
∂n
∂n
2�� = ∫ (−�
Gf dV
∂G
∂U
+ U ) dS + ∫
Gf dV
∂n
∂n
−��
Student ID P46047158
Content Derive the boundary Integral Equation
HOMEWORK
Problem
Let U be a function in the region D with the boundary B. U satisfies the following
equation:
Or
ΔU x, y = f x, y
ΔU r⃗ = f r⃗
U is an unknown function need to be determined.
Green’s function G associated with Laplace’s operator has the form as follow:
ΔG r⃗, ⃗⃗⃗⃗
r =
In the polar coordinates, we have:
ΔG r⃗, r⃗⃗⃗⃗ =
∂
r ∂r
r
∂G
∂r
⇒r
⇒
∂G
∂r
∂G
∂r
r⃗ − ⃗⃗⃗⃗
r
= 0 where r = |r⃗ − ⃗⃗⃗⃗|
r
=C
=
r
⇒ G = C lnr + D
Choose C=1 and D = 0. The Green’s function becomes:
Green’s Second Identity
Where
G(r⃗, r⃗⃗⃗⃗) = ln r
∫ (U
∂V
∂U
− V ) dS = ∫
∂n
∂n
UΔV − VΔU dV
B is a piecewise smooth contour enclosing domain D
�⃗⃗ is a normal unit vector
U, V are functions which must have continuous second order partial derivatives
Boundary Integral Equation for two domain problems
Let U be an unknown function satisfying the equation below:
ΔU r⃗ = f r⃗
G be Green’s function associated with Laplace’s operator:
ΔG r⃗, ⃗⃗⃗⃗
r =
r⃗ − ⃗⃗⃗⃗
r
Assume that U exists on the domain D with the boundary B and r⃗⃗⃗⃗ ∈ D
Green’s function is infinite at r⃗ = ⃗⃗⃗⃗
r so we enclose r⃗⃗⃗⃗ by a very small circle D�.
In the region D-D Green’s second identity becomes
∫
∫
+��
+��
(G
(G
∫ (�
∂G
∂U
− U ) dS = ∫
GΔU − UΔG dV
∂n
∂n
−��
∂U
∂G
− U ) dS = ∫
Gf dV
∂n
∂n
−��
∂U
∂G
∂U
∂G
Gf dV
− U ) dS + ∫ (G
− U ) dS = ∫
∂n
∂n
∂n
∂n
−��
��
Set � → 0, we obtain:
lim[∫
�
�→
∂
∂n
−U
∂G
∂n
dS + ∫�
�
G
∂
∂n
−U
∂G
∂n
We have three terms which need to be evaluate:
lim ∫
�→
−��
lim ∫� �
�→
�
�→
�
lim ∫� �
Gf dV − lim ∫
Gf dV = ∫
��
��
��
�→
�
−��
dS] = lim ∫
�→
f ln dV
=∫
Gf dV − lim ∫ f ln � dθ
=∫
Gf dV − lim � ��� ∫ fdθ
�→
�� = lim ∫�
�→
�
�→
�
��
��
�� = lim ∫� �
��
= lim ∫
�→
�
�
�→
��� �� = lim ∫
�
��
−��
�→
� ��
��� �� = lim ∫
�→
−��� = −2�� ⃗⃗⃗⃗
�
Plug back to the Green’s second identity, we obtain:
��
�
���� �� = 0
�
����
−��
���
Gf dV
∫ (�
∂U
∂G
− U ) dS + 2πU = ∫
∂n
∂n
2�� = ∫ (−�
Gf dV
∂G
∂U
+ U ) dS + ∫
Gf dV
∂n
∂n
−��