Introduction to Simulation - Lecture 22
Integral Equation Methods
Jacob White
Thanks to Deepak Ramaswamy, Michal Rewienski,
Xin Wang and Karen Veroy
Outline
Integral Equation Methods
Exterior versus interior problems
Start with using point sources
Standard Solution Methods in 2-D
Galerkin Method
Collocation Method
Issues in 3-D
Panel Integration
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Interior Versus Exterior Problems
Interior
Exterior
∇ 2T = 0
outside
∇ 2T = 0
inside
Temperature
known on surface
“Temperature in a Tank”
Temperature
known on surface
“Ice Cube in a Bath”
What is the heat flow?
Heat Flow
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∂T
∫
∂n
surface
Thermal
= conductivity
Exterior Problem in Electrostatics
potential
+
v
-
∇2Ψ = 0
Outside
Ψ is given on Surface
What is the capacitance?
Capacitance
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Dielectric
= Permitivity
∂Ψ
∫
∂n
surface
Drag Force in a Microresonator
Courtesy of Werner Hemmert, Ph.D. Used with permission.
Resonator
Computed Forces
Bottom View
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Discretized Structure
Computed Forces
Top View
What is common about these problems.
Exterior Problems
Drag Force in MEMS device - fluid (air) creates drag.
Coupling in a Package - Fields in exterior create coupling
Capacitance of a Signal Line - Fields in exterior.
Quantities of Interest are on the surface
MEMS device - Just want surface traction force
Package - Just want coupling between conductors
Signal Line - Just want surface charge.
Exterior Problem is linear and space-invariant
MEMS - Exterior Stokes Flow equation (linear).
Package - Maxwell’s equations in free space (linear).
Signal Line - Laplace’s equation in free space (linear).
But problems are geometrically very complex!
Exterior Problems
Surface
Why not use Finite-Difference
or FEM methods
2-D Heat Flow Example
T = 0 at ∞
But, must
truncate the
mesh
∂T
Only need
on the surface, but T is computed everywhere
∂n
Must truncate the mesh, ⇒ T (∞) = 0 becomes T ( R ) = 0
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Green’s Function
Laplace’s Equation
(
In 2-D
2
2
( x − x0 ) + ( y − y0 )
)
If u = log
∂ 2u ∂ 2u
then 2 + 2 = 0 for all ( x, y ) ≠ ( x0 , y0 )
∂x
∂y
In 3-D
If u =
1
( x − x0 ) + ( y − y0 ) + ( z − z0 )
2
2
2
∂ 2u ∂ 2u ∂ 2u
then 2 + 2 + 2 = 0 for all ( x, y, z ) ≠ ( x0 , y0 , z0 )
∂x
∂y
∂z
Proof: Just differentiate and see!
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Laplace’s Equation
in 2-D
Simple Idea
u is given on surface
Surface
( x0 , y0 )
Let u = log
(
∂ 2u ∂ 2u
+ 2 = 0 outside
2
∂x
∂y
( x − x0 ) + ( y − y0 )
2
∂ 2u ∂ 2u
+ 2 = 0 outside
2
∂x
∂y
2
)
Problem Solved
Does not match boundary conditions!
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Simple Idea
Laplace’s Equation
in 2-D
“More Points”
u is given on surface
∂ 2u ∂ 2u
+ 2 = 0 outside
2
∂x
∂y
( x2 , y2 )
( x1 , y1 )
n
Let u = ∑ ωi log
i =1
(
( xn , yn )
( x − xi ) + ( y − yi )
2
2
) = ∑ω G ( x − x , y − y )
n
i =1
i
i
i
Pick the ωi ' s to match the boundary conditions!
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Simple Idea
Laplace’s Equation
in 2-D
“More Points Equations”
(x , y )
t1
t1
Source Strengths selected
to give correct potential at
test points.
( x2 , y2 )
( x1 , y1 )
( xn , yn )
 G ( xt − x1 , yt − y1 ) L L G ( xt − xn , yt − yn )  ω   Ψ ( xt , yt ) 
1
1
1
1
1
1

 1 

M
O
M
M

 M  

=



M 
M
O
M
M

  

G x − x , y − y L L G x − x , y − y  ω n   Ψ x , y 
1
1
tn
tn
tn
n
tn
n 
tn
tn 


(
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)
(
)
(
)
Computational results using points approach
Circle with Charges r=9.5
Potentials on the Circle
n=20
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r
R=10
n=40
Laplace’s Equation
in 2-D
Integral Formulation
Limiting Argument
Want to smear point charges onto surface
Results in an Integral Equation
Ψ ( x ) = ∫ G ( x, x′ ) σ ( x′ ) dS ′
surface
How do we solve the integral equation?
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Laplace’s Equation
in 2-D
Basis Function Approach
Basic Idea
n
Represent σ ( x ) = ∑ ωi ϕ i ( x )
{
i =1
Basis Functions
Example Basis
Represent circle with straight lines
Assume σ is constant along each line
The basis functions are “on” the surface
Can be used to approximate the density
May also approximate the geometry
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Laplace’s Equation
in 2-D
Basis Function Approach
Geometric Approximation is
not new.
Piecewise Straight surface basis
Triangles for 2-D FEM
Functions approximate the circle approximate the circle too!
Ψ ( x) =
∫
approx
surface
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n
G ( x, x′ ) ∑ ωiϕi ( x′ ) dS ′
i =1
Laplace’s Equation
in 2-D
x1
xn
ln
l1
x2
l2
Ψ ( x) =
∫
Basis Function Approach
Piecewise Constant Straight
Sections Example.
1) Pick a set of n Points on the
surface
2) Define a new surface by
connecting points with n lines.
3) Define ϕ i ( x ) = 1 if x is on line li
otherwise, ϕ i ( x ) = 0
n
n
i =1
i =1
G ( x, x′ ) ∑ ωiϕi ( x′ ) dS ′ = ∑ ωi
approx
surface
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G ( x, x′ ) dS ′
∫
line l
i
How do we determine the ω i ' s ?
Basis Function Approach
Laplace’s Equation
in 2-D
R ( x) ≡ Ψ ( x) −
∫
Residual Definition and
minimization
n
G ( x, x′ ) ∑ ωiϕi ( x′ ) dS ′
approx
surface
i =1
We will pick the ω i ' s to make R ( x ) small.
General Approach: Pick a set of test functions
φ1 ,K , φn , and force R ( x ) to be orthogonal to the set
∫ φ ( x )R ( x ) dS = 0
i
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for all i.
Basis Function Approach
Laplace’s Equation
in 2-D
Residual minimization using
test functions
∫ φ ( x ) R ( x ) dS = ∫ φ ( x ) Ψ ( x ) dS − ∫ ∫
i
i
n
φi ( x ) G ( x, x′ ) ∑ ω jϕ j ( x′ ) dS ′dS = 0
j =1
approx
surface
We will generate different methods by chosing the φ1 ,K , φn ,
(
)
Collocation: φi ( x ) = δ x − xti (point-matching)
Galerkin Method: φi ( x ) = ϕ i ( x ) (basis = test)
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Basis Function Approach
Laplace’s Equation
in 2-D
Collocation
Collocation: φi ( x ) = δ ( xi ) (point-matching)
∫ δ ( x − x ) R ( x ) dS = R ( x ) = Ψ ( x ) − ∫
ti
ti
( )
n
⇒ Ψ xti = ∑ ω j
j =1
(
G xti , x′
ti
approx
surface
∫
(
n
) ∑ω ϕ ( x′) dS ′ = 0
j =1
j
)
G xti , x′ ϕ j ( x′ ) dS ′
approx
surface
1444424444
3
Ai , j
 A1,1 L L A1,n  ω1   Ψ ( xt1 ) 


 M O



M  M

 = M 
 M
O M  M   M 


  
 An ,1 L L An ,n  ω n   Ψ xtn 
( )
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j
Laplace’s Equation
in 2-D
xn l
n
xt1
l1
l2
x2
Basis Function Approach
Centroid Collocation for
Piecewise Constant Bases
( )
n
Ψ xti = ∑ ω j
j =1
∫
(
)
G xti , x′ ϕ j ( x′ ) dS ′
approx
surface
Collocation point in
line center
 A1,1 L L A1,n  ω1   Ψ ( xt1 ) 


 M O



M  M

 = M 
 M
O M  M   M 


  
 An ,1 L L An ,n  ω n   Ψ xtn 
( )
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( )
n
Ψ xti = ∑ ω j
j =1
∫ G ( x , x′) dS ′
ti
line j
1442443
Ai , j
Laplace’s Equation
in 2-D
( )
n
Ψ xti = ∑ ω j
j =1
Basis Function Approach
Centroid Collocation
Generates a nonsymmetric A
∫ G ( x , x′) dS ′
ti
line j
1442443
Ai , j
xt1
xt2
l1
A1,2 =
l2
∫ G ( x , x′) dS ′ ≠ ∫ G ( x
t1
line 2
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t2
line1
, x′ ) dS ′ = A2,1
Laplace’s Equation
in 2-D
Basis Function Approach
Galerkin
Galerkin: φi ( x ) = ϕ i ( x ) (test=basis)
∫ ϕ ( x ) R ( x ) dS = ∫ ϕ ( x ) Ψ ( x ) dS − ∫ ∫
i
n
ϕi ( x ) G ( x, x′ ) ∑ ω jϕ j ( x′ ) dS ′dS = 0
i
∫
n
ϕi ( x ) Ψ ( x ) dS = ∑ ω j
approx
surface
j =1
14442444
3
bi
 A1,1 L L
 M O

 M
O

 An ,1 L L
approx
surface
∫ ∫
G ( x, x′ ) ϕi ( x ) ϕ j ( x′ ) dS ′dS
approx approx
surface surface
1444444
424444444
3
Ai , j
A1,n  ω1   b1 
M   M   M 
 = 
M  M   M 
   
An ,n  ω n  bn 
If G ( x, x′) = G ( x′, x) then Ai , j =A j ,i
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j =1
A is symmetric
Basis Function Approach
Laplace’s Equation
in 2-D
ln
l1 xn
l2
Galerkin for Piecewise
Constant Bases
x2
n
∫ Ψ ( x ) dS = ∑ω ∫ ∫ G ( x, x′) dS ′dS
linei
14243
bi
j =1
j
linei line j
144424443
Ai , j
 A1,1 L L A1,n  ω1   b1 
 M O
   
M

 M  =  M 
 M
O M  M   M 

   
 An ,1 L L An ,n  ω n  bn 
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3-D Laplace’s
Equation
Basis Function Approach
Piecewise Constant Basis
Integral Equation: Ψ ( x ) =
∫
surface
Discretize Surface into
Panels
1
σ ( x′ ) dS ′
x − x′
n
Represent σ ( x ) ≈ ∑ ωi ϕ i ( x )
{
i =1
Basis Functions
ϕ j ( x ) = 1 if x is on panel j
Panel j ϕ ( x ) = 0 otherwise
j
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3-D Laplace’s
Equation
Put collocation points at
panel centroids
Basis Function Approach
Centroid Collocation
( )
n
Ψ xci = ∑ ω j
xci Collocation
point
j =1
G(x
∫
panel j
ci
)
, x′ dS ′
14442444
3
Ai , j
 A1,1 L L A1,n  ω1   Ψ ( xc1 ) 


 M O



M  M

 = M 
 M
O M  M   M 


  
 An ,1 L L An ,n  ω n   Ψ xcn 


( )
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Basis Function Approach
3-D Laplace’s
Equation
Calculating Matrix Elements
xci Collocation
z
point
y
Ai , j =
x
∫
panel j
1
dS ′
xci − x′
Panel j
One point
quadrature
Approximation
Four point
quadrature
Approximation
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Panel Area
Ai , j ≈
xci − xcentroid j
4
0.25* Area
j =1
xci − x po int j
Ai , j ≈ ∑
Basis Function Approach
3-D Laplace’s
Equation
Calculating “Self-Term”
xci Collocation
z
point
y
Ai ,i =
x
∫
panel i
1
dS ′
xci − x′
Panel i
One point
quadrature
Approximation
Ai ,i =
∫
panel i
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Ai ,i
Panel Area
≈
xci − xci
1
424
3
0
1
dS ′ is an integrable singularity
xci − x′
Basis Function Approach
3-D Laplace’s
Equation
Calculating “Self-Term”
Tricks of the trade
z
xci Collocation
y
point
Panel i
x
∫
Ai ,i =
panel i
Disk of radius R
surrounding
collocation point
∫
Integrate in two Ai ,i =
disk
pieces
Disk Integral has
singularity but has
analytic formula
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∫
disk
1
dS ′
xci − x′
1
1
dS ′ +
dS ′
∫
xci − x′
rest of panel xci − x′
R 2π
1
dS ′ = ∫
xci − x′
0
∫
0
1
rdrdθ = 2π R
r
Basis Function Approach
3-D Laplace’s
Equation
Calculating “Self-Term”
Other Tricks of the trade
z
xci Collocation
y
point
x
Panel i
Ai ,i =
∫
panel i
1
dS ′
xci − x′
1
424
3
Integrand is singular
1) If panel is a flat polygon, analytical formulas exist
2) Curve panels can be handled with projection
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Basis Function Approach
3-D Laplace’s
Equation
Galerkin (test=basis)
n
ϕ ( x ) Ψ ( x ) dS = ∑ ω ∫ ∫ ϕ ( x ) G ( x, x′ )ϕ ( x′ ) dS ′dS
∫144
2443
144444244444
3
i
bi
j =1
j
i
j
Ai , j
For piecewise constant Basis
n
1
dS ′dS
Ψ ( x ) dS ′ = ∑ ω j ∫
∫14
∫
panel
i
panel
j
x − x′
243 j =1
14444244443
bi
Ai , j
 A1,1 L L A1,n  ω1   b1 
 M O
   
M

 M  =  M 
 M
O M  M   M 

   
 An ,1 L L An ,n  ω n  bn 
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3-D Laplace’s
Equation
Basis Function Approach
Problem with dense matrix
Integral Equation Method Generate Huge
Dense Matrices
 A1,1 L L A1,n  ω1   Ψ ( xc1 ) 


 M O



M  M

 = M 
 M
O M  M   M 


  
 An ,1 L L An ,n  ω n   Ψ xcn 
( )
Gaussian Elimination Much Too Slow!
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Summary
Integral Equation Methods
Exterior versus interior problems
Start with using point sources
Standard Solution Methods
Collocation Method
Galerkin Method
Next Time Æ “Fast” Solvers
Use a Krylov-Subspace Iterative Method
Compute MV products Approximately