Transcript I. Overview - Computer Engineering Group

Finite Difference Method
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
For conductor exterior, solve Laplacian
equation
k
l
In 2D:
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i
m
j
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Uniform Grid
i, j+1
i–1, j
i+1, j
i, j
i, j–1
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Basic Properties

Consider two conductors
–Q

+Q
S
Let v=f(Q). From Gauss’ law
if we double the amount of charge, E will also double
Really?
since the equation is linear
 Therefore, v and Q are linearly related, or Q=Cv
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Multiple Conductors

Consider conductors 1, 2, …, n
Q1
Qn
Q2

Apply the above argument for every pair of
conductors i and j
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Capacitance Matrix
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BEM Review


Partition conductor surfaces into panels
Build coefficient matrix P, where
and G is Green’s function, such as
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
Solve linear system Pq=v
Add charges to get capacitance
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Make It Faster
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Discretization: O(n)
Compute P: O(n2) O(n)
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Since P is size nn, P can not be
constructed explicitly
Solve Pq=v: O(n3) O(n)
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Iterative methods
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Fast Multipole Methods
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N-body problem: Given n particles in 3D
space, compute all forces between the
particles
Fast multipole algorithms
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Appel 85
Rokhlin 86, Greengard & Rokhlin 87
O(n) time
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Basic Idea of Multipole
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A cluster of charges at distance can
be approximated by a single charge
Reduce operations from k2 to k
Form all clusters recursively in O(n)
time — hard part!
potential
k charges
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at k points
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Solve Ax=b Iteratively

Approximate Ax–b=0:
Initial solution x
Compute Ax
If Ax–b > t/b,
modify x

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Bottleneck: Matrix-vector product Ax
A is not used elsewhere
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Example: Jacobi Method
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Example: Jacobi Method
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Transformation
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Iterations
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Ax = b  Dx=Dx–Ax+b  x = (I–D–1A)x+D–1b
x(i+1)= (I–D–1A) x(i)+D–1b
x(0) = 0, x(1) = D–1b, x(2) = (I–D–1A) x(1)+D–1b, …
If diagonal dominate, then the method
converges
Better iterative methods exist that converge
under weaker conditions
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Fast Algorithm HiCap
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Conductor surface refinement:


Approximate P and store it in a hierarchical
data structure of size O(n)

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Adaptively partition conductor surfaces into small
panels according to a user supplied threshold 
The data structure permits O(n) time matrixvector product Px for any n-vector x
Solve linear system Pq=v using iterative
methods
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Adaptive Panel Partition

If interaction between Ai Aj > , refine
Ai and Aj. Otherwise, record Pij in P.
A
C
C
B
E
F G
H
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1
2
I
M N
L
J
J
3
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Representation of Matrix P

P is stored as links in a hierarchical data
structure
A
B
D
F
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H
C
I
E
G
J
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Example

If area/dist  1, refine the panel
A
2
C
4
H
1/7
1/5
B
C
1/5
I
J
1/3
B
1
4
I
J
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Example (cont’d)

If area/dist  1, refine the panel
A
2
4
1
H
1/7
1/5
C
B
C
1/5
I
J
E
F G
M N
L
4
J
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F
E
G
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A
Full 8x8
B
D
matrix P:
B
H
K
E
D
E
A
C
K
I
L
H
J
M
N
I
C
1/4.6
1/4.6
1/5.5
J
L
A
Implicitly
stored P:
B
B
D
H
K
E
D
E
A
C
K
I
L
H
J
I
C
1/5
J
L
Properties of P

P positive, symmetric, positive definite



If fully expanded, P is size nn
P can be approximated by O(n) block
entries, where n is the number of panels


Positive definite: xPx > 0 for all x
This is because each panel interacts with
constant number of other panels
The block entries allow O(n) time matrixvector product Px for any x
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Mat-Vec Pq, Step 1
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Compute charge for all panels
A
B
D
F
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C
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K
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Mat-Vec Pq, Step 2
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Compute potential for all panels
A
B
D
F
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C
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E
G
J
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Mat-Vec Pq, Step 3
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Distribute potential to leaf panels
A
B
D
F
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C
I
E
G
J
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M
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Solving Linear Systems

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Use fast iterative methods GMRES
Each iteration requires a matrix-vector
product Pq that can be completed in
O(n) time
Solution obtained in 10-20 iterations,
regardless of n
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Error and Time Complexity


Error of approximation is controlled
by 
Time complexity is O(n) because step
takes O(n) time
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Multi-layer Dielectric
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Kernel independent
methods
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Multi-layer Green’s function
Kernel dependent methods
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=8.0
m3
=4.0
Discretize dielectric-dielectric m2 =3.9 m2 m2
interface
=4.1
Introduce interface variables
m1
and modify linear system
Expensive
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Other Dielectric Problems

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Conformal dielectric
Voids
Air gap
m3
m2
m2
m1
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Comparison of Methods
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FastCap: O(n)
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Random Walk
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Kernel independent
Singular Value Decomposition: O(nlogn)
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Kernel independent, QuickCap
Pre-corrected FFT: O(nlogn)
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Kernel dependent (1/r)
Kernel independent, Assura RCX
HiCap: O(n)
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Kernel independent
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