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## Finite Difference Method

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### 8/11/09. ELEN 689. 1. Finite Difference Method. For conductor exterior, solve Laplacian equation ... 8/11/09. ELEN 689. 14. Adaptive Panel Partition. If ... – PowerPoint PPT presentation

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Title: Finite Difference Method

1
Finite Difference Method
• For conductor exterior, solve Laplacian equation
• In 2D

k
m
i
l
j
2
Uniform Grid
i, j1
i1, j
i, j
i1, j
i, j1
3
Basic Properties
• Consider two conductors
• Let vf(Q). From Gauss law
• if we double the amount of charge, E will also
double since the equation is linear
• Therefore, v and Q are linearly related, or QCv

Q
Q
S
Really?
4
Multiple Conductors
• Consider conductors 1, 2, , n
• Apply the above argument for every pair of
conductors i and j

Q1
Qn
Q2
5
Capacitance Matrix
6
BEM Review
• Partition conductor surfaces into panels
• Build coefficient matrix P, where
• and G is Greens function, such as
• Solve linear system Pqv
• Add charges to get capacitance

7
Make It Faster
• Discretization O(n)
• Compute P O(n2) ?O(n)
• Since P is size n?n, P can not be constructed
explicitly
• Solve Pqv O(n3) ?O(n)
• Iterative methods

8
Fast Multipole Methods
• N-body problem Given n particles in 3D space,
compute all forces between the particles
• Fast multipole algorithms
• Appel 85
• Rokhlin 86, Greengard Rokhlin 87
• O(n) time

9
Basic Idea of Multipole
• 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 at k points
k charges
10
Solve Axb Iteratively
• Approximate Axb0
• Bottleneck Matrix-vector product Ax
• A is not used elsewhere

Initial solution x Compute Ax
If Axb gt t/b, modify x
11
Example Jacobi Method
12
Example Jacobi Method
• Transformation
• Ax b ? DxDxAxb ? x (ID1A)xD1b
• Iterations
• x(i1) (ID1A) x(i)D1b
• x(0) 0, x(1) D1b, x(2) (ID1A) x(1)D1b,
• If diagonal dominate, then the method converges
• Better iterative methods exist that converge
under weaker conditions

13
Fast Algorithm HiCap
• Conductor surface refinement
• Adaptively partition conductor surfaces into
small panels according to a user supplied
threshold ?
• Approximate P and store it in a hierarchical data
structure of size O(n)
• The data structure permits O(n) time
matrix-vector product Px for any n-vector x
• Solve linear system Pqv using iterative methods

14
• If interaction between Ai Aj gt ?, refine Ai and
Aj. Otherwise, record Pij in P.

C
C
A
E
B
F
G
M
N
L
I
H
J
J
15
Representation of Matrix P
• P is stored as links in a hierarchical data
structure

A
H
C
B
I
J
D
E
K
L
N
G
M
F
16
Example
• If area/dist ? 1, refine the panel

A
H
2
1/7
1/5
1/5
C
C
B
I
J
4
1/3
B
1
I
4
J
17
Example (contd)
• If area/dist ? 1, refine the panel

A
H
2
1/7
1/5
1/5
C
C
B
I
J
4
E
F
G
D
E
K
1
L
M
N
L
4
N
G
M
J
F
18
A
H
Full 8x8 matrix P
B
I
J
C
E
K
D
L
D
B
E
A
C
1/4.6
M
K
N
1/4.6
I
1/5.5
L
H
J
19
A
H
Implicitly stored P
B
I
J
C
E
K
D
L
D
B
E
A
C
1/5
K
I
L
H
J
20
Properties of P
• P positive, symmetric, positive definite
• Positive definite xPx gt 0 for all x
• If fully expanded, P is size n?n
• P can be approximated by O(n) block entries,
where n is the number of panels
• This is because each panel interacts with
constant number of other panels
• The block entries allow O(n) time matrix-vector
product Px for any x

21
Mat-Vec Pq, Step 1
• Compute charge for all panels

A
H
B
C
I
J
D
E
K
L
N
F
G
M
22
Mat-Vec Pq, Step 2
• Compute potential for all panels

A
H
B
C
I
J
D
E
K
L
N
G
M
F
23
Mat-Vec Pq, Step 3
• Distribute potential to leaf panels

A
H
B
C
I
J
D
E
K
L
N
G
M
F
24
Solving Linear Systems
• 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

25
Error and Time Complexity
• Error of approximation is controlled by ?
• Time complexity is O(n) because step takes O(n)
time

26
Multi-layer Dielectric
• Kernel independent methods
• Multi-layer Greens function
• Kernel dependent methods
• Discretize dielectric-dielectric interface
• Introduce interface variables and modify linear
system
• Expensive

?8.0
m3
?4.0
m2
m2
m2
?3.9
?4.1
m1
27
Other Dielectric Problems
• Conformal dielectric
• Voids
• Air gap

m3
m2
m2
m1
28
Comparison of Methods
• FastCap O(n)
• Kernel dependent (1/r)
• Random Walk
• Kernel independent, QuickCap
• Pre-corrected FFT O(nlogn)
• Kernel independent
• Singular Value Decomposition O(nlogn)
• Kernel independent, Assura RCX
• HiCap O(n)
• Kernel independent