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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
Adaptive Panel Partition
  • 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
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