04/19/23 ELEN 689 1

Finite Difference Method For conductor exterior, solve Laplacian

equation

In 2D:il m

k

j

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Uniform Grid

i, ji–1, j i+1, j

i, j+1

i, j–1

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Basic Properties Consider two conductors

Let v=f(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 Q=Cv

–Q +Q S

Really?

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Multiple Conductors Consider conductors 1, 2, …, n

Apply the above argument for every pair of conductors i and j

Q1Q2 Qn

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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

Solve linear system Pq=v Add charges to get capacitance

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Make It Faster Discretization: O(n) Compute P: O(n2) O(n)

Since P is size nn, P can not be constructed explicitly

Solve Pq=v: O(n3) O(n) Iterative methods

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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

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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!k chargespotential

at k points

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Solve Ax=b Iteratively Approximate Ax–b=0:

Bottleneck: Matrix-vector product Ax A is not used elsewhere

Initial solution x

Compute Ax

If Ax–b > t/b,

modify x

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Example: Jacobi Method

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Example: Jacobi Method Transformation

Ax = b Dx=Dx–Ax+b x = (I–D–1A)x+D–1b Iterations

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 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 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

H

C

B

I

J

C

EF G

M NL

J

1 2 3 4 5

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Representation of Matrix P

A

D

G

H

CB

E

F

I J

K L

M N

P is stored as links in a hierarchical data structure

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Example If area/dist 1, refine the panel

C

B

I

J

A H

CB I J4

1

4

2

1/5

1/71/5

1/3

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Example (cont’d) If area/dist 1, refine the panel

C

EF G

M NL

J

A

D

G

H

CB

E

F

I J

K L

M N

4

1

4

2

1/5

1/71/5

A

B C

D E

H

I J

K L

A

B

C

D

E

H

I

J

L

K NM 1/4.6

Full 8x8

matrix P:

1/5.51/4.6

A

B C

D E

H

I J

K L

A

B

C

D

E

H

I

J

L

K1/5

Implicitly

stored P:

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Properties of P P positive, symmetric, positive definite

Positive definite: xPx > 0 for all x If fully expanded, P is size nn 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

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

A

B C

D E

F G

H

I J

K L

M N

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Mat-Vec Pq, Step 2

A

B C

D E

F G

H

I J

K L

M N

Compute potential for all panels

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Mat-Vec Pq, Step 3 Distribute potential to leaf panels

A

B C

D E

F G

H

I J

K L

M N

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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

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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 Kernel independent

methods Multi-layer Green’s function

Kernel dependent methods Discretize dielectric-dielectric

interface Introduce interface variables

and modify linear system Expensive

m2m2 m2

m1

m3

=3.9

=8.0

=4.0

=4.1

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Other Dielectric Problems Conformal

dielectric Voids Air gap

m1

m3

m2 m2

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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