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PPT – Finite Difference Method PowerPoint presentation | free to download - id: ec101-Y2NjN

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Finite Difference Method

- For conductor exterior, solve Laplacian equation
- In 2D

k

m

i

l

j

Uniform Grid

i, j1

i1, j

i, j

i1, j

i, j1

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?

Multiple Conductors

- Consider conductors 1, 2, , n
- Apply the above argument for every pair of

conductors i and j

Q1

Qn

Q2

Capacitance Matrix

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

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

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

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

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

Example Jacobi Method

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

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

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

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

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

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

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

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

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

Mat-Vec Pq, Step 1

- Compute charge for all panels

A

H

B

C

I

J

D

E

K

L

N

F

G

M

Mat-Vec Pq, Step 2

- Compute potential for all panels

A

H

B

C

I

J

D

E

K

L

N

G

M

F

Mat-Vec Pq, Step 3

- Distribute potential to leaf panels

A

H

B

C

I

J

D

E

K

L

N

G

M

F

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

Error and Time Complexity

- Error of approximation is controlled by ?
- Time complexity is O(n) because step takes O(n)

time

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

Other Dielectric Problems

- Conformal dielectric
- Voids
- Air gap

m3

m2

m2

m1

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