Discretization and Symmetry Rob F. Remis and J orn T. Zimmerling - - PowerPoint PPT Presentation

Discretization and Symmetry

Rob F. Remis and J¨

• rn T. Zimmerling

DCSE Fall School, Delft, November 4 – 8, 2019 1

Introduction

Objective

Discretize Maxwell’s equation to formulate model order reduction using linear algebra Symmetry in the discrete domain Application of this theory to imaging

2

Instantaneously Reacting Material: 1D

One-dimensional Maxwell equation ∂y ∂y

• +

σ

• +

ε µ

• ∂t

Ez Hy

• = −

Jext

z

• ,

(1) succinctly written as [D + S + M∂t] f = − q, (2) Discretization: differential operators & functions → matrices and vectors

3

Maxwell on a grid

δ1 δ2 δ3 δQ+1 y = 0 y = L Ez(y0) Ez(y1) Ez(y2) Ez(y3) Ez(yQ) Ez(yQ+1) Hx(ˆ y1) Hx(ˆ y2) Hx(ˆ y3) Hx(ˆ yQ) Hx(ˆ yQ+1) ˆ δ1 ˆ δ2 ˆ δn

4

Maxwell on a grid

δ1 δ2 δ3 δQ+1 y = 0 y = L Ez(y0) Ez(y1) Ez(y2) Ez(y3) Ez(yQ) Ez(yQ+1) Hx(ˆ y1) Hx(ˆ y2) Hx(ˆ y3) Hx(ˆ yQ) Hx(ˆ yQ+1) ˆ δ1 ˆ δ2 ˆ δn

We discretize the equation ∂yHx|y=yq + σ(yq)Ez(yq, t)εr(yq)∂tEz(yq, t) = −J ext

z

(yq, t),

• n primary grid

for q = 1, 2, ..., Q and ∂yEz|y=ˆ

yq + µ(ˆ

yq)∂tHx(ˆ yq, t) = 0

• n the dual grid

for q = 1, 2, ..., Q + 1.

5

Discrete Equations

We finally arrive at hx(ˆ yq+1, t) − hx(ˆ yq, t) ˆ δy;q + σ(yq)ez(yq, t) + ε(yq)∂tez(yq, t) = −jext

z

(yq, t), for q = 1, 2, ..., Q and ez(yq, t) − ez(yq−1, t) δy;q + µ(ˆ yq)∂thx(ˆ yq, t) = 0, for q = 1, 2, ..., Q + 1. Where are the boundary conditions?

6

System Formulation

We collect the FD approximation in the vectors ez = [ez(y1, t), ez(y2, t), ..., ez(yQ, t)]T, hx = [hx(ˆ y1, t), hx(ˆ y2, t), ..., hx(ˆ yQ+1, t)]T. The source vector jext

z

is defined in a similar manner. In addition we introduce the differentiation matrices ˆ Y and Y as

ˆ Y =          −ˆ δ−1

y;1

ˆ δ−1

y;1

−ˆ δ−1

y;2

ˆ δ−1

y;2

· · · · · · −ˆ δ−1

y;Q

ˆ δ−1

y;Q

         , Y =            δ−1

y;1

−δ−1

y;2

δ−1

y;2

−δ−1

y;3

δ−1

y;3

· · · · −δ−1

y;Q

δ−1

y;Q

−δ−1

y;Q+1

           .

Note that ˆ Y ∈ RQ×(Q+1) and Y ∈ R(Q+1)×Q.

7

System Formulation 2

We also introduce the diagonal medium matrices Mσ = diag(σ(y1), σ(y2), ..., σ(yQ)), Mε = diag(ε(y1), ε(y2), ..., ε(yQ)), Mµ = diag(µ(ˆ y1), µ(ˆ y2), ..., µ(ˆ yQ+1)).

8

System Formulation 2

We also introduce the diagonal medium matrices Mσ = diag(σ(y1), σ(y2), ..., σ(yQ)), Mε = diag(ε(y1), ε(y2), ..., ε(yQ)), Mµ = diag(µ(ˆ y1), µ(ˆ y2), ..., µ(ˆ yQ+1)). Leading to the discrete system

• ˆ

Y Y

• +

Mσ

• +

Mε Mµ

• , ∂t

ez hx

• = −

jext

z

• r more succinctly

[D + S + M∂t] f(t) = −q(t).

9

What happens in higher dimensions?

Every field component (Hx, Hy, Ez) gets defined on ints own grid ∂xHy and ∂yHx are evaluated on the primary grid of Ez We sort these 2D fields into a single vector We still obtain a form similar to [D + S + M∂t] f(t) = −q(t).

10

Properties and Symmetry summary

To summarize we find DTW = −WD, aswell with δ− = diag(I, −I) DTW = δ−WD. This symmetry is useful to preserve in reduced order modeling.

11

Can we image based on finite-differences?

Assume a active array imaging configuration

?

ΩIm ΩArray

Can we reconstruct the impedance z(x) =

• µ

ε in 1D from

boundary measurements u(0, s) in the Laplace (s) domain? vx(x, s) + s 1 z(x)u(x, s) = 0 ux(x, s) + sz(x)v(x, s) = 0 (3) v(0, s) = −1 u(L, s) = 0,

12

Can we image based on finite-differences?

We want to reconstruct z(x) from measuring u(x) Define the data as [2m-1]/[2m] rational function Λ(s) = u(0, s) v(0, s) = −u(0, s) =

m

• j=1

rj s + ζj + ¯ rj s + ¯ ζj (4) m residues rj and poles ζj

13

Can we image based on finite-differences?

Can we link this rational function to a FD discretization?

δ1 δ2 δ3 δn ˆ δ1 y = 0 y = L u2 u3 un+1 v2 v3 vn+1 ˆ δ2 ˆ δ3 ˆ δn

Discretizing on this grid leads to vj+1 − vj ˆ δj + s 1 zFD

j

uj = 0 uj − uj−1 δj−1 + sˆ zFD

j

vj = 0 v1(s) = −1 un+1(s) = 0. with zj = z(yj), ˆ zj = z(ˆ yj), uj = u(yj) and vj = v(ˆ yj)

14

Compare spectral and FD discretization

We have two representations for the same data. The FD data after sorting the field as [u1, v2, u2, v3, . . . ] reads

ΛFD(iω) = −eT

1

               

1 ˆ δ1

. . .

1 δ1

− 1

δ1

. . . ... ... ... . . . . . . − 1

ˆ δn 1 ˆ δn

. . .

1 δn

        + s       

1 z1

ˆ z1 ...

1 zn

−ˆ zn+1               

−1

e1,

and the data can be written as

ΛFD(iω) = eT

1

              ζ1 ζ1 ... ζm ζm        + s        1 1 ... 1 1              

−1 

      y1 y1 . . . y1 y1        ,

If we take m = n we can find a linear algebraic transform directly from Λ(s) to Λ(s)FD and link the poles ζj and residues rj to zj and ˆ zj.

15

Compare spectral and FD discretization

The grid steps δj and ˆ δj are known and medium independent. The impedance can be found from the data (here m = 20)

16