pml for the fdtd

8/10/2019 pml for the fdtd

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Construction of PML

Mark Fischer

Construction of PML - Mark Fischer p. 1


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Contents

1 Introduction

Non-Maxwellian PML

2 Maxwellian PML

Anisotropic absorber as a PML

3

PML using differential forms

Maxwells equations in differential forms formulation

Equivalence of Maxwellian and non-Maxwellian PMLs New PML formulations



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Introduction

Non-Maxwellian PML using complex coordinate stretching

xi xi = xi

0

si(xi)dxi

with

si(xi) =ai(x

i) +ii(xi)/

= /x

1

/x2

/x3

=

1s1 /x

1

1s2/x2

1s3/x3

Modified Maxwell equation (frequency domain):

E = 0 H = 0

E = i H H = i E



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We have seen that this complex coordinate stretching

offers a PML with great accuracy. can be easily used for 1, 2 or 3 dimensions. involves a modification of Maxwells equations! cant be implemented easily in existing FEM code.



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We have seen that this complex coordinate stretching

offers a PML with great accuracy. can be easily used for 1, 2 or 3 dimensions. involves a modification of Maxwells equations! cant be implemented easily in existing FEM code.

Solution: use material constants and to provide the needed additional degrees

of freedom.

Maxwellian PML



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Maxwellian PML

Maxwells equations in time harmonic form:

E = 0

H = 0

E = i H MH H = i E+EE

withM :magnetic conductivity E :electric conductivity

= 0

x+ xEi

0 0

0 y+ yEi

0

0 0 z+ zEi

= 0

x+ xMi

0 0

0 y+ y

Mi 0

0 0 z+ zMi



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Nessecary condition for a PML:

Impedance matching : Z=

=Z0 =

00

0

=

0= =

a 0 00 b 00 0 c

with some complex numbers a,b,c



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Nessecary condition for a PML:

Impedance matching : Z=

=Z0 =

00

0

=

0= =

a 0 00 b 00 0 c

with some complex numbers a,b,c

Maxwells equations reduce to

E = 0

H = 0 E = i0 H H = i0 E



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This equations lead to plane waves

E(r, t) = Eei(krt)

H(r, t) = Hei(k

rt)

with the dispersion relation

k2xbc +

k2yac +

k2zab =k

20 =200,

which is the equation of an ellipsoid

kx = k0bc sin cosky = k0

ac sin sin

kz = k0ab cos



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Example

x

z

i

r t

0,0

,

free space PML

Ei, H

i

Er, H

r

Et, H

t

Dispersion relation:

kx = k0

bc sin

ky = 0

kz = k0ab cos

Ei, Hi eik0(sin ix+cos iz)Er,

Hr r e

ik0(sin rx+cos rz)

Et, Ht t eik0(bc sin tx+

ab cos tz)



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Example

x

z

i

r t

0,0

,

free space PML

Ei, H

i

Er, H

r

Et, H

t

Dispersion relation:

kx = k0

bc sin

ky = 0

kz = k0ab cos

Ei, Hi eik0(sin ix+cos iz)Er,

Hr r e

ik0(sin rx+cos rz)

Et, Ht t eik0(bc sin tx+

ab cos tz)

Continuity of the solutions on interface: Ei+Er =Et andHi+Hr =Ht

Phase matching yields a generalization of Snells law

sin i = sin r =bc sin t



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TM and TE waves

x

z

i

r

t

Ei

Er

Et

Hi

Hr

Ht

i

r

t

Ei

Er

Et

Hi

Hr H

t

TM TE



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TM and TE waves

x

z

i

r

t

Ei

Er

Et

Hi

Hr

Ht

i

r

t

Ei

Er

Et

Hi

Hr H

t

TM TE

Reflection coefficients (using continuity of the solutions on the interface):

rTM =

bacos t cos i

cos i+bacos trTE =

cos i

bacos t

cos i+bacos tConstruction of PML - Mark Fischer p. 9


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Imposing bc= 1 and a=b

the interface will be perfectly reflectionless for any frequency, angle of incidence

and polarization.We now writea =b= 1c =+i

Et(r, t) = Eek0cos tzeik0(sin tx+ cos tz)eit

wave length in absorber rate of decay in absorber

penetration depth= 1

k0cos t


Ph i l I i


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Physical Interpretation

optical axis

k-surface

interface

uniaxial crystal optical axis perpendicular to

interface

electric conductivityE =0S magnetic conductivityM =0S

S=

0 0

0 0

0 0 2+2

z - component is negative Jz = 2+2Ez

dependent sources in the material!


S M lli PML


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Summary Maxwellian PML

We now have found a PML formulation

that uses an anisotropic material as an absorbing layer. that is similar but not equal to the techniques showed before.

that is easy to implement in existingfrequency-domaincode.


S M lli PML


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Problems and Questions remaining:

Generalization to other geometries (e.g. cylindrical, spherical coordinates)? Link between the 2 PML formulations? Are there other PML formulations?


S M lli PML


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Problems and Questions remaining:

Generalization to other geometries (e.g. cylindrical, spherical coordinates)? Link between the 2 PML formulations? Are there other PML formulations?

Next Step:

Electromagnetics with differential forms


PML i diff ti l f


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PML using differential forms

Non-Maxwellian PML formulation:

xi xi = xi

0

si(xi)dxi

with

si(xi) =ai(x

i) +ii(xi)/

Re-Interpretation:

mapping on complex coordinates change of metric

gij =ijgij =gkl xk

xixl

xj =

(s1)2 0 0

0 (s2)2 0

0 0 (s3)2


G l C


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General Case

Consider the general orthogonal curvilinear case(u1, u2, u3) gij is given in terms of the Lam coefficientshi: gij =h2i (u1, u2, u3) ij

Chooseu3 to be analytically continued: u3

u3 = u

3

0 s()d

gij =

(h1)2 0 0

0 (h2)2 0

0 0 (h3)2

withh1/2 =h1/2(u1, u2,u3)and h3 =sh3(u

1, u2,u3).


Mapping forms to ectors


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Mapping forms to vectors

Given a metricgij = (hi)2 ij there is a natural isomorphism mapping 1-forms to vectors

= idu

i gij

=

i

hi u

i

2-forms to axial vectors

= idu[i+1]

du[i+2] gij

= i

h[i+1]h[i+2] ui

withui the unit vector inui direction and[i] i mod 3fori = 3and [3] = 3.


Maxwells equations


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Maxwells equations

Maxwells Equations using differential forms (no sources!)

dE = iB

dH =

iD

dD = 0

dB = 0

E,H : el., magn. field intensity 1-forms D,B : el., magn. flux density 2-forms d : exterior derivative, metric independent

d acts on 1-forms (=vectors) : curl

on 2-forms (=axial vectors) : div


C tit ti P t


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Constitutive Parameters

For differential forms, the constitutive parameters are given in terms of Hodge staroperators:

D = eE

B = hH

The Hodge Star operator

establishes in the 3D case a natural isomorphism between the 1-forms E, Hand the 2-forms D, B.

depends on the metric for the euclidean metric is given thoughdx = dydz,dy=dxdz and

dz=dxdy.



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Expressing the electric and magnetic 1-forms in terms of (u1,u2,u3)

E=Eihidui H=Hihidu

i

The flux 2-forms become

D=e(Eihidui) =

j

ijEjh[i+1]h[i+2]du[i+1] du[i+2]

B =h(Hihidui) =j

ijHjh[i+1]h[i+2]du[i+1] du[i+2]

NB: the star operator depends on the metric!


Change of metric


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Change of metric

Maxwells equations under a change on the metric

dE = iB

dH =

iD

dD = 0

dB = 0

same as before.

D = eE

B = hH

modified operatorse/h defined by new metric.

ThePMLin the diff. forms language isuniqueand unifies the various PMLformulations.

The different formulations can be derived by a simplechoice on how to mapthe forms to vector quantities.


The Maxwellian PML Formulation


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The Maxwellian PML Formulation

Map from forms to corresponding dual vector quantities governed by originalmetric tensor(gij):

E= Eihidui

(gij)

Em =Em

i ui =

hi

hiEiui

D (gij) Dm =Dmi ui =

j

hkhlhkhl

ij Ejui

Modified constitutive tensors are given through

Dm =PML Em

with

(PML)ij =h[i+1]h[i+2]h[i+1]h[i+2]

ijhj

hj


Example


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Example

free space:

h1 = 1

h2 = 1

h3 = 1

i

r

t

x

z

PMLfree space

z z = z

0 s()d

Inside PML:

h1 = 1

h2 = 1

h3 = s(z)

ij =0 ij (PML)ii = h[i+1]h[i+2]h[i+1]h[i+2]

0hi

hi


Example


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Example

free space:

h1 = 1

h2 = 1

h3 = 1

i

r

t

x

z

PMLfree space

z z = z

0 s()d

Inside PML:

h1 = 1

h2 = 1

h3 = s(z)

ij =0 ij (PML)ii = h[i+1]h[i+2]h[i+1]h[i+2]

0hi

hi

PML =0 s(z) 0 00 s(z) 0

0 0 1s(z)

In accordance with the result derived before!


Non-Maxwellian PML Formulation


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Non-Maxwellian PML Formulation

Map from forms to corresponding dual vector quantities governed by modified,complex metric tensor(gij):

E= Eihidui (gij)

Ec =Eci u

i = Eiui

D (gij) Dc =Dci ui =

j

ij Ejui

In contrary to the Maxwellian formulation, we obtain that

the constitutive relations stay the same: Dc = Ec and Bc = Hc Maxwells equations are modified to add additional degrees of freedom. In

the Cartesian case, we obtain

E = 0 E = i H H = 0 H = i E


New Classes of PML


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New Classes of PML

Other choices of metrics(gij)are also possible: e.g. hybridizations:

(gij) =(gij) +(gij) (gij) =

3k=1(gik)

(gkj)

The second choice leads to

E(,) = E(,)i u

i =h1ihi

Eiui

D(,) = D(,)i u

i =j

h1[i+1]h1

[i+2]

h[i+1]h[i+2]

ij Ejui

and a permittivity

(,)ij =

h1[i+1]h1[i+2]

h[i+1]h[i+2]

ijhj

h1j


Summary


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Summary

E H D B

Em

Hm

Dm

Bm

Maxwellian PML

E H D B

E H D B~ ~ ~ ~

non-Maxwellian,complex space PML

Ec

Hc

Dc

Bc

[gij]Original Maxwellian fields

[gij]

[gij]

[gij]

~

[gij]~

PML: changeon the metric

vectors forms

E H D B

Others PMLs

[gij]^

(isomorphisms)

(differing by metric factors) (unique)

(,)(,)(,)(,)


Conclusion


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Conclusion

1

Change of variables change of constitutiveparameters

No change on Maxwells equations! tedious calculation

2

Differential forms provide

a method independent of the field equations. an elegant way to generalize a PML for different

geometries. a unique formulation for a PML.

(different formulations correspond to different mappings)


pml for the fdtd

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