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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
Construction of PML - Mark Fischer p. 2
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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
Construction of PML - Mark Fischer p. 3
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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.
Construction of PML - Mark Fischer p. 4
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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
Construction of PML - Mark Fischer p. 4
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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
Construction of PML - Mark Fischer p. 5
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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
Construction of PML - Mark Fischer p. 6
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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
Construction of PML - Mark Fischer p. 6
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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
Construction of PML - Mark Fischer p. 7
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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)
Construction of PML - Mark Fischer p. 8
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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
Construction of PML - Mark Fischer p. 9
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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
Construction of PML - Mark Fischer p. 10
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!
Construction of PML - Mark Fischer p. 11
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.
Construction of PML - Mark Fischer p. 12
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.
Problems and Questions remaining:
Generalization to other geometries (e.g. cylindrical, spherical coordinates)? Link between the 2 PML formulations? Are there other PML formulations?
Construction of PML - Mark Fischer p. 12
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.
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
Construction of PML - Mark Fischer p. 12
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
Construction of PML - Mark Fischer p. 13
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).
Construction of PML - Mark Fischer p. 14
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.
Construction of PML - Mark Fischer p. 15
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
Construction of PML - Mark Fischer p. 16
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.
Construction of PML - Mark Fischer p. 17
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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!
Construction of PML - Mark Fischer p. 18
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.
Construction of PML - Mark Fischer p. 19
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
Construction of PML - Mark Fischer p. 20
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
Construction of PML - Mark Fischer p. 21
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!
Construction of PML - Mark Fischer p. 21
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
Construction of PML - Mark Fischer p. 22
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
Construction of PML - Mark Fischer p. 23
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)
(,)(,)(,)(,)
Construction of PML - Mark Fischer p. 24
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)
Construction of PML - Mark Fischer p. 25