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Spring 2003Prof. Tim Warburtontimwar@math.unm.edu

MA557/MA578/CS557Lecture 30

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Special Edition of ANUM on Absorbing Boundary Conditions

Applied Numerical Mathematics

Volume 27, Issue 4, Pages 327-560 (August 1998)Special Issue on Absorbing Boundary ConditionsEdited by Eli Turkel

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Structure of PML Regions

* * 0x y x y

* *0, , 0x x y y

* *0, , 0x x y y

*

*

0

, 0y y

x x

*

*

0

, 0y y

x x

* *, , , 0y y x x

* *, , , 0y y x x * *, , , 0y y x x

* *, , , 0y y x x

PEC

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How Thick Does the PML Region Need To Be

• Suppose we consider the region in blue [a,a+delta] • And we set:

PEC

x=a

* *, 0n

x x y yx aC

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Good Papers on PML Stability

• Eliane Becache, Peter G. Petropoulosy and Stephen D. Gedney, “On the long-time behavior of unsplit Perfectly Matched Layers”.

• E.Turkel, A. Yefet, “Absorbing PML Boundary Layers for Wave-Like Equations”, Applied Numerical Mathematics, Volume 27, pp 533-557, 1998.

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Today• Last class we examined Berenger’s split field, PML, TE Maxwell’s equations.

• Berenger introduced anisotropic dissipative terms which allow plane waves to pass into an absorbing region without reflection.

• However – Abarbanel, Gottlieb, and Hesthaven later showed that the split PML may suffer explosive instability due to the fact that it is only weakly well posed:

• S. Abarbanel, D. Gottlieb and J. S. Hesthaven, “Long Time Behavior of the Perfectly Matched Layer Equations in Computational Electromagnetics”, Journal of Scientific Computing,vol. 17, no. 1-4, pp. 405-422, 2002.

• Yet – even later, Becache and Joly showed that the split PML has at worst a linearly growing solution in the late-time.

• E. Becache and P. Joly, “On the analysis of Berenger’s Perfectly Matched Layers for Maxwell’s equations”, Mathematical Modelling and Numerical Analysis, vol. 36, no. 1, pp.87-119, 2002.

• ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4164.pdf

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Catalogue of Some PMLs

• There are three well known PML formulations.

1) Berenger’s split PML

2) Ziolkowski’s PML based on a Lorentz material.

3) Abarbanel & Gottlieb’s mathematically derived PML.

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Recall Berenger’s Split PML

z

*

*

, are defined so that H

:zx zy zx zy

zx zyxy x

zx zyyx y

yzxx zx

zy xy zy

H H H H

and

H HE Et y

H HEE

t xEH H

t xH E Ht y

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Recall: Wave Speeds

• In the previous notation we looked at eigenvalues of linear combination of the flux matrices:

• The eigenvalues computed by Matlab:

• i.e. 0,0,1,-1 under constraint on(alpha,beta)

• So for Lax-Friedrichs wetake

0 0 0 0 0 0 1 1 0 00 0 1 1 0 0 0 0 0 0

, 0 1 0 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0

A B C

1

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Eigenvectors of C• Using Matlab we can determine the eigenvectors of C

• So C does not have a full space of eigenvectors, which in turn means that C can not be diagonalized.

• So the split PML equations are hyperbolic but only weakly well posed.

2 2

2 2

2 2

00

, , 1

1

where = a

b ba aa ab b

b

a aa a

a

i.e.

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Ziolkowski’s PML

• Ziolkowski proposed a method based on a physical polarized absorbing Lorenz material.

• R. W. Ziolkowski, “Time-derivative Lorentz material model-based absorbing boundary condition” IEEE Trans. Antennas Propagat., vol. 45, pp. 1530-1535, Oct. 1997.

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Lorentz Material Model Based PML

• Introduce 3 new auxiliary variables K,Jx,Jy:

y x

x y

x y

x zx

y zy

x

x z

y z

y

y z

zz x

E Ht yE Ht x

EH Et

K

J Ht yJ Ht xK Ht

E

E

Hy x

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Ziolkowski’s Lorentz Material Model Based PML

• Modification proposed by Abarbanel and Gottlieb:

x z

y

xx y

z

y

y x

x

x

yx y x

x y z

y

xzx y z

E Ht yE Ht x

EEHt y x

P EtP

Et

E

E

H

K Ht

K

x x x x

y y y y

P J EP J E

Set:

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Reconfigured Lorentz Material Model Based PML

• Note that now the corrections are all lower order terms:

x z

y

xx y

z

y

y x

x

x

yx y x

x y z

y

xzx y z

E Ht yE Ht x

EEHt y x

P EtP

Et

E

E

H

K Ht

K

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Abarbanel and Gottlieb’s PML

• Abarbanel and Gottlieb proposed a mathematically derived PML.

• Like the Ziolkowski’s PML it is constructed by adding lower order terms and auxiliary variables.

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Abarbanel & Gottlieb’s PML

2

2

y x y y

x y x x

xx y

yy x

xx x y

y

x z

y z

yxyxx y

z

y y x

E Ht yE Ht x

EEHt

E P

E P

dd Q Qdx

P EtP

EtQ Q E

d

tQ

Q E

y y

t

x

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Converting Berenger To An Unsplit PML

• It is possible to start from the Berenger split PML and return to the Maxwell’s TE equations with additional, lower order terms.

• See:

E.Turkel, A. Yefet, “Absorbing PML Boundary Layers for Wave-Like Equations”, Applied Numerical Mathematics, Volume 27, pp 533-557, 1998.

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Berenger To Robust PML

• We will start with the Berenger PML equations:

z

*

*

H zx zy

x zy x

y zx y

yzxx zx

zy xy zy

H H

E H Et yE H Et x

EH Ht xH E Ht y

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Berenger To Robust PML

• Next we Fourier transform in time:

ˆˆ ˆ

ˆˆ ˆ

ˆˆ ˆ

ˆˆ ˆ

zx y x

zy x y

yzx x zx

xzy y zy

Hi E Ey

Hi E ExE

i H HxEi H Hy

ˆ i tf e f t dt

z

*

*

H zx zy

x zy x

y zx y

yzxx zx

zy xy zy

H H

E H Et yE H Et x

EH Ht xH E Ht y

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Berenger To Robust PML

• Gather like terms:

ˆˆ 0

ˆˆ 0

ˆˆ 0

ˆˆ 0

zy x

zx y

yx zx

xy zy

Hi Ey

Hi ExE

i HxEi Hy

ˆˆ ˆ

ˆˆ ˆ

ˆˆ ˆ

ˆˆ ˆ

zx y x

zy x y

yzx x zx

xzy y zy

Hi E Ey

Hi E ExE

i H HxEi H Hy

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Berenger To Robust PML

• Multiply Hzx, Hzy terms with new factors:

ˆˆ 0

ˆˆ 0

ˆˆ 0

ˆˆ 0

zy x

zx y

yy x zx y

xx y zy x

Hi Ey

Hi ExE

i i H ixEi i H iy

ˆˆ 0

ˆˆ 0

ˆˆ 0

ˆˆ 0

zy x

zx y

yx zx

xy zy

Hi Ey

Hi ExE

i HxEi Hy

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Berenger To Robust PML• Eliminate split variables:

ˆˆ 0

ˆˆ 0

ˆˆ 0

ˆˆ 0

zy x

zx y

yy x zx y

xx y zy x

Hi Ey

Hi ExE

i i H ixEi i H iy

ˆˆ1 0

ˆˆ1 0

ˆˆˆ 1 1 0

y zx

x zy

y yx xy x z

Hi Ei y

Hi Ei x

EEi i H i ii y i x

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Berenger To Robust PML• Expand out Hz terms in 3rd equation:

ˆˆ1 0

ˆˆ1 0

ˆˆˆ 1 1 0

y zx

x zy

y yx xy x z

Hi Ei y

Hi Ei x

EEi i H i ii y i x

2

ˆˆ1 0

ˆˆ1 0

ˆˆˆ 1 1 0

y zx

x zy

y yx xx y x y z

Hi Ei y

Hi Ei x

EEi i H i ii y i x

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Berenger To Robust PML• Expand out Hz terms in 3rd equation:

• Also divide 3rd by i*w:

2

ˆˆ1 0

ˆˆ1 0

ˆˆˆ 1 1 0

y zx

x zy

y yx xx y x y z

Hi Ei y

Hi Ei x

EEi i H i ii y i x

ˆˆ1 0

ˆˆ1 0

ˆˆˆ 1 1 0

y zx

x zy

x y y yx xx y z

Hi Ei y

Hi Ei x

EEi Hi i y i x

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Berenger To Robust PML• Create Auxiliary variables and substitute into PML

ˆˆ1 0

ˆˆ1 0

ˆˆˆ 1 1 0

y zx

x zy

x y y yx xx y z

Hi Ei y

Hi Ei x

EEi Hi i y i x

1ˆ ˆ:

1ˆ ˆ:

1ˆ ˆ:

x x

y y

z z

P Ei

P Ei

Q Hi

+

ˆˆ1 0

ˆˆ1 0

ˆ ˆ ˆ ˆˆ ˆ 0

y zx

x zy

x x x y y yx y z x y z

Hi Ei y

Hi Ei x

E P E Pi H Q

y x

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Berenger To Robust PML• Inverse Fourier transform:

ˆˆ1 0

ˆˆ1 0

ˆ ˆˆ 0

ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆˆ ˆ

y zx

x zy

x yx y z

x x

y y

z

yx

z

xx z

y y

Hi Ei y

Hi Ei x

E Ei H

y x

i P E

i P E

P PQ

i Q H

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Berenger To Robust PML• Inverse Fourier transform:

0

0

0x

x zy x

y zx y

yxzx y z

y yx x

xx

yy

z

z

y

z

E HEt yE HEt x

EEH Ht y x

P EtP

Et

H

Q

Qt

PP

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Berenger To Robust PML• Change of variables:

• Manipulate equations:

x x x x

y y y y

x z

E E P

E E P

E H

y x x x x y x

x

x

y y y y x y

x y z x y z

xx x x

yy y y

zz

z

y z

y xz

E P

E P

H Q

P E PtP

E Ht y

E Ht xE EH

t

E P

x

H

y

tQt

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Comments

• Notice that the additional auxiliary variables Px,Py,Qz are defined as solutions of ODEs.

• Corrections to TE Maxwell’s are linear corrections in Ex,Ey,Hz,Px,Py,Qz strongly hyperbolic equations well posed.

• Technically, one should verify that this is still a PML.

y x x x x y x

x

x

y y y y x y

x y z x y z

xx x x

yy y y

zz

z

y z

y xz

E P

E P

H Q

P E PtP

E Ht y

E Ht xE EH

t

E P

x

H

y

tQt

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Surce Term Stability

• We can verify that the source matrix is a non-positive matrix:

Eigenvalues are:0,0, , , ,x x y y

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Eigenvectors

Full set of vectors (at least in the corners):

0 0 0 00 0 0 0

0 0 0 0, , , , ,

0 0 1 0 1 01 0 0 0 0 10 1 0 1 0 0

y x x

y x x

x y

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Message on Derivation of a General PML

• After reviewing the literature it appears that there is a certain art to constructing a PML for a given set of PDEs.

• For a possible generic approach see:

• Hagstrom et al:

• http://www.math.unm.edu/~hagstrom/papers/aero.ps• http://www.math.unm.edu/~hagstrom/papers/newpml.ps