1 Spring 2003 Prof. Tim Warburton MA557/MA578/CS557 Lecture 30.
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Transcript of 1 Spring 2003 Prof. Tim Warburton MA557/MA578/CS557 Lecture 30.
2
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
3
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
4
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
5
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.
6
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
7
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.
8
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
9
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
10
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.
11
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.
12
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
13
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:
14
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
15
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.
16
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
17
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.
18
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
19
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
20
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
21
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
22
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
23
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
24
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
25
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
26
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
27
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
28
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
29
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
30
Surce Term Stability
• We can verify that the source matrix is a non-positive matrix:
Eigenvalues are:0,0, , , ,x x y y
31
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
32
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