MA557/MA578/CS557. Lecture 30. Prof. Tim Warburton. Spring

Transcription

1 MA557/MA578/CS557 Lecture 3 Spring 23 Prof. Tim Warburton timwar@math.unm.edu 1

2 Special dition of ANUM on Absorbing Boundar Conditions Applied Numerical Mathematics Volume 27, Issue 4, Pages (August 1998) Special Issue on Absorbing Boundar Conditions dited b li Turkel 2

3 * *,,, = = * *, * *,,, Structure of PML Regions * * = =,, * * = = = = PC * * = =,, * *,,, = = * *, * 3 *,,,

4 ow Thick Does the PML Region Need To Be Suppose we consider the region in blue [a,a+delta] And we set: * a * = = C, = = δ n δ PC 4 =a

5 Good Papers on PML Stabilit liane Becache, Peter G. Petropoulos and Stephen D. Gedne, On the long-time behavior of unsplit Perfectl Matched Laers..Turkel, A. Yefet, Absorbing PML Boundar Laers for Wave-Like quations, Applied Numerical Mathematics, Volume 27, pp ,

6 Toda Last class we eamined Berenger s split field, PML, T Mawell s equations. Berenger introduced anisotropic dissipative terms which allow plane waves to pass into an absorbing region without reflection. owever Abarbanel, Gottlieb, and esthaven later showed that the split PML ma suffer eplosive instabilit due to the fact that it is onl weakl well posed: S. Abarbanel, D. Gottlieb and J. S. esthaven, Long Time Behavior of the Perfectl Matched Laer quations in Computational lectromagnetics, Journal of Scientific Computing,vol. 17, no. 1-4, pp , 22. Yet even later, Becache and Jol showed that the split PML has at worst a linearl growing solution in the late-time.. Becache and P. Jol, On the analsis of Berenger s Perfectl Matched Laers for Mawell s equations, Mathematical Modelling and Numerical Analsis, vol. 36, no. 1, pp , 22. ftp://ftp.inria.fr/inria/publication/publi-pdf/rr/rr-4164.pdf 6

7 Catalogue of Some PMLs There are three well known PML formulations. 1) Berenger s split PML 2) Ziolkowski s PML based on a Lorent material. 3) Abarbanel & Gottlieb s mathematicall derived PML. 7

8 Recall Berenger s Split PML ( ) ( ) ( ), are defined so that = + and : + = t + + = t + = t = t * * 8

9 Recall: Wave Speeds In the previous notation we looked at eigenvalues of linear combination of the flu matrices: 1 1 β β 1 1 α α A =, B = C= 1 α 1 β The eigenvalues computed b Matlab: i.e.,,1,-1 under constraint on (alpha,beta) So for La-Friedrichs we take λ =1 9

10 igenvectors of C Using Matlab we can determine the eigenvectors of C i.e. ba ba a a a, a, 2 2 a a b b 1 where a = a + b 2 2 So C does not have a full space of eigenvectors, which in turn means that C can not be diagonalied. So the split PML equations are hperbolic but onl weakl well posed. 1

11 Ziolkowski s PML Ziolkowski proposed a method based on a phsical polaried absorbing Loren material. R. W. Ziolkowski, Time-derivative Lorent material model-based absorbing boundar condition I Trans. Antennas Propagat., vol. 45, pp , Oct

12 Lorent Material Model Based PML Introduce 3 new auiliar variables K,J,J: t = t t + = ( + ) + = J = t J =+ t K = + t K 12

13 Set: Ziolkowski s Lorent Material Model Based PML Modification proposed b Abarbanel and Gottlieb: P = J + P = J + = t + = t t + = P t P t K t = = =+ ( + ) K 13

14 Reconfigured Lorent Material Model Based PML Note that now the corrections are all lower order terms: = t + = t t + = P t P t K t = = =+ ( + ) K 14

15 Abarbanel and Gottlieb s PML Abarbanel and Gottlieb proposed a mathematicall derived PML. Like the Ziolkowski s PML it is constructed b adding lower order terms and auiliar variables. 15

16 Abarbanel & Gottlieb s PML = t t + = 2 P 2 P d d + = Q Q t d d P = t P t Q t Q t = + + = Q = Q 16

17 Converting Berenger To An Unsplit PML It is possible to start from the Berenger split PML and return to the Mawell s T equations with additional, lower order terms. See:.Turkel, A. Yefet, Absorbing PML Boundar Laers for Wave-Like quations, Applied Numerical Mathematics, Volume 27, pp ,

18 Berenger To Robust PML We will start with the Berenger PML equations: = t + = t t = t ( ) = + + = * * 18

19 Berenger To Robust PML Net we Fourier transform in time: i t ( ) ( ) ω f ω e f t dt = = t + = t t = t ( ) = + + = * * = + = + = = 19

20 Berenger To Robust PML Gather like terms: = + = + = = i + = i + + = i + + = i + = ( ω ) ( ω ) ( ω ) ( ω ) 2

21 Berenger To Robust PML Multipl, terms with new factors: i + = i + + = i + + = i + = ( ω ) ( ω ) ( ω ) ( ω ) ( ) + = ( ) + + = = = ( )( ) ( ) ( )( ) ( ) 21

22 Berenger To Robust PML liminate split variables: ( ) + = ( ) + + = = = ( )( ) ( ) ( )( ) ( ) 1 + = = = ( )( ) 22

23 Berenger To Robust PML pand out terms in 3 rd equation: 1 + = = = ( )( ) 1 + = = = ( 2 ( ) ( )( ) ) 23

24 Berenger To Robust PML pand out terms in 3 rd equation: 1+ = = ( 2 ( ) ( )( ) ) = Also divide 3 rd b i*w: 1 + = = ( ) ( ) = 24

25 Berenger To Robust PML Create Auiliar variables and substitute into PML 1 P : = i ω 1 P : = i ω 1 Q : = i ω = = ( ) ( ) = (( ) ( )) 1 + = = ( ) ( ) P P Q + = 25

26 Berenger To Robust PML Inverse Fourier transform: (( ω) ( )) 1 + = = ( ) ( ) P P + + i Q + = P = P = Q = 26

27 Berenger To Robust PML Inverse Fourier transform: + = t + + = t ( P ) ( + P ) + ( ) Q t = P t P t Q t = = = 27

28 Change of variables: Manipulate equations: Berenger To Robust PML t = + P = + P = t + t + P t P t Q t ( ) ( ) = P ( ) ( ) = P ( ) = + Q = P = P = 28

29 Comments Notice that the additional auiliar variables P,P,Q are defined as solutions of ODs. Corrections to T Mawell s are linear corrections in,,,p,p,q strongl hperbolic equations well posed. Technicall, one should verif that this is still a PML. t t + t + P t P t Q t ( ) ( ) = P ( ) ( ) = P ( ) = + Q = P = P = 29

30 Surce Term Stabilit We can verif that the source matri is a non-positive matri: igenvalues are:,,,,, 3

31 igenvectors Full set of vectors (at least in the corners): +,,,,,

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 PDs. For a possible generic approach see: agstrom et al: