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MA557/MA578/CS557 Lecture 30

MA557/MA578/CS557 Lecture 30. Spring 2003 Prof. Tim Warburton timwar@math.unm.edu. Special Edition of ANUM on Absorbing Boundary Conditions. Structure of PML Regions. PEC. How Thick Does the PML Region Need To Be. Suppose we consider the region in blue [a,a+delta] And we set:. PEC.

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MA557/MA578/CS557 Lecture 30

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  1. MA557/MA578/CS557Lecture 30 Spring 2003 Prof. Tim Warburton timwar@math.unm.edu

  2. Special Edition of ANUM on Absorbing Boundary Conditions

  3. Structure of PML Regions 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

  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. • Berenger’s split PML • Ziolkowski’s PML based on a Lorentz material. • Abarbanel & Gottlieb’s mathematically derived PML.

  8. Recall Berenger’s Split PML

  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

  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. 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:

  13. Ziolkowski’s Lorentz Material Model Based PML • Modification proposed by Abarbanel and Gottlieb: Set:

  14. Reconfigured Lorentz Material Model Based PML • Note that now the corrections are all lower order terms:

  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

  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:

  19. Berenger To Robust PML • Next we Fourier transform in time:

  20. Berenger To Robust PML • Gather like terms:

  21. Berenger To Robust PML • Multiply Hzx, Hzy terms with new factors:

  22. Berenger To Robust PML • Eliminate split variables:

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

  24. Berenger To Robust PML • Expand out Hz terms in 3rd equation: • Also divide 3rd by i*w:

  25. Berenger To Robust PML • Create Auxiliary variables and substitute into PML +

  26. Berenger To Robust PML • Inverse Fourier transform:

  27. Berenger To Robust PML • Inverse Fourier transform:

  28. Berenger To Robust PML • Change of variables: • Manipulate equations:

  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.

  30. Surce Term Stability • We can verify that the source matrix is a non-positive matrix: Eigenvalues are:

  31. Eigenvectors 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 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

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