1 / 32

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.

olivia-owen
Télécharger la présentation

MA557/MA578/CS557 Lecture 30

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related