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1 Spring 2003 Prof. Tim Warburton MA557/MA578/CS557 Lecture 29

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2 Maxwell’s 2D TM Equations We established Maxwell’s TM equations as: We added PEC boundary conditions (say suitable for a domain bounded by a superconducting material):

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3 Exterior Domain Now suppose we wish to consider the case of an electromagnetic wave bouncing off a scatterer: PEC Incident field Scattered field

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4 Exterior Domain We know which boundary conditions to apply at the PEC scatterer. We do not know what to do at an artificial domain boundary. We do suppose that there are no waves incident from infinity. PEC Incident field Scattered field Artificial radiation boundary condition

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5 Perfectly Matched Layer (PML) We will consider the PML approach proposed by Berenger (J. Comp. Phys. 114: , 1994). Berenger’s approach: –Create a region at the border of the domain where dissipative terms are introduced into the pde. –Make sure that plane waves entering the absorbing region are not reflected at the interface between the absorbing region and the free space region. –Make sure that there is no reflection at the interface for any angle of incident and any frequency. –Try to make the fields decay by orders of magnitude in the PML region (reduces backscatter into the domain).

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6 TE Mode Maxwell’s The subset of Maxwell’s equations considered in Berenger’s paper involve (Ex,Ey,Hz): Note the dissipative terms on the rhs. If the spatial terms were not there then the solutions would decay as:

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7 Stage 1: Split the magnetic field Hz First Berenger split the Hz field into two coefficients and made the dissipative terms anisotropic:

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8 Wave Structures In Matrix form:

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9 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 we take

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10 Structure of PML Regions PEC

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11 Split Equations Look for traveling plane wave solutions, Where alpha,beta are unknown complex numbers. E0, phi, and frequency all specified We now seek plane wave solutions for the split equations (parameterized by direction)

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12 + Conditions on alpha, beta, Hzx0, Hzy0

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13 Looking for waves traveling into the domain Basic algebra

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14 Special Case We did not specify yet how the fields are defined. Suppose we set:

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15 Special Case Recall form of plane wave. + We can repeat this for the other 3 fields.

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16 Interpretation of Solutions We derived the solutions of the TE Maxwell equations with dissipative terms assuming there is a plane wave solution. What we find is for solutions traveling into the PML layer that the influence of the absorption terms only modifies the plane wave inside the layer.

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17 Traveling wave solution Dissipation in layer

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18 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

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19 Effectiveness of PML All plane waves travelling in the direction (cos(phi),sin(phi)) decay as: After traversing the pml region from a to a+delta the field will decay by a factor of: For the sigma we chose:

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20 Terminating the PML If we terminate the far boundary of the PML with a reflecting PEC boundary condition then the solution will decay on the way to the boundary and back into the domain. i.e. the reflection coefficient for the PML will be: i.e. if you want the solution to decay by a factor of 100 for an n’th order PML region then the width delta is determined approximately by: Notice if phi = pi/2 (i.e. the wave is traveling parallel to the region) then the solution does not decay in the pml region. However, it will decay when it reaches the upper pml region..

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21 Papers and Web Notes Prime source: A Perfectly Matched Layer for the Absorption of Electromagnetic Waves, Journal of Computational Physics Volume 114, Pages , Notes: Some related papers: Misc: Optimizing the perfectly matched layer, Computer Methods in Applied Mechanics and Engineering, Volume 164, Issues 1-2, October 1998, Pages

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22 Solving with the USEMe PDE Wizard

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23 Comparing with and without PML Demo..

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24 Next Class We will discuss possible instabilities created by splitting the magnetic field. Investigate unsplit alternative PMLs. Visit PML in the context of DG.

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