Common use of Convolutional Perfectly Matched Layer Clause in Contracts

Convolutional Perfectly Matched Layer. ‌ One of the greatest challenges of the FDTD method has been the generation of an efficient and accurate solution of electromagnetic wave interaction problems in unbounded regions. For such problems, an absorbing boundary condition (ABC) must be introduced at the outer lattice boundary to simulate the extension of the lattice to infinity. A number of analytical techniques have been discussed to achieve this goal in [86–91]. An alternate approach to achieve an ABC is to terminate the outer boundary of the space lattice in an absorbing material medium. This is analogous to the physical treatment of the walls of an anechoic chamber. Ideally, the absorbing medium is only a few lattice cells thick, reflectionless to all impinging waves over their full frequency spectrum, highly absorbing, and effective in the near field of a source or a scatterer. An early attempt at implementing such an absorbing material boundary condition was reported by Holland [86], who used a conventional lossy dispersionless absorbing medium. The difficulty with this tactic is that such an absorbing layer is matched only to normally incident plane waves. As a result, the entire category of lossy material ABCs had only limited electromagnetic application. In 1994, a breakthrough in this area was created by ▇▇▇▇▇▇▇▇’▇ introduction of a highly effective absorbing-material ABC designated the perfectly matched layer (PML) [92]. The innovation of ▇▇▇▇▇▇▇▇’▇ PML is that plane waves of arbitrary incidence, polarisation and frequency are matched at the boundary. Perhaps of equal importance is that the PML can be used as an absorbing boundary to terminate domains comprised of inhomogeneous, dispersive, anisotropic and even non-linear media, which was previously not possible with analytically derived ABCs. In this thesis, a more efficient implementation is applied as previously published by ▇▇▇▇▇ and ▇▇▇▇▇▇ in [93] based on a recursive convolution technique. This has since been referred to as the CPML formulation. The CPML is based on the stretched- coordinate form of ▇▇▇▇▇▇▇’▇ equations as proposed in (2.2). One example is illustrated below ϵ ∂t Ex + σ Ex = sy (t ) ∗ ∂y Hz − sz (t ) ∗ ∂z Hy . (2.7) where the choice of the complex stretching variable will be as proposed by ▇▇▇▇▇▇▇▇ and ▇▇▇▇▇▇. Specifically, it is assumed that α jωϵ si = κi + σi , i = x, y, or z (2.8) Finally, according to the conclusion in [93], the CPML formulation is generated as follows En+1 − En En+1 + En xi+1/2,j,k − /H xi+1/2,j,k + σ xi+1/2,j,k xi+1/2,j,k − /H = ∆t n+1 2 zi+1/2,j+1/2,k n+1 2 zi+1/2,j−1/2,k − n+1 2 yi+1/2,j,k+1/2 n+1 2 yi+1/2,j,k−1/2 + (2.9) κy ∆y κz ∆z N −1 m=0 Z0y (m) n m+1 2 zi+1/2,j+1/2,k n m+1 2 − − /H zi+1/2,j−1/2,k ∆y N −1 + m=0 Z0y (m) n m+1 2 yi+1/2,j,k+1/2 n m+1 2 − − /H yi+1/2,j,k−1/2 . ∆z Fortunately, z0i (m) can be performed recursively using the recursive convolution method [94, 95]. The set of auxiliary expressions ψi is introduced, and Equation 2.9 is imple- mented as another form En+1 − En En+1 + En xi+1/2,j,k − /H xi+1/2,j,k + σ xi+1/2,j,k xi+1/2,j,k − /H = ∆t n+1 2 zi+1/2,j+1/2,k n+1 2 zi+1/2,j−1/2,k − n+1 2 yi+1/2,j,k+1/2 n+1 2 yi+1/2,j,k−1/2 + (2.10) κy ∆y +ψn+1/2 κz ∆z + ψn+1/2 where ψn+1/2 = by ψn−1/2 exyi+1/2,j,k exzi+1/2,j,k + ay (Hz − Hz )/∆y exyi+1/2,j,k ψ / n+1 2 exzi+1/2,j,k exyi+1/2,j,k = bz ψ − /