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chap6_V3 - Chapter 6 Absorbing Boundary Conditions As see...

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Chapter 6 Absorbing Boundary Conditions As see from the previous chapter, exact 2D radiation boundary conditions can be established on a circle by using cylindrical harmonics, or on a arbitrary surface using the surface integral equation. However, these exact radiation boundary conditions make part of the the impedance matrix (the part that relates to the surface unknowns) dense, thus significantly increasing the computational cost. This dense matrix is due to nonlocal radiation of sources on the outer boundary. Therefore, it is desirable to have a radiation boundary condition that can retain the sparsity of the impedance matrix of the FEM. Such approximate radiation boundary conditions are usually known as the absorbing boundary conditions (ABCs). An absorbing boundary condition approximates the radiation boundary condition required for an unbounded problem so that negligible erroneous reflections occur at the computational edge when an unbounded domain is truncated by the ABC. In the previous chapter, we already learn the exact radiation boundary conditions, either on a circular outer boundary or the integral equation at an arbitrary outer boundary, although they make the impedance matrix dense for the part associated with boundary unknowns. In this Chapter, the Bayliss-Turkel ABC, a local approximate ABC, and the perfectly matched layer (PML) ABC are presented to overcome this difficulty. The PML boundary conditions can be designed to yield no reflections at the boundary interface, but to make it usable for the FEM and other partial differential equation methods (such as the FDTD), one must truncate the PML medium to a finite thickness; thus, there will be reflections from the outer boundary. From this viewpoint, the PML ABC is also approximate. However, such approximate ABC will preserve the sparseness of the FEM system matrix. 6.1 The Bayliss-Turkel ABC The Bayliss-Turkel ABC is based on the far-zone radiation condition. In 2D, Sommerfeld radiation condition for the scattered field E s z at ρ → ∞ is lim ρ →∞ ∂E s z ∂ρ = - jk b E s z (6.1) in a background medium with the wavenumber k b . This is a local condition, but it holds only at ρ → ∞ . For a finite distance ρ , this condition is only approximate. Since the definition of the far field zone is with respect to the wavelength, it is more convenient to define a normalized length ˜ ρ = k b ρ (6.2) so that the radiation condition can be rewritten as lim ˜ ρ →∞ ∂E s z ρ, φ ) ˜ ρ = - jE s z (6.3) 105
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CHAPTER 6. ABSORBING BOUNDARY CONDITIONS For a finite value of ˜ ρ , this is only approximately valid, and will give rise to some approximation errors. Below we will use this normalized length ˜ ρ to obtain a better approximation than (6.2) for a finite distance from the scatterer.
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