Nonstandard Curvilinear Implementation

The construction of the higher order nonstandard unsplit-field PMLs in curvilinear meshes initiates from the division of a 3-D space, Q, into two areas (separated with a nonplanar interface f) such that Q = f cs U 2pml where f2cs refers to the computational domain and 2pml is the area occupied by the PML under research. Considerable in the procedure is the extension of the stretched-coordinate theory to nonstandard models [30]. This is performed through the following steps  

To exhibit the advantages of the above conventions and without loss of generality, the most frequendy encountered case of spherical coordinates (r, 0, cp) is analyzed. In this particular system, the usual PML arrangements suffer from several weaknesses, especially from an accuracy and convergence outlook. It is mentioned that the proposed method has also been applied - in an analogous way - to other coordinate systems with a very satisfactory performance. 

Let us study the derivation of the FDTD-PML expressions under a Maxwellian manner. Unfortunately, such complicated curvilinear environments create highly dispersive reflections that form bands of transmitted modes growing spatially instead of being damped in the layer. For the goals of the approach, an inhomogeneous, isotropic, and lossless dielectric medium is examined in the frequency domain. Thus, 

FIGURE 4.4 The higher order nonstandard PML in spherical coordinates 

Equations (4.17) and (4.18) constitute a causal hyperbolic system, according to the mathematical analysis of [26], which allows the physical implementation of the unsplit PML. Hence, these curvilinear absorbers do not generally create wave modes that grow linearly in time and for the majority of the problems are not vulnerable to perturbations. In the expressions written 

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Next, analysis proceeds to the iris-coupled elliptical resonator of Figure 7.5(a). Herein, the curvilinear nonstandard operators, (3.70)—(3.81) for M = 2 and L = 2, are implemented... 

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