A comparative Analysis of Cylindrical CFS-PML ABC for Finite Volume Simulations in the Frequency Domain

A comparative analysis of two cylindrical complexfrequency-shifted perfectly matched layers (CFS-PML) absorbing boundary condition (ABC) for bi-dimensional (2-D) finite-volume (FV) simulations in the frequency domain is presented. The impact of CFS-PML parameters on the wave absorption, as well as on the condition number of the associated system matrix is investigated by comparing the performance of two PML loss profiles, viz., polynomial and geometric grading. FV-CFS-PML results are validated against analytical solution. Numerical results show that inclusion of a CFS-PML within the FV computational domain increases the condition number of the system matrix and therefore the use of CFS-PML 3-D FV simulations is limited.


I. INTRODUCTION
One of the biggest challenges in the computational electromagnetics modeling is the efficient and accurate solution of electromagnetic fields in unbounded problems.In the analysis of large scale problems, it is essential to employ iterative solvers for the resulting associated sparse linear system.In general, the convergence of these solvers becomes poorer as the condition number of the system matrix increases and, it may not be achieved in many cases.Furthermore, in order to simulate unbounded problems in both lossless and low-loss media, an absorbing boundary condition (ABC) must be constructed to eliminate spurious reflections from computational boundaries.The perfectly matched layers (PML) ABC has been shown to be very effective for discrete methods [1]- [5].
However, some papers in the literature report that the use of PML in frequency domain methods, such as the finite-volume and finite-element, increases the condition number and consequently, the solution can be plagued by convergence problems [6]- [8].It should be note that the problem associated with the condition number is not really predicated on frequency domain methods but rather whether the method produces require the solution of a large system matrix or not.Time domain methods such as finite-volume time-domain (FVTD) and finite-element time-domain (FETD)  can be also affected by the condition number [9].On the other hand, some frequency domain methods such as the numerical mode matching (NMM) can produce much smaller matrices and be less affected by the condition number issue [10].
A three-dimensional (3-D) finite volume (FV) algorithm has been developed and successfully applied to simulate electromagnetic (EM) well-logging tool response in high-loss geophysical formations [11]- [14].In low-loss media, however, its application implies increasing the computational domain.To save memory requirements and CPU time, a PML must be incorporated in the outermost cells of the grid in order to absorb outgoing waves.
The degradation of the condition number of FV system matrices after the implementation of the PMLs in the computational domain was first investigated in [15], where coaxial waveguides terminated by PMLs in the longitudinal direction were analyzed in terms of its loss parameters and number of layers.In [16], a similar study was done but with coaxial waveguides terminated by longitudinal complex-frequency-shifted (CFS) PMLs [17].
In this paper, a 2-D FV algorithm is applied to a coaxial waveguide backed by a cylindrical CFS-PML in both longitudinal and radial directions.This geometry mimics a FV computational domain with a metallic mandrel around the z axis, which is common in EM well-logging tools simulations.
Here, the main objective is to analyze the effect of the CFS-PML parameters on the condition number of the FV system matrix taking also into account the numerical discretization error.We assess this by comparing the performance of two PML loss profiles, viz., polynomial and geometric grading.
Furthermore, the numerical reflection coefficient is also investigated.The FV-CFS-PML technique is validated against analytical solution showing very good agreement.Numerical results show that the inclusion of a CFS-PML within the FV computational domain increases substantially the condition number of the system matrix.Therefore, unless a well-conditioned CFS-PML is developed, the use of CFS-PML 3-D FV simulations is limited since it requires the use of a direct method for solving the FV system matrix.

A. Finite Volume Technique
In the FV technique, the physical space is decomposed into small volumes and the partial differential equations (PDE) are integrated over each volume.The present FV technique is based on a staggered-grid scheme developed in cylindrical coordinates to better conform to the majority of welllogging tool geometries and to avoid staircasing errors [18].The computational grid is uniform in both the longitudinal z-direction and the radial  -direction.Ampere's law is integrated over faces of the dual grid  ~.For any surface   S with boundary S  , this gives ( AoP46 where  and  are the conductivity, and permittivity of the medium, respectively.s J  is the electric current density (impressed source).The electric and magnetic fields are the unknowns to be determined.Dirichlet boundary conditions are assumed at the computational boundaries.Discrete equations are obtained by evaluating the above over each face of the dual grid.By using integration dual faces S ~ perpendicular to the  -direction (Fig. 1a), we arrive at: where i, k, refer to a primal grid nodal indexing,   and z  are the cylindrical grid spatial increments in the  -and z-directions, respectively; 0 I is the current source amplitude, and The magnetic field can be eliminated from (2) using Faraday's law: where  is the permeability of the medium.By using integration primary faces S perpendicular to the  -and z-directions (Fig. 1b and Fig. 1c), we arrive at: Substituting (4a) and (4b) into (2), a discrete linear system [A][X]=[B] is obtained, where [A] is a complex non-Hermitian matrix, [X] is the vector of (electric field) degrees of freedom (DoFs), and [B] is the discrete source representation.

B. Cylindrical Perfectly Matched Layers
A cylindrical PML is incorporated in the outermost cells of the grid in order to absorb outgoing waves.This is done by modifying the constitutive parameters inside the PML region, and hence it does not require any modification on Maxwell's equation themselves.In cylindrical coordinates, the PML constitutive tensors that are matched to a homogeneous nondispersive medium characterized by constitutive parameters  and  , are given by

AoP48
In the above,  ~ is the analytic continuation of the radial coordinate to a complex variable domain, and  s and z s are frequency-dependent complex stretching variables.Here, two types of PML are implemented for comparison purposes, viz., the standard PML and the CFS-PML.The stretching variables  s and z s are defined as: are functions of position only [11].Note that in the standard PML the parameters   and z  are set equal to zero.Inside the CFS-PML region, ordinary longitudinal outgoing eigenmodes are transformed to , where and  stands for  or z ; and similarly for radial eigenfunctions in terms of Hankel functions.Hence, the transformed eigenmodes exhibit exponential decay inside the CFS-PML so as to reduce spurious reflections from the grid terminations.However, in the lowfrequency limit 0   , the behavior does not exhibit induced attenuation.To circumvent this, is scaled to be maximum at the inner CFS-PML interface, and minimum at the grid termination.Here, the following scaling is adopted: where d is the CFS-PML thickness and m is a taper profile for both real and imaginary parts of the stretching variables.
Once, the design of an effective CFS-PML requires balancing the theoretical reflection error, and the numerical discretization error; several profiles have been suggested for grading   in the context of CFS-PML.The most successful use a polynomial or geometric variation of the CFS-PML loss.
Here, the following polynomial and geometric grading are adopted [1].
is the CFS-PML conductivity at the outer boundary,  is the incidence angle over CFS-PML, and    R is the theoretical reflection error given by where 0   is the CFS-PML conductivity at its surface, g is the scaling factor, and   is the FV space increment.

A. Validation
In order to validate the present FV-CFS-PML method, the algorithm is applied to a lossless coaxial waveguide backed by a cylindrical CFS-PML in both the longitudinal and radial directions, as illustrated in Fig. 2(a) e 2(b), respectively.In all simulations performed here, unless mentioned otherwise, the  attenuation.It should be noted that the design of an effective CFS-PML requires balancing the theoretical reflection error and the numerical discretization error.Normally, the CFS-PML conductivity profile is chosen as large as possible to minimize the theoretical reflection error.
Unfortunately, if the CFS-PML conductivity profile is too large, the discretization error due to the FV approximation dominates, and the numerical (actual) reflection error is potentially orders of magnitude higher than what equation (10c) predicts.This problem is more pronounced in our case because the cell size is too small.To investigate the impact of CFS-PML parameters on the condition number (CN) of the FV system matrix, some of the input data have their values modified.In this study, the computational domain is discretized using a   Moreover, it can be noted that both profiles (polynomial and geometric) show similar behavior.Fig. 7 shows the CN as a function of max

Fig. 1 .
Fig. 1.Unit cell of the staggered grid scheme for spatial discretization of electromagnetic fields on the cylindrical grid.(a) Dual faces perpendicular to the -direction.(b) Primary faces perpendicular to the -direction.(c) Primary faces perpendicular to the -direction.

Fig. 4 .
Fig. 4. Numerical Reflection Coefficient X Theoretical reflection coefficient for different values of max   .In both profiles,

Fig. 5
Fig. 5 shows the COEFN as a function of max  
cell size is uniform in both directions.For the longitudinal PML case, the domain was discretized by using as a function of COEF for different values of max   .For both the longitudinal and the radial CFS-PML case, it is observed that for as the COEF increases.
plays no role in the CN.

Fig. 6 .
Fig. 6.Condition Number X Theoretical Reflection Coefficient for different values of max   .In both profiles, NPML=6 and do require such step and A comparative Analysis of Cylindrical CFS-PML ABC for Finite Volume Simulations in the Frequency Domain Ana Paula M. Lima and Marcela S. Novo