Comparison and Application in 3D Matlab-based Finite-Difference Frequency-Domain Method

Qiuzhao Dong(NU), He Zhan(NU), Ann Morgenthler and Carey Rapapport(NU) (contact: qzdong@ece.neu.edu,rapapport@ece.neu.edu)

This work was supported in part by CenSSIS, the Center for Subsurface Sensing and Imaging Systems, under the Engineering Research Centers Program of the National Science Foundation (Award Number EEC-9986821)

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AbstractThe FDFD electromagnetic model computes wave scattering by directly discretizing

Maxwell’s equations along with specifying the material characteristics in the scattering volume. No boundary conditions are need except for the outer grid termination absorbing boundary. We use a sparse matrix Matlab code with generalized minimum residue (GMRES) Krylov subspace iterative method to solve the large sparse matrix equation, along with the Perfectly Matched Layer (PML) absorbing boundary condition. The PML conductivity profile employs the empirical optimal value from[4-6].

The sparse Matlab-based model is about 100 times faster than a previous Fortran-based code implemented on the same Alpha-class supercomputer. The 3D FDFD model is easily manipulated; it can handle all types of layer-based geometries if the target region is less than 25% of the total computational space.

Several cases have been investigated. The scattered electromagnetic fields due to spherical and elliptic mine-like TNT targets buried in simulated Bosnian soil are computed and compared to reference solutions. The field distribution of dipole in the half space is computed and compared to the half-space Born approximation forward Modeling[8].

Opportunities for Technology Transfer• Conventional techniques versus our approach

- Scalar Helmholtz wave equation in frequency domain are well computed with different boundary condition and inhomogeneous media in 2D ; 3D Fortran-based FDFD modeling is time&memory –consuming with simple geometries;

- 2D Matlab-based FDFD methods deal with complicated geometries and isotropic, dispersive media;

- Our approach about 3D Matlab-based FDFD method is a valuable forward modeling for layered 3D inhomogeneous, dispersive media and high frequencies in reasonable memory and computational time.

• Technology Transfer

- The general purpose of this research is detecting the subsurface targets according to their EM properties. This model can be applied to the well-logging in the oil field by the induction (or resistivity) coupling voltage. The geometry for well logging is commonly anisotropic multi-layered & multi-faulted structure, which is suitable for the proposed model .

- This model can be also applied to other fields such as mine detection and tumor detection with the corresponding high and low frequencies.

3D FDFD Modeling� 3D matlab-based FDFD (finite difference frequency domain) method :

-- Based on the general Maxwell’s equations, the wave equation is

where µ= µ0.-- Equipped with the popular PML (perfectly matched layer) ABC (absorbing boundary

conditions.-- Employing the Yee cellgeometry as the grid structure of finite difference method.

�� The applying mathematical methodThe applying mathematical method

The method finally leads to solving the problem of matrix equation: Ax=B; where A is the coefficient matrix, B is the source column matrix and x is the unknown. A is a very large sparse matrix. Therefore the problem is suitable for the Krylove subspace iterative methods. One of them, GMRES (Generalized minimum residue method), is employed after optimalizing the structure of matrix A by multiplying the assisted matrix and doing some permutations.

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ImprovementThe new modified model reduces the computational time (CPU time) to ~1/10 of the previous one. For the grid size with 97x97x85 along x y and z axes and 161 total iterative number, the CPU time of the previous model is around 15 hours, the modified one is only about 1.5 hours. The operative memory decreases to less half (3/7) of the previous model, for example, with the restart=30 and the same grid size as above, the memory is ~5 G for the modified one, but 12G for the old one.

Note: restart is the value of the inner iterative number in GMRES method, it is roughly linear to the necessary memory and slightly relative to the CPU time.

Applicatio

I. Simulation for Sphere TargetsSimulation for Sphere Targets

Geometry and Applied parameters:The TNT scatterer is buried 5 cm under the surface with the shape of sphere: x2+y2+z2=(5cm)2;The operating frequency is 960MHz;Bosnian soil with relative dielectric constant ε=9.19(1+i0.014);97x97x85 grid points along x, y & z axis;The normally incident plane wave with x polarization;

Analysis:

The comparison between modified FDFD method and SAMM method agree very well. In the modified FDFD model, the restart=20, the total iterative number is 241, the CPU time is about 123 minutes (it is around 20 hours previously) , the relative residue goes down to 0.07 (the previous one is around 0.12), the operative memory is 3.7G (the previous is around 10G). Therefore, in this case, the modified FDFD method is indeed improved considering the CPU time, the memory even the performance from the previous model..

II.II. Simulation of elliptic targetsSimulation of elliptic targets

Geometry and Applying parameters:The TNT scatterer is buried 5cm under the surface with the shape of ellipse: 25x2 +25y2+49z 2=(35/2cm)2;The operating frequency is 960MHz;Bosnian soil with relative dielectric constant ε=9.19(1+i0.014);97x97x85 grid points along x, y & z axis;The normally incident plane wave with x polarization;

Conclusion:the CPU time and memory used in the modified FDFD model are much less than these of the previous

model. It is practicable in some sort.

Future Works

• Optimize the algorithm, parallelize the Matlab code to further reduce the CPU time, make it more applicable;

• Apply the complicated geometry in the code, such as rough surface.• Implement the anisotropic media in 2D & 3D FDFD model.

References[1] J. Berenger, “A Perfectly matched layer for the absorption of electromagnetic waves,”J. Computat. Phys., vol. 114, pp.185-200,Oct,1994;

[2] E. Marengo, C. Rappaport and E. Miller, “Optimum PML ABC Conductivity Profile in FDFD”,in review IEEE Transactions on Magnetics, 35,1506-1509, (1999)

[3] S. Winton and C. Rappaport,”Specifying PML Conductivities by Considering Numerical Reflection Dependencies”,IEEE Transactions on Antenna and Propogation, september,2000

[4] S. Winton and C. Rappaport,”Pfrofiling the Perfectly Matched Layer to Improve Large Angle Performance”, IEEE Transactions on Antenna and Propogation, Vol 48,No. 7,July,2000

[5] C. Rappaport, M. Kilmer, and Eric Miller, “Accuracy considerations in using the PML ABC with FDFD Helmholtz equation computation,” Int. J. Numer. Modeling, Vol 13, pp. 471-482,Sept. 2001.

[6] Morgenthaler A.W, Rappaport C.M, “Scattering from lossy dielectric objects buried beneath randomly rough ground: validating the semi-analytic mode matching algorithm with 2-D FDFD“, IEEE Transactions on Geoscience and Remote Sensing, : Volume: 39 page(s): 2421 - 2428 ,Nov. 2001.

[7] Carey M. Rappaport, Qiuzhao Dong, Emmett Bishop, A. Morgenthaler, M. Kilmer, “ Finite Difference Frequency Domain (FDFD) Modeling of Two Dimensional TE Wave Propagation” , URSI Symposium Conference Proceedings, to appear 2004.

[8] He zhan, “Forward Modeling and Shape-based inversion for Geophysics Problem”, PHD thesis,August,2006.

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The magnitude and phase distribution of Ex components at plane x=0 ,y=0 and z=0 from FDFD and SAMM

III. Comparison to BAFM method

Geometry and Parameters:Interface located at z=0 with air above;The operating frequency is 1GHz;Wet soil with relative dielectric constant ε=20+i1.06;The dipole with z polarization located 5cm below the interface;

The magnitude and phase distribution of Ex components at plane x=0 ,y=0

and z=0 from FDFD and SAMM