Vol. 154
The Time-Harmonic Discontinuous Galerkin Method as a Robust Forward Solver for Microwave Imaging Applications
Progress In Electromagnetics Research, Vol. 154, 1-21, 2015
Novel microwave imaging systems require flexible forward solvers capable of incorporating arbitrary boundary conditions and inhomogeneous background constitutive parameters. In this work we focus on the implementation of a time-harmonic Discontinuous Galerkin Method (DGM) forward solver with a number of features that aim to benefit tomographic microwave imaging algorithms: locally varying high-order polynomial field expansions, locally varying high-order representations of the complex constitutive parameters, and exact radiating boundary conditions. The DGM formulated directly from Maxwell's curl equations facilitates including both electric and magnetic contrast functions, the latter being important when considering quantitative imaging with magnetic contrast agents. To improve forward solver performance we formulate the DGM for time-harmonic electric and magnetic vector wave equations driven by both electric and magnetic sources. Sufficient implementation details are provided to permit existing DGM codes based on nodal expansions of Maxwell's curl equations to be converted to the wave equation formulations. Results are shown to validate the DGM forward solver framework for transverse magnetic problems that might typically be found in tomographic imaging systems, illustrating how high-order expansions of the constitutive parameters can be used to improve forward solver performance.
Ian Jeffrey Nicholas Geddert Kevin Brown Joe LoVetri , "The Time-Harmonic Discontinuous Galerkin Method as a Robust Forward Solver for Microwave Imaging Applications," Progress In Electromagnetics Research, Vol. 154, 1-21, 2015.

1. Hesthaven, J. S. and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis and Applications, Springer, New York, 2008.

2. Hesthaven, J. S. and T. Warburton, "Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell's equations," J. Comput. Phys., Vol. 181, No. 1, 186-221, 2002.

3. Jeffrey, I., "Finite-volume simulations of Maxwell's equations on unstructured grids,", Ph.D. Dissertation, University of Manitoba, 2011.

4. Liu, M., K. Sirenko, and H. Bagci, "An efficient discontinous Galerkin finite element method for highly accurate solution of Maxwell's equations," IEEE Trans. Antennas. Propag., Vol. 60, No. 8, 3992-3998, 2012.

5. Shi, Y. and C.-H. Liang, "Simulations of the left-handed medium using discontinuous Galerkin method based on the hybrid domains," Progress In Electromagnetics Research, Vol. 63, 171-191, 2006.

6. Buffa, A. and I. Perugia, "Discontinuous Galerkin approximation of the Maxwell eigenproblem," SIAM J. Numer. Anal., Vol. 44, No. 5, 2198-2226, 2006.

7. Warburton, T. and M. Embree, "The role of the penalty in the local discontinuous Galerkin method for Maxwell's eigenvalue problem," Comput. Method Appl. Mech. Eng., Vol. 195, No. 25, 3205-3223, 2006.

8. Li, L., S. Lanteri, and R. Perrussel, "A hybridizable discontinuous Galerkin method for solving 3D time-harmonic Maxwell's equations," Numerical Mathematics and Advanced Applications, 119-128, Springer, Berlin Heidelberg, 2013.

9. Bouajaji, M. E. and S. Lanteri, "High order discontinuous Galerkin method for the solution of 2D time-harmonic Maxwell's equations," Appl. Math. Comput., Vol. 219, No. 13, 7241-7251, 2013.

10. Arnold, D. N., F. Brezzi, B. Cockburn, and L. D.Marini, "Unified analysis of discontinuous Galerkin methods for elliptic problems," SIAM J. Numer. Anal., Vol. 39, No. 5, 1749-1779, 2002.

11. Gabard, G., "Discontinuous Galerkin methods with plane waves for time-harmonic problems," J. Comput. Phys., Vol. 225, No. 2, 1961-1984, 2007.

12. Perugia, I., D. Schötzau, and P. Monk, "Stabilized interior penalty methods for the time-harmonic Maxwell equations," Comput. Method Appl. Mech. Eng., Vol. 191, No. 41, 4675-4697, 2002.

13. Lohrengel, S. and S. Nicaise, "A discontinuous Galerkin method on refined meshes for two-dimensional time-harmonic Maxwell equations in composite materials," J. Comput. Appl. Math., Vol. 206, No. 1, 27-54, 2007.

14. Hiptmair, R., A. Moiola, and I. Perugia, "Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations," Math. Comput., Vol. 82, No. 281, 247-268, 2013.

15. Jin, J., The Finite Element Method in Electromagnetics, Wiley, New York, 2002.

16. Dolean, V., H. Fol, S. Lanteri, and R. Perrussel, "Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods," J. Comput. Appl. Math., Vol. 218, No. 2, 435-445, 2008.

17. Zakaria, A., I. Jeffrey, and J. LoVetri, "Full-vectorial parallel finite-element contrast source inversion method," Progress In Electromagnetics Research, Vol. 142, 463-483, 2013.

18. Zakaria, A., A. Baran, and J. LoVetri, "Estimation and use of prior information in FEM-CSI for biomedical microwave tomography," IEEE Antennas Wireless Propag. Lett., Vol. 11, 1606-1609, 2012.

19. Bellizzi, G., O. M. Bucci, and I. Catapano, "Microwave cancer imaging exploiting magnetic nanoparticles as contrast agent," IEEE Trans. Biomed. Eng., Vol. 58, No. 9, 2528-2536, 2011.

20. Hanson, G. W. and A. B. Yakovlev, Operator Theory for Electromagnetics: An Introduction, Springer, New York, 2002.

21. Bonnet, P., X. Ferrieres, B. L. Michielsen, P. Klotz, and J. L. Roumiguires, Finite-volume Time Domain Method, 307-367, Academic Press, San Diego, 1999.

22. Sankaran, K., C. Fumeaux, and R. Vahldieck, "Cell-centered finite-volume-based perfectly matched layer for time-domain Maxwell system," IEEE Trans. Microw. Theory Techn., Vol. 54, No. 3, 1269-1276, 2006.

23. Dosopoulos, S. and J.-F. Lee, "Interior penalty discontinuous Galerkin finite element method for the time-dependent first order Maxwell's equations," IEEE Trans. Antennas Propag., Vol. 58, No. 12, 4085-4090, 2010.

24. Pearson, L., R. Whitaker, and L. Bahrmasel, "An exact radiation boundary condition for the finite-element solution of electromagnetic scattering on an open domain," IEEE Trans. Magn., Vol. 25, No. 4, 3046-3048, 1989.

25. Firsov, K. D. and J. LoVetri, "FVTD-integral equation hybrid for Maxwell's equations," Int. J. Numer. Model. El., Vol. 21, No. 1-2, 29-42, 2008.

26. Ziolkowski, R. W., N. K. Madsen, and R. C. Carpenter, "Three-dimensional computer modeling of electromagnetic fields: A global lookback lattice truncation scheme," J. Comput. Phys., Vol. 50, No. 3, 360-408, 1983.

27. Shanker, B., M. Lu, A. A. Ergin, and E. Michielssen, "Plane-wave time-domain accelerated radiation boundary kernels for FDTD analysis of 3-D electromagnetic phenomena," IEEE Trans. Antennas Propag., Vol. 53, No. 11, 3704-3716, 2005.

28. Harrington, R. F., Time-harmonic Electromagnetic Fields, 2nd Ed., Wiley-Interscience, New York, 2001.

29. Geuzaine, C. and J.-F. Remacle, "GMSH: A 3-D finite-element mesh generator with built-in pre- and post-processing facilities," Int. J. Numer. Meth. Eng., Vol. 79, No. 11, 1309-1331, 2009.

30. Jeffrey, I., J. Aronsson, M. Shafieipour, and V. Okhmatovski, "Error controllable solutions of large-scale problems in electromagnetics: MLFMA-accelerated locally corrected Nyström solutions of CFIE in 3D," IEEE Antennas Propag. Mag., Vol. 55, No. 3, 294-308, 2013.

31. Lazebnik, M., et al., "A large-scale study of the ultrawideband microwave dielectric properties of normal, benign and malignant breast tissues obtained from cancer surgeries," Phys. Med. Biol., Vol. 52, No. 20, 6093, 2007.

32. Burfeindt, M. J., et al., "MRI-derived 3-D-printed breast phantom for microwave breast imaging validation," IEEE Antennas Wireless Propag. Lett., Vol. 11, 1610-1613, 2012.