volume | PIER Journals

Abstract

In this paper, we present a new numerical anisotropy optimization method for the three-dimensional (3D) radial point interpolation method (RPIM) in lossy media. Instead of evaluating the parameters of the artificial anisotropy or the scaling factors along the selected axes, as it is usually done in classical optimization algorithms, once the analytical expressions of these parameters have been determined, they are assigned at each node through their shape functions. By adaptive factor, we mean that its value varies in such a way to cancel the discrepancy between numerical and exact wavenumbers at each node. Doing such optimization at each node is indeed being possible during the calculation of these parameters by the RPIM dispersion relation. Therefore the numerical anisotropy is no longer optimized by averaging over the entire Cartesian grid but in each node direction. The RPIM numerical anisotropy adaptive optimization method (AOM) in lossy media is presented, and the theoretical adaptive factors are given as functions of nodes positions. Our results show that the numerical errors of the dispersion and the anisotropy are considerably reduced, after being optimized with the AOM. The proposed AOM scheme is applied for a 3D rectangular cavity in order to test its validity and evaluate the accuracy of the numerical results of our approach.

1. Ciarlet, P. G., Handbook of Numerical Analysis, Numerical Methods in Electromagnetics, Vol. XIII, Elsevier, 2005.

2. Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, 2000.

3. Liu, G. R. and Y. T. Gu, An Introduction to Meshfree Methods and Their Programming, Springer, 2005.

4. Kaufmann, T., Y. Yu, C. Engstrom, Z. Chen, and C. Fumeaux, "Recent developments of the meshless radial point interpolation method for time-domain electromagnetics," Int. J. Numer. Model., Vol. 25, 468-489, 2012.
doi:10.1002/jnm.1830

5. Kaufmann, T., C. Engstrom, and C. Fumeaux, "Residual-based adaptive refinement for meshless eigenvalue solvers," International Conference in Electromagnetics on Advanced Applications, 244-247, IEEE, Sydney, Australia, 2010.

6. Alaa, G., E. Francomanob, A. Tortoricib, E. Toscanob, and F. Violaa, "Smoothed particle electromagnetics: A Mesh-free solver for transients," Journal of Computational and Applied Mathematics, Vol. 191, 194-205, 2006.
doi:10.1016/j.cam.2005.06.036

7. Schweitzer, M. A., A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations, Springer, 2003.
doi:10.1007/978-3-642-59325-3

8. Atluri, S. N. and S. Shen, "The Meshless Local Petrov-Galerkin (MLPG) method: A simple & less-costly alternative to the finite element and boundary element methods," CMES, Vol. 3, No. 1, 11-51, 2002.

9. Juntunen, J. S. and T. D. Tsiboukis, "Reduction of numerical dispersion in FDTD method through artificial anisotropy," IEEE Transactions on Microwave Theory and Techniques, Vol. 48, No. 4, Apr. 2000.
doi:10.1109/22.842030

10. Wang, S. and F. L. Teixeira, "Dispersion-relation-preserving FDTD algorithms for large-scale three-dimensional problems," IEEE Trans. on Ant. and Prop., Vol. 51, No. 8, 1818-1828, Aug. 2003.
doi:10.1109/TAP.2003.815435

11. Nehrbass, J. W., J. O. Jevtic, and R. Lee, "Reducing the phase error for finite-difference methods without increasing the order," IEEE Trans. on Ant. and Prop., Vol. 46, No. 8, 1194-1201, Aug. 1998.
doi:10.1109/8.718575

12. Cole, J. B., "A high-accuracy realization of the Yee algorithm using non-standard finite differences," IEEE Transactions on Microwave Theory and Techniques, Vol. 45, 991-996, Jun. 1997.
doi:10.1109/22.588615

13. Forgy, E. A. and W. C. Chew, "A time-domain method with isotropic dispersion and increased stability on an overlapped lattice," IEEE Trans. on Ant. and Prop., Vol. 50, 983-996, Jul. 2002.

14. Yu, Y. and Z. Chen, "A 3-D radial point interpolation method for meshless time-domain modeling," IEEE Transactions on Microwave Theory and Techniques, Vol. 57, No. 8, 2015-2020, Aug. 2009.
doi:10.1109/TMTT.2009.2025450

15. Lee, J. F., R. Lee, and A. Cangellaris, "Time-domain finite-element methods," IEEE Trans. on Ant. and Prop., Vol. 45, No. 3, 430-442, Mar. 1997.
doi:10.1109/8.558658

16. Naamen, H. and T. Aguili, "Unconditional stability analysis of the 3D-radial point interpolation method and Crank-Nicolson scheme," Progress In Electromagnetics Research M, Vol. 68, 119-131, 2018.

17. Jose, A. P., G. Ana, G. Oscar, and V. Angel, "Analysis of two alternative ADI-FDTD formulations for transverse-electric waves in lossy materials," IEEE Trans. on Ant. and Prop., Vol. 57, No. 7, 2047-2054, Jul. 2009.

18. Naamen, H., A. B. H. Hamouda, and T. Aguili, "Anisotropy analysis of the 3D-radial point interpolation method in lossy media," Progress In Electromagnetics Research C, Vol. 128, 207-218, 2023.

19. Harrington, R. F., Time Harmonic Electromagnetic Fields, IEEE Press Series on Electromagnetic Wave Theory, Wiley-Interscience, 2001.
doi:10.1109/9780470546710

20. Koh, I. S., H. Kim, J. Mi. Lee, J. G. Yook, and C. S. Pil, "Novel explicit 2-D FDTD scheme with isotropic dispersion and enhanced stability," IEEE Trans. on Ant. and Prop., Vol. 54, No. 11, Nov. 2006.

21. Fu, W. and E. L. Tan, "Stability and dispersion analysis for ADI-FDTD method in lossy media," IEEE Trans. on Ant. and Prop., Vol. 55, No. 4, Apr. 2007.
doi:10.1109/TAP.2007.893378

22. Pozar, D. M., Microwave Engineering, 3rd Ed., Wiley, New York, 2005.

Dr. Hichem Naamen

Dr. Ajmi Ben Hadj Hamouda

Dr. Taoufik Aguili