volume | PIER Journals

Abstract

⋅A p-adaptive discontinuous Galerkin time-domain method is developed to obtain high-order solutions to electromagnetic scattering problems. A novel feature of the proposed method is the use of divergence error to drive the p-adaptive method. The nature of divergence error is explored, and that it is a direct consequence of the act of discretization is established. Its relation with relative truncation error is formed which enables the use of divergence error as an inexpensive proxy to truncation error. Divergence error is used as an indicator to dynamically identify and assign spatial operators of varying accuracy to substantial regions in the computational domain. This results in a reduced computational cost compared to a comparable discontinuous Galerkin time-domain solution using uniform degree piecewise polynomial bases throughout. Numerical results are presented to show performance of the proposed divergence error based p-adaptive solutions. It is shown that an accuracy similar to that of uniformly higher order solutions is obtained in terms of the scattering width, using fewer degrees of freedom.

1. Yee, K., "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Transactions on Antennas and Propagation, Vol. 14, No. 3, 302-307, 1966.

2. Hagness, S. C., A. Taflove, and S. D. Gedney, "Finite-difference time-domain methods," Handbook of Numerical Analysis, Vol. 13, No. 4, 199-315, 2005.

3. Shankar, V., A. H. Mohammadian, and W. F. Hall, "A time-domain, finite-volume treatment for the Maxwell equations," Electromagnetics, Vol. 10, No. 1-2, 127-145, 1990.

4. Hesthaven, J. S. and T. Warburton, Nodal Discontinuous Galerkin Methods, Texts in Applied Mathematics, Vol. 54, Springer, New York, NY, 2008.

5. Assous, F., et al., "On a finite-element method for solving the three-dimensional Maxwell equations," Journal of Computational Physics, Vol. 109, No. 2, 222-237, 1993.

6. Munz, C.-D., et al., "Divergence correction techniques for Maxwell solvers based on a hyperbolic model," Journal of Computational Physics, Vol. 161, No. 2, 484-511, 2000.

7. Cockburn, B., F. Li, and C.-W. Shu, "Locally divergence-free discontinuous Galerkin methods for the Maxwell equations," Journal of Computational Physics, Vol. 194, No. 2, 588-610, 2004.

8. Li, F. and C.-W. Shu, "Locally divergence-free discontinuous galerkin methods for MHD equations," Journal of Scientific Computing, Vol. 22, 413-442, 2005.

9. Chandrashekar, P., "A global divergence conforming DG method for hyperbolic conservation laws with divergence constraint," Journal of Scientific Computing, Vol. 79, No. 1, 79-102, 2019.

10. Li, F. and L. Xu, "Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations," Journal of Computational Physics, Vol. 231, No. 6, 2655-2675, 2012.

11. Yakovlev, S., L. Xu, and F. Li, "Locally divergence-free central discontinuous Galerkin methods for ideal MHD equations," Journal of Computational Science, Vol. 4, No. 1-2, 80-91, 2013.

12. Cioni, J. P., L. Fezoui, and H. Steve, "A parallel time-domain maxwell solver using upwind schemes and triangular meshes," IMPACT of Computing in Science and Engineering, Vol. 5, No. 3, 215-247, 1993.

13. Kompenhans, M., et al., "Adaptation strategies for high order discontinuous Galerkin methods based on Tau-estimation," Journal of Computational Physics, Vol. 306, 216-236, December 2016.

14. Rueda-Ramírez, A. M., et al., "Truncation error estimation in the p-anisotropic discontinuous Galerkin spectral element method," Journal of Scientific Computing, Vol. 78, No. 1, 433-466, 2019.

15. Kompenhans, M., et al., "Comparisons of p-adaptation strategies based on truncation- and discretisation-errors for high order discontinuous Galerkin methods," Computers & Fluids, Vol. 139, 36-46, 2016.

16. Chatterjee, A., "A multilevel numerical approach with application in time-domain electromagnetics," Communications in Computational Physics, Vol. 17, No. 3, 703-720, 2015.

17. Chatterjee, A. and R. S. Myong, "Efficient implementation of higher-order finite volume time-domain method for electrically large scatterers," Progress In Electromagnetics Research B, Vol. 17, 233-254, 2009.

18. Moxey, D., et al., "Towards p-adaptive spectral/hp element methods for modelling industrial flows," Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016, ed. by M. L. Bittencourt, N. A. Dumont, and J. S. Hesthaven, 63{79, Springer International Publishing, Cham, 2017.

19. Abarbanel, S. and D. Gottlieb, "On the construction and analysis of absorbing layers in CEM," Applied Numerical Mathematics, Vol. 27, No. 4, 331-340, 1998.

20. Tόth, G., "The ∇ ⋅ B = 0 constraint in shock-capturing magnetohydrodynamics codes," Journal of Computational Physics, Vol. 161, No. 2, 605-652, 2000.

21. Brackbill, J. U. and D. C. Barnes, "The effect of Nonzero ∇ ⋅ B on the numerical solution of the magnetohydrodynamic equations," Journal of Computational Physics, Vol. 35, No. 3, 426-430, 1980.

22. Naranjo Alvarez, S., et al., "The virtual element method for resistive magnetohydrodynamics," Computer Methods in Applied Mechanics and Engineering, Vol. 381, 113815, 2021.

23. Chris Fragile, P., et al., "Divergence-free magnetohydrodynamics on conformally moving, adaptive meshes using a vector potential method," Journal of Computational Physic: X,, Vol. 2, 100020, 2019.

24. Wheatley, V., H. Kumar, and P. Huguenot, "On the role of Riemann solvers in discontinuous Galerkin methods for magnetohydrodynamics," Journal of Computational Physics, Vol. 229, No. 3, 660-680, 2010.

25. Ramshaw, J. D., "A method for enforcing the solenoidal condition on magnetic field in numerical calculations," Journal of Computational Physics, Vol. 52, No. 3, 592-596, 1983.

26. Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd Ed., Artech House, 1995.

27. Fraysse, F., J. De Vicente, and E. Valero, "The estimation of truncation error by τ-estimation revisited," Journal of Computational Physics, Vol. 231, No. 9, 3457-3482, 2012.

28. Syrakos, A., et al., "Numerical experiments on the efficiency of local grid refinement based on truncation error estimates," Journal of Computational Physics, Vol. 231, No. 20, 6725-6753, 2012.

29. Zienkiewicz, O. C., R. L. Taylor, and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann, 2013.

30. Hernández, A., et al., "An adaptive meshing automatic scheme based on the strain energy density function," Engineering Computations, Vol. 14, No. 6, 604-629, 1997.

31. Davies, R. W., K. Morgan, and O. Hassan, "A high order hybrid finite element method applied to the solution of electromagnetic wave scattering problems in the time domain," Computational Mechanics, Vol. 44, No. 3, 321-331, 2009.

32. Ledger, P. D., et al., "Arbitrary order edge elements for electromagnetic scattering simulations using hybrid meshes and a PML," International Journal for Numerical Methods in Engineering, Vol. 55, No. 3, 339-358, 2002.

33. Young, J. W. and J. C. Bertrand, "Multiple scattering by two cylinders," Journal of the Acoustical Society of America, Vol. 58, No. 6, 1190-1195, 1975.

Dr. Apurva Tiwari

Prof. Avijit Chatterjee