A Divergence-free Meshless Method for Transient Vector Wave Equations
Keywords:
Divergence free, meshless, RPIM, vector radial basis function (RBF)Abstract
With the implementation of the vector radial basis function (RBF), which is theoretically divergence free, we propose a meshless method for solving the transient vector wave equation. Unlike the conventional radial point interpolation meshless (RPIM) method based on the scalar RBF that solves electric field and magnetic field components separately with scalar wave equations, the proposed method solves the vector wave equation directly. Therefore, the long-existing technical challenge of the source in the traditional RPIM method is resolved due to the direct solution of the vector wave equation. In addition, the stability condition of the proposed method is presented. At last, several numerical experiments are conducted to validate the accuracy of the proposed solver.
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References
V. Cingoski, N. Miyamoto, and H. Yamashita, “Element-free Galerkin method for electromagnetic field computations,” IEEE Trans. Magnetics, vol. 34, pp. 3236-3239, 1998.
S. A. Viana and R. C. Mesquita, “Moving least square reproducing kernel method for electromagnetic field computation,” IEEE Trans. Magnetics, vol. 35, pp. 1372-1375, 1999.
G. Ala, E. Francomano, A. Tortorici, E. Toscano, and F. Viola, “Smoothed particle electromagnetics: A mesh-free solver for transients,” J. Comput. and Applied Math., vol. 191, pp. 194-205, 2006.
T. Kaufmann, C. Fumeaux, and R. Vahldieck, “The meshless radial point interpolation method for time-domain electromagnetics,” IEEE MTT-S International Microwave Symposium Digest, pp. 61-64, 2008.
Y. Yu and Z. Chen, “A 3-D radial point interpolation method for meshless time-domain modeling,” IEEE Trans. Microw. Theory Tech., vol. 57, pp. 2015-2020, 2009.
Y. Yu and Z. Chen, “Towards the development of an unconditionally stable time-domain meshless method,” IEEE Trans. Microw. Theory Tech., vol. 58, pp. 578-586, 2018.
B. Jiang, J. Wu, and L. A. Povinelli, “The origin of spurious solutions in computational electromagnetics,” J. Comput. Physics, vol. 125, pp. 104-123, 1996.
T. Kaufmann, C. Engstrom, C. Fumeaux, and R. Vahldieck, “Eigenvalue analysis and longtime stability of resonant structures for the meshless radial point interpolation method in time domain,” IEEE Trans. Microw. Theory and Tech., vol. 58 , pp. 3399-3408, 2010.
S. Lowitzsch, “Matrix-valued radial basis functions: Stability estimates and applications,” Advances Comput. Math., vol. 23, pp. 299-315, 2005.
S. Lowitzsch, “A density theorem for matrix-valued radial basis functions,” Numerical Algorithms, vol. 39, pp. 253-256, 2005.
S. Lowitzsch, “Approximation and Interpolation Employing Divergence-free Radial Basis Functions with Applications,” Ph.D. Dissertation, Texas A&M University, 2002.
H. Wendland, “Divergence-free kernel methods for approximating the Stokes problem,” Society for Industrial and Applied Mathematics Journal on Numerical Analysis, vol. 47, pp. 3158-3179, 2009.
C. P. McNally, “Divergence-free interpolation of vector fields from point values–exact ∇·B=0 in numerical simulations,” Monthly Notices of the Royal Astronomical Society: Letters, vol. 413, pp. L76-L80, 2011.
S. Yang, Z. Chen, Y. Yu, and S. Ponomarenko, “A divergence-free meshless method based on the vector basis function for transient electromagnetic analysis,” IEEE Trans. Microw. Theory Tech., vol. 62, pp. 1409-1416, 2014.
J. Zhang, J. Lu, Y. Liu, and S. Yang, “A vector meshless method for transient vector wave equations,” 2018 International Appl. Comp. Electro. Society (ACES) Symp., Aug. 2018.
D. Jiao and J. Jin, “A general approach for the stability analysis of the time-domain finite-element method for electromagnetic simulations,” IEEE Trans. Antena. Propag., vol. 50, pp. 1624-1632, 2002.
D. Smithe, J. Cary, and J. Carlsson, “Divergence preservation in the ADI algorithms for electromagnetics,” J. Comput. Phys., vol. 228, pp. 7289-7299, 2009.
O. Vlasiuk, T. Michaels, N. Flyer, and B. Fornberg, “Fast high-dimensional node generation with variable density,” Computers & Mathematics with Applications, vol. 76, no. 7, pp. 1739-1757, 2018.
