A method for calculating the Casimir force between large, complex 3D objects is presented. Difficulties have previously arisen in broadband multiscale calculation using CEM methods. To expand the range of problems that can be calculated, we use an integral equation, domain decomposition method (DDM) and argument principle to derive the Casimir force formula. The broadband integral equation DDM, which is the augmented equivalence principle algorithm (A-EPA), allows for an efficient broadband solution of large, complex objects. A-EPA subdivides a complex problem into separate smaller subproblems that are later recombined into a reduced matrix. This yields a reduced number of unknowns for complex structures making them feasible with modest computer resources. We demonstrate the advantages of the A-EPA by simulating large, finite, 3D, unaligned corrugated plates, which have previously only been modeled approximately as infinite plates using 2D techniques.
Phillip R. Atkins,
Weng Cho Chew,
Lin E. Sun,
Li Jun Jiang,
"Casimir Force for Complex Objects Using Domain Decomposition Techniques," Progress In Electromagnetics Research,
Vol. 149, 275-280, 2014. doi:10.2528/PIER14102112
1. Atkins, P. R., "A study on computational electromagnetics problems with applications to Casimir force calculations,", Ph.D. thesis, University of Illinois at Urbana-Champaign, 2013. doi:10.2528/PIER13082105
2. Atkins, P. R., Q. I. Dai, W. E. I. Sha, and W. C. Chew, "Casimir force for arbitrary objects using the argument principle and boundary element methods," Progress In Electromagnetics Research, Vol. 142, 615-624, 2013. doi:10.1016/S0370-1573(01)00015-1
3. Bordag, M., U. Mohideen, and V. M. Mostepanenko, "New developments in the Casimir effect," Phys. Rep., Vol. 353, 1-205, 2001.
4. Chew, W. C., J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, Berlin, 2001. doi:10.1109/8.237620
5. Chew, W. C. and C. C. Lu, "The use of Huygens equivalence principle for solving the volume integral equation of scattering," IEEE Trans. Antennas Propag., Vol. 41, No. 7, 897-904, 1993. doi:10.1109/8.384194
6. Chew, W. C. and C. C. Lu, "The use of Huygens equivalence principle for solving 3-d volume integral equation of scattering," IEEE Trans. Antennas Propag., Vol. 43, No. 5, 500-507, 1995.
7. Fraysse, V., L. Giraud, S. Gratton, and J. Langou, "A set of GMRES routines for real and complex arithmetics on high performance computers,", Technical report, CERFACS Technical Report TR/PA/03/3, 2003.
8. Lambrecht, A. and V. N. Marachevsky, "New geometries in the Casimir effect: Dielectric gratings," J. Phys. Conf. Ser., Vol. 161, 1-8, 2009. doi:10.1137/S0895479895281484
9. Lehoucq, R. B. and D. C. Sorensen, "Deflation techniques for an implicitly restarted Arnoldi iteration," SIAM. J. Matrix Anal. & Appl., Vol. 17, No. 4, 789-821, 1996.
10. Lehoucq, R. B., D. C. Sorensen, and C. Yang, "ARPACK Users’ Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods," SIAM, 1998.
11. Li, M. K., "Studies on applying the equivalence principle algorithm on multiscale problems," Ph.D. thesis, University of Illinois at Urbana-Champaign , 2007. doi:10.1109/TAP.2006.888453
12. Li, M. K. and W. C. Chew, "Wave-field interaction with complex structures using equivalence principle algorithm," IEEE Trans. Antennas Propag., Vol. 55, No. 1, 130-138, 2007. doi:10.1109/TAP.2008.926785
13. Li, M. K. and W. C. Chew, "Multiscale simulation of complex structures using equivalence principle algorithm with high-order field point sampling scheme," IEEE Trans. Antennas Propag., Vol. 56, No. 8, 2389-2397, 2008. doi:10.1002/mop.21777
14. Li, M. K., W. C. Chew, and Li J. Jiang, "A domain decomposition scheme based on equivalence theorem," Microwave and Opt. Tech. Lett., Vol. 48, No. 9, 1853-1857, 2006.
15. Ma, Z. H., "Fast methods for low frequency and static EM problems,", Ph.D. thesis, The University of Hong Kong, 2013. doi:10.1103/PhysRevD.80.085021
16. Rahi, S. J., T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, "Scattering theory approach to electrodynamic Casimir forces," Phys. Rev. D, Vol. 80, 085021, 2009. doi:10.1109/TAP.1982.1142818
17. Rao, S. M., D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propag., Vol. 30, 409-418, 1982.
18. Homer Reid, M. T., A. W. Rodriguez, J. White, and S. G. Johnson, "Efficient computation of Casimir interactions between arbitrary 3d objects," Phys. Rev. Lett., Vol. 103, 2009.
19. Homer Reid, M. T., J. White, and S. G. Johnson, "Computation of Casimir interactions between arbitrary three-dimensional objects with arbitrary material properties," Phys. Rev. A, Vol. 84, 2011. doi:10.1137/0907058
20. Saad, Y. and M. H. Schultz, "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM J. Sci. Stat. Comput., Vol. 7, No. 3, 856-869, 1986.
21. Sun, L., "An enhanced volume integral equation method and augmented equivalence principle algorithm for low frequency problems,", Ph.D. thesis, University of Illinois at Urbana-Champaign, 2010.