volume | PIER Journals

Abstract

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.

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 Google Scholar

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 Google Scholar

3. Bordag, M., U. Mohideen, and V. M. Mostepanenko, "New developments in the Casimir effect," Phys. Rep., Vol. 353, 1-205, 2001. Google Scholar

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 Google Scholar

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. Google Scholar

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. Google Scholar

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 Google Scholar

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. Google Scholar

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. Google Scholar

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 Google Scholar

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 Google Scholar

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 Google Scholar

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. Google Scholar

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 Google Scholar

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 Google Scholar

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. Google Scholar

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. Google Scholar

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 Google Scholar

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. Google Scholar

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. Google Scholar

Dr. Phillip R. Atkins

Professor Weng Cho Chew

Dr. Maokun Li

Dr. Lin E. Sun

Dr. Zu-Hui Ma

Dr. Li Jun Jiang