×

The stability of integral equation time-domain scattering computations for three-dimensional scattering; similarities and differences between electrodynamic and elastodynamic computations. (English) Zbl 1012.78009

Authors’ abstract: Time-domain integral equation analyses are prone to instabilities, in a range of applications areas including acoustics, electrodynamics and elastodynamics, and a variety of retrospective averaging schemes have been proposed to improve matters. In this paper, we investigate stability behaviour, in parallel, in electrodynamic and elastodynamic cases. It is observed empirically that the tendency to instability is increased as the treatment becomes more nearly explicit. The timestepping procedure is recast in a recursive matrix formulation, and it is shown that it is the eigenvalues of this large matrix which determine the long-term stability behaviour. This treatment is then extended to cover general averaging schemes, allowing the likely effectiveness of such schemes to be assessed. Simple modelling rules, which for all practical purposes will ensure stability, are presented.

MSC:

78A45 Diffraction, scattering
78M15 Boundary element methods applied to problems in optics and electromagnetic theory
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Rynne, IMA Journal of Applied Mathematics 35 pp 297– (1985)
[2] Rynne, Electromagnetics 6 pp 129– (1986)
[3] Smith, Electromagnetics 10 pp 439– (1990)
[4] Rao, IEEE Transactions on Antennas and Propagation 39 pp 56– (1991)
[5] Vechinski, IEEE Transactions on Antennas and Propagation 40 pp 661– (1992)
[6] Vechinski, Journal of the Optical Society of America 11 pp 1458– (1994)
[7] Dodson, Applied Computational Electromagnetics Society Journal 13 pp 291– (1997)
[8] Integral equation time domain computation of large scattering problems. Ph.D. Thesis, Mechanical Engineering Department, Imperial College, University of London, 1998.
[9] A numerically stable method of moments time domain model. Proceedings of the 12th Annual Review of Progress in Applied Computational Electromagnetics 1996; 1238-1247.
[10] A plane wave algorithm for the fast analysis of transient electromagnetic scattering phenomena. Presented at 14th Annual Review of Progress in Applied Computational Electromagnetics, Monterey, CA, 1998.
[11] Chew, IEEE Transactions on Antennas and Propagation 45 pp 533– (1997)
[12] Transient analysis of acoustic scattering using marching on in time with plane wave time domain algorithm. Presented at 14th Annual Review of Progress in Applied Computational Electromagnetics, Monterey, CA, 1988.
[13] Dodson, Parallel Computing 25 pp 925– (1999)
[14] Dodson, IEEE Antennas and Propagation Magazine 40 pp 12– (1998)
[15] Dodson, Applied Computational Electromagnetics Society Journal 13 pp 131– (1998)
[16] Andrews, Bulletin of the Seismological Society of America 84 pp 1184– (1994)
[17] Antes, Finite element Analysis and Design 1 pp 313– (1985)
[18] Cole, Bulletin of the Seismological Society of America 68 pp 1331– (1978)
[19] Mack, PAGEOPH 139 pp 763– (1992)
[20] Siebrits, International Journal for Numerical Methods in Engineering 37 pp 3229– (1994)
[21] Boundary Elements in Dynamics. Computational Mechanics Publications: Southampton, 1993.
[22] Ahmad, International Journal for Numerical Methods in Engineering 26 pp 891– (1987)
[23] Boundary element method implementation for three dimensional transient elastodynamics. In Developments in Boundary Element Methods, (eds.). Elsevier: London, 1986; 29-65. · Zbl 0586.73136
[24] Banerjee, Earthquake Engineering and Structural Dynamics 14 pp 933– (1986)
[25] Wang, International Journal for Numerical Methods in Engineering 33 pp 1737– (1992)
[26] The stability properties of time domain elastodynamic boundary element methods. In Boundary Element Methods, vol. 17, Computational Mechanics Publications: Southampton, 1995. · Zbl 0840.73074
[27] Siebrits, International Journal for Blasting and Fragmentation 1 pp 305– (1997)
[28] Peirce, International Journal for Numerical Methods in Engineering 40 pp 319– (1997)
[29] Peirce, Numerical Methods for Partial Differential Equations 12 pp 585– (1996)
[30] Birgisson, International Journal for Numerical Methods in Engineering 46 pp 871– (1999)
[31] Bluck, International Journal for Numerical Methods in Engineering 39 pp 1419– (1996)
[32] Bluck, IEEE Transactions on Antennas and Propogation 45 pp 894– (1997)
[33] Elastodynamics: Volume II Linear Theory. Academic Press: New York, 1975.
[34] Hirose, Engineering Analysis with Boundary Elements 8 pp 146– (1991)
[35] Rizos, Computational Mechanics 15 pp 249– (1994)
[36] Alekseyeva, Computational Mechanics 18 pp 147– (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.