The Journal of the Acoustical Society of Korea (한국음향학회지)

The Acoustical Society of Korea (한국음향학회)
권호 표지
DOI QR코드

DOI QR Code

On the Solution Method for the Non-uniqueness Problem in Using the Time-domain Acoustic Boundary Element Method

시간 영역 음향 경계요소법에서의 비유일성 문제 해결을 위한 방법에 관하여

  • Received : 2011.09.21
  • Accepted : 2011.11.15
  • Published : 2012.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

The time-domain solution from the Kirchhoff integral equation for an exterior problem is not unique at certain eigen-frequencies associated with the fictitious internal modes as happening in frequency-domain analysis. One of the solution methods is the CHIEF (Combined Helmholtz Integral Equation Formulation) approach, which is based on employing additional zero-pressure constraints at some interior points inside the body. Although this method has been widely used in frequency-domain boundary element method due to its simplicity, it was not used in time-domain analysis. In this work, the CHIEF approach is formulated appropriately for time-domain acoustic boundary element method by constraining the unknown surface pressure distribution at the current time, which was obtained by setting the pressure at the interior point to be zero considering the shortest retarded time between boundary nodes and interior point. Sound radiation of a pulsating sphere was used as a test example. By applying the CHIEF method, the low-order fictitious modes could be damped down satisfactorily, thus solving the non-uniqueness problem. However, it was observed that the instability due to high-order fictitious modes, which were beyond the effective frequency, was increased.

Kirchhoff 적분식을 이용하여 외부 음향 문제의 시간 영역 응답을 계산하는 경우, 주파수영역 해석과 마찬가지로 가상적인 내부 음향 모드에 기인한 비유일성 문제가 발생한다. 이를 해결하는 방법들 중의 하나로서 CHIEF(Combined Helmholtz Integral Equation Formulation) 방법이 쓰이는데, 이는 몇몇 내부 수음점의 응답을 0으로 추가하여 구속하는 조건을 부가하는 기법이다. 이 기법은 주파수 영역 경계요소법에서는 간편한 수식 때문에 많이 사용되고 있지만, 시간 영역에서는 사용된 예가 없다. 본 연구에서는 대상체 내부의 가상 수음점과 경계 표면의 절점들간의 최소 거리에 대한 지연시간을 고려하여, 계산하고자 하는 미지수인 현재 시간의 경계 표면 음장을 구속함으로써, 시간 영역 해석에 적합하도록 CHIEF 방법을 수식화하였다. 예제로서, 반지름 방향으로 진동하는 구의 음향 방사 문제를 다루었다. CHIEF 방법을 적용함에 따라 저차의 내부 음향 모드에 기인한 비유일성 문제를 해결할 수 있었고, 비요동 모드에 의한 수치적 불안정성을 피할 수 있었다. 그러나, 유효주파수 밖에 남은 내부 음향의 고차모드들에 의한 수치적 불안정성은 증가하였다.

Keywords

References

  1. K. M. Mitzner, "Numerical solution for transient scattering from a hard surface-retarded potential techniques," J. Acoust. Soc. Am., vol. 42, no. 2, pp. 391-397, 1967. https://doi.org/10.1121/1.1910590
  2. R. P. Shaw, "Transient acoustic scattering by a free (pressure release) sphere," J. Sound Vib., vol. 20, no. 3, pp. 321-331, 1972. https://doi.org/10.1016/0022-460X(72)90613-X
  3. P. H. L. Groenenboom, "Wave propagation phenomena," Progress in Boundary Element Methods 2, Pentech Press, London, Chap. 2, 1983.
  4. W. J. Mansur and C. A. Brebbia, "Formulation of boundary element method for transient problems governed by the scalar wave equation," Appl. Math. Model., vol. 6, no. 4, pp. 307-311, 1982. https://doi.org/10.1016/S0307-904X(82)80039-5
  5. S. J. Dodson, S. P. Walker, and M. J. Bluck, "Implicitness and stability of time-domain integral equation scattering analyses," Appl. Comput. Electron., vol. 13, no. 3, pp. 291-301, 1997.
  6. P. D. Smith, "Instabilities in time marching methods for scattering: cause and rectification," Electromagnetics, vol. 10, no. 4, pp. 439-451, 1990. https://doi.org/10.1080/02726349008908256
  7. H. Wang, D. J. Henwood, P. J. Harris, and R. Chakrabarti, "Concerning the cause of instability in time stepping boundary element methods applied to the exterior acoustic problem," J. Sound Vib., vol. 305, no. 1-2, pp. 289-297, 2007. https://doi.org/10.1016/j.jsv.2007.03.083
  8. T. W. Wu (ed.), Boundary Element Acoustics, WIT Press, Southampton, Chaps. 2, 4, 2000.
  9. A. J. Burton and G. F. Miller, "The application of integral equation methods to the numerical solution of some exterior boundary-value problems," in Proc. the Royal Society of London series A, vol. 323, pp. 201-210, 1971. https://doi.org/10.1098/rspa.1971.0097
  10. A. A. Ergin, B. Shanker, and E. Michielssen, "Analysis of transient wave scattering from rigid bodies using a Burton-Miller approach," J. Acoust. Soc. Am., vol. 106, no. 5, pp. 2396-2404, 1999. https://doi.org/10.1121/1.428076
  11. D. J. Chappell, P. J. Harris, D. Henwood, and R. Chakrabarti, "A stable boundary element method for modeling transient acoustic radiation," J. Acoust. Soc. Am., vol. 120, no. 1, pp. 76-80, 2006.
  12. J. M. Parot and C. Thirard, "A numerical algorithm to damp instabilities of a retarded potential integral equation," Eng. Anal. Bound. Elem., vol. 35, no. 4, pp. 691-699, 2011. https://doi.org/10.1016/j.enganabound.2010.11.002
  13. H. A. Schenck, "Improved integral formulation for acoustic radiation problems," J. Acoust. Soc. Am., vol. 44, no. 1, pp. 41-58, 1968. https://doi.org/10.1121/1.1911085
  14. A. F. Seybert and T. K. Rengarajan, "The use of CHIEF to obtain uniqueness solutions for acoustic radiation using boundary integral equations," J. Acoust. Soc. Am., vol. 81, no. 5, pp. 1299-1306 (1987). https://doi.org/10.1121/1.394535
  15. J. W. Kim and D. J. Lee, "Optimized compact finite difference scheme with maximum resolution," AIAA J., vol. 34, no. 5, pp. 887-893, 1996. https://doi.org/10.2514/3.13164
  16. H. Lomax, T. H. Pulliam and D. W. Zingg, Fundamentals of Computational Fluid Dynamics, Springer, New York, Chap. 3, 2001.
  17. B. N. Datta, Numerical Linear Algebra and Application, Society for Industrial and Applied Mathematics, Philadelphia, Chap. 8, 2009.
  18. H.-W. Jang and J.-G. Ih, "Treatment of the impedance boundary condition for a stable calculation of the time-domain acoustic BEM," in Proc. 6th Forum Acusticum, pp. 217-222, June 2011.
  19. R. J. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, Philadelphia, Chap. 10, 2007.
  20. D. T. Blackstock, Fundamentals of Physical Acoustics, Wiley, New York, Chap. 10, 2000.
  21. R. J. Allemang, "The modal assurance criterion -Twenty years of use and abuse," Sound Vib., pp. 14-21, Aug. 2003.
  22. J. M. Parot, C. Thirard, and C. Puillet, "Elimination of a non-osillatory instability in a retarded potential integral equation," Eng. Anal. Bound. Elem., vol. 31, no. 2, pp. 133-151, 2007. https://doi.org/10.1016/j.enganabound.2006.09.014