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Eigenvalue Analysis of a Membrane Using the Multiscale Adaptive Wavelet-Galerkin Method

멀티스케일 적응 웨이블렛-갤러킨 기법을 이용한 박막 고유치 문제 해석

  • 이용섭 (서울대학교 대학원 기계항공공학부) ;
  • 김윤영 (서울대학교 기계항공공학부 멀티스케일 창의연구단)
  • Published : 2004.03.01

Abstract

Since the multiscale wavelet-based numerical methods allow effective adaptive analysis, they have become new analysis tools. However, the main applications of these methods have been mainly on elliptic problems, they are rarely used for eigenvalue analysis. The objective of this paper is to develop a new multiscale wavelet-based adaptive Galerkin method for eigenvalue analysis. To this end, we employ the hat interpolation wavelets as the basis functions of the finite-dimensional trial function space and formulate a multiresolution analysis approach using the multiscale wavelet-Galerkin method. It is then shown that this multiresolution formulation makes iterative eigensolvers very efficient. The intrinsic difference-checking nature of wavelets is shown to play a critical role in the adaptive analysis. The effectiveness of the present approach will be examined in terms of the total numbers of required nodes and CPU times.

Keywords

Multiscale;Multiresolution;Interpolation Wavelets;Adaptive Scheme;Eigenvalue

References

  1. Sehmi, N. S., 1989, Large Order Stuructural Eigenanalysis Techniques, Ellis Horwood Limited, Chichester, England.
  2. Zienkiewicz, O. C. and Zhu, J. Z., 1992, 'The Superconvergent Patch Recovery and a Posteriori Error Estimates. Part 1:The Recovery Technique,' Int. J. Num. Meth. Eng., Vol. 33, pp. 1331-1364. https://doi.org/10.1002/nme.1620330702
  3. Zienkiewicz, O. C.and Zhu, J. Z., 1992, 'The Superconvergent Patch Recovery and a Posteriori Error Estimates. Part 2:Error Estimates and Adaptivity,' Int. J. Num. Meth. Eng., Vol. 33, pp. 1365-1382. https://doi.org/10.1002/nme.1620330703
  4. Kim,Y.Y. and Yoon, G. H., 2000, 'Multi-Resolution, Multi-Scale Topology Optimization - A New Paradigm,' Int. J. Solids Structures, Vol. 37, pp. 5529-5559. https://doi.org/10.1016/S0020-7683(99)00251-6
  5. Bathe, K. -J., 1996, Finite Element Procedures, Prebtice-Hall, New Jersey, USA.
  6. Kim, Y. Y. and Jang, G. W., 2001, 'Han Interpolation Wavelet-Based Multi-Scale Galerkin Method for Thin-Walled Box Beam Analysis,' Int. J. Num. Meth. Eng., Vol. 53, pp. 1575-1592. https://doi.org/10.1002/nme.352
  7. Christon, M. A., Roach, D. W., 2000, 'The Numerical Performance of Wavelet for PDEs:the Multi-Scale Finite Element,' Computational Mechanics, Vol. 25, pp. 230-244. https://doi.org/10.1007/s004660050472
  8. Yoon Young Kim, Gang-Won Jang and Jae Eun Kim, 2002, 'Multiscale Wavelet-Galerkin Method in General Two-Dimensional Problems,' Trans. of the KSME, A, Vol. 26, No. 5, pp. 939-951.
  9. Bertoluzza, S. and Naldi, G., 1996, 'An Adaptive Collocation Method for the Numerical Solutions of Partial Differential Equations,' Appl. Comput. Harmon. Anal., Vol. 3, pp. 1-9. https://doi.org/10.1006/acha.1996.0001
  10. Cohen,A. and Masson, R., 1999, 'Wavelet Methods for Second-Order Elliptic Problem, Preconditioning, and Adaptivitys,' SIAM Journal on Scientific Computing, Vol. 21, pp. 1006-1026. https://doi.org/10.1137/S1064827597330613
  11. Dahmen, W., 2001, 'Wavelet Methods for PDEs-some Recent Developments,' J. Comput. Appl. Math, Vol. 128, pp. 133-185. https://doi.org/10.1016/S0377-0427(00)00511-2