A spline wavelets element method for frame structures vibration
A spline wavelets element method that combines the versatility of the finite element method with the accuracy of spline functions approximation and the multiresolution strategy of wavelets is proposed for frame structures vibration analysis. Instead of exploring orthogonal wavelets for specific differential operators, the spline wavelets are applied directly in finite element implementation for general differential operators. Although lacking orthogonality, the “two-scale relations” of spline functions and its corresponding wavelets from multiresolution analysis are employed to facilitate the elemental matrices manipulation by constructing two transform matrices under the constraint of finite domain of elements. In the actual formulation, the segmental approach for spline functions is provided to simplify the computation, much as conventional finite element procedure does. The assembled system matrices at any resolution level are reusable for the furthur finer resolution improvement. The local approximation and hiararchy merits make the approach competitive especially for higher mode vibration analysis. Some examples are studied as verification and demonstration of the approach.
KeywordsVibration Analysis Spline Function Multiresolution Analysis Elemental Matrice Finite Domain
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- Briggs, W. L.; Henson, Van Emden 1993: Wavelets and multigrid. SIAM J. Sci. Comp. 14: 506–510Google Scholar
- Chui, C. K. 1992: An introduction to wavelets. New York: Academic PressGoogle Scholar
- Deubechies, I. 1989: Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41: 909–996Google Scholar
- Daubechies, I. 1992: Ten lectures on wavelets. Philadelphia: CBMS-NSF Regional Conference Series in Applied Mathematics #61, SIAMGoogle Scholar
- Glowinski, R.; Lawton, W. M.; Ravachol, M.; Tenenbaum, E. 1989: Wavelets solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. Ma.: Preprint, AWARE Inc.Google Scholar
- Jara-Almonte, J.; Mitchell, L. D. 1991: A hybrid eigenproblem formulation using the finite element method; Part II examples. Int. J. of Anal. and Exp. Modal Anal. 6: 117–130Google Scholar
- Lorentz, R. A.; Madych, W. R. 1991: Spline wavelets for ordinary differential equations. Gesellschaft fuer Mathematik und Datenverarbeitung MBH: (PB92-109727/HDM) Report No. GMD-562Google Scholar
- Magrab, E. B. 1979: Vibration of elastic structural members. The Netherlands: Sijthoff and NoordhoffGoogle Scholar
- Meyer, Y. 1991: Wavelets and operators. Translated by D. H. Salinger. Cambridge University PressGoogle Scholar
- Strang, G. 1989: Wavelets and dilation equations: abrief introduction. SIAM Review 31: 614–627Google Scholar
- Szabò, B.; Babuška, I. 1991: Finite element analysis. New York: John Wiley and SonsGoogle Scholar
- Xu, J. C.; Shann, W. C. 1991: Galerkin-wavelets methods for two-point boundary value problems. Report No. AM 85, University of Penn. StateGoogle Scholar