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Multiple scale finite element methods. (English) Zbl 0758.73049
Summary: New temporal and spatial discretization methods are developed for multiple scale structural dynamic problems. The concept of fast and slow time scales is introduced for the temporal discretization. The required time step is shown to be dependent only on the slow time scale, and therefore, large time steps can be used for high frequency problems. To satisfy the spatial counterpart of the requirement on time step constraint, finite-spectral elements and finite wave elements are developed. Finite-spectral element methods combine the usual finite elements with the fast convergent spectral functions to obtain a faster convergence rate; whereas, finite wave elements are developed in parallel to the temporal shifting technique. Therefore, the spatial resolution is increased substantially. These methods are especially applicable to structural acoustics and linear space structures. Numerical examples are presented to illustrate the effectiveness of these methods.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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[1] and , Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1976.
[2] Belytschko, Int. j. numer. methods eng. 12 pp 1575– (1979)
[3] and , ’A review of recent developments in time integration’, State-of-the-Art Surveys on Computational Mechanics, ASME, New York, 1989, pp. 185-199.
[4] Belytschko, Comp. Methods Appl. Mech. Eng.
[5] and , Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. · Zbl 0417.34001
[6] The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, N.J., 1974. · Zbl 0375.65052
[7] Fix, SIAM Rev. 18 pp 460– (1976)
[8] and , Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, New York, 1977.
[9] The Finite Element Method, Prentice-Hall, Englewood Cliffs, N.J., 1987.
[10] Hughes, J. Appl. Mech. 45 pp 371– (1978) · Zbl 0392.73076
[11] Hughes, J. Appl. Mech. 45 pp 375– (1978) · Zbl 0392.73077
[12] Igusa, J. Eng. Mech. ASCE 111 pp 20– (1985)
[13] Liu, Int. j. numer. methods eng. 19 pp 125– (1983)
[14] Liu, Comp. Struct. 17 pp 371– (1983)
[15] and , Innovative Methods for Nonlinear Problems, Pineridge Press, Swansea, U.K., 1984.
[16] Liu, Comp. Struct. 19 pp 521– (1984)
[17] Liu, Int. j. numer. methods eng. 20 pp 1581– (1984)
[18] Perturbation Methods, Wiley Interscience, New York, 1969.
[19] Patera, J. Comp. Phys. 54 pp 468– (1984)
[20] Ramirez, Eng. Comp. 5 pp 205– (1989)
[21] Sackman, Eng. Struct. 1 pp 179– (1979)
[22] ’Medium frequency linear vibrations of anisotropic elastic structures’, Rech. Aerosp., 1983.
[23] and , Private communication, ONERA, France, 1987.
[24] and , An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, N.J., 1973.
[25] Zhang, J. Eng. Mech. ASCE 115 pp 1515– (1989)
[26] Zhang, Comp. Methods Appl. Mech. Eng.
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