A new algorithm for solving some mechanical problems. (English) Zbl 1013.74081

Summary: This paper uses discrete singular convolution algorithm for solving certain mechanical problems. Benchmark mechanical systems, including plate vibrations and incompressible flows, are employed to illustrate the robustness and to test the accuracy of the present algorithm.


74S30 Other numerical methods in solid mechanics (MSC2010)
74M25 Micromechanics of solids
74H45 Vibrations in dynamical problems in solid mechanics
74K20 Plates
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI


[1] Ablowitz, M.J.; Herbst, B.M.; Schober, C., On the numerical solution of the sine – gordon equation, J. comput. phys., 126, 299-314, (1996) · Zbl 0866.65064
[2] Lanczos, C., Trigonometric interpolation of empirical and analytical functions, J. math. phys., 17, 123-199, (1938) · Zbl 0020.01301
[3] Cooley, J.W.; Tukey, J.W., An algorithm for the machine calculation of complex Fourier series, Math. comput., 19, 297-301, (1965) · Zbl 0127.09002
[4] Finlayson, B.A.; Scriven, L.E., The method of weighted residuals – a review, Appl. mech. rev., 19, 735-748, (1966)
[5] Orszag, S.A., Comparison of pseudospectral and spectral approximations, Stud. appl. math., 51, 253-259, (1972) · Zbl 0282.65083
[6] Fornberg, B., On a Fourier method for the integration of hyperbolic equations, SIAM J. numer. anal., 12, 509-528, (1975) · Zbl 0349.35003
[7] Forsythe, G.E.; Wasow, W.R., Finite-difference methods for partial differential equations, (1960), Wiley New York · Zbl 0099.11103
[8] Isaacson, E.; Keller, H.B., Analysis of numerical methods, (1966), Wiley New York · Zbl 0168.13101
[9] Zienkiewicz, O.C., The finite element method in engineering science, (1971), McGraw-Hill London · Zbl 0237.73071
[10] Desai, C.S.; Abel, J.F., Introduction to the finite element methods, (1972), Van Nostrand Reinhold New York
[11] Oden, J.T., The finite elements of non-linear continua, (1972), McGraw-Hill New York · Zbl 0235.73038
[12] Nath, B., Fundamentals of finite elements for engineers, (1974), Athlone Press London
[13] Fenner, R.T., Finite element methods for engineers, (1975), Imperial College Press London
[14] Cheung, Y.K., Finite strip methods in structural analysis, (1976), Pergamon Press Oxford · Zbl 0375.73073
[15] Rao, S.S., The finite element method in engineering, (1982), Pergamon Press New York · Zbl 0472.73083
[16] Reddy, J.N., Energy and variational methods in applied mechanics, (1984), Wiley New York · Zbl 0635.73017
[17] Wei, G.W., Discrete singular convolution for the fokker – planck equation, J. chem. phys., 110, 893-8942, (1999)
[18] S. Guan, C.-H. Lai, G.W. Wei, Boundary controlled nanoscale morphology in a circular domain, Physica D, submitted, 1999
[19] Wei, G.W., Discrete singular convolution method for the sine – gordon equation, Physica D, 137, 247-259, (2000) · Zbl 0944.35087
[20] Wei, G.W., Solving quantum eigenvalue problems by discrete singular convolution, J. phys. B, 33, 343-359, (2000)
[21] Schwartz, L., Théore des distributions, (1951), Hermann Paris
[22] L.W. Qian, G.W. Wei, A note on regularized Shannon’s sampling formulae, J. Approx. Theory, submitted, 1999
[23] Wei, G.W.; Zhang, D.S.; Kouri, D.J.; Hoffman, D.K., Lagrange distributed approximating functionals, Phys. rev. lett., 79, 775-779, (1997)
[24] Wei, G.W., Quasi-wavelets and quasi interpolating wavelets, Chem. phys. lett., 296, 215-222, (1998)
[25] Narita, Y., Application of a series-type method to vibration of orthotropic rectangular plates with mixed boundary conditions, J. sound vibr., 77, 345-355, (1981) · Zbl 0477.73066
[26] Bartlett, C.C., The vibration and buckling of a circular plate clamped on part of its boundary and simply supported on the remainder, J. mech. appl. math., 16, 431-440, (1963) · Zbl 0121.19504
[27] Keer, L.M.; Stahl, B., Eigenvalue problems of rectangular plates with mixed boundary conditions, J. appl. mech., 39, 513-520, (1972) · Zbl 0235.73025
[28] Hirano, Y.; Okazaki, K., Vibration of a circular plate having partly clamped or partly simply supported edges, Bull. jpn soc. mech. engrg., 19, 610-618, (1976)
[29] Fan, S.C.; Chueng, Y.K., Flexural free vibrations of rectangular plates with complex supported conditions, J. sound vibr., 93, 81-94, (1984)
[30] Eastep, F.E.; Hemming, F.G., Natural frequency of circular plates with partially free partially clamped edges, J. sound vibr., 84, 152-159, (1982)
[31] Mizusawa, T.; Kaijita, T., Vibration and buckling of rectangular plates with non-uniform elastic constraints in rotation, Int. J. solid struct., 23, 45-55, (1986) · Zbl 0601.73067
[32] Liew, K.M.; Hung, K.C.; Lam, K.Y., On the use of substructure method for vibration analysis of rectangular plates with discontinuous boundary conditions, J. sound vibr., 163, 451-462, (1993) · Zbl 0925.73374
[33] Liew, K.M.; Hung, K.C.; Lim, M.K., On the use of domain decomposition method for vibration of symmetric laminates having discontinuities at the same edge, J. sound vibr., 178, 243-264, (1994)
[34] Chia, C.Y., Non-linear vibration anisotropic rectangular plates with non-uniform edge constraints, J. sound vibr., 101, 539-550, (1985)
[35] Leissa, A.W.; Laura, P.A.A.; Gutierrez, R.H., Vibrations of rectangular plate with non-uniform elastic edge supports, J. appl. mech., 47, 281-292, (1979) · Zbl 0416.73054
[36] Liew, K.M.; Wang, C.M., Vibration analysis of plates by the pb2 rayleigh – ritz method: mixed boundary conditions reentrant corners and internal curved supports, Mech. struct. mech., 20, 281-292, (1992)
[37] A. Leissa, Vibration of Plates, Published for the Acoustical Society of America through the American Institute of Physics, 1993 · Zbl 0268.73033
[38] G.N. Watson, Theory of Bessel functions, Cambridge, 1966 · Zbl 0174.36202
[39] Bell, J.B.; Colella, P.; Glaz, H.M., A second-order projection method for the incompressible navier – stokes equations, J. comput. phys., 85, 257-283, (1989) · Zbl 0681.76030
[40] E, W.; Shu, C.W., A numerical resolution study of high-order essentially non-oscillatory schemes applied to incompressible flow, J. comput. phys., 110, 39-46, (1994) · Zbl 0790.76055
[41] Lou, Z.J.; Ferraro, R., A parallel incompressible flow solver package with a parallel multigrid elliptic kernel, J. comput. phys., 125, 225-243, (1996) · Zbl 0848.76049
[42] Yang, H.H.; Seymour, B.R.; Shizgal, B.D., A Chebyshev pseudospectral multi-domain method for steady flow past a cylinder, Computers fluids, 23, 829-851, (1994) · Zbl 0817.76064
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.