×

zbMATH — the first resource for mathematics

Generalized multiple scale reproducing kernel particle methods. (English) Zbl 0896.76069
An approach to unify reproducing kernel methods and an extension to include time and spatial shifting are proposed. The groundwork is set by revisiting the Fourier analysis of discrete systems. The multiresolution concept, its significance in devising the reproducing kernel methods and its discrete counterpart, reproducing kernel particle methods, are explained. An edge detection technique based on multiresolution analysis is developed. This wavelet approach, together with particle methods, gives rise to a straightforward \(h\)-adaptivity algorithm. By using this framework, a Hermite reproducing kernel method is also proposed, and its relation to wavelet methods is presented. It is also shown that the new approach generalizes existing kernel methods, and it can easily be degenerated into other widely used methods. Finally, multiple-scale methods based on frequency and wave number shifting techniques are presented, together with a stability analysis for Newmark time-integration schemes for the low frequency equation.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
74S30 Other numerical methods in solid mechanics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Amarantuga, K.; Williams, J.R.; Qian, S.; Weis, J., Wavelet-Galerkin solutions for one-dimension partial differential equations, Inter. J. numer. methods engrg., 37, 2703-2716, (1994) · Zbl 0813.65106
[2] Babuska, I.; Melenk, J.M., The partition of unity finite element method, University of maryland technical note BN-1185, (1995) · Zbl 0949.65117
[3] Belytschko, T.; Gu, L.; Lu, Y.Y., Fracture and crack growth by efg methods, Model. simul. mater. sci., 2, 519-534, (1994)
[4] Belytschko, T.; Krongauz, Y.; Fleming, M.; Organ, D.; Liu, W.K., Smoothing and accelerated computations in the element free Galerkin method, J. comput. appl. math., (1996), to appear · Zbl 0862.73058
[5] Belytschko, T.; Lu, Y.Y.; Gu, L., Element free Galerkin methods, Inter. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077
[6] Belytschko, T.; Lu, Y.Y.; Gu, L., A new implementation of the element free Galerkin method, Comput. methods appl. mech. engrg., 113, 397-414, (1994) · Zbl 0847.73064
[7] Bracewell, R.N., The Fourier transform and its applications, (1986), McGraw-Hill · Zbl 0068.16503
[8] Cartwright, M., Fourier methods for mathematicians scientists and engineers, (1990), Ellis Horwood Limited · Zbl 0717.42001
[9] Chen, J.S.; Pan, C.; Liu, W.K., Reproducing kernel particle methods for large deformation analysis of nonlinear structures, Comput. methods appl. mech. engrg., (1996), submitted
[10] Chen, Y., Multiresolution, multiple field and generalized reproducing kernel particle methods, ()
[11] Chui, Charles K., An introduction to wavelets, (1992), Academic Press · Zbl 0925.42016
[12] Chui, Charles K.; Montefusco, Laura; Puccio, Luigia, Wavelets: theory, algorithms and applications, (1994), Academic Press · Zbl 0816.00025
[13] Daubechies, I., ()
[14] Duarte, C.A.; Oden, J.T., hp clouds—a meshless method to solve boundary-value problems, TICAM report 95-05, (1995)
[15] Gade, S.; Herlusen, H., Windows to fft analysis, Sound vib., 14, (1988)
[16] Gingold, R.A.; Monaghan, J.J., Smooth particle hydrodynamics: theory and application to non-spherical stars, Mon. not. roy. astron. soc., 181, 375-389, (1977) · Zbl 0421.76032
[17] Glowinski, R.; Lawton, M.; Ravachol, M.; Tenenbaum, E., Wavelet solutions of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension, () · Zbl 0799.65109
[18] Harris, F.J., On the use of windows for harmonic analysis with the discrete Fourier transform, Ieee, 66, 1, (1978)
[19] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewood Cliffs, NJ
[20] Keller, J.B.; Givoli, D., Exact non-reflecting boundary conditions, J. comput. phys., 82, 172-192, (1989) · Zbl 0671.65094
[21] Körner, T.W., Exercises for Fourier analysis, (1993), Cambridge University Press · Zbl 0649.42001
[22] Körner, T.W., Fourier analysis, (1993), Cambridge University Press · Zbl 0649.42001
[23] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least-squares methods, Math. comput., 37, 141-158, (1981) · Zbl 0469.41005
[24] Lee, D.T.; Sheu, S.M.; Shen, C.F., Geosheet: A distributed visualization tool for geometric algorithms, (), Preliminary version appeared in
[25] Libersky, L.; Petschek, A.G., Smooth particle hydrodynamics with strength of materials, () · Zbl 0791.76066
[26] Liu, W.K., An introduction to wavelet reproducing kernel particle methods, USACM bull., 8, 1, 3-16, (1995)
[27] Liu, W.K.; Adee, J.; Jun, S.; Liu, W.K.; Adee, J.; Jun, S., Reproducing kernel and wavelet particle methods for elastic and plastic problems, (), 175-190
[28] Liu, W.K.; Chang, C.T.; Chen, Y.; Uras, R.A., Multiresolution reproducing kernel particle methods in acoustic problems, acoustics, vibrations, and rotating machines, Asme de, Vol. 84-2, 881-900, (1995), Part B
[29] Liu, W.K.; Chen, Y., Wavelet and multiple scale reproducing kernel methods, Int. J. numer. methods fluids, 21, 901-931, (1995) · Zbl 0885.76078
[30] Liu, W.K.; Chen, Y.; Chang, C.T.; Belytschko, T., Advances in multiple scale kernel particle methods, Comput. mech., 18-2, 73-111, (1996) · Zbl 0868.73091
[31] Liu, W.K.; Chen, Y.; Jun, S.; Chen, J.S.; Belytschko, T.; Pan, C.; Uras, R.A.; Chang, C.T., Overview and applications of the reproducing kernel particle methods, Archives of comput. methods in engineering; state of the art reviews, 3, 3-80, (1996)
[32] W.K. Liu, W. Hao, Y. Chen and J. Gosz, Multiresolution reproducing kernel particle methods, in preparation. · Zbl 0893.73078
[33] Liu, W.K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. numer. methods engrg., 38, 1655-1679, (1995) · Zbl 0840.73078
[34] Liu, W.K.; Jun, S.; Sihling, D.T.; Chen, Y.; Hao, W., Multiresolution reproducing kernel particle method for computational fluid dynamics, (), and also to appear in · Zbl 0880.76057
[35] Liu, W.K.; Jun, S.; Zhang, Y.F., Reproducing kernel particle methods, Int. J. numer. methods fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[36] W.K. Liu, S. Li and T. Belytschko, Moving least-square kernel galerkin method (i) methodology and convergence, Comput. Methods Appl. Mech. Engrg., in press. · Zbl 0883.65088
[37] Liu, W.K.; Oberste-Brandenburg, C., Reproducing kernel and wavelet particle methods, (), 39-56
[38] Liu, W.K.; Zhang, Y.; Ramirez, M.R., Multiple scale finite element methods, Int. J. numer. methods engrg. fluids, 32, 969-990, (1991) · Zbl 0758.73049
[39] Monaghan, J.J., An introduction to smooth particle hydrodynamics, Comput. phys. communicat., 48, 89-96, (1988) · Zbl 0673.76089
[40] Nayroles, B.; Touzot, G.; Villon, P., Diffuse approximation and diffuse elements, Comput. mech., 10, 307-318, (1992) · Zbl 0764.65068
[41] Oñate, E.; Idelsohn, S.; Zienkiewicz, O.C., Finite point methods in computational mechanics, Int. J. numer. methods engrg., (1995)
[42] A.D. Poularikas and S. Seely, Signals and Systems, 2nd edition (PWS-KENT Publishing Company)
[43] Preparata, F.P.; Shamos, M.I., Computational geometry: an introduction, (1985), Springer-Verlag · Zbl 0759.68037
[44] Qian, S.; Weiss, J., Wavelet and the numerical solution of partial differential equations, J. comput. phys., 106, 155-175, (1993) · Zbl 0771.65072
[45] Shapriro, H.S.; Silverman, R.A., Alias-free sampling of random noise, Ieee, 8, 2, (1960) · Zbl 0121.14204
[46] Shaw, L., Spectral estimates from nonuniform samples, IEEE, trans. audio electroacoust., 1, (1971)
[47] Shodja, H.M.; Mura, T.; Liu, W.K.; Shodja, H.M.; Mura, T.; Liu, W.K., Multiresolution analysis of a micromechanical model, (), 33-54
[48] Strang, G., Wavelets and dilation equations: a brief introduction, SIAM rev., 31, 4, 614-627, (1989) · Zbl 0683.42030
[49] Sulsky, D.; Chen, Z.; Schreyer, H.L., The application of a material-spatial numerical method to penetration, (), 91-102
[50] Williams, J.R.; Amaratunga, K., Introduction to wavelets in engineering, Int. J. numer. methods engrg., 37, 2365-2388, (1994) · Zbl 0812.65144
[51] Yagawa, G.; Yamada, T.; Kawai, T., Some remarks on free mesh method: A kind of meshless finite element method, () · Zbl 0894.73182
[52] Yao, K.; Thomas, J.B., On some stability and interpolatory properties of nonuniform sampling expansion, IEEE trans. circuit theory, 14, 4, (1967)
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.