×

zbMATH — the first resource for mathematics

Reproducing kernel hierarchical partition of unity. I: Formulation and theory. II: Applications. (English) Zbl 0945.74079
From the summary: We develop the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation, together with its discretized counterpart. To form such hierarchical partition, we introduce a class of basic wavelet functions. Then, based upon the built-in consistency conditions, we derive differential consistency conditions for the hierarchical kernel functions. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. The new hierarchical basis is an intrinsic pseudo-spectral basis, which remains as a partition of unity in a local region, because the discrete wavelet kernels form a ‘partition of nullity’.
In part II, the hierarchical reproducing kernels are used as a multiple scale basis to compute numerical solutions of Helmholtz equation, to solve a model equation of wave propagation, and to simulate shear band formation in elasto-viscoplastic materials.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74J10 Bulk waves in solid mechanics
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Belytschko, Computer Methods in Applied Mechanics and Engineering 139 pp 3– (1996)
[2] Liu, Computer Methods in Applied Mechanics and Engineering 139 pp 91– (1996)
[3] Melenk, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996)
[4] Duarte, Computer Methods in Applied Mechanics and Engineering 139 pp 237– (1996)
[5] Duarte, Numerical Methods for Partial Differential Equations 12 pp 673– (1996)
[6] Hierarchical partition of unity methods. In 4th U.S. National Congress on Computational Mechanics. San Francisco, CA, 1997.
[7] Zienkiewicz, Computers and Structures pp 53– (1982)
[8] Hierarchical bases and the finite element method. Acta Numerica; Cambridge University Press: Cambridge, 1996;1-43.
[9] Lancaster, Mathematics of Computation 37 pp 141– (1980)
[10] Liu, Computer Methods in Applied Mechanics and Engineering 143 pp 422– (1997)
[11] Wave Packet and their Bifurcations in Geophysical Fluid Dynamics; Springer: New York, 1990.
[12] Othogonormal wavelet packet bases. Preprint, Yale University, New Haven, CT, 1990.
[13] Chui, SIAM Journal on Mathematical Analysis 24 pp 712– (1993)
[14] Duval-Destin, SIAM Journal on Mathematics Analysis 24 pp 739– (1993) · Zbl 0878.42026
[15] An Introduction to Wavelets; Academic Press: Boston, 1992. · Zbl 0925.42016
[16] Ten Lectures on Wavelets. Philadelphia, PA: Society for industrial and Applied Mathematics, 1992.
[17] Finite Element Analysis; Wiley: New York, 1991.
[18] Grossmann, SIAM Journal of Mathematical Analysis 15 pp 723– (1984)
[19] Wavelets and Operators; Cambridge University Press: Cambridge, 1992. The French edition was published in 1990 under the name Ondelettes et Operateurs.
[20] Wavelets: Algorithms & Applications; SIAM: Philadelphia, 1993.
[21] A Friendly Guide to Wavelets; Birkhäuser: Boston, 1994. · Zbl 0839.42011
[22] Daubechies, SIAM Journal on Mathematical Analysis 24 pp 499– (1993)
[23] Beylkin, Communications on Pure and Applied Mathematics XLIV pp 141– (1991)
[24] Computational Methods for Integral Equations; Cambridge University Press: New York, 1985. · Zbl 0592.65093
[25] Liu, International Journal of Numerical Methods in Fluids 20 pp 1081– (1995)
[26] Battle, Communications in Mathematical Physics 110 pp 601– (1987)
[27] Lemarié, Journal Mathematiques Pures et Appliques 67 pp 227– (1988)
[28] Liu, International Journal of Numerical Methods in Fluids 21 pp 901– (1995)
[29] Li, International Journal for Numerical Methods in Engineering (1998)
[30] Farwig, Journal of Computational and Applied Mathematics 16 pp 79– (1986)
[31] Moving least square reproducing kernel methods. Ph.D. Thesis, McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, May 1997. · Zbl 0883.65088
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.