Abstract
A novel approach to multiresolution analysis based on reproducing kernel particle methods (RKPM) and wavelets is presented. The concepts of reproducing conditions, discrete convolutions, and multiple scale analysis are described. By means of a newly proposed semidiscrete Fourier analysis, RKPM is further elaborated in the frequency domain, and the interpolation estimate and the convergence of Galerkin solutions are given. The elimination of a mesh, combined with the properties of the dilation and translation of a window function, multiresolution analysis, wavelet-based error estimators, and edge detection brings about a new generation of hp adaptive methods. In addition, this class of multiple scale reproducing kernel particle methods is particularly suitable for problems with large deformations, high gradients, and high modal density. The current application areas of RKPM include structural acoustics, structural dynamics, elastic-plastic deformation, computational fluid dynamics and hyperelasticity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Attaway, S. W.; Heinstein, M. W.; Mello, F. J.; Swegle, J. W. 1993: Coupling of Smooth Particle Hydrodynamics With Pronto, preprint
Babuska, I.; Melenk, J. M. 1995: The Partition of Unity Finite Element Method, Univ. of Maryland, Technical Note BN-1185
Beklkin, G. R.; Coifman, I.; Daubechies, S.; Mallat, Y.; Meyer, L.; Raphael, Ruskai, B.; (eds) 1992: Wavelets and Their Applications, Cambridge, MA
Belytschko, T.; Lu, Y. Y.; Gu, L. 1994a: Element Free Galerkin Methods, International Journal for Numerical Methods in Engineering, 37: 229–256
Belytschko, T.; Lu, Y. Y.; Gu, L. 1994b: Fracture and crack growth by element free Galerkin methods, Modelling Simulation Science Engineering, 2: 519–534
Belytschko, T.; Lu, Y. Y.; Gu, L.; Tabbara, M. 1995a: Element-Free Galerkin Methods for Static and Dynamic Fracture, International Journal Solids Structures, Vol. 32, No. 17/18, pp. 2547–2570
Belytschko, T.; Krongauz, Y.; Fleming, M.; Organ, D.; Liu, W. K. 1995b: Smoothing and Accelerated Computations in the Element Free Galerkin Method, submitted to Journal of Computational Applied Mathematics
Chui, C. K. 1992: An Introduction to Wavelets, Academic Press
Chen, Y. J.; Liu, W. K.; Chang, C. T.; Uras, R. A. 1995: Multiresolution Reproducing Kernel Particle Method for Structural Acoustics, submitted to Journal of Computational Acoustics
Daubechies, I.; 1992: Ten Lectures on Wavelets, CBMS/NSF Series in Applied Mathematics, No. 61, SIAM Publication
Duarte, C. A.; Oden, J. T. 1995: Hp Clouds-A Meshless Method to Solve Boundary-Value Problems, TICAM Report 95-05
Gingold, R. A.; Monaghan, J. J. 1977: Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars, Mon. Not. Roy. Astron. Soc., 181: 375–389
Johnson, G. R.; Peterson, E. H.; Stryrk, R. A. 1993: Incorporation of an SPH Option into the EPIC code for a Wide Range of High Velocity Impact Computations, preprint
Libersky, L.; Petschek, A. G. 1990: Smooth Particle Hydrodynamics with Strength of Materials, Proceedings of The next Free-Lagrange Conference, Jackson Lake Lodge, Moran, Wyoming, June 3–7
Liu, W. K.; Zhang, Y.; Ramirez, M. R. 1991: Multiple Scale Finite Element Methods, International Journal for Numerical Methods in Engineering, 32: 969–990
Liu, W. K.; Haeussermann, U. 1992: Multiple Temporal and Spatial Scale Methods, New Methods in Transient Analysis, P. Smolinski, W. K. Liu, G. Hulbert and K. Tamma, eds., PVP 246/AMD 143, ASME, pp. 51–64
Liu, W. K.; Oberste-Brandenburg, C. 1993a: Reproducing Kernel and Wavelet Particle Methods, Aerospace Structures: Nonlinear Dynamics and System Response, Eds. Cusumano, J. P., Pierre, C., and Wu, S. T., AD 33, ASME, pp. 39–56
Liu, W. K.; Adee, J.; Jun, S. 1993b: Reproducing Kernel Particle Methods for Elastic and Plastic Problems, Advanced Computational Methods for Material Modeling, Eds. Benson, D. J., and Asaro, R. A., AMD 180 and PVP 268, ASME, pp. 175–190
Liu, W. K.; Chen, Y. J. 1995: Wavelet and Multiple Scale Reproducing Kernel Methods, International Journal for Numerical Methods in Fluids, vol. 21, pp. 901–932
Liu, W. K. 1995: Feature Article on An Introduction to Wavelet Reproducing Kernel Particle Methods, USACM Bulletin, Vol. 8, No. 1, pp. 3–16
Liu, W. K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T. 1995a: Reproducing Kernel Particle Methods for Structural Dynamics, International Journal for Numerical Methods in Engineering, vol. 38, pp. 1655–1679
Liu, W. K.; Jun, S.; Zhang, Y. F. 1995b: Reproducing Kernel Particle Methods, International Journal of Numerical Methods in Fluids, vol. 20, pp. 1081–1106
Liu, W. K.; Chen, Y.; Jun, S.; Chen, J. S.; Belytschko, T.; Pan, C.; Uras, R. A.; Chang, C. T.; 1996a: Overview and Applications of the Reproducing Kernel Particle Methods, Archives of Computational Methods in Engineering, State of the Art Reviews, vol. 3, 3–80, 1996
Liu, W. K.; Jun, S.; Sihling, D. T.; Chen, Y.; Hao, W. 1995d: Multiresolution Reproducing Kernel Particle Method for Computational Fluid Dynamics, presented at the Third US-Japan Symposium on Finite Element Methods in Large-Scale Computational Fluid Dynamics
Liu, W. K.; Li, S.; Belytschko, T. 1995e: Moving Least Square Kernel Galerkin Method (I) Methodology and Convergence, submitted to Computer Methods in Applied Mechanics and Engineering.
Liu, W. K.; Chen, Y.; Uras, R. A. 1995f: Enrichment of the Finite Element Method With the Reproducing Kernel Particle Method, Current Topics in Computational Mechanics, Eds. J. F. Cory, Jr., and J. L. Gordon, ASME PVP-Vol. 305, pp. 253–258
Liu, W. K.; Hao, W.; Chen, Y.; and Gosz, J. 1996b: “Multiresolution Reproducing Kernel Particle Methods” Submitted to Academic Press Series. “Wavelet Analysis and Applications” edited by W. Pahmen, A. Kurdila and P. Oswald, “Multiresolution Analysis and Wavelets for Numerical Solution of Partial Differential Equations”
Lu, Y. Y.; Belytschko, T.; Gu, L. 1994: A New Implementation of the Element Free Galerkin Method, Computer Methods in Applied Mechanics and Engineering, 113, 397–414
Mallat, S. 1994: Multi-resolution Approximations and Wavelet Orthogonal Bases of L2(R), Trans. Amer. Math. Soc., 315: 69–87
Monaghan, J. J. 1988: An Introduction to SPH, Comp. Phys. Comm., 48, 89–96
Nayroles, B.; Touzot, G.; Villon, P. 1992: Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements, Computational Mechanics, Vol. 10, pp. 307–318
Oñate, E.; Idelsohn, S.; Zienkiewicz, O. C. July, 1995: Finite Point Methods in Computational Mechanics, International Center for Numerical Methods in Engineering
Shodja, H. M.; Mura, T.; Liu, W. K. 1995: Multiresolution Analysis of a Micromechanical Model, Computational Methods in Micromechanics, ASME AMD 212/MD 62, eds. S. Ghosh and M. Ostoja-Starzewski, pp. 33–54
Strang, G. 1989: Wavelets and Dilation Equations: a Brief Introduction, SIAM Review, 31: 614–627
Sulsky, D.; Chen, Z.; Schreyer, H. L. 1992: The Application of a Material-Spatial Numerical Method to Penetration, New Methods in Transient Analysis, eds. Smolinski, P., Liu, W. K., Hulbert, G. and Tamma, K., ASME, PVP Vol. 246/AMD Vol. 143, pp. 91–102
Williams, J. R.; Amaratunga, K. 1994: Introduction to Wavelets in Engineering, International Journal for Numerical Methods in Engineering, Vol. 37, 2365–2388
Yagawa, G.; Yamada, T.; Kawai, 1995: Some Remarks on Free Mesh Method: A kind of Meshless Finite Element Methods, To be presented at ICES-95
Author information
Authors and Affiliations
Additional information
Communicated by S. N. Atluri, 21 January 1996
Dedicated to the 10th anniversary of Computational Mechanics
Rights and permissions
About this article
Cite this article
Liu, W.K., Chen, Y., Chang, C.T. et al. Advances in multiple scale kernel particle methods. Computational Mechanics 18, 73–111 (1996). https://doi.org/10.1007/BF00350529
Issue Date:
DOI: https://doi.org/10.1007/BF00350529