Journal ArticleDOI
Reproducing kernel particle methods
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TLDR
A new continuous reproducing kernel interpolation function which explores the attractive features of the flexible time-frequency and space-wave number localization of a window function is developed and is called the reproducingkernel particle method (RKPM).Abstract:
A new continuous reproducing kernel interpolation function which explores the attractive features of the flexible time-frequency and space-wave number localization of a window function is developed. This method is motivated by the theory of wavelets and also has the desirable attributes of the recently proposed smooth particle hydrodynamics (SPH) methods, moving least squares methods (MLSM), diffuse element methods (DEM) and element-free Galerkin methods (EFGM). The proposed method maintains the advantages of the free Lagrange or SPH methods; however, because of the addition of a correction function, it gives much more accurate results. Therefore it is called the reproducing kernel particle method (RKPM). In computer implementation RKPM is shown to be more efficient than DEM and EFGM. Moreover, if the window function is C∞, the solution and its derivatives are also C∞ in the entire domain. Theoretical analysis and numerical experiments on the 1D diffusion equation reveal the stability conditions and the effect of the dilation parameter on the unusually high convergence rates of the proposed method. Two-dimensional examples of advection-diffusion equations and compressible Euler equations are also presented together with 2D multiple-scale decompositions.read more
Citations
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Journal ArticleDOI
Meshless methods: An overview and recent developments
TL;DR: Meshless approximations based on moving least-squares, kernels, and partitions of unity are examined and it is shown that the three methods are in most cases identical except for the important fact that partitions ofunity enable p-adaptivity to be achieved.
Journal ArticleDOI
The Partition of Unity Method
Ivo Babuška,Jens Markus Melenk +1 more
TL;DR: In this article, a new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved, which can therefore be more efficient than the usual finite element methods.
Journal ArticleDOI
Cracking particles: A simplified meshfree method for arbitrary evolving cracks
Timon Rabczuk,Ted Belytschko +1 more
TL;DR: A new approach for modelling discrete cracks in meshfree methods is described, in which the crack can be arbitrarily oriented, but its growth is represented discretely by activation of crack surfaces at individual particles, so no representation of the crack's topology is needed.
Journal ArticleDOI
A stabilized conforming nodal integration for Galerkin mesh-free methods
TL;DR: In this paper, a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integrations, where an integration constraint is introduced as a necessary condition for a linear exactness in the mesh-free Galerkin approximation.
Journal ArticleDOI
Mechanics of carbon nanotubes
TL;DR: The theoretical predictions and the experimental techniques that are most often used for the challenging tasks of visualizing and manipulating these tiny structures are reviewed and the computational approaches taken, including ab initio quantum mechanical simulations, classical molecular dynamics, and continuum models are outlined.
References
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Journal ArticleDOI
Smoothed particle hydrodynamics: Theory and application to non-spherical stars
R. A. Gingold,Joseph J Monaghan +1 more
Journal ArticleDOI
A numerical approach to the testing of the fission hypothesis.
TL;DR: A finite-size particle scheme for the numerical solution of two-and three-dimensional gas dynamical problems of astronomical interest is described and tested in this article, which is then applied to the fission problem for optically thick protostars.
Journal ArticleDOI
Element‐free Galerkin methods
Ted Belytschko,Y. Y. Lu,L. Gu +2 more
TL;DR: In this article, an element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems, where moving least-squares interpolants are used to construct the trial and test functions for the variational principle.
Journal ArticleDOI
Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations
A. N. Brooks,Thomas J. R. Hughes +1 more
TL;DR: In this article, a new finite element formulation for convection dominated flows is developed, based on the streamline upwind concept, which provides an accurate multidimensional generalization of optimal one-dimensional upwind schemes.
Book
An introduction to wavelets
TL;DR: An Overview: From Fourier Analysis to Wavelet Analysis, Multiresolution Analysis, Splines, and Wavelets.
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