Meshless methods: An overview and recent developments

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.

Reproducing kernel particle methods

A local symmetric weak form (LSWF) for linear potential problems is developed, and a truly meshless method, based on the LSWF and the moving least squares approximation, is presented for solving potential problems with high accuracy. The essential boundary conditions in the present formulation are imposed by a penalty method. The present method does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the “energy”. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. No post-smoothing technique is required for computing the derivatives of the unknown variable, since the original solution, using the moving least squares approximation, is already smooth enough. Several numerical examples are presented in the paper. In the example problems dealing with Laplace & Poisson's equations, high rates of convergence with mesh refinement for the Sobolev norms ||·||0 and ||·||1 have been found, and the values of the unknown variable and its derivatives are quite accurate. In essence, the present meshless method based on the LSWF is found to be a simple, efficient, and attractive method with a great potential in engineering applications.

A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics

Smoothed particle hydrodynamics (SPH) is a meshfree particle method based on Lagrangian formulation, and has been widely applied to different areas in engineering and science. This paper presents an overview on the SPH method and its recent developments, including (1) the need for meshfree particle methods, and advantages of SPH, (2) approximation schemes of the conventional SPH method and numerical techniques for deriving SPH formulations for partial differential equations such as the Navier-Stokes (N-S) equations, (3) the role of the smoothing kernel functions and a general approach to construct smoothing kernel functions, (4) kernel and particle consistency for the SPH method, and approaches for restoring particle consistency, (5) several important numerical aspects, and (6) some recent applications of SPH. The paper ends with some concluding remarks.

Smoothed Particle Hydrodynamics (SPH): an Overview and Recent Developments

Soon after the discovery of carbon nanotubes, it was realized that the theoretically predicted mechanical properties of these interesting structures–including high strength, high stiffness, low density and structural perfection–could make them ideal for a wealth of technological applications. The experimental verification, and in some cases refutation, of these predictions, along with a number of computer simulation methods applied to their modeling, has led over the past decade to an improved but by no means complete understanding of the mechanics of carbon nanotubes. We review the theoretical predictions and discuss the experimental techniques that are most often used for the challenging tasks of visualizing and manipulating these tiny structures. We also outline the computational approaches that have been taken, including ab initio quantum mechanical simulations, classical molecular dynamics, and continuum models. The development of multiscale and multiphysics models and simulation tools naturally arises as a result of the link between basic scientific research and engineering application; while this issue is still under intensive study, we present here some of the approaches to this topic. Our concentration throughout is on the exploration of mechanical properties such as Young’s modulus, bending stiffness, buckling criteria, and tensile and compressive strengths. Finally, we discuss several examples of exciting applications that take advantage of these properties, including nanoropes, filled nanotubes, nanoelectromechanical systems, nanosensors, and nanotube-reinforced polymers. This review article cites 349 references. @DOI: 10.1115/1.1490129#

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Mechanics of carbon nanotubes

Multiple-scale Kernel Particle methods are proposed as an alternative and/or enhancement to commonly used numerical methods such as finite element methods. The elimination of a mesh, combined with the properties of window functions, makes a particle method suitable for problems with large deformations, high gradients, and high modal density. The Reproducing Kernel Particle Method (RKPM) utilizes the fundamental notions of the convolution theorem, multiresolution analysis and window functions. The construction of a correction function to scaling functions, wavelets and Smooth Particle Hydrodynamics (SPH) is proposed. Completeness conditions, reproducing conditions and interpolant estimates are also derived. The current application areas of RKPM include structural acoustics, elastic-plastic deformation, computational fluid dynamics and hyperelasticity. The effectiveness of RKPM is extended through a new particle integration method. The Kronecker delta properties of finite element shape functions are incorporated into RKPM to develop a Cm kernel particle finite element method. Multiresolution and hp-like adaptivity are illustrated via examples.

Overview and applications of the reproducing Kernel Particle methods

Multiple scale methods based on reproducing kernel and wavelet analysis are developed. These permit the response of a system to be separated into different scales. These scales can be either the wave numbers corresponding to spatial variables or the frequencies corresponding to temporal variables, and each scale response can be examined separately. This complete characterization of the unknown response is performed through the integral window transform, and a space-scale and time-frequency localization process is achieved by dilating the flexible multiple scale window function. An error estimation technique based on this decomposition algorithm is developed which is especially useful for local mesh refinement and convergence studies. This flexible space-scale window function can be constructed to resemble the well-known unstructured multigrid and hp-adaptive finite element methods. However, the multiple scale adaptive refinements are performed simply by inserting nodes into the highest wavelet scale solution region and at the same time narrowing the window function. Hence hp-like adaptive refinements can be performed without a mesh.

Wavelet and multiple scale reproducing kernel methods

Generalized multiple scale reproducing kernel particle methods

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.

Advances in multiple scale kernel particle methods

Multiresolution analysis based on the reproducing kernel particle method (RKPM) is developed for computational fluid dynamics. An algorithm incorporating multiple-scale adaptive refinement is introduced. The concept of using a wavelet solution as an error indicator is also presented. A few representative numerical examples are solved to illustrate the performance of this new meshless method.

Yijung Chen

Papers

Overview and applications of the reproducing Kernel Particle methods

Wavelet and multiple scale reproducing kernel methods

Generalized multiple scale reproducing kernel particle methods

Advances in multiple scale kernel particle methods

Multiresolution reproducing kernel particle method for computational fluid dynamics