Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

An iterative meshfree method for the elliptic monge–ampère equation in 2D

Abstract: The elliptic Monge–Ampere equation is a fully nonlinear partial differential equation, which originated in geometric surface theory and has been widely applied in dynamic meteorology, elasticity, geometric optics, image processing, and others. The numerical solution of the elliptic Monge–Ampere equation has been a subject of increasing interest recently. In this article, we provide an iterative meshfree method for the elliptic Monge–Ampere equation. The nonlinear equation is simplified as series of Poisson equations with the iterative sequence. Then, the Kansa's method is used to solve these linear equations. We prove the convergence of this method. We also present some numerical experiments to demonstrate the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1507–1517 , 2014

Radial Basis Function Method for the Eigen Analysis of Waveguide

Abstract: Traditional mesh-based methods in eigen analysis of waveguide meet two big challengers.One comes from the mesh generation,which is a big difficulty especially in complex geometry.The other originates from the local linear approximation,whose approximation accuracy is poor.So radial basis function method was introduced in this paper,which is a kind of meshless methods with collocation type.Ideas on how to solve eigenvalue problems and detailed implementation procedure were presented with corresponding discrete equations.Comparing to finite element method in the analyzing rectangular,round and L-shaped waveguide,the method suggested in this paper can be realized with low computation and high precision.In addition,the radial basis function method can be implemented more convenient and also can be extended into high dimension problems easily.

Analysis of a crack problem via RKPM and GRKPM and a note on particle volume

Abstract: Analysis of a crack problem via RKPM and GRKPM and a note on particle volume Mani Khezri1, Alireza Hashemian1, Hossein M. Shodja1,2,3 Summary Meshless methods using kernel approximation like reproducing kernel particle method (RKPM) and gradient RKPM (GRKPM) generally use a set of particles to discretize the subjected domain. One of the major steps in discretization procedure is determination of associated volumes particles. In a non-uniform or irregular configuration of particles, determination of these volumes comprises some difficulties. This paper presents a straightforward numerical method for determination of related volumes and conducts a survey on influence of different assumption about computing the volume for each particle. Stress intensity factor (SIF) as a major representing parameter in fracture of solids is calculated by employing meshless methods for an edge-cracked plate under first mode loading condition which is one of the benchmark problems in fracture mechanics. The obtained results are compared using an analysis in terms of dilation parameter.

A multi-scale particle method for mean field equations: the general case

Abstract: A multi-scale meshfree particle method for macroscopic mean field approximations of generalized interacting particle models is developed and investigated. The method is working in a uniform way for large and small interaction radii. The well resolved case for large interaction radius is treated, as well as underresolved situations with small values of the interaction radius. In the present work we extend the approach from [39] for porous media type limit equations to a more general case, including in particular hyperbolic limits. The method can be viewed as a numerical transition between a DEM-type method for microscopic interacting particle systems and a meshfree particle method for macroscopic equations. We discuss in detail the numerical performance of the scheme for various examples and the potential gain in computation time. The latter is shown to be particularly high for situations near the macroscopic limit. There are various applications of the method to problems involving mean field approximations in swarming, tra?c, pedestrian or granular fow simulation.

The forward and inversion analysis of high rock-fill dam during construction period using the node-based smoothed point interpolation method

Abstract: The meshfree node-based smoothed point interpolation method (NS-PIM) is extended to the forward and inversion analysis of a high gravelly soil core rock-fill dam during construction periods.,As one member of the meshfree methods, the NS-PIM has the advantages of “softer” stiffness and adaptability to large deformations which is quite indispensable for the stability analysis of rock-fill dams. In this work, the present method contains a reconstruction procedure to deal with the existence or nonexistence of the construction layers. After verifying the validity of the NS-PIM method for nonlinear elastic model during construction period, the convergence features of the NS-PIM and FEM methods are further investigated with different mesh schemes. Furthermore, the NS-PIM and FEM methods are applied for the forward analysis of a high gravelly soil core rock-fill dam and the convergence features under complex stress conditions are also studied using the rock-fill dam model. Finally, the NS-PIM method is used to calculate the Duncan–Chang parameters of the deep overburden under the high gravelly soil core rock-fill dam based on the back-propagation neural network method.,The results show that: the NS-PIM solution for construction analysis still possesses the property of upper bound solution even under complex stress conditions and can provide comparatively more conservative results for safety evaluation. Furthermore, it can be used to evaluate the accuracy of results and mesh quality together with the FEM solution which has the property of lower bound solution; the inversion analysis in this work provides a set of material parameters for the deep overburden under high rock-fill dam during construction period and the calculated results show good agreement with the measured displacement values and it is feasible to apply the NS-PIM to the forward and inversion analysis of high rock-fill dams on deep overburden during construction periods.,In further study, the feasibility of three-dimensional problems, elastic–plastic problems, contact problems and multipoint inversion can still be probed in the NS-PIM solution for the forward and inversion analysis of high rock-fill dams on deep overburden.,This paper introduced a method for the forward and inversion analysis of high rock-fill dams during construction period using the NS-PIM solution. The property of upper bound solution ensures that the NS-PIM can provide more conservative results for safety evaluation. The inversion analysis in this work provides a set of material parameters for the deep overburden under high rock-fill dam during construction periods.,First, the analysis from forward to inversion for high rock-fill dams during construction period using the NS-PIM solution is accomplished in this work. A procedure dealing with the existence or nonexistence of the construction layers is also developed for the construction analysis. Second, it is confirmed in this work that the NS-PIM still possesses the property of upper bound solution even under complex stress conditions (the forward analysis of high rock-fill dams during construction period). Thus, more conservative results can be provided for safety evaluation. Furthermore, it can be used to evaluate the accuracy of results and mesh quality together with the FEM solution which has the property of lower bound solution. Third, the calculated material parameters of the deep overburden in this work can be used for further studies of the high rock-fill dam.