This paper first provides a common framework for partial differential equation problems in both strong and weak form by rewriting them as generalized interpolation problems. Then it is proven that any well–posed linear problem in strong or weak form can be solved by certain meshless kernel methods to any prescribed accuracy. AMS Subject Classification: 35A25, 65N35

A new method for the numerical simulation of the so-called soliton dynamics arising in a nonlinear Schrödinger equation in the semi-classical regime is proposed. For the time discretization a classical fourth-order splitting method is used. For the spatial discretization, however, a meshfree method is employed in contrast to the usual choice of (pseudo) spectral methods. This approach allows to...

The repeated replacement method (RRM) is a Lagrangian meshfree method which we have previously applied to the Euler equations for compressible fluid flow. In this paper we present new enhancements to RRM, and we apply the enhanced method to both linear and nonlinear elasticity. We compare the results of ten test problems to those of analytic solvers, to demonstrate that RRM can successfully sim...

This paper is concerned with the boundary particle method (BPM), a new boundary-only radial basis function collocation schemes. The method is developed based on the multiple reciprocity principle and applying either high-order nonsingular general solutions or singular fundamental solutions as the radial basis function. Like the multiple reciprocity BEM (MR-BEM), the BPM does not require any inn...

Meshfree finite difference methods for the Poisson equation approximate the Laplace operator on a point cloud. Desirable are positive stencils, i.e. all neighbor entries are of the same sign. Classical least squares approaches yield large stencils that are in general not positive. We present an approach that yields stencils of minimal size, which are positive. We provide conditions on the point...

This paper describes a multiresolution approach to field modeling that can be used with any meshfree or mesh based method for adaptive solution refinement. The refined solution is represented as a superposition of a coarse (unrefined) solution and a sequence of refinements that provide additional degrees of freedom with higher spatial or functional resolution. Each refinement is treated as a so...

This paper presents a new particle method in LS-DYNA for the severe deformation and failure analyses of solid mechanics problems. The new formulation is first established following a standard meshfree Galerkin approach for a solving of the partial differential equation of linear elastic problems. A smoothed displacement field is introduced to the Galerkin formulation and leads to a regularized ...

Given a dynamical system, the problem of reconstructing the attraction basins can effectively be solved by means of meshfree interpolation tools. We propose a novel approach to calculate separatrix manifolds by detecting points via an efficient bisection-like scheme. Such data are then interpolated by means of the implicit Partition of Unity (PU) method. The detection algorithm works for dynami...

The accuracy of the conventional finite element (FE) approximation for analysis acoustic propagation is always characterized by an intractable numerical dispersion error. With aim enhancing performance FE acoustics, a coupled FE-Meshfree method based on triangular elements proposed in this work. In new element, required local built using point interpolation mesh-free techniques with polynomial-...