Meshfree methods

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

All Well--Posed Problems have Uniformly Stable and Convergent Discretizations

Abstract: This paper considers a large class of linear operator equations, including linear boundary value problems for partial differential equations, and treats them as linear recovery problems for objects from their data. Well-posedness of the problem means that this recovery is continuous. Discretization recovers restricted trial objects from restricted test data, and it is well-posed or stable, if this restricted recovery is continuous. After defining a general framework for these notions, this paper proves that all well-posed linear problems have stable and refinable computational discretizations with a stability that is determined by the well-posedness of the problem and independent of the computational discretization. The solutions of discretized problems converge when enlarging the trial spaces, and the convergence rate is determined by how well the full data of the object solving the full problem can be approximated by the full data of the trial objects. %of the discretization. This allows very simple proofs of convergence rates for generalized finite elements, symmetric and unsymmetric Kansa-type collocation, and other meshfree methods like Meshless Local Petrov-Galerkin techniques. It is also shown that for a fixed trial space, weak formulations have a slightly better convergence rate than strong formulations, but at the expense of numerical integration. Since convergence rates are reduced to those coming from Approximation Theory, and since trial spaces are arbitrary, this also covers various spectral and pseudospectral methods. All of this is illustrated by examples.

A three-dimensional adaptive analysis using the meshfree node-based smoothed point interpolation method (NS-PIM)

Abstract: In this paper, a three-dimensional (3-D) adaptive analysis procedure is proposed using the meshfree node-based smoothed point interpolation method (NS-PIM) Previous study has shown that the NS-PIM works well with the simplest four-node tetrahedral mesh, which is easy to be implemented for complicated geometry In contrast to the displacement-based FEM providing lower bound solutions, the NS-PIM possesses the attractive property of providing upper bound solutions in strain energy norm In the present adaptive procedure, a novel error indicator is devised for NS-PIM settings, which evaluates the maximum difference of strain energy values among four nodes in each of the tetrahedral cells A simple h-type local refinement scheme is adopted and coupled with 3-D mesh automatic generator based on Delaunay technology Numerical results indicate that the adaptive refinement procedure can effectively capture the stress concentration and solution singularities, and carry out local refinement automatically The present adaptive procedure achieves much higher convergence in strain energy solution compared to the uniform refinement, and obtains upper bound solution in strain energy efficiently for force driven problems

Two meshless methods based on local radial basis function and barycentric rational interpolation for solving 2D viscoelastic wave equation

Abstract: In this paper, 2D viscoelastic wave equation is solved numerically both on regular and irregular domains. For spatial approximation of viscoelastic wave equation two meshless methods based on local radial basis function and barycentric rational interpolation are proposed. Both of the spatial approximation methods do not need mesh, node connectivity or integration process on local subdomains so they are truly meshless. For local radial basis function method we used an existing algorithm in literature to choose an acceptable shape parameter. Time marching is performed with fourth order Runge Kutta method. L 2 and L ∞ error norms for some test problems are reckoned to indicate efficiency and performance of the proposed two methods. Also, stability of the methods is discussed. Acquired results confirm the applicability of the proposed methods for 2D viscoelastic wave equation. We have performed some comparisons between the proposed two methods in the sense of accuracy and computational cost. From the comparisons, we have observed that performance of the barycentric rational interpolation in the sense of accuracy is slightly better than the performance of local radial basis function however computational cost of the local radial basis function is less than the computational cost of barycentric rational interpolation.

A meshfree method for incompressible fluid dynamics problems

Abstract: We show that meshfree variational methods may be used for the solution of incompressible fluid dynamics problems using the R-function method (RFM). The proposed approach constructs an approximate solution that satisfies all prescribed boundary conditions exactly using approximate distance fields for portions of the boundary, transfinite interpolation, and computations on a non-conforming spatial grid. We give detailed implementation of the method for two common formulations of the incompressible fluid dynamics problem: first using scalar stream function formulation and then using vector formulation of the Navier-Stokes problem with artificial compressibility approach. Extensive numerical comparisons with commercial solvers and experimental data for the benchmark back-facing step channel problem reveal strengths and weaknesses of the proposed meshfree method.