Meshfree methods

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

The least-squares meshfree method

Abstract: A new efficient meshfree method is presented in which the first-order least-squares method is employed instead of the Galerkin's method. In the meshfree methods based on the Galerkin formulation, the source of many difficulties is in the numerical integration. The current method, in this respect, has different characteristics and is expected to remove some of the integration-related problems. It is demonstrated through numerical examples that the present formulation is highly robust to integration errors. Therefore, numerical integration can be performed with great ease and effectiveness using very simple algorithms. Copyright © 2001 John Wiley & Sons, Ltd.

Collocated discrete least‐squares (CDLS) meshless method: Error estimate and adaptive refinement

Abstract: Meshless methods are new approaches for solving partial differential equations. The main characteristic of all these methods is that they do not require the traditional mesh to construct a numerical formulation. They require node generation instead of mesh generation. In other words, there is no pre-specified connectivity or relationships among the nodes. This characteristic make these methods powerful. For example, an adaptive process which requires high computational effort in mesh-dependent methods can be very economically solved with meshless methods. In this paper, a posteriori error estimate and adaptive refinement strategy is developed in conjunction with the collocated discrete least-squares (CDLS) meshless method. For this, an error estimate is first developed for a CDLS meshless method. The proposed error estimator is shown to be naturally related to the least-squares functional, providing a suitable posterior measure of the error in the solution. A mesh moving strategy is then used to displace the nodal points such that the errors are evenly distributed in the solution domain. Efficiency and effectiveness of the proposed error estimator and adaptive refinement process are tested against two hyperbolic benchmark problems, one with shocked and the other with low gradient smooth solutions. These experiments show that the proposed adaptive process is capable of producing stable and accurate results for the difficult problems considered. Copyright © 2007 John Wiley & Sons, Ltd.

Second-order convex maximum entropy approximants with applications to high-order PDE

Abstract: We present a new approach for second order maximum entropy (max-ent) meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for non-uniform node sets, and prescribes robustly feasible constraints for the entropy maximization program defining the approximants. We examine the performance of the proposed approximation scheme in the numerical solution by a direct Galerkin method of a number of partial differential equations (PDEs), including structural vibrations, elliptic second order PDEs, and fourth order PDEs for Kirchhoff-Love thin shells and for a phase field model describing the mechanics of biomembranes. The examples highlight the ability of the method to deal with non-uniform node distributions, and the high accuracy of the solutions. Surprisingly, the first order meshfree max-ent approximants with large supports are competitive when compared to the proposed second order approach in all the tested examples, even in the higher order PDEs.