Meshfree methods

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

Weak-form Based Meshfree Methods: Element Free Galerkin Method (EFG) - An Overview

Abstract: In this paperan overview of Meshfree methods is given. Various types of Meshfree methods and their historical development are discussed. Weak form and strong form based Meshfree methods are provided and finally a description of element free Galerkin method (EFG) is given along with a summary of recent contributions in this Meshfree method. Implementation strategies, challenges and notable case studies regarding application of Meshfree methods are also discussed in detail.

Meshfree finite differences for vector Poisson and pressure Poisson equations with electric boundary conditions

Abstract: We demonstrate how meshfree finite difference methods can be applied to solve vector Poisson problems with electric boundary conditions. In these, the tangential velocity and the incompressibility of the vector field are prescribed at the boundary. Even on irregular domains with only convex corners, canonical nodal-based finite elements may converge to the wrong solution due to a version of the Babuska paradox. In turn, straightforward meshfree finite differences converge to the true solution, and even high-order accuracy can be achieved in a simple fashion. The methodology is then extended to a specific pressure Poisson equation reformulation of the Navier-Stokes equations that possesses the same type of boundary conditions. The resulting numerical approach is second order accurate and allows for a simple switching between an explicit and implicit treatment of the viscosity terms.

Influence Study of the Influence Domain to Numerical Simulation Results with Meshless Method

Abstract: Meshless method calculation accuracy is influenced by many factors, in which influence domain and node distribution are the most important. Due to the restrictions of the meshless methods themselves, their respective influence factors are different. In the paper, the advantages and disadvantages of the collocation method, the meshless method based on the local weak formulation and collocation (MWS), the meshless radial basis interpolation method based on global weak formulation (RPIM) and the weighted least squares meshless method (MWLS) are discussed by comparing the average error of nodes value in different influence domain radius. The results show that the accuracy of the MVS method is higher, but not stable; the radial basis interpolation method based on global weak formulation (RPIM) is a relatively stable method, but needs a large amount of calculation; better results can be obtained using the collocation with a small amount of the polynomial basis function added, simple and practicable.

A meshless method coupling peridynamics with corrective smoothed particle method for predicting material failure

Abstract: In this paper, we developed a meshless framework that couples peridynamics (PD) and the corrective smoothed particle method (CSPM) to simulate the complex damage and fracturing process of structures. This framework retains the advantages of PD in simulating material failure, and solves the problem that PD cannot simulate structures with irregular grids, and it also reduces the interface effect. Furthermore, the coupled PD-CSPM overcomes the challenge that stress boundary cannot be directly applied on CSPM-based models. In addition, PD-CSPM is capable of eliminating the limitation of Poisson's ratio imposed by the bond-based PD method. By simulating the deformation and failure behaviors of structures, the performance of PD-CSPM is verified and validated by the experimental observation and the numerical results obtained using other methods (the classic finite element method (FEM), extended finite element method (XFEM), and phase field method). Using the developed method, we successfully simulated fracture and failure behaviors of structures with irregular grids.

Higher order meshless approximation applied to Finite Difference and Finite Element methods in selected thermomechanical problems

Abstract: This paper is focused on the application of higher order meshless schemes in a numerical analysis of selected thermomechanical problems. Numerical investigation is based upon meshless Finite Difference method as well as its combinations with Finite Element method, performed at two different levels of analysis. The first variant assumes conjugation of element and meshless approximation schemes, while in the second one, the problem domain is divided into several disjoint subdomains, with a parallel and independent operating of coupled methods in each subdomain. The most important advantage of the applied approximation technique is no requirements of modification or enhancement of the existing discretization model. Therefore, high approximation orders may be assumed without providing new nodes, elements and degrees of freedom, maintaining the entire numerical model as simple as possible. This approach is especially convenient in coupled multi-field 2D and 3D problems, for instance stationary and non-stationary thermoelastic ones. In those problems, standard higher order approximation techniques may lead to complex numerical models caused by the rapid growth of a number of degrees of freedom and ill-conditioned schemes. The proposed approach is derived for three dimensional non-stationary thermoelastic problems. Moreover, it is examined on variety of 2D and 3D benchmark examples and engineering applications. Both solution accuracy and convergence rate are taken into account. Obtained results are very promising as they reflect the competitiveness of the approach comparing to other commonly applied higher order approaches.