Meshfree methods

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

A Meshless Approach in Solution of Multiscale Solidification Modeling

Abstract: This paper describes an overview of a new meshless solution procedure for calculation of one-domain coupled macroscopic heat, mass, momentum and species transfer problems as well as phase-field concepts of grain evolution. The solution procedure is defined on the macro [1] as well as on the micro levels [2] by a set of nodes which can be non-uniformly distributed. The domain and boundary of interest are divided into overlapping influence areas. On each of them, the fields are represented by the multiquadrics radial basis functions (RBF) collocation on a related sub-set of nodes. The time-stepping is performed in an explicit way. All governing equations are solved in their strong form, i.e. no integrations are performed. The polygonisation is not present and the formulation of the method is practically independent of the problem dimension. The solution can be easily and efficiently adapted in node redistribution and/or refinement sense, which is of utmost importance when coping with fields exhibiting sharp gradients. The concept and the results of the multiscale solidification modeling with the new approach are compared with the classical mesh-based [3] approach. The method turns out to be extremely simple to code and accurate, inclusion of the complicated physics can easily be looked over. The coding in 2D or 3D is almost identical.

Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion.

Abstract: Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme which is stable. In addition, there is a large computational cost for assembling the stiffness matrix of the nonlocal problem because high order Gaussian quadrature is usually needed to evaluate the integral. We thus provide a remedy to the problem by introducing a quasi-discrete nonlocal diffusion operator for which no numerical quadrature is further needed after applying the RK collocation scheme. The quasi-discrete nonlocal diffusion operator combined with RK collocation is shown to be convergent to the correct local diffusion problem by taking the limits of nonlocal interaction and spatial resolution simultaneously. The theoretical results are then validated with numerical experiments. We additionally illustrate a connection between the proposed technique and an existing optimization based approach based on generalized moving least squares (GMLS).

Optimal transportation meshfree method in geotechnical engineering problems under large deformation regime

Abstract: Meshfree methods have been demonstrated as suitable and strong alternatives to the more standard numerical schemes such as finite elements or finite differences. Moreover, when formulated in a Lagrangian approach, they are appropriate for capturing soil behavior under high‐strain levels. In this paper, the optimal transportation meshfree method has been applied for the first time to geotechnical problems undergoing large deformations. All the features employed in the current methodology (ie, F‐bar, explicit viscoplastic integration, and master‐slave contact) are described and validated separately. Finally, the model is applied to the particular case of shallow foundations by using von Mises and Drucker‐Prager yield criteria to find the load at failure in the. The presented methodology is demonstrated to be robust and accurate when solving this type of problems.

Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

Abstract: This paper presents three boundary meshless methods for solving prob- lems of steady-state and transient heat conduction in nonlinear functionally graded materials (FGMs). The three methods are, respectively, the method of fundamen- tal solution (MFS), the boundary knot method (BKM), and the collocation Trefftz method (CTM) in conjunction with Kirchhoff transformation and various variable transformations. In the analysis, Laplace transform technique is employed to han- dle the time variable in transient heat conduction problem and the Stehfest nu- merical Laplace inversion is applied to retrieve the corresponding time-dependent solutions. The proposed MFS, BKM and CTM are mathematically simple, easy- to-programming, meshless, highly accurate and integration-free. Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are con- sidered, and the results are compared with those from meshless local boundary integral equation method (LBIEM) and analytical solutions to demonstrate the effi- ciency of the present schemes.

Consistent and stable meshfree Galerkin methods using the virtual element decomposition

Abstract: Summary Over the past two decades, meshfree methods have undergone significant development as a numerical tool to solve partial differential equations (PDEs). In contrast to finite elements, the basis functions in meshfree methods are smooth (nonpolynomial functions), and they do not rely on an underlying mesh structure for their construction. These features render meshfree methods to be particularly appealing for higher-order PDEs and for large deformation simulations of solid continua. However, a deficiency that still persists in meshfree Galerkin methods is the inaccuracies in numerical integration, which affects the consistency and stability of the method. Several previous contributions have tackled the issue of integration errors with an eye on consistency, but without explicitly ensuring stability. In this paper, we draw on the recently proposed virtual element method, to present a formulation that guarantees both the consistency and stability of the approximate bilinear form. We adopt maximum-entropy meshfree basis functions, but other meshfree basis functions can also be used within this framework. Numerical results for several two-dimensional and three-dimensional elliptic (Poisson and linear elastostatic) boundary-value problems that demonstrate the effectiveness of the proposed formulation are presented. Copyright © 2017 John Wiley & Sons, Ltd.