Meshfree methods

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

A locking-free meshfree curved beam formulation with the stabilized conforming nodal integration

Abstract: A locking-free meshfree curved beam formulation based on the stabilized conforming nodal integration is presented. Motivated by the pure bending solutions of thin curved beam, a meshfree approximation is constructed to represent pure bending mode without producing parasitic shear and membrane deformations. Furthermore, to obtain the exact pure bending solution (bending exactness condition), the integration constraints corresponding to the Galerkin weak form are derived. A nodal integration with curvature smoothing stabilization that satisfies the integration constraints is proposed under the Galerkin weak form for shear deformable curved beam. Numerical examples demonstrate that the resulting meshfree formulation can exactly reproduce pure bending mode with arbitrary dicretizations, and the method is stable and free of shear and membrane locking. Computational efficiency and accuracy are achieved simultaneously in the proposed formulation

Assessment of global and local meshless methods based on collocation with radial basis functions for parabolic partial differential equations in three dimensions

Abstract: A comparison of the performance of the global and the local radial basis function collocation meshless methods for three dimensional parabolic partial differential equations is performed in the present paper. The methods are structured with multiquadrics radial basis functions. The time-stepping is performed in a fully explicit, fully implicit and Crank–Nicolson ways. Uniform and non-uniform node arrangements have been used. A three-dimensional diffusion–reaction equation is used for testing with the Dirichlet and mixed Dirichlet–Neumann boundary conditions. The global methods result in discretization matrices with the number of unknowns equal to the number of the nodes. The local methods are in the present paper based on seven-noded influence domains, and reduce to discretization matrices with seven unknowns for each node in case of the explicit methods or a sparse matrix with the dimension of the number of the nodes and seven non-zero row entries in case of the implicit method. The performance of the methods is assessed in terms of accuracy and efficiency. The outcome of the comparison is as follows. The local methods show superior efficiency and accuracy, especially for the problems with Dirichlet boundary conditions. Global methods are efficient and accurate only in cases with small amount of nodes. For large amount of nodes, they become inefficient and run into ill-conditioning problems. Local explicit method is very accurate, however, sensitive to the node position distribution, and becomes sensitive to the shape parameter of the radial basis functions when the mixed boundary conditions are used. Performance of the local implicit method is comparatively better than the others when a larger number of nodes and mixed boundary conditions are used. The paper represents an extension of our recently made similar study in two dimensions.

A comparison of three explicit local meshless methods using radial basis functions

Abstract: In this paper, three kinds of explicit local meshless methods are compared: the local method of approximate particular solutions (LMAPS), the local direct radial basis function collocation method (LDRBFCM) which are both first presented in this paper, and the local indirect radial basis function collocation method (LIRBFCM). In all three methods, the time discretization is performed in explicit way, the multiquadric radial basis functions (RBFs) are used to interpolate either initial temperature field and its derivatives or the Laplacian of the initial temperature field. The five-noded sub-domains are used in localization. Numerical results of simple diffusion equation with Dirichlet jump boundary condition are compared on uniform and random node arrangement, the accuracy and stabilities of these three local meshless methods are asserted. One can observe that the improvement of the accuracy with denser nodes and with smaller time steps for all three methods. All methods provide a similar accuracy in uniform node arrangement case. For random node arrangement, the LMAPS and the LDRBFCM perform better than the LIDRBFCM.

The natural radial element method

Abstract: SUMMARY In this work an innovative numerical approach is proposed, which combines the simplicity of low-order finite elements connectivity with the geometric flexibility of meshless methods. The natural neighbour concept is applied to enforce the nodal connectivity. Resorting to the Delaunay triangulation a background integration mesh is constructed, completely dependent on the nodal mesh. The nodal connectivity is imposed through nodal sets with reduce size, reducing significantly the test function construction cost. The interpolations functions, constructed using Euclidian norms, are easily obtained. To prove the good behaviour of the proposed interpolation function several data-fitting examples and first-order partial differential equations are solved. The proposed numerical method is also extended to the elastostatic analysis, where classic solid mechanics benchmark examples are solved. Copyright © 2013 John Wiley & Sons, Ltd.

Spectral meshless radial point interpolation (SMRPI) method to two‐dimensional fractional telegraph equation

Abstract: H. Ammari In this article, an innovative technique so-called spectral meshless radial point interpolation (SMRPI) method is proposed and, as a test problem, is applied to a classical type of two-dimensional time-fractional telegraph equation defined by Caputo sense for (1 < α≤2). This new methods is based on meshless methods and benefits from spectral collocation ideas, but it does not belong to traditional meshless collocation methods. The point interpolation method with the help of radial basis functions is used to construct shape functions, which play as basis functions in the frame of SMRPI method. These basis functions have Kronecker delta function property. Evaluation of high-order derivatives is not difficult by constructing operational matrices. In SMRPI method, it does not require any kind of integration locally or globally over small quadrature domains, which is essential of the finite element method (FEM) and those meshless methods based on Galerkin weak form. Also, it is not needed to determine strict value for the shape parameter, which plays an important role in collocation method based on the radial basis functions (Kansa's method). Therefore, computational costs of SMRPI method are less expensive. Two numerical examples are presented to show that SMRPI method has reliable rates of convergence. Copyright © 2015 John Wiley & Sons, Ltd.