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Meshfree methods
About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.
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TL;DR: In this article, a new numerical quadrature technique, partition of unity quadratures (PUQ), for Galerkin meshless methods is presented. But it is not a meshless method based on finite covering.
Abstract: Numerical quadrature is an important ingredient of Galerkin meshless methods. A new numerical quadrature technique, partition of unity quadrature (PUQ), for Galerkin meshless methods was presented. The technique is based on finite covering and partition of unity. There is no need to decompose the physical domain into small cell. It possesses remarkable integration accuracy. Using Element-free Galerkin methods as example, Galerkin meshless methods based on PUQ were studied in detail. Meshing is always not required in the procedure of constitution of approximate function or numerical quadrature, so Galerkin meshless methods based on PUQ are “truly” meshless methods.
6 citations
TL;DR: In this article, a spectral meshless radial point interpolation method is proposed to degenerate parabolic equations arising from the spatial diffusion of biological populations and satisfactory agreements is archived, which is based on collocation methods with mesh-free techniques as a background.
Abstract: A kind of spectral meshless radial point interpolation method is proposed to degenerate parabolic equations arising from the spatial diffusion of biological populations and satisfactory agreements is archived. This method is based on collocation methods with mesh-free techniques as a background. In contrast to the finite-element method and those meshless methods based on Galerkin weak form, such as element-free Galerkin, there is no integration tools in this approach. Furthermore, some numerical experiments are given to validate the accuracy of the results and stability of the present method.
6 citations
TL;DR: This procedure is for the first time investigated to simulate the non-linear behavior of structures with quasi-brittle materials, as well as the lack of the Kronecker-delta property and numerical integration problems.
Abstract: Meshfree techniques, such as hp -Clouds and Element Free Galerkin Methods, have been used as attractive alternatives to finite element method, due to the flexibility in constructing conforming approximations. These approximations can present high regularity, improving the description of the state variables used in physically non-linear problems. On the other hand, some drawbacks can be highlighted, as the lack of the Kronecker-delta property and numerical integration problems. These drawbacks can be overcome by using a C k , k arbitrarily large, partition of unity (PoU) function, built over a finite element mesh, but with the approximate characteristic of the meshfree methods. Here, this procedure is for the first time investigated to simulate the non-linear behavior of structures with quasi-brittle materials. The smeared crack model is adopted and numerical results, obtained with different kinds of polynomial enrichments, are compared with the experimental results.
6 citations
TL;DR: In this paper, the effects of numerical integration on Galerkin meshless methods for solving elliptic partial differential equations with Neumann boundary conditions were studied, and specific numerical schemes, diagonal elements correction and background mesh integration, that satisfy the zero row sum condition were identified.
Abstract: In this paper, we study effects of numerical integration on Galerkin meshless methods for solving elliptic partial differential equations with Neumann boundary conditions. The shape functions used in the meshless methods reproduce linear polynomials. The numerical integration rules are required to satisfy the so-called zero row sum condition of stiffness matrix, which is also used by Babuska et al. (Int. J. Numer. Methods Eng. 76:1434–1470, 2008). But the analysis presented there relies on a certain property of the approximation space, which is difficult to verify. The analysis in this paper does not require this property. Moreover, the Lagrange multiplier technique was used to handle the pure Neumann condition. We have also identified specific numerical schemes, diagonal elements correction and background mesh integration, that satisfy the zero row sum condition. The numerical experiments are carried out to verify the theoretical results and test the accuracy of the algorithms.
6 citations
TL;DR: The implementation of a parallel algorithm that solves a computational geometry problem arising in meshfree methods and can be used to efficiently compute the mass and stiffness matrices for the meshfree Galerkin method.
Abstract: We describe the implementation of a parallel algorithm that solves a computational geometry problem arising in meshfree methods. We solve the following problem: Given a collection of N d-rectangles Si and Q points xk in Rd, compute Ik = { i | xk \in Si }. This task is necessary for computing values of the meshfree basis functions Psii (xk) and can be used to efficiently compute the mass and stiffness matrices for the meshfree Galerkin method.
6 citations