Meshfree methods

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

Meshfree point collocation method for elasticity and crack problems

Abstract: A generalized diffuse derivative approximation is combined with a point collocation scheme for solid mechanics problems. The derivatives are obtained from a local approximation so their evaluation is computationally very efficient. This meshfree point collocation method has other advantages: it does not require special treatment for essential boundary condition nor the time-consuming integration of a weak form. Neither the connectivity of the mesh nor differentiability of the weight function is necessary. The accuracy of the solutions is exceptional and generally exceeds that of element-free Galerkin method with linear basis. The performance and robustness are demonstrated by several numerical examples, including crack problems. Copyright © 2004 John Wiley & Sons, Ltd.

Buckling analysis of sandwich plates with functionally graded skins using a new quasi-3D hyperbolic sine shear deformation theory and collocation with radial basis functions

Abstract: A hyperbolic sine shear deformation theory is used for the linear buckling analysis of functionally graded plates. The theory accounts for through-the-thickness deformations. The buckling governing equations and boundary conditions are derived using Carrera's Unified Formulation and further interpolated by collocation with radial basis functions. The collocation method is truly meshless, allowing a fast and simple discretization of equations in the domain and on the boundary. A numerical investigation has been conducted considering and neglecting the thickness stretching effects on the buckling of sandwich plates with functionally graded skins. Numerical results demonstrate the high accuracy of the present approach.

A local radial basis functions—Finite differences technique for the analysis of composite plates

Abstract: Radial basis functions are a very accurate means of solving interpolation and partial differential equations problems. The global radial basis functions collocation technique produces ill-conditioning matrices when using multiquadrics, making the choice of the shape parameter a crucial issue. The use of local numerical schemes, such as finite differences produces much better conditioned matrices. However, finite difference schemes are limited to special grids. For scattered points, a combination of finite differences and radial basis functions would be a possible solution. In this paper, we use a higher-order shear deformation plate theory and a radial basis function—finite difference technique for predicting the static behavior of thin and thick composite plates. Through numerical experiments on square and L-shaped plates, the accuracy and efficiency of this collocation technique is demonstrated, and the numerical accuracy and convergence are thoughtfully examined. This technique shows great potential to solve large engineering problems without the issue of ill-conditioning.

Novel adaptive meshfree integration techniques in meshless methods

Abstract: SUMMARY We propose two strategies of novel adaptive numerical integration based on mapping techniques for solving the complicated problems of domain integration encountered in meshfree methods. Several mapping methods are presented in detail that map a complex integration domain to much simpler ones, for example, squares, triangles or circles. The techniques described in the paper can be applied to both global and local weak forms, and the highly nonlinear meshfree integrands are evaluated with controlled accuracy. The necessity of the clumsy procedure of background mesh or cell structures used for integration purpose in existing meshfree methods is avoided, and many meshfree methods that require the domain integration can now become ‘truly meshfree’. Various numerical examples in two dimensions are considered to demonstrate the applicability and the effectiveness of the proposed methods and it shows that the accuracy is improved significantly. Their obtained results are compared with analytical solutions and other approaches and very good agreements are found. Additionally, some three-dimensional cases applied by the present methods are also examined. Copyright © 2012 John Wiley & Sons, Ltd.