Meshfree methods

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

Assessment of rpim shape parameters for solution accuracy of 2d geometrically nonlinear problems

Abstract: This study discussed the effects of shape parameters on the radial point interpolation method (RPIM) accuracy in 2D geometrically nonlinear problems. Four finite deformation problems with compressible Neo-Hookean material are numerically solved with the RPIM algorithm using the multi-quadric (MQ) radial basis function. Both regular and irregular node distributions are used. Their displacements and Cauchy stresses are compared for different values of shape parameters and monomial basis. It is found that the shape parameters proposed for linearly elastic problems (q = 1.03, αc = 4) can still be applicable to 2D geometrically nonlinear problems but careful selections should be made for the calculation of stress. For example, when q is used as 1.75 with irregular node distributions, stresses can be calculated more precisely.

Discontinuous Radial Basis Function Approximations for Meshfree Methods

Abstract: Meshfree methods with discontinuous radial basis functions and their numerical implementation for elastic problems are presented We study the following radial basis functions: the multiquadratic (MQ), the Gaussian basis functions and the thin-plate basis functions These radial basis functions are combined with step function enrichments directly or with enriched Shepard functions The formulation is coupled with level set methods and requires no explicit representation of the discontinuity Numerical results show the robustness of the method, both in accuracy and convergence

Natural Element Method Applied to Electromagnetic Problems

Abstract: In this paper, the natural element method (NEM), which is part of the meshless methods family, is used to solve electromagnetic problems. It is an interesting alternative to the finite element method. This method is based on the Voronoi diagram and the natural neighbors. These two concepts and the construction of the shape function are presented. The NEM shape functions are interpolant and linear on the boundary of the studied domain. This property allows a natural coupling with the finite element method. A 2D electrostatic example is studied. Results are compared to those obtained by FE method in order to illustrate the interest and the relevance of the new method. A second example of an electric machine is proposed in order to illustrate the ability of the NEM to be naturally coupled with FEM.

Refinement computations of electromagnetic fields using FE and meshless methods

Abstract: A refinement algorithm for electromagnetic field computations using a combination of finite element and meshless methods is introduced. Bridging scales are used to separate the finite element and meshless shape functions to make the refinement hierarchical and to uphold the mathematical properties such as consistency and linear independence for all the bases. To facilitate the application of the proposed algorithm, details about the node addition, requirements for the node distribution, and relationships between the finite element and meshless shape functions, as well as the determination of the stop criterion are also fully addressed. Primary numerical results are reported to demonstrate and validate the applicability and advantages of the proposed algorithm over traditional ones.