A bridging transition technique for the combination of meshfree method with finite element method in 2D solids and structures

Abstract

For certain continuum problems, it is desirable and beneficial to combine two different methods together in order to exploit their advantages while evading their disadvantages. In this paper, a bridging transition algorithm is developed for the combination of the meshfree method (MM) with the finite element method (FEM). In this coupled method, the MM is used in the sub-domain where the MM is required to obtain high accuracy, and the FEM is employed in other sub-domains where FEM is required to improve the computational efficiency. The MM domain and the FEM domain are connected by a transition (bridging) region. A modified variational formulation and the Lagrange multiplier method are used to ensure the compatibility of displacements and their gradients. To improve the computational efficiency and reduce the meshing cost in the transition region, regularly distributed transition particles, which are independent of either the meshfree nodes or the FE nodes, can be inserted into the transition region. The newly developed coupled method is applied to the stress analysis of 2D solids and structures in order to investigate its' performance and study parameters. Numerical results show that the present coupled method is convergent, accurate and stable. The coupled method has a promising potential for practical applications, because it can take advantages of both the MM and FEM when overcome their shortcomings.

A new coupling technique for the combination of wavelet-Galerkin method with finite element method in solids and structures

“…Coupling methods have been successfully used in the combination of FEM with the boundary element method (BEM) , FEM with meshless methods (MMs) , isogeometric method (IGM) with MMs , IGM with IGM , and MM/BEM and MM/FEM/BEM .…”

“…Moreover, the high‐order continuity of smooth basis functions near the transition elements will be lost. The methods based on modified weak forms can achieve connection of displacement through a common boundary (i.e., an interface curve for a 2D problem and an interface surface for a 3D problem), called the interface. In addition, the high‐order compatibility can be implemented when a transition (bridging) region is used between two methods , while the penalty factors or the extra variables related to Lagrange multipliers should be introduced into solution system.…”

“…The methods based on modified weak forms can achieve connection of displacement through a common boundary (i.e., an interface curve for a 2D problem and an interface surface for a 3D problem), called the interface. In addition, the high‐order compatibility can be implemented when a transition (bridging) region is used between two methods , while the penalty factors or the extra variables related to Lagrange multipliers should be introduced into solution system. In , the algorithms based on the polynomial reconstruction condition are developed to construct the mixed basis functions in transition region, which can effectively connect different basis functions.…”