ABSTRACT In this paper, the influence of involved nodes at a computational node on the stability, accuracy and efficiency of various commonly used weak‐ and strong‐form numerical methods, such as the Galerkin finite element method (FEM), the boundary element method (BEM), the finite volume method (FVM), the meshfree method (MFM) and the free element method (FrEM), is deeply investigated. It is found that the number of influence nodes for a computational node is heavily related to the mentioned three aspects. In view of this phenomenon, a new collocation technique, called Auxiliary Collocation Points enhanced strong‐form Free Element Method (ACPFrEM), is proposed in this paper by introducing some auxiliary collocation points around a collocation node in FrEM. This technique can make all nodes of the forming the free element for a point have contributions to the system of equations. As a result, ACPFrEM can dramatically improve the stability and accuracy of the strong‐form FrEM (SFrEM), even better than the weak‐form FrEM (WFrEM) and FEM, while it can maintain the computation efficiency as good as the SFrEM. The most distinct feature of the proposed method is that although a number of auxiliary points are added around each collocation node, ACPFrEM does not increase the number of the total nodes associated with the final system of equations. The number and distribution of the auxiliary points at a collocation node will be investigated in detail and a few numerical examples on thermal and mechanical problems of isotropic and functionally graded materials are examined for the stability, accuracy and efficiency of the proposed method.
Gao et al. (Wed,) studied this question.