Aspects of the use of orthogonal basis functions in the element‐free Galerkin method

The element‐free Galerkin (EFG) method is probably the most widely used meshless method at present. In the EFG method, shape functions are derived from a moving least‐squares approximation using a polynomial basis, a calculation involving the inversion of a small matrix. A new implementation of the EFG method was published soon after the original where an alternative approach using an orthogonal basis was proposed to avoid matrix inversion in the formulation of the shape functions. In this paper we revisit this topic and show that the difficulties associated with the use of a polynomial basis remain present in the orthogonal case. We also show that certain terms in the derivative expressions are omitted in the new implementation of the EFG, which can lead to errors. Finally, we propose a new approach that avoids inversion while maintaining accuracy.