In recent decades, numerical methods have been widely used to solve solid mechanics problems. One of the most important issues in the Continuum mechanics' field is the singularity and the stress intensification in its adjacency. Element Free Galerkin (EFG) method is one of the most powerful and well-known numerical methods for solving solid mechanics' problems, which has been proved for its proper accuracy, good continuity of response, supporting large deformations, and appropriate stability of the solution process. So far, a large number of Studies have been done based on EFG method to solve fracture mechanics' problems, in order to correctly estimate the stress and displacement fields in the vicinity of the singular point. The present research examines EFG problems with weak singularities in their physical domain, suffering from discontinuity in the derivatives of the potential fields. Most of the well-known numerical methods, including EFG, use smooth bases to estimate the unknown fields of the problem. These bases are unable to reproduce the response in the singularity affected zone. Adding some enrichment bases that include the singularity properties according to the type of the problem can improve the answer quality in that zone. Conventionally, deriving such bases requires solution of an analytical problem having similar geometry to the main problem, in order to accurately estimate the singularity order of the problem. What is followed in this research is the automatic extraction of the enrichment bases in the process of solving problems, without knowledge of its singularity order. For this purpose, the homogeneous form of the PDE is approximately satisfied using the weighted residual integration. Consequently, the Equilibrated Singular Basis Functions (EqSBFs) are produced. Primary basis functions used in the weighted integral are an expansion of polynomials in the radial direction and Fourier series in the angular direction. In order to add the ability to solve problems with weak singularities, by using a special transformation, the equations are rewritten in a transformed polar coordinate system, so that the singularity effect can be reproduced. New functions, named Equilibrated Singular Basis Functions, are added to the smooth bases of the EFG and lead to the enrichment of the mentioned method. In this research, various problems will be analyzed to eva‎luate the proposed method, some of which having analytical solution and some do not. Validation for real scaled problems without analytical solution is done using COMSOL commercial software. The presented results for the mentioned problems will show that enrichment by the proposed method will lead to a reasonable estimation of the stress and displacement fields in the singularity-dominated zone, and improve the quality of the answer. Meanwhile, this enrichment is easy to apply and can be used while preserving the important properties and advantages of the EFG method.

نيما نورمحمدي، بيژن برومند

فرشيد مسيبي، سعيد صرامي