Vol.:(0123456789) 1 3 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2018) 40:541 https://doi.org/10.1007/s40430-018-1459-z TECHNICAL PAPER Accelerated element‑free Galerkin method for analysis of fracture problems Sahil Garg 1  · Mohit Pant 1 Received: 27 October 2016 / Accepted: 16 October 2018 © The Brazilian Society of Mechanical Sciences and Engineering 2018 Abstract The work presents a modifed form of element-free Galerkin method (EFGM) based on a procedure which allows selection of equal number of nodes in the support domain. This procedure is further improved by selecting domain nodes depending on the region of the problem geometry. The optimum number of support domain nodes is obtained by performing an opti- mization of selectable EFGM parameters namely total number of nodes in problem geometry, Gauss quadrature and random or desired number of nodes in support domain. Taguchi L-16 orthogonal array is used to obtain optimized values of these parameters. The optimized EFGM parameters provide good accuracy with lesser number of nodal points and reduce the computational time of conventional EFGM by an average of 85%. Interaction integral technique has been used in order to extract the stress intensity factors for the simulated problems. The worth of so presented accelerated element-free Galerkin method (AEFG) is established by simulating various cases of fracture problems and the results so obtained are concurrent with those available in literature. Keywords EFGM · Fracture · Optimization · Computational time 1 Introduction Computational methods provide a myriad of applications by saving a lot of engineering efort, cost and time in today’s technological world. The purpose of using computational tools is to save cost and time by giving up the need of pro- totype testing. The earliest development of computational methods for analysis involved the fnite element method (FEM) widely used for a variety problems related to design and analysis [1–5]. Since FEM is a mesh-based method, it has limitations in meshing moving boundary problems, moreover the distorted and low quality meshes produced erroneous results consequentially requiring hard labor and excessive time. Further improvements of procedures for mesh-based methods led to the development of meshless methods (MMs). In MMs, approximation is built with the help of nodal points only, and these methods are able enough to remove the difculties associated with mesh- based methods. EFGM [6] is a type of meshless method widely used for analysis of a variety problems [7–14]. The simulations car- ried out by using EFGM are dependent on predefned EFG parameters like total number of nodes in the problem geom- etry, type of Gauss quadrature selected, number of nodes in support domain, the use of polynomials for defning weight functions and enforcement of essential boundary conditions. EFGM scores over other meshless methods used for fracture analysis by removing the need of remeshing and redistri- bution of nodal arrangements. EFGM also provides higher rates of convergence, higher adaptivity and can handle large material distortions easily [15]. The application of EFGM to the feld of fracture mechanics [16–21] has provided the researchers with a versatile tool to perform the crack analysis with ease. EFGM has proved its worth in a variety of appli- cations [22] by solving complex 3D problems due to its abil- ity to couple with other methods [23–26]. Since EFGM has established itself as a premier method in the feld of design and analysis by delivering accurate results, the further stud- ies should be concerned in the development of EFG; such Technical Editor: Paulo de Tarso Rocha de Mendonça, Ph.D. * Sahil Garg sahil.garg1017@gmail.com Mohit Pant mohitpant.iitr@gmail.com 1 Deptartment of Mechanical Engineering, National Institute of Technology, Hamirpur, H.P. 177005, India