Isfahan University of Technology Journal of Computational Methods in Engineering 2228-7698 44 2 2025 11 22 On the Use of Error Weight Function in the Numerical Solution of Nonlocal Problems Using the Finite Element Method On the Use of Error Weight Function in the Numerical Solution of Nonlocal Problems Using the Finite Element Method 151 165 3721 10.47176/jcme.44.2.1062 FA Ali Gerami Department of Mechanical Engineering, Isfahan University of Technology Mohammad Silani Department of Mechanical Engineering, Isfahan University of Technology Mahdi Javanbakht Department of Mechanical Engineering, Isfahan University of Technology Journal Article 2025 08 30 Various functions can serve as weight functions in the numerical solution of nonlocal problems, including the error function. The value of the error function at a given point depends on the nonlocal parameter and the distance from the target point. To reduce computational costs in numerical procedures, interactions are often neglected between the points with a distance above a specific radius (interaction radius). Consequently, a relationship between the radius and the nonlocal parameter in the error function is established to determine the optimal interaction radius. Due to the widespread utilization of the finite element method in the solution of non-local problems, and considering application of the integral value of the weight function in two-phase kernels, the effect of element size on the calculation of the integral value is investigated. The findings indicate that, due to discretization, the error in calculating the integral value of the weight function increases as the ratio of the element size to the nonlocal parameter grows. This can result in an integral value exceeding 1, thereby violating the normalization condition of the weight function. Various functions can serve as weight functions in the numerical solution of nonlocal problems, including the error function. The value of the error function at a given point depends on the nonlocal parameter and the distance from the target point. To reduce computational costs in numerical procedures, interactions are often neglected between the points with a distance above a specific radius (interaction radius). Consequently, a relationship between the radius and the nonlocal parameter in the error function is established to determine the optimal interaction radius. Due to the widespread utilization of the finite element method in the solution of non-local problems, and considering application of the integral value of the weight function in two-phase kernels, the effect of element size on the calculation of the integral value is investigated. The findings indicate that, due to discretization, the error in calculating the integral value of the weight function increases as the ratio of the element size to the nonlocal parameter grows. This can result in an integral value exceeding 1, thereby violating the normalization condition of the weight function. https://jcme.iut.ac.ir/article_3721_9e406957d45fcb6c6f38c2ada7bace91.pdf