Optimization of Compatible Meshfree Quadrature Rule for Nonlocal Problems with Applications to Peri Dynamics Study

Abstract

The optimization of a compatible meshfree quadrature rule for nonlocal problems with applications to Peridynamics is investigated in this study. Peridynamics is a nonlocal continuum mechanics theory that models material behavior based on interactions between material points within a finite neighbourhood. In Peridynamics, the accurate evaluation of nonlocal integral equations is crucial for obtaining reliable and efficient solutions. Traditional numerical integration methods, such as Gaussian quadrature, are not directly applicable to nonlocal problems due to their local nature. Hence, the development of a compatible meshfree quadrature rule that can effectively handle nonlocal interactions is of great importance. The objective it to optimize a quadrature rule that accurately captures the nonlocal interactions in Peridynamics while maintaining computational efficiency. The optimization process involves the selection of appropriate quadrature points and weights that minimize the quadrature error and maximize the computational efficiency. Various optimization techniques, such as genetic algorithms, particle swarm optimization, or machine learning algorithms, are explored to search for an optimal quadrature rule. The optimized quadrature rule is then applied to several Peridynamics problems, including fracture mechanics, material failure, and dynamic response analysis. The performance of the optimized quadrature rule is evaluated by comparing the results with those obtained using traditional quadrature methods and analytical solutions, when available. The accuracy, stability, and computational efficiency of the optimized quadrature rule are analysed and discussed. The findings provide valuable insights into the development and optimization of compatible meshfree quadrature rules for nonlocal problems, particularly in the context of Peridynamics. The optimized quadrature rule offers improved accuracy in capturing nonlocal interactions and reduces the computational cost compared to traditional methods. The applications of the optimized quadrature rule to various Peridynamics problems demonstrate its effectiveness and reliability.

The implications of it extend beyond Peridynamics and can be applicable to other nonlocal models in the field of computational mechanics. The optimized quadrature rule has the potential to enhance the accuracy and efficiency of numerical simulations involving nonlocal phenomena, enabling more reliable predictions of material behavior and structural response. The findings of this contribute to the advancement of numerical methods for nonlocal problems and provide a foundation for further research and developments in the field of compatible meshfree quadrature rules for nonlocal problems.