On efficient numerical integration for peridynamic modeling

Abstract

Peridynamics (PD) stands out as a promising approach for addressing problems featuring cracks or strong discontinuities, surpassing methods grounded in classical continuum mechanics (CCM) within such contexts. Nonetheless, PD exhibits a heightened computational demand compared to CCM-based approaches, attributed to the nonlocal nature of the PD approaches and the application of a single-point integration rule in its standard discretization. This paper introduces an improved class of numerical quadratures to mitigate the latter challenge. Additionally, a reduced form of these numerical quadratures is proposed to diminish computational costs. A systematic approach for comparing different reduced patterns is presented. To extend the applicability of these numerical quadratures to regions near boundaries or cracks, an extended weighted least squares (WLS)-based numerical integration scheme is introduced. The performance of the proposed schemes is investigated through a series of numerical examples. The findings indicate that the proposed approach demonstrates strong performance across various challenges, such as crack propagation in 3D domains and multiphysical scenarios. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2025.