Discrepancy-informed quadrature strategy for the nonlocal macro-meso-scale consistent damage model

The nonlocal macro-meso damage (NMMD) model has shown promising results in simulating the fracture process of materials. However, due to the inherent limitations of the nonlocal methods, its stability depends on whether the number of elements/nodes within the nonlocal region is sufficient. This paper proposes a discrepancy-informed quadrature strategy for NMMD to address its inherent limitations. Concretely, two discrepancies are defined to evaluate the uniformity of family point distribution within the nonlocal domain, followed by the introduction of a discrepancy-informed nonlocal quadrature strategy. The proposed strategy refines this by discretizing the nonlocal integral domain into finite circles for 2D or spheres for 3D and the discrete family points are uniformly arranged in each layer. This new quadrature strategy ensures relatively uniform distribution of pair points in the nonlocal space of each real material point, thus significantly enhancing the robustness and computational efficiency of the NMMD model. Numerical examples corroborate the strategy in accurately simulating different fracture modes, achieving an average computational speedup of approximately three times. This discrepancy-informed quadrature strategy effectively addresses the element size constraints of the original NMMD quadrature strategy.