Efficient modeling strategies utilizing stiffness matrix formulation for non-ordinary state-based peridynamics

Purpose This paper derives and implements a matrix-based formulation for Non-Ordinary state-based (NOSB) correspondence Peridynamics (PD) to overcome the computational limitations of traditional pointwise summation approaches and provides an open-source implementation within the PeriLab framework. Methods A linearized stiffness-matrix representation of the NOSB formulation is developed, incorporating an integrated zero-energy mode stabilization algorithm. To enhance computational efficiency, Guyan reduction is employed for model condensation, and hybrid solver strategies are introduced that combine linear static analysis with Velocity-Verlet time integration, with or without condensation of far-field elastic regions. Results The matrix-based formulation demonstrates equivalent accuracy to pointwise implementations. The hybrid strategies achieve speedups of up to 20 × compared to pure explicit time integration for a dogbone tensile specimen, with 95% of the savings attributable to the elimination of the quasi-static loading phase. Validation on a double cantilever beam confirms that crack initiation times agree within 0.1% across all solver variants, and that condensed far-field regions do not compromise fracture prediction accuracy provided the active PD domain fully encompasses the process zone. Conclusion The proposed hybrid strategies are particularly effective for problems with extended quasi-static loading phases followed by localized damage evolution. All implementations are provided as open-source code to facilitate reproducibility and further development by the research community.