Mathematical Modelling and Optimization Methods in Geomechanically Informed Blast Design: A Systematic Literature Review - Fabian Leon; Luis Rojas; Alvaro Peña; Paola Moraga; Pedro Robles; Blanca Gana; Jose García (Mathematics)
Background: Rock–blast design is a canonical inverse problem that joins elastodynamic partial differential equations (PDEs), fracture mechanics, and stochastic heterogeneity. Objective: Guided by the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) protocol, a systematic review of mathematical methods for geomechanically informed blast modelling and optimisation is provided. Methods: A Scopus–Web of Science search (2000–2025) retrieved 2415 records; semantic filtering and expert screening reduced the corpus to 97 studies. Topic modelling with Bidirectional Encoder Representations from Transformers Topic (BERTOPIC) and bibliometrics organised them into (i) finite-element and finite–discrete element simulations, including arbitrary Lagrangian–Eulerian (ALE) formulations; (ii) geomechanics-enhanced empirical laws; and (iii) machine-learning surrogates and multi-objective optimisers. Results: High-fidelity simulations delimit blast-induced damage with ≤0.2 m mean absolute error; extensions of the Kuznetsov–Ram equation cut median-size mean absolute percentage error (MAPE) from 27% to 15%; Gaussian-process and ensemble learners reach a coefficient of determination (<semantics>R2>0.95</semantics>) while providing closed-form uncertainty; Pareto optimisers lower peak particle velocity (PPV) by up to 48% without productivity loss. Synthesis: Four themes emerge—surrogate-assisted PDE-constrained optimisation, probabilistic domain adaptation, Bayesian model fusion for digital-twin updating, and entropy-based energy metrics. Conclusions: Persisting challenges in scalable uncertainty quantification, coupled discrete–continuous fracture solvers, and rigorous fusion of physics-informed and data-driven models position blast design as a fertile test bed for advances in applied mathematics, numerical analysis, and machine-learning theory.