The Numerical Manifold Method (NMM) is an effective numerical approach for simulating discontinuous deformations in rock mechanics and geotechnical engineering. The penalty method is straightforward to implement within NMM, but it presents a significant challenge in selecting an appropriate penalty parameter. To enhance contact calculation accuracy and minimize reliance on penalty parameters, an augmented Lagrangian method (ALM) is introduced, which effectively combines the advantages of both the Lagrangian multiplier method and the penalty method. Furthermore, to mitigate redundant constraints and the repetitive computations arising from non-convergent Open-Close iterations, the contact detection algorithm in NMM is refined, and a synchronous AL-O-C iteration scheme is implemented. The performance of the augmented Lagrangian enhanced NMM is evaluated against the standard NMM through several benchmark examples, with comparisons focused on computational efficiency and contact accuracy. The results demonstrate that the proposed method can operate effectively with smaller penalty values, thereby achieving a superior balance between computational precision and efficiency.
Li et al. (Thu,) studied this question.