Acceleration of a projected gradient algorithm for the Bingham flow problem by rigidity enforcement with penalty

A new modification to the classical projected gradient method is introduced for the Bingham model in viscoplastic flows. In these flows, large regions may exist where the velocity field is rigid, i.e. where it has a zero symmetric gradient. The proposed approach aims to accelerate convergence by estimating a current approximate rigid zone during iterations while maintaining a simple algorithm structure. Local rigidity is enforced through a penalty procedure, along with splitting the projected gradient step into two local components: a rigid step and a fluid step. Convergence analysis is conducted in a theoretical case where the rigid zone is assumed to be already localised, showing that the penalty method allows for larger rigid steps. A practical analysis is also provided on two classical benchmarks. In the first benchmark, the flow solution is explicitly known, enabling careful examination of solver validation and convergence rate. In the second benchmark, the flow solution is more complex, and the robustness and efficiency of the algorithm are better assessed. In both cases, the number of iterations required to achieve a prescribed accuracy is significantly reduced compared to the classical projected gradient method when both the rigid descent step and penalty value are appropriately adjusted. Improvements are observed even with moderate penalty parameters, thereby avoiding stability issues.