Unified drag-force temporal discretization in the u-U-p fractional step method

The u - U - p formulation, which utilizes solid displacement, fluid displacement, and pressure as fundamental unknowns, is a preferred approach for computing the dynamic response of fluid-saturated geomaterials. In the past, the implicit monolithic solution was the conventional approach for finite element analysis. However monolithic solution encounters significant computational challenges due to their large-scale linear equation system and pronounced material nonlinearity involved. As a viable alternative, the semi-explicit–implicit fractional step method (FSM) demonstrates strong adaptability for rapid dynamic problems and can substantially reduce computational complexity compared to the monolithic solution. Nevertheless, the temporal discretization of the drag force within the FSM directly impacts the selection of time step sizes and computational accuracy. This study proposes a unified temporal discretization framework for the drag force in the u - U - p formulation. Depending on the choice of time factors, this method can be categorized into one explicit and three implicit schemes. Notably, a new implicit scheme significant improvements in computational accuracy over other schemes as verified by numerical examples. This unified drag-force fractional step method (UDF-FSM) strictly adheres to the process of explicit velocity prediction, implicit pressure solution, and explicit velocity correction, applicable to both high and low permeability problems.