A Fast Time-Discontinuous Peridynamic Method for Stress Wave Propagation Problems in Saturated Porous Media

When employing numerical methods to simulate the behavior of saturated porous media under impact load, the Gibbs phenomenon often occurs, significantly compromising the accuracy of numerical calculations. To improve the accuracy of numerical solutions, this study introduces a fast time-discontinuous peridynamic (TDPD) method for simulating the propagation of fluid pressure and solid stress waves within saturated porous media. In this method, the classical u-p and u-U models are reformulated from spatial differential equations into integral equations to facilitate the simulation of fracture problems. Subsequently, the basic field variables are independently interpolated in the temporal domain, with the introduction of jump terms representing the discontinuities of variables between adjacent time steps. Then an integral weak form in the temporal domain of the spatially discrete governing equations is constructed and the basic formula of the TDPD is derived. Additionally, an efficient approximation method that directly uses the internal force to calculate its time derivative is further proposed. These characteristics ensure that the TDPD can effectively and efficiently capture the inherent sharp gradient features of wave propagation in solid phase and fluid while controlling spurious numerical oscillations. Several representative numerical examples demonstrate that the TDPD can obtain more accurate results compared with conventional peridynamic solution schemes. Moreover, the TDPD can also be viewed as a novel time integration technique, holding substantial potential for high-precision numerical solutions of hyperbolic equations in diverse physical contexts.