A time-discontinuous peridynamic method for coupled thermomechanical and transient heat conduction problems

Spurious numerical oscillations frequently arise when solving hyperbolic differential equations under impact loading using numerical methods. These oscillations, often referred to as the Gibb's phenomenon, resulting in significant disparities between numerical and analytical solutions. To mitigated these discrepancies and improve the accuracy of numerical solutions, this study presents a time-discontinuous peridynamic method (TDPD) for simulating the propagation heat and stress waves in transient heat conduction and coupled thermomechanical problems. In this method, the non-Fourier heat conduction model is reformulated from spatial differential equations into integral equations to simulate transient heat conduction. Additionally, the basic equations for weakly and fully coupled thermomechanical problems within the peridynamics framework are provided separately by combining the Fourier heat conduction model with the dynamic equation. 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. Furthermore, an integral weak form in the temporal domain of the spatially discrete governing equations is constructed and the basic formula of TDPD is derived. These characteristics ensure that TDPD can effectively capture the sharp gradient features inherent in heat and stress wave propagation while controlling spurious numerical oscillations. Several representative numerical examples demonstrate that TDPD yields more accurate results compared to conventional peridynamic solution schemes. Moreover, 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.