Concurrent semi-Lagrangian reproducing kernel formulation and stability analysis

The semi-Lagrangian Reproducing Kernel (SL RK) has been successfully applied to simulations of extreme deformation problems thanks to intensive advancements in meshfree methods in recent decades. The SL RK shape function is constructed centered at a material point with a fixed spatial radius, and therefore it is a time-dependent function. When the SL RK is introduced in the weak form of the equation of motion, the time derivatives of SL RK lead to convective terms in the semi-discrete equation, significantly reducing the computational efficiency and accuracy. This paper presents a novel approximation scheme in which displacement, velocity, and acceleration are approximated by the same SL RK at a time instant. Consequently, the proposed concurrent SL RK formulation avoids the need to compute the convective terms, offering an effective numerical solution for high-strain rate dynamic problems. Furthermore, an eigenvalue analysis is performed on the semi-discrete equation of motion to investigate the temporal stability of the concurrent SL RK formulation. A comprehensive study of the eigenvalue and critical time step is conducted to identify the effective parameters for the stability of the system. Several manufactured dynamic problems are studied to validate the analytical estimate. The estimated analytical eigenvalue and the critical time step are consistent with the numerical result.