Stability in Lagrangian and Semi-Lagrangian Reproducing Kernel Discretizations Using Nodal Integration in Nonlinear Solid Mechanics.

Stability analyses of Lagrangian and Semi-Lagrangian Reproducing Kernel (RK) approximations for nonlinear solid mechanics are performed. It is shown that the semi-Lagrangian RK discretization yields a convective term resulting from the non-conservative coverage of material points under the kernel support. The von Neumann stability analysis shows that the discrete equations of both Lagrangian and semi-Lagrangian discretizations are stable when they are integrated using stabilized conforming nodal integration. On the other hand, integrating the semi-Lagrangian discretization with a direct nodal integration yields an unstable discrete system which resembles the tensile instability in SPH. Under the framework of semi-Lagrangian discretization, it is shown that the inclusion of convective term yields a more stable discrete system compared to the semi-Lagrangian discretization without convective term as was the case in SPH. [ABSTRACT FROM AUTHOR]