Stable particle methods based on Lagrangian kernels

A large deformation particle method based on the Krongauz-Belytschko corrected-gradient meshfree method with Lagrangian kernels is developed. In this form, the gradient is corrected by a linear transformation so that linear completeness is satisfied. For the test functions, Shepard functions are used; this guarantees that the patch test is met. Lagrangian kernels are introduced to eliminate spurious distortions of the domain of material stability. A mass allocation scheme is developed that captures correct reflection of waves without any explicit application of traction boundary conditions. In addition, the Lagrangian kernel versions of various forms of smooth particle methods (SPH), including the standard forms and the Randles-Libersky modification are presented and studied. Results are obtained for a variety of problems that compare this method to standard forms of SPH, the Randles-Libersky correction and large deformation versions of the element-free Galerkin method.