An alternative updated Lagrangian formulation for finite particle method

The Finite Particle Method (FPM) is a typical meshfree particle method, which is developed on the basis of standard smooth particle hydrodynamics (SPH) method. Like other collocational meshfree particle methods, the conventional updated Lagrangian FPM with Eulerian kernels suffers from tension instability problem. To tackle this problem, an updated Lagrangian formulation with Lagrangian kernels for the FPM method is proposed. With this formulation, the configuration is updated for each increment step, but the initial neighboring particles for particle approximation are kept. To avoid saving the information of the initial configuration in the numerical procedure, a step-by-step transformation scheme is proposed for updating the kernels with the configuration. Numerical examples are provided to demonstrate that the proposed numerical method is (i) effective in eliminating tension instability problem; (ii) can provide as good solution as a total Lagrangian scheme at a similar computational cost; and (iii) has potentials in solving fluid-solid interaction problems and problems with very large distortion.