Semi-Lagrangian reproducing kernel formulation and application to modeling earth moving operations

Lagrangian formulation has traditionally been introduced for solids under large deformation. Earth-moving operations on the other hand, involve excessive material deformation and damage that cannot be modeled by the Lagrangian formulation. In this paper, a Semi-Lagrangian reproducing kernel (RK) formulation is introduced for modeling earth-moving operations. von Neumann stability analyses of Lagrangian and Semi-Lagrangian Reproducing Kernel (RK) formulations are first performed. The analysis results show that Semi-Lagrangian weak form integrated by a direct nodal integration (DNI), resembling Smoothed Particle Hydrodynamics (SPH), leads to a tensile instability that is inconsistent with the material stability. On the contrary, Semi-Lagrangian RK weak form integrated using the stabilized conforming nodal integration (SCNI) exhibits consistent numerical stability and material stability. The stable time steps for Lagrangian and Semi-Lagrangian RK formulations using central difference are also estimated and an enhanced stability is observed when the weak forms are integrated by SCNI compared to that using DNI or one-point Gauss quadrature. Earth moving simulation is performed to demonstrate the applicability of the proposed Semi-Lagrangian RK formulations to excessive material deformation and fragment problems.

[1]  Wing Kam Liu,et al.  Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures , 1996 .

[2]  R. P. Ingel,et al.  An approach for tension instability in smoothed particle hydrodynamics (SPH) , 1995 .

[3]  T. Belytschko,et al.  Stable particle methods based on Lagrangian kernels , 2004 .

[4]  Hui-Ping Wang,et al.  A Lagrangian reproducing kernel particle method for metal forming analysis , 1998 .

[5]  Jiun-Shyan Chen,et al.  Non‐linear version of stabilized conforming nodal integration for Galerkin mesh‐free methods , 2002 .

[6]  Jiun-Shyan Chen,et al.  A stabilized conforming nodal integration for Galerkin mesh-free methods , 2001 .

[7]  J. Monaghan.  SPH without a Tensile Instability , 2000 .

[8]  T. Belytschko,et al.  Nodal integration of the element-free Galerkin method , 1996 .

[9]  Ted Belytschko,et al.  Stability Analysis of Particle Methods with Corrected Derivatives , 2002 .

[10]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[11]  Jiun-Shyan Chen,et al.  Lagrangian Meshfree Formulation for Analysis of Geotechnical Materials , 2001 .

[12]  Martin W. Heinstein,et al.  An analysis of smoothed particle hydrodynamics , 1994 .

[13]  Ted Belytschko,et al.  A unified stability analysis of meshless particle methods , 2000 .

[14]  S. Attaway,et al.  Smoothed particle hydrodynamics stability analysis , 1995 .

[15]  Sivakumar Kulasegaram,et al.  Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations , 2000 .

[16]  Jiun-Shyan Chen,et al.  Stability in Lagrangian and Semi-Lagrangian Reproducing Kernel Discretizations Using Nodal Integration in Nonlinear Solid Mechanics , 2007 .