Semi-Lagrangian reproducing kernel particle method for fragment-impact problems P.C. Guan a, b , S.W. Chi c , J.S. Chen c, * , T.R. Slawson d , M.J. Roth d a Computation and Simulation Center, National Taiwan Ocean University, Keelung, Taiwan b Department of Systems Engineering & Naval Architecture, National Taiwan Ocean University, Keelung, Taiwan c Civil & Environmental Engineering Department, University of California Los Angeles (UCLA), Los Angeles, USA d U.S. Army Engineer Research and Development Center, Vicksburg, MS, USA article info Article history: Received 26 December 2010 Received in revised form 24 May 2011 Accepted 3 August 2011 Available online 29 September 2011 Keywords: Semi-Lagrangian Reproducing kernel particle method Penetration Meshfree abstract Fragment-impact problems exhibit excessive material distortion and complex contact conditions that pose considerable challenges in mesh based numerical methods such as the finite element method (FEM). A semi-Lagrangian reproducing kernel particle method (RKPM) is proposed for fragment-impact modeling to alleviate mesh distortion difficulties associated with the Lagrangian FEM and to minimize the convective transport effect in the Eulerian or Arbitrary Lagrangian Eulerian FEM. A stabilized non- conforming nodal integration with boundary correction for the semi-Lagrangian RKPM is also proposed. Under the framework of semi-Lagrangian RKPM, a kernel contact algorithm is introduced to address multi-body contact. Stability analysis shows that temporal stability of the kernel contact algo- rithm is related to the velocity gradient between two contacting bodies. The performance of the proposed methods is examined by numerical simulation of penetration processes. Published by Elsevier Ltd. 1. Introduction Structures subjected to fragment-impact conditions exist in many construction, mining, oil drilling, and defense applications. The mechanics in these types of problems involve extremely high strains and strain rates, large degree of material damage and frag- mentation, and complex contact conditions. The solutions of these problems are typically rough (non smooth), leading to ill- conditioning in the discrete equations of numerical methods. A typical example is the mesh distortion or even entanglement in the Lagrangian finite element method (FEM) which results in poor solution accuracy, instability, and solution divergence. Moving mesh and Arbitrary Lagrangian Eulerian (ALE) FEMs [14,15] were proposed to overcome mesh distortion and entanglement diffi- culties. However, the convective transport effects in these methods yield non self-adjoint systems and require numerical techniques such as streamline upwinding or Petrov-Galerkin methods [2,3,14] to remedy the associated numerical spurious oscillations. Further, to avoid mesh entanglement induced numerical instability, algo- rithms such as employment of erosion models [17] have been introduced in finite elements. This introduces numerical diffusion of material damage and leads to artificial material degradation inconsistent with the physical material damage. In FEMs, the quality of approximation is strongly tied to the geometry of mesh topology. For nearly one and a half decades, there has been an increasing effort in developing a class of numerical methods to weaken the interrelationship between the quality of approximation and the topology of mesh connectivity among discrete points. These methods are collectively called the meshfree methods. The reproducing kernel particle method (RKPM) [16,8] is one such method where the approximation functions are constructed entirely based on the nodal positions of a set of discrete points without the use of nodal connectivity information. This reduces the strong dependency between approximation accuracy and mesh quality, alleviating the mesh distortion type difficulties in large deformation analysis. This unique property allows RKPM to model extremely large deforma- tion problems using a total Lagrangian formulation [8], and it was successfully applied to elastomeric and metal forming applications [9e11,20], among others. It has been shown [8] that the Lagrangian kernels maintain the same coverage of neighboring particles and thus kernel stability for reproducing conditions is conserved. Similar argument has been made in [23] and shown that the tensile instability does not occur when using Lagrangian kernel. For large deformation problems involving severe material damage and separation, the deformation gradient is no longer * Corresponding author. E-mail address: jschen@seas.ucla.edu (J.S. Chen). Contents lists available at SciVerse ScienceDirect International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng 0734-743X/$ e see front matter Published by Elsevier Ltd. doi:10.1016/j.ijimpeng.2011.08.001 International Journal of Impact Engineering 38 (2011) 1033e1047