INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2015; 102:839–866 Published online 28 July 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4728 A level set enhanced natural kernel contact algorithm for impact and penetration modeling Sheng-Wei Chi 1 , Chung-Hao Lee 2 , Jiun-Shyan Chen 3, * ,† and Pai-Chen Guan 4 1 Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago, IL 60607, USA 2 Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, TX 78712, USA 3 Department of Structural Engineering, University of California, San Diego, CA 92093, USA 4 Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung, Taiwan SUMMARY A natural kernel contact (NKC) algorithm under the framework of the semi-Lagrangian reproducing kernel particle method (semi-Lagrangian RKPM) is proposed to model multi-body contact with specific consideration for impact and penetration modeling. The NKC algorithm utilizes the interaction of the semi- Lagrangian kernel functions associated with contacting bodies to serve as the non-penetration condition. The effects of friction are represented by introducing a layer of the friction-like elasto-plasticity material between contacting bodies. This approach allows the frictional contact conditions and the associated kinematics to be naturally embedded in the semi-Lagrangian RKPM inter-particle force calculation. The equivalence in the Karush–Kuhn–Tucker conditions between the proposed NKC algorithm and the conventional contact kinematic constraints as well as the associated state variable relationships are identified. A level set method is further introduced in the NKC algorithm to represent the contact surfaces without pre-defined potential contact surfaces. The stability analysis performed in this work shows that temporal stability in the semi- Lagrangian RKPM with NKC algorithms is related to the velocity gradient between contacting bodies. The proposed methods have been verified by several benchmark problems and applied to the simulation of impact and penetration processes. Copyright © 2014 John Wiley & Sons, Ltd. Received 31 March 2014; Revised 27 May 2014; Accepted 27 May 2014 KEY WORDS: natural kernel contact; level set; meshfree; reproducing kernel; semi-Lagrangian 1. INTRODUCTION Mesh-based computational methods, such as finite element methods (FEMs) and finite difference (FD) methods, have been widely used in modeling scientific and engineering problems. Because of the need of a quality mesh in construction of the approximation functions, this type of meth- ods, however, encounters mesh-related issues in penetration modeling, where materials undergo extremely large deformation, damage, and separation. Special techniques, such as element erosion [1, 2], hp-adaptivity [3], and extrinsic enrichment of the approximation functions [4, 5], have been commonly adopted to address these issues. Nevertheless, other issues related to mass conservation and surface erosion due to element erosion and the effectiveness of modeling high-density crack formation remain. In addition, contact surface evolution is also arduous for penetration modeling in the conventional contact algorithms [6–9]. The need of defining the contact surfaces a priori in the Lagrange multiplier method [10–12] and its variations, such as the perturbed Lagrange [9, 13] and the augmented Lagrange [14, 15], leads to loss of generality for surface evolution. Although *Correspondence to: Jiun-Shyan Chen, Department of Structural Engineering, University of California, San Diego, CA 92093, USA. † E-mail: js-chen@ucsd.edu Copyright © 2014 John Wiley & Sons, Ltd.