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SUMMARYA 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 semiLagrangian 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 semiLagrangian 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.

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In this work, the weighted reproducing kernel collocation method (weighted RKCM) is introduced to solve the inverse Cauchy problems governed by both homogeneous and inhomogeneous second-order linear partial differential equations. As the inverse Cauchy problem is known for the incomplete boundary conditions, how to numerically obtain an accurate solution to the problem is a challenging task. We first show that the weighted RKCM for solving the inverse Cauchy problems considered is formulated in the least-squares sense. Then, we provide the corresponding error analysis to show how the errors in the domain and on the boundary can be balanced with proper weights. The numerical examples demonstrate that the weighted discrete systems improve the accuracy of solutions and exhibit optimal convergence rates in comparison with those obtained by the traditional direct collocation method. It is shown that neither implementation of regularization nor implementation of iteration is needed to reach the desired accuracy. Further, the locality of reproducing kernel approximation gets rid of the ill-conditioned system.