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When high-order numerical approximation method is applied to model the propagation of shock wave or discontinuity, it usually creates unstable unreal numerical oscillations around the discontinuous regions. In this research, we propose a non-oscillation meshfree scheme based on reproducing kernel particle method (RKPM) which can maintain the accuracy and minimize the oscillation in the modeling of shock wave propagation. In the proposed method, the original influence domain of high-order RK approximation is divided into several subdomains. Then we apply low-order RK approximation within each subdomain. Instead of directly using the discrete particles to build the numerical approximation, we consider that the high-order approximation is constructed by the summation of those low-order approximations multiplied by a local weight function. By adjusting these local weights with the "smoothness indicator", we can determine the "effect" of the corresponding subdomain and the discrete particles inside this subdomain. Therefore, the subdomain containing discontinuity would not participate in the high-order approximation, and the numerical oscillation is automatically suppressed. The proposed method does not need artificial viscosity or numerical damping to stabilize the solution. Several benchmark problems with shock wave propagation are tested. The results show that the proposed method can maintain high-order accuracy without numerical oscillation.

Slope stability analyses are often performed using Limit Equilibrium Methods (LEMs) and Finite Element Method (FEM). However, these methods can only model the slope condition up to the point of failure. Meshfree methods, which do not require a mesh or a grid in the simulation process, have the potential to model the post-failure slope behavior as mesh tangling would not occur to cause numerical instability and non-convergence. Hence, while retaining the benefits of conventional numerical schemes, meshfree method can be more advantageous when problems with large deformation are encountered. In this paper, Semi-Lagrangian Reproducing Kernel Particle Method (SLRKPM), which is a type of meshfree method, is extended to analyze geomechanics problems such as the stability of a slope and post failure slope behavior. The results from SLRKPM agree well with those from convention methods (LEMs and FEM) in terms of factor-of-safety and failure surface. In addition, SLRKPM is able to simulate the slope failure process and successfully capture the development of shear band. This proves that SLRKPM has a significant advantage over FEM when dealing with problems involving large deformation and failure of geomaterials.

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.

This paper proposed the new Isoparametric Reproducing Kernel Particle Method (IsoRKPM) for modeling nonlinear plate and shell deformation problems. The Reproducing Kernel shape functions are constructed on a two-dimensional parent domain. Following the concept of isoparametric mapping, the RK shape functions are directly used to approximate the plate geometry. A High Order Nodal Integration (HONI) is developed to integrate the Galerkin weak form of the Mindlin plate equilibrium equations. The proposed IsoRKPM with HONI is applied to solve several benchmark problems. Both modal and convergence analyses show that HONI provides more stable and accurate solutions than the nodal integration method.

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.

This paper employs the reproducing kernel (RK) approximation for evaluation of field theory-based incompatibility tensor in a polycrystalline plasticity simulation. The modulation patterns, which is interpreted as mimicking geometrical-type dislocation substructures, are obtained based on the proposed method. Comparisons are made using FEM and RK based approximation methods among different support sizes and other evaluation conditions of the strain gradients. It is demonstrated that the evolution of the modulation patterns needs to be accurately calculated at each time step to yield a correct physical interpretation. The effect of the higher order strain derivative processing zone on the predicted modulation patterns is also discussed.

This work introduces the weighted collocation method with reproducing kernel approximation to solve the inverse Laplace equations. As the inverse problems in consideration are equipped with over-specified boundary conditions, the resulting equations yield an overdetermined system. Following our previous work, the weighted collocation method using a least-squares minimization has shown to solve the inverse Cauchy problems efficiently without using techniques such as iteration and regularization. In this work, we further consider solving the inverse problems of Laplace type and introduce the Shepard functions to deal with singularity. Numerical examples are provided to demonstrate the validity of the method.

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.