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In this work, we study the reproducing kernel (RK) collocation method for the peridynamic Navier equation. We first apply a linear RK approximation on both displacements and dilatation, then back-substitute dilatation, and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit and also to the local limit as nonlocal interactions vanish. The stability is shown by comparing the collocation scheme with the standard Galerkin scheme using Fourier analysis. We then apply the RK collocation to the quasi-discrete peridynamic Navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed. The analysis is carried out on a special family of rectilinear Cartesian grids for the RK collocation method with a designated kernel with finite support. We assume the Lam{e} parameters satisfy $lambda geq mu$ to avoid adding extra constraints on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.
Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition
In this paper we present an asymptotically compatible meshfree method for solving nonlocal equations with random coefficients, describing diffusion in heterogeneous media. In particular, the random diffusivity coefficient is described by a finite-dim
The Gaussian kernel plays a central role in machine learning, uncertainty quantification and scattered data approximation, but has received relatively little attention from a numerical analysis standpoint. The basic problem of finding an algorithm fo
The overarching goal of this work is to develop an accurate, robust, and stable methodology for finite deformation modeling using strong-form peridynamics (PD) and the correspondence modeling framework. We adopt recently developed methods that make u
The overarching goal of this work is to develop an accurate, robust, and stable methodology for finite deformation modeling using strong-form peridynamics (PD) and the correspondence modeling framework. We adopt recently developed methods that make u