Peridynamic Simulations of Infinite Regions Using a Perfectly Matched Layer
Absorbing boundary conditions for non-local methods (peridynamics) will be presented. Peridynamics is a non-local method in which the force term of the momentum equation is evaluated as the convolution of the displacement with a suitable micro-force funct
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Peridynamic Simulations of Infinite Regions Using a Perfectly Matched Layer Raymond A. Wildman and George A. Gazonas
Abstract Absorbing boundary conditions for non-local methods (peridynamics) will be presented. Peridynamics is a non-local method in which the force term of the momentum equation is evaluated as the convolution of the displacement with a suitable micro-force function, or kernel. Though the method involves integration, it is based on a differential equation formulation and requires specialized absorbing boundary conditions to simulate infinite regions. Applications include simulation indentation experiments, which can provide verification/validation of the method. Perfectly matched layers (PML) are absorbing boundary regions that are reflection-less at their interface and decay impinging waves exponentially. Here, PML are applied to peridynamics by treating the kernel as the convolution of the displacement with the second derivative of a nascent Dirac delta distribution, such as the appropriately defined Gaussian distribution. In this sense, the peridynamics formulation can be split into a coupled auxiliary field formulation, suitable for a PML application.
29.1
Introduction
Peridynamics, a non-local formulation of elastodynamics, was introduced as a method that more easily incorporates discontinuities such as cracks and damage in a numerical simulation [1]. The formulation treats an elastic solid as a collection of nodes, connected by bonds that break if stretched past some predefined limit. While most peridynamics work has focused on simulating problems with free or fixed boundary conditions, there are applications in which the simulation of an infinite medium may be useful, such as wave or crack propagation in a half-space. Absorbing boundary conditions are a way of simulating an infinite medium by absorbing any impinging waves at the computational boundaries so they do not reflect back into the simulation. A PML is such an absorbing boundary, and was originally introduced for electromagnetic simulations [2, 3]. PMLs differ from traditional absorbing boundary conditions in that they are an absorbing layer, placed between the computational region of interest and the truncation of the grid or mesh. They can also be thought of as an artificial, anisotropic absorbing material, which is why the flexibility of a state-based peridynamics is necessary [4]. PMLs have two important qualities: First, waves in a PML decay exponentially, and second, in their analytic form, the interface of a PML and the computational region is reflectionless. Since their introduction, PMLs have been extended to many different types of media [5, 6], different numerical methods [7, 8], and different fields [9–11]. This paper implements a peridynamic formulation of elastodynamics in one (1D) and two dimensions (2D) and terminates the boundary with a PML. As is discussed, the use of a PML is facilitated with an auxiliary field formulation, derived from state-based peridynamics for 2D. In 1D, the auxiliary formulation of peridyn
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