Dirichlet absorbing boundary conditions for classical and peridynamic diffusion-type models
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ORIGINAL PAPER
Dirichlet absorbing boundary conditions for classical and peridynamic diffusion-type models Arman Shojaei1
· Alexander Hermann1 · Pablo Seleson2 · Christian J. Cyron1,3
Received: 7 April 2020 / Accepted: 2 July 2020 © The Author(s) 2020
Abstract Diffusion-type problems in (nearly) unbounded domains play important roles in various fields of fluid dynamics, biology, and materials science. The aim of this paper is to construct accurate absorbing boundary conditions (ABCs) suitable for classical (local) as well as nonlocal peridynamic (PD) diffusion models. The main focus of the present study is on the PD diffusion formulation. The majority of the PD diffusion models proposed so far are applied to bounded domains only. In this study, we propose an effective way to handle unbounded domains both with PD and classical diffusion models. For the former, we employ a meshfree discretization, whereas for the latter the finite element method (FEM) is employed. The proposed ABCs are time-dependent and Dirichlet-type, making the approach easy to implement in the available models. The performance of the approach, in terms of accuracy and stability, is illustrated by numerical examples in 1D, 2D, and 3D. Keywords Peridynamic diffusion model · Absorbing boundary conditions · Nonlocal diffusion · Corrosion · Unbounded domain
1 Introduction This manuscript has been co-authored by UT-Battelle, LLC, under Contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doepublic-access-plan).
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Arman Shojaei [email protected] Alexander Hermann [email protected] Pablo Seleson [email protected] Christian J. Cyron [email protected]
1
Institute of Materials Research, Materials Mechanics, Helmholtz-Zentrum Geesthacht, Max-Planck-Str. 1, 21502 Geesthacht, Germany
2
Computer Science and Mathematics Division, Oak Ridge National Laboratory, One Bethel Valley Road, P.O. Box 2008, MS-6211, Oak Ridge, TN 37831-6211, USA
3
Institute of Continuum and Materials Mechanics, Hamburg University of Technology, Eissendorfer Str. 42, 21073 Hamburg, Germany
Peridynamics (PD) is a recent nonlocal theory that has received a widespread attention in computational mechanics. It has been widely exploited to solve various problems in mechanics and physics. The theory was originally introduced by Silling [56] and Silling et al. [58] to handle material failure in solid structures, which is not an easy task for the classical continuum mechanics (CCM) theory. In fact, the original formulation of PD introduces an equation of motion in solid mechanics based on integro-differe
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