Abstract
We study some nonlocal models related to peridynamics. These models are parameterized by a horizon that serves as a length scale measuring the range of nonlocal interactions. We focus on robust numerical schemes that can approximate both nonlocal models and their local limits as the horizon parameter vanishes. A representative linear model is used as an illustration. We show the lack of robustness of some standard numerical methods and describe a remedy to get asymptotically compatible schemes by utilizing elements of the recently developed nonlocal vector calculus and nonlocal calculus of variations. Such findings may be useful to ongoing research on modeling and simulations of nonlocal and multiscale problems.
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Acknowledgements
The authors would like to thank Max Gunzburger, Rich Lehoucq, Tadele Mengesha, Michael Parks, Stewart Silling, Florin Bobaru, John Foster, Erdogan Madenci and John Mitchell for their discussions on the subject. This work is supported in part by the U.S. NSF grant DMS-1318586, and AFOSR MURI center for material failure prediction through peridynamics.
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Du, Q., Tian, X. (2015). Robust Discretization of Nonlocal Models Related to Peridynamics. In: Griebel, M., Schweitzer, M. (eds) Meshfree Methods for Partial Differential Equations VII. Lecture Notes in Computational Science and Engineering, vol 100. Springer, Cham. https://doi.org/10.1007/978-3-319-06898-5_6
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DOI: https://doi.org/10.1007/978-3-319-06898-5_6
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