Recent Progress in Mathematical and Computational Aspects of Peridynamics
Recent developments in the mathematical and computational aspects of the nonlocal peridynamic model for material mechanics are provided. Based on a recently developed vector calculus for nonlocal operators, a mathematical framework is constructed that has proved useful for the mathematical analyses of peridynamic models and for the development of finite element discretizations of those models. A specific class of discretization algorithms referred to as asymptotically compatible schemes is discussed; this class consists of methods that converge to the proper limits as grid sizes and nonlocal effects tend to zero. Then, the multiscale nature of peridynamics is discussed including how, as a single model, it can account for phenomena occurring over a wide range of scales. The use of this feature of the model is shown to result in efficient finite element implementations. In addition, the mathematical and computational frameworks developed for peridynamic simulation problems are shown to extend to control, coefficient identification, and obstacle problems.
KeywordsPeridynamics Nonlocal vector calculus Variational forms Multiscale methods Finite element method Optimal control Obstacle problems
MD: supported by the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4). Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.
QD: supported in part by the US NSF grant DMS-1558744, the AFOSR MURI Center for Material Failure Prediction Through Peridynamics, and the OSD/ARO/MURI W911NF-15-1-0562 on Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications.
MG: supported by the US NSF grant DMS-1315259, US Department of Energy Office of Science grant DE-SC0009324, US Air Force Office of Scientific Research grant FA9550-15-1-0001, and DARPA Equips program through the Oak Ridge National Laboratory.
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