Recent progress in mathematical and computational aspects of peridynamics

Abstract

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.