A nonlocal vector calculus and nonlocal models for diffusion and mechanics
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Date and time:Ìý
Friday, March 22, 2013 - 1:00pm
Abstract:Ìý
Based on notions for nonlocal fluxes between volumes and nonlocal balance laws and a nonlocal vector calculus we have developed, we introduce nonlocal models for diffusion and the nonlocal peridynamics continuum model for mechanics. A feature of the nonlocal problems that has important practical consequences are that constraints, e.g., of Dirichlet type, are applied over volumes and not along bounding surfaces. A brief review of the nonlocal calculus is given, including definitions of nonlocal divergence, gradient, curl operators and derivations of a nonlocal Gauss theorem and Green’s identities.
Through appropriate limiting processes, relations between the nonlocal operators and their differential counterparts are established. The nonlocal calculus is used to define weak formulations of the nonlocal diffusion and mechanics problems which are then analyzed, showing, for example, that unlike elliptic partial differential equations, these problems do not necessary result in the smoothing of data. We briefly consider finite element methods for nonlocal problems, focusing on solutions containing jump discontinuities; in this setting, discontinuous Galerkin methods are conforming and nonlocal problems can lead to optimally accurate approximations.