An algorithm for imposing local boundary conditions in peridynamic models on arbitrary domains

Abstract

Imposing local boundary conditions in nonlocal/peridynamic models is often desired/needed. Fictitious nodes methods (FNMs) are commonly used techniques for this purpose but they are limited, in general, to domains with simple geometry. FNMs also mitigate the well-known peridynamic surface/skin effect at boundaries/surfaces. Here, we introduce a general algorithm that automatically locates mirror nodes for fictitious nodes in the mirror-based FNM, without requiring an explicit mathematical description of the boundary. The algorithm is based on computing a nonlocal gradient, at fictitious nodes, to determine the “generalized” normal direction to the boundary of a domain with arbitrary geometry. We test several FNMs on peridynamic diffusion problems with or without singularities, that exist in the corresponding local models, along the boundaries. We find that the mirror-based FNM works best in agreeing with the classical solutions. We then test the new algorithm with this FNM for diffusion problems in domains with complex geometries, including one with intersecting cracks. The algorithm is general and should work for any type of peridynamic model, including those for problems with moving boundaries and growing cracks, for which enforcement of local-type boundary conditions is desired. Since high accuracy is critical near boundaries of arbitrary shape (including corners, notches, crack tips) in a variety of problems, the new algorithm has potential for high impact.

Convergence Analysis and Numerical Studies for Linearly Elastic Peridynamics with Dirichlet-Type Boundary Conditions

“…The approach is investigated numerically and it shows better agreement with classical solutions, while also being able to handle non-smooth boundaries, and even crack lines. Similar as the mirror FNM idea in [63] and the vanishing horizon idea in [14], in [42] the authors propose two methods for dealing with boundary conditions in one-dimensional bond-based peridynamics model. The Extended Domain Method (EDM) adds a layer to the domain Ω on which an odd reflection of the classical solution is imposed.…”

“…While for a local problem data on the domain's surface can be easily provided by experimentalists through surface measurements, in the nonlocal setting one must provide volumetric data for the boundary collar, which may be difficult (or even impossible) to obtain. Thus, practioners must introduce ad-hoc methods for prescribing both Dirichlet and Neumann-type boundary conditions for nonlocal problem [8,14,16,31,48,58,62,63], and in this work we mainly focus on the Dirichlet-type volume constraints. Most commonly, the convergence of nonlocal solutions to classical counterparts has been studied for homogeneous Dirichlet-type boundary conditions in second order ( [36,37]), or higher order ( [43]) problems.…”

“…The naive FNM was later extended to the Taylor FNM by using Taylor expansion (to linear terms) for points in the boundary collar [52] and the mirror-based FNM by reflecting the values on interior collar nodes to their corresponding mirror nodes for domains with simple geometries [3,4,30,40]. Recently, the mirror-based FNM was further extended to more general domains in [63] for nonlocal diffusion problems, where the authors propose a novel…”

A Peridynamic Model for Crevice Corrosion Damage

“…The thickness is required because of the special way that nonlocal boundary conditions are defined in PD problems. Details on nonlocal boundary conditions are available in [58,59]. The rest of the boundaries (including the symmetry line) are free boundaries and the no-flux conditions is naturally satisfied.…”

A novel and effective way to impose boundary conditions and to mitigate the surface effect in state‐based Peridynamics

“…In addition to that, we show how to implement the fictitious node method in state‐based Peridynamics with a fictitious layer of thickness (instead of as in Reference 23) via the truncated Taylor series up to a general order n . The surface effect is mitigated by the presence of the fictitious nodes and Dirichlet boundary conditions are applied similarly to what is done in References 31,35. On the other hand, we propose an innovative method to impose Neumann boundary conditions in a “peridynamic way”, namely by means of the peridynamic concept of force flux .…”