A review of Local-to-Nonlocal coupling methods in nonlocal diffusion and nonlocal mechanics

Abstract

Local-to-Nonlocal (LtN) coupling refers to a class of methods aimed at combining nonlocal and local modeling descriptions of a given system into a unified coupled representation. This allows to consolidate the accuracy of nonlocal models with the computational expediency of their local counterparts, while often simultaneously removing additional nonlocal modeling issues such as surface effects. The number and variety of proposed LtN coupling approaches have significantly grown in recent year, yet the field of LtN coupling continues to grow and still has open challenges. This review provides an overview of the state-of-the-art of LtN coupling in the context of nonlocal diffusion and nonlocal mechanics, specifically peridynamics. We present a classification of LtN coupling methods and discuss common features and challenges. The goal of this review is not to provide a preferred way to address LtN coupling but to present a broad perspective of the field, which would serve as guidance for practitioners in the selection of appropriate LtN coupling methods based on the characteristics and needs of the problem under consideration.

A general and fast convolution-based method for peridynamics: Applications to elasticity and brittle fracture

“…We kept the horizon fixed 𝛿 = 0.1 cm, and changed the number of nodes in the 𝑥 1 and 𝑥 2 directions. Six simulations were conducted with (𝑁 1 , 𝑁 2 ) = (2 7 , 2 6 ), (2 8 , 2 7 ), (2 10 , 2 9 ), (2 10 , 2 10 ), (2 11 , 2 10 ), and (2 12 , 2 11 ), leading to the m-factors: (𝑚 1 , 𝑚 2 ) ≈ (2.5, 3), (5,6), (10,12), (10,24), (20,24), and (40, 48) respectively. The case with (𝑚 1 , 𝑚 2 ) ≈ (10, 24) is chosen to test whether discretization anisotropy influences the results.…”

“…Various attempts have been made to reduce the cost of peridynamic simulations. Coupling the local theory with PD is one approach that uses a local model for parts of the domain, and uses the PD model only at locations near cracks/damage as necessary [10,11]. This approach does not work well for problems in which damage/cracks are (or become) widely distributed throughout the domain, such as in problems like impact fragmentation, etc.…”

Coupling local and nonlocal equations with Neumann boundary conditions

“…From a mathematical point of view, interesting problems arise from coupling local and nonlocal models, see [4,5,14,15,19,22,23,26,29] and references therein. As previous examples of coupling approaches between local and nonlocal regions we refer the reader to [2,4,5,14,15,16,19,22,23,24,26,29,30,31,32] the survey [17] and references therein. Previous strategies treat the coupling condition as an optimization problem (the goal is to minimize the mismatch of the local and nonlocal solutions in a common overlapping region).…”

Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase field fracture by nonlocal operator method

“…In the spirit of nonlocality, PD has been extended in many directions, for example, dual-horizon PD [44,45], peridynamic plate/shell theory [46,47,48,49], mixed peridynamic Petrov-Galerkin method for compressible and incompressible hyperelastic material [50,51], phase field based peridynamic damage model for composite structures [52], wave dispersion analysis of PD [53], damage mechanism in PD [54], coupling scheme for state-based PD and FEM [55,56], higher-order peridynamic material models for elasticity [57], to list a few.…”