An energy-based coupling approach to nonlocal interface problems.

Capodaglio, Giacomo; D’Elia, Marta; Bochev, Pavel B.; Gunzburger, Max

doi:10.2172/1618813

Cited by 10 publications

(15 citation statements)

References 27 publications

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“…Ω = ∪ k Ω k , Ω k ∩ Ω l = ∅, and λ(x) = λ k , µ(x) = µ k for x ∈ Ω k . Discussions on the mathematical properties of this heterogeneous system can be found in, e.g., [58]. Specifically, when λ(x) and µ(x) may vary for each material point x, we propose the following formulation:…”

Section: Composite Materials With Discontinuous Materials Propertiesmentioning

confidence: 99%

An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture

Yu,

You,

Trask

2021

Preprint

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Meshfree discretizations of state-based peridynamic models are attractive due to their ability to naturally describe fracture of general materials. However, two factors conspire to prevent meshfree discretizations of state-based peridynamics from converging to corresponding local solutions as resolution is increased: quadrature error prevents an accurate prediction of bulk mechanics, and the lack of an explicit boundary representation presents challenges when applying traction loads. In this paper, we develop a reformulation of the linear peridynamic solid (LPS) model to address these shortcomings, using improved meshfree quadrature, a reformulation of the nonlocal dilitation, and a consistent handling of the nonlocal traction condition to construct a model with rigorous accuracy guarantees. In particular, these improvements are designed to enforce discrete consistency in the presence of evolving fractures, whose a priori unknown location render consistent treatment difficult. In the absence of fracture, when a corresponding classical continuum mechanics model exists, our improvements provide asymptotically compatible convergence to corresponding local solutions, eliminating surface effects and issues with traction loading which have historically plagued peridynamic discretizations. When fracture occurs, our formulation automatically provides a sharp representation of the fracture surface by breaking bonds, avoiding the loss of mass. We provide rigorous error analysis and demonstrate convergence for a number of benchmarks, including manufactured solutions, free-surface, nonhomogeneous traction loading, and composite material problems. Finally, we validate simulations of brittle fracture against a recent experiment of dynamic crack branching in soda-lime glass, providing evidence that the scheme yields accurate predictions for practical engineering problems.

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Section: Composite Materials With Discontinuous Materials Propertiesmentioning

confidence: 99%

An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture

Yu,

You,

Trask

2021

Preprint

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“…In this paper we focus on the prohibitively expensive computational cost, particularly apparent when the ratio between horizon and discretization length becomes large. Moreover, we explore the treatment of virtual nonlocal interfaces, which, for nonlocal problems, is still in its infancy [5,8]. Specifically, we propose a new substructuring-based domain decomposition (DD) approach for meshfree discretizations of nonlocal equations that resembles non-overlapping substructuring DD approaches for PDEs.…”

Section: Introductionmentioning

confidence: 99%

“…Note that even though Euclidean neighborhoods are the standard choice in nonlocal modeling, some recent works have considered the use of square neighborhoods, for which the associated nonlocal problem is well posed[8,28].…”

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confidence: 99%

A FETI approach to domain decomposition for meshfree discretizations of nonlocal problems

Xu,

Glusa,

D'Elia

et al. 2021

Preprint

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We propose a domain decomposition method for the efficient simulation of nonlocal problems. Our approach is based on a multi-domain formulation of a nonlocal diffusion problem where the subdomains share "nonlocal" interfaces of the size of the nonlocal horizon. This system of nonlocal equations is first rewritten in terms of minimization of a nonlocal energy, then discretized with a meshfree approximation and finally solved via a Lagrange multiplier approach in a way that resembles the finite element tearing and interconnect method. Specifically, we propose a distributed projected gradient algorithm for the solution of the Lagrange multiplier system, whose unknowns determine the nonlocal interface conditions between subdomains. Several two-dimensional numerical tests illustrate the strong and weak scalability of our algorithm, which outperforms the standard approach to the distributed numerical solution of the problem. This work is the first rigorous numerical study in a two-dimensional multi-domain setting for nonlocal operators with finite horizon and, as such, it is a fundamental step towards increasing the use of nonlocal models in large scale simulations.

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“…However, the utilization of nonlocal models in applications that could benefit from their improved predictive capabilities is hindered by several modeling and numerical challenges. These include the unresolved treatment of nonlocal interfaces [2,12], the nontrivial prescription of nonlocal volume constraints (the nonlocal counterpart of boundary conditions) [15,23], and the fact that computational costs attendant to the use of nonlocal problems may become prohibitive as the extent of the nonlocal interactions increases; see, e.g., [17,22] for variational methods and [13,Chapter 7] for mesh-free methods. Other critical challenges are related to the uncertain nature of model parameters; in fact, modeling parameters such as δ and those characterizing the kernel, applied forces, and/or sources can be non-measurable, sparse, and/or subject to noise.…”

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confidence: 99%

“…This work is part of a comprehensive effort by the authors to fill the theoretical and practical gaps in the current understanding of nonlocal interfaces (both physical ones and those created by DD solution algorithms) by developing a rigorous nonlocal interface theory for nonlocal diffusion (see the preliminary work [12]), including pure fractional diffusion, and nonlocal mechanics. Our ultimate goal is to design efficient and scalable DD solvers to unlock the full potential of nonlocal models.…”

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confidence: 99%

A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions

Capodaglio¹,

D’Elia²,

Gunzburger³

et al. 2020

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A rigorous mathematical framework is provided for a substructuring-based domaindecomposition approach for nonlocal problems that feature interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation problems is used in which a computational domain is subdivided into non-overlapping subdomains. In the nonlocal setting, this approach is substructuring-based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains. Key results include the equivalence between the global, singledomain nonlocal problem and its multi-domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal domain-decomposition methods.

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An energy-based coupling approach to nonlocal interface problems.

Cited by 10 publications

References 27 publications

An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture

An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture

A FETI approach to domain decomposition for meshfree discretizations of nonlocal problems

A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions

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