191 research outputs found
Variational Theory and Domain Decomposition for Nonlocal Problems
In this article we present the first results on domain decomposition methods
for nonlocal operators. We present a nonlocal variational formulation for these
operators and establish the well-posedness of associated boundary value
problems, proving a nonlocal Poincar\'{e} inequality. To determine the
conditioning of the discretized operator, we prove a spectral equivalence which
leads to a mesh size independent upper bound for the condition number of the
stiffness matrix. We then introduce a nonlocal two-domain variational
formulation utilizing nonlocal transmission conditions, and prove equivalence
with the single-domain formulation. A nonlocal Schur complement is introduced.
We establish condition number bounds for the nonlocal stiffness and Schur
complement matrices. Supporting numerical experiments demonstrating the
conditioning of the nonlocal one- and two-domain problems are presented.Comment: Updated the technical part. In press in Applied Mathematics and
Computatio
A quasinonlocal coupling method for nonlocal and local diffusion models
In this paper, we extend the idea of "geometric reconstruction" to couple a
nonlocal diffusion model directly with the classical local diffusion in one
dimensional space. This new coupling framework removes interfacial
inconsistency, ensures the flux balance, and satisfies energy conservation as
well as the maximum principle, whereas none of existing coupling methods for
nonlocal-to-local coupling satisfies all of these properties. We establish the
well-posedness and provide the stability analysis of the coupling method. We
investigate the difference to the local limiting problem in terms of the
nonlocal interaction range. Furthermore, we propose a first order finite
difference numerical discretization and perform several numerical tests to
confirm the theoretical findings. In particular, we show that the resulting
numerical result is free of artifacts near the boundary of the domain where a
classical local boundary condition is used, together with a coupled fully
nonlocal model in the interior of the domain
On the stability of the generalized, finite deformation correspondence model of peridynamics
A class of peridynamic material models known as constitutive correspondence
models provide a bridge between classical continuum mechanics and peridynamics.
These models are useful because they allow well-established local constitutive
theories to be used within the nonlocal framework of peridynamics. A recent
finite deformation correspondence theory (Foster and Xu, 2018) was developed
and reported to improve stability properties of the original correspondence
model (Silling et al., 2007). This paper presents a stability analysis that
indicates the reported advantages of the new theory were overestimated.
Homogeneous deformations are analyzed and shown to exibit unstable material
behavior at the continuum level. Additionally, the effects of a particle
discretization on the stability of the model are reported. Numerical examples
demonstrate the large errors induced by the unstable behavior. Stabilization
strategies and practical applications of the new finite deformation model are
discussed
Numerical Methods for the Nonlocal Wave Equation of the Peridynamics
In this paper we will consider the peridynamic equation of motion which is
described by a second order in time partial integro-differential equation. This
equation has recently received great attention in several fields of Engineering
because seems to provide an effective approach to modeling mechanical systems
avoiding spatial discontinuous derivatives and body singularities. In
particular, we will consider the linear model of peridynamics in a
one-dimensional spatial domain. Here we will review some numerical techniques
to solve this equation and propose some new computational methods of higher
order in space; moreover we will see how to apply the methods studied for the
linear model to the nonlinear one. Also a spectral method for the spatial
discretization of the linear problem will be discussed. Several numerical tests
will be given in order to validate our results
Cohesive Dynamics and Brittle Fracture
We formulate a nonlocal cohesive model for calculating the deformation state
inside a cracking body. In this model a more complete set of physical
properties including elastic and softening behavior are assigned to each point
in the medium. We work within the small deformation setting and use the
peridynamic formulation. Here strains are calculated as difference quotients.
The constitutive relation is given by a nonlocal cohesive law relating force to
strain. At each instant of the evolution we identify a process zone where
strains lie above a threshold value. Perturbation analysis shows that jump
discontinuities within the process zone can become unstable and grow. We derive
an explicit inequality that shows that the size of the process zone is
controlled by the ratio given by the length scale of nonlocal interaction
divided by the characteristic dimension of the sample. The process zone is
shown to concentrate on a set of zero volume in the limit where the length
scale of nonlocal interaction vanishes with respect to the size of the domain.
In this limit the dynamic evolution is seen to have bounded linear elastic
energy and Griffith surface energy. The limit dynamics corresponds to the
simultaneous evolution of linear elastic displacement and the fracture set
across which the displacement is discontinuous. We conclude illustrating how
the approach developed here can be applied to limits of dynamics associated
with other energies that - converge to the Griffith fracture energy.Comment: 38 pages, 4 figures, typographical errors corrected, removed section
7 of previous version and added section 8 to the current version, changed
title to Cohesive Dynamics and Brittle Fracture. arXiv admin note: text
overlap with arXiv:1305.453
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