1 Introduction
Nonlocal models can accurately describe physical phenomena arising from longrange forces at the microscale and mesoscale. Such phenomena cannot be accounted for by partial differential equations (PDEs) where interaction is limited to points that are in direct contact with each other. As a result, mathematically nonlocal models are represented by integral operators, which are better suited to capture interactions occurring across a distance.
Examples of applications where longrange forces are essential for predictive simulations can be found in a diverse spectrum of scientific applications such as anomalous subsurface transport benson2000application ; schumer2001eulerian ; schumer2003multiscaling ; delgoshaie2015non , fracture mechanics silling2000reformulation ; ha2011characteristics ; littlewood2010simulation , image processing buades2010image ; gilboa2007nonlocal ; gilboa2008nonlocal ; lou2010image , magnetohydrodynamics schekochihin2008mhd , multiscale and multiphysics systems askari2008peridynamics ; alali2012multiscale
bates1999integrodifferential ; fife2003some , and stochastic processes meerschaert2011stochastic ; d2017nonlocal ; burch2014exit .Although research on nonlocal models has recently intensified, there is still a lack of a mathematically rigorous and physically consistent nonlocal interface (NLI) theory. Examples of NLI research are few and far between in the literature with alali2015peridynamics ; seleson2013interface being perhaps the only published work in this field. However, the NLI formulations in these papers do not provide a rigorous theoretical framework because they do not address existence and uniqueness of solutions to NLI problems, nor do they establish formal convergence to local limits.
The absence of such a framework has hampered the wider adoption of nonlocal models in applications that require proper handling of material interfaces, such as multimaterial simulations, fluidstructure interaction, and contact, to name a few. The main goal of this work is to fill this theoretical gap by providing a rigorous NLI theory that would establish a much needed foundation for the further development and application of nonlocal models in science and engineering applications.
Nonlocal interface problems are challenging for a number of reasons rooted in the need to treat a local physical interface arising from material discontinuities in nonlocal terms. In particular, one of the key technical challenges is to discover appropriate nonlocal transmission conditions capable of providing physically consistent wellposed NLI problems that converge to classical formulations as the characteristic parameter of the nonlocal model goes to zero.
Our strategy is motivated by an energybased description of classical local interface problems in which the local energy of the coupled system is minimized subject to constraints modeling the physics of the local interface. Here we follow this template to obtain mathematically rigorous NLI formulations by minimizing the nonlocal energy of the system subject to constraints describing the physics of the nonlocal interface. In so doing, we obtain nonlocal transmission conditions and wellposed NLI problems that converge to their local counterparts at the local limit.
The resulting mathematical NLI framework is validated through numerical experiments that confirm the theoretical predictions, provide additional insights on the NLI model and suggest followup research directions that will be pursued in forthcoming work.
The paper is structured as follows: in Section 2, we review a classical energybased formulation of a local interface problem, which provides the template for its nonlocal counterpart. The fundamentals of nonlocal problems are highlighted in Section 3, where we introduce a nonlocal volume constrained problem. The nonlocal interface problem is introduced in Section 4, and local limits are derived in Section 5. Section 6 presents our numerical results.
2 Local interface problem
In this section, we review a classical energybased formulation of local interface problems for secondorder elliptic PDEs. Let and be two disjoint open and bounded subsets of , , with boundaries and , respectively, and such that .
Let . The interface between the domains is denoted by and is defined as . To facilitate the link between local and nonlocal interface problems, let us also set and . Note that due to the configuration of the domains, and . Schematics of the domains for the local interface problem are given in Figure 1.
2.1 Local energy minimization principle
Consider the following (local) energy functional
(1)  
where the functions and represent the different material properties of the two domains and are assumed to be positive and bounded from below. The functions , are known.
We obtain a particular instance of a local interface problem by choosing a specific constrained minimization setting for (1). To this end, let us define the following energy spaces, for
(2)  
Note that, due to the configuration of the domains depicted in Figure 1,
. Tensor product spaces are then defined as
and .Minimization Principle 2.1.
Given , , , , and , find such that
subject to the constraints
(3) 
.
The second constraint in (3), i.e., the continuity of the states across the interface, is a modeling assumption about the physics of the interface, which gives rise to a specific flavor of a local interface problem. This constraint is a particular case of a general coupling condition given by , where and are some given functions. To avoid unnecessary technical details that are not essential for our purposes we do not consider this more general type of interface conditions in this paper.
2.2 Weak formulation
2.3 Strong formulation
As usual, we derive the strong form of the interface problem from the weak formulation (4) by assuming that and are sufficiently regular. Collecting terms, integrating by parts, and taking into account that yields
(5)  
Because and are arbitrary on and , respectively, we may first set arbitrary on , on , and on and then set arbitrary on , on , and on to obtain from (5) the strong forms on , , of the subdomain equations. Substituting these equations back into (5) leaves us with
Using that on we then recover from this equation the flux continuity condition . Thus, the strong (PDE) form of the local interface problem corresponding to the Minimization Principle (2.1) is given by
(6)  
(7)  
(8)  
(9)  
(10) 
We note that the strong form of the interface problem contains the flux continuity condition (10) that was not explicitly present in Minimization Principle 1. This condition is a consequence of (9) that, as already mentioned, is a modeling assumption about the physics of the interface. Interfaces for which both the jumps in the state and in the normal flux are zero across the interface are known as perfect interfaces Javili_14_CMAME . In contrast, interfaces for which one or both of these quantities are discontinuous across the interface are known as imperfect; see, e.g. Javili_14_CMAME .
3 Nonlocal volume constrained problems
In this section, we review the fundamentals of nonlocal volume constrained problems. Let be an open and bounded subset of . Given a positive real number , we define the interaction domain associated with as follows
(11) 
Note that depends on , even though it is not written explicitly. Figure 2 shows an example of a twodimensional domain and its interaction region.
3.1 Nonlocal energy minimization principle
In this work we use an energybased characterization of nonlocal volume constrained problems which mirrors the Dirichlet principle for the gradient operator. Specifically, we seek the states of the nonlocal model as suitably constrained minimizers of the following nonlocal energy functional:
(12) 
The function is referred to as the kernel and is required to satisfy
(13) 
Let us define the following function spaces
(14)  
Minimization Principle 3.1.
Given , , and , find such that
subject to on .
We refer to the constraint in the Minimization Principle 3.1 as a Dirichlet volume constraint because it generalizes the standard (local) Dirichlet boundary condition.
3.2 Weak formulation
The necessary optimality condition of the Minimization Principle 3.1 is given by the following variational equation: find such that on and
(15) 
for all .
3.3 Strong formulation
To state the strong form of (15) we recall the nonlocal diffusion operator
(16) 
and the nonlocal Green’s identity Du_13_MMMAS
(17)  
Using (17) and the fact that on one can transform (15) into the following equation
(18) 
Since is arbitrary on one then easily obtains the strong form of the nonlocal volume constrained problem
(19)  
4 Nonlocal interface problem
In this section we derive a mathematically rigorous formulation of a nonlocal interface problem that provides a physically consistent extension of the perfect local interface problem in Section 2. This nonlocal counterpart of a perfect interface problem represents the main contribution of the present work.
We retain the domain configuration from Section 2, i.e., and are disjoint bounded subsets of such that , and . In the nonlocal interface problem, the domains interact with each other through regions that have nonzero measure. These regions are defined in different ways depending on the relative location of and and on two parameters and . Specifically, for and , we introduce:

: the (external) interaction domain of .

: the region of that interacts with when .

: the region of that interacts with when .

: the analogue of on .

: the analogue of on .
Figures 3, 4, and 5 illustrate the geometric configuration for the nonlocal interface problem and the various subdomains involved.
We also introduce the set if . Note that by definition and for any and . The kernel function is defined for all as follows:
(20) 
where is the indicator function on the set . Examples of explicit definitions for the functions will be given in the local limits analysis.
4.1 Nonlocal energy minimization principle
Following the energybased description of the local perfect interface problem we start with defining the nonlocal energy of the system as follows
(21) 
Using that and are disjoint, we split the energy in two parts, associating the first to and the second to , i.e.
(22)  
Let us define the following function spaces
(23)  
The constrained space is defined as
(24) 
We also introduce the tensor product spaces and .
Minimization Principle 4.1.
Given , and , find such that
subject to the constraints
(25) 
.
4.2 Weak formulation
4.3 Strong formulation
The strong form of the nonlocal interface problem can be obtained from the weak form as follows. We begin by denoting the double integral terms in (26) by , , , and , respectively, so that the weak equation assumes the form
(27) 
To transform these terms we will use the following identity
(28) 
where and are two generic subsets of . For the first term in equation (27), using that on , we have
(29)  
For , we use again that on to obtain:
(30)  
A few comments on the last equality: in the first term we used that on , whereas in the second term the outer integral is nonzero only on , thus we used the condition on . Note that the inner integral in the second term is nonzero only on , because the region of influence is determined a ball of radius , from the definition of . Again in the second term, used the fact that on and substituted with in the inner integral. For the term labeled in equation (27) we use the same reasoning as for and obtain:
(31)  
For the last term in equation (27), labeled , we proceed as for to get:
(32)  
To obtain the strong form of the nonlocal interface problem, we now collect all contributions from , , and and exploit the fact that and are independent of each other and arbitrary in and . The resulting nonlocal subdomain problems are given by
(33) 
and
(34) 
respectively. Next, we isolate terms in (33) and (34) that do not interact with and respectively. After substituting (20), we obtain the following two equations:
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