Variational theory and domain decomposition for nonlocal problems

Abstract

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 Poincaré 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. Dirichlet boundary value problem( 1.2) Let n and d denote the dimensions of the function space and the spatial domain, respectively. Ω ⊂ R d is a bounded domain, BΩ is given in (2.1), b is given, and u(x) ∈ R n is prescribed for x ∈ R d \Ω. We prescribe the value of u(x) outside Ω and not just on the boundary of Ω, owing to the nonlocal nature of the problem.Nonlocal models are useful where classical (local) models cease to be predictive. Examples include porous media flow [18,19,20], turbulence [21], fracture of solids, stress fields at dislocation cores and cracks tips, singularities present at the point of application of concentrated loads (forces, couples, heat, etc.), failure in the prediction of short wavelength behavior of elastic waves, microscale heat transfer, and fluid flow in microscale channels [22]. These are also cases where microscale fields are nonsmooth. Consequently, nonlocal models are also useful for multiscale modeling. Recent examples of nonlocal multiscale modeling include the upscaling of molecular dynamics to nonlocal continuum mechanics [23], and development of a rigorous multiscale method for the analysis of fiber-reinforced composites capable of resolving dynamics at structural length scales as well as the length scales of the reinforcing fibers [24]. Progress towards a nonlocal calculus is reported in [25]. Development and analysis of a nonlocal diffusion equation is reported in [26,27,28]. Theoretical developments for general class of integro-differential equation related to the fractional Laplacian are presented in [29,30,31]. Mathematical and numerical analysis for linear nonlocal peridynamic boundary problems appears in [32,33]. We discuss in §2 some specific contexts where the nonlocal operator L appears, and the assumptions placed upon L by those interpretations.

On the Role of the Influence Function in the Peridynamic Theory

“…As a prime example, peridynamic solid mechanics models admit discontinuous displacement fields, even in their strong forms, making them suitable for modeling material failure and fracture (Ha and Bobaru (2010)). Mathematical investigations of the peridynamic theory have lead to the development of a nonlocal calculus (Gunzburger and Lehoucq (2010)), rigorous analysis of peridynamic operators , Du and Zhou (2010), Emmrich and Weckner (2007)), and development of the first nonlocal domain decomposition theory (Aksoylu and Parks (2009)). …”

A Convergent Adaptive Finite Element Algorithm for Nonlocal Diffusion and Peridynamic Models

“…More theoretical studies of related volume-constraint problems can be found in [19,30,20]. Within the context of PD-based nonlocal models, there have been a variety of numerical methods implemented for their approximations including finite difference, finite element, quadrature, and particle-based methods [3,9,15,24,27,29,35,38,42]. Given the ability of nonlocal PD models to simulate cracks or fractures, an adaptive method is a natural way to reduce the computational cost.…”

On Nonlocal Modeling in Continuum Mechanics

“…It is also worth mentioning that higher-order spatial partial derivatives are a common approach used to introduce a length scale into a governing equation, which is necessary for modeling phenomena at different geometric scales (Seleson et al 2009). Capturing nonlocality via domain decomposition and variational calculus techniques is shown in (Di Paola et al 2010;Aksoylu, Parks 2011).…”