Tadele Mengesha

Results on Nonlocal Boundary Value Problems

In this article, we provide a variational theory for nonlocal problems where nonlocality arises due to the interaction in a given horizon. With this theory, we prove well-posedness results for the weak formulation of nonlocal boundary value problems with Dirichlet, Neumann, and mixed boundary conditions for a class of kernel functions. The motivating application for nonlocal boundary value problems is the scalar stationary peridynamics equation of motion. The well-posedness results support practical kernel functions used in the peridynamics setting. We also prove a spectral equivalence estimate which leads to a mesh size independent upper bound for the condition number of an underlying discretized operator. This is a fundamental conditioning result that would guide preconditioner construction for nonlocal problems. The estimate is a consequence of a nonlocal Poincaré-type inequality that reveals a horizon size quantification. We provide an example that establishes the sharpness of the upper bound in the spectral equivalence.

The bond-based peridynamic system with Dirichlet-type volume constraint

In this paper, the bond-based peridynamic system is analysed as a non-local boundary-value problem with volume constraint. The study extends earlier works in the literature on non-local diffusion and non-local peridynamic models, to include non-positive definite kernels. We prove the well-posedness of both linear and nonlinear variational problems with volume constraints. The analysis is based on some non-local Poincare-type inequalities and the compactness of the associated non-local operators. It also offers careful characterizations of the associated solution spaces, such as compact embedding, separability and completeness. In the limit of vanishing non-locality, the convergence of the peridynamic system to the classical Navier equations of elasticity with Poisson ratio ¼ is demonstrated.

NONLOCAL KORN-TYPE CHARACTERIZATION OF SOBOLEV VECTOR FIELDS

A new nonlocal characterization of Sobolev vector fields in the spirit of Korns inequality is obtained. As an application of this result, a nonlocal means of identification of rigid motions is given. A nonlocal characterization of vector fields with bounded deformation is also presented. A compactness criteria is proved for bounded sequences of vector fields in Lp.

On the variational limit of a class of nonlocal functionals related to peridynamics

In this paper we study variational problems on a bounded domain for a nonlocal elastic energy of peridynamic-type which result in nonlinear systems of nonlocal equations. The well-posedness of variational problems is established via a careful study of the associated energy spaces. In the event of vanishing nonlocality we establish the convergence of the nonlocal energy to a corresponding local energy using the method of -convergence. Building upon existing techniques, we prove an Lp-compactness result (on bounded domains) based on near-boundary estimates that is used to study the variational limit of minimization problems subject to various volumetric constraints. For energy functionals in suitable forms, we find the corresponding limiting energy explicitly. As a special case, the classical Navier-Lame potential energy is realized as a limit of linearized peridynamic energy offering a rigorous connection between the nonlocal peridynamic model to classical mechanics for small uniform strain.

Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization

We examine the composition of the L ∞ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory. We identify local representation formulas that in the fine phase limit provide upper bounds on the limit superior of the L ∞ norms of gradient fields. The local representation formulas are expressed in terms of the weak limit of the gradient fields and local corrector problems. The upper bounds may diverge according to the presence of rough interfaces. We also consider the fine phase limits for layered microstructures and for sufficiently smooth periodic microstructures. For these cases we are able to provide explicit local formulas for the limit of the L ∞ norms of the associated sequence of gradient fields. Local representation formulas for lower bounds are obtained for fields corresponding to continuously graded periodic microstructures as well as for general sequences of oscillatory coefficients. The representation formulas are applied to problems of optimal material design.