Asymptotic Compatibility of Parameterized Nonlocal Optimal Control and Design Problems

This dissertation studies two types of nonlocal optimization problems motived by peridynamics, which is a contemporary family of nonlocal models used for describing physical phenomena in solids that lead to inherent discontinuities in the materials. The first type of problem studied is an optimal control problem where we vary the force applied to a fixed material to reach a desired displacement as closely as possible; the second type of problem is an optimal design problem where the force is fixed but we may choose the material, to again yield a desired displacement of said material.

The relation to the classical Navier-Lamé model of linear elasticity is made apparent for both types of problems by studying our problems in the limit as the horizon, or nonlocal modeling parameter, vanishes. Then, both problems are discretized with the finite element method, and convergence of solutions for the corresponding families of the discrete problems is demonstrated.

However, the highlight of this dissertation is new asymptotic compatibility results in both the optimal control and optimal design settings, which show that we may approximate classical elasticity problems with nonlocal discrete problems for any sequences of the horizon and discrete parameters simultaneously taken to approach zero. The case of optimal control problems is studied for peridynamics models. Meanwhile, for optimal design problems, both nonlocal conductivity and peridynamics contexts are discussed, and are presented as applications of a new, more general framework for the asymptotic compatibility of parameterized optimal design problems.

Finally, simulations using the C++ finite element libraries deal.ii and FreeFEM++ are included, and graphical and tabular results are presented to support the theoretical results.