Degree

Doctor of Philosophy (PhD)

Department

Department of Mathematics

Document Type

Dissertation

Abstract

This thesis focuses on constrained optimization problems with constraints on the state variables. When the constraints involve partial differential equations or variational inequalities, the optimization problem is also known as Mathematical Programs with Equilibrium Constraints. First, we applied active-set properties of optimal solutions to transform variational inequality constraints into partial differential equation constraints and devised an active-set method which allowed us to solve the optimization problems using the adjoint approach. We extended our approach to evolution problems with constraints on the trajectory of the state variable, such as the irreversibility condition in fracture mechanics. We implemented a gradient descent algorithm using finite element analysis. The numerical results of the active-set method showed an excellent agreement with the classical penalty approach, with significant improvements in performance and accuracy. For the evolution optimization problem, our numerical results matched analytical solutions.

Committee Chair

Bourdin, Blaise

DOI

10.31390/gradschool_dissertations.5707

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