An optimal shape design problem for plates

TL;DR

This paper develops a finite element-based method for optimizing the shape of plates by controlling their thickness, incorporating regularization, error bounds, and a semismooth Newton solver, with numerical validation.

Contribution

It introduces a novel approach combining variational discretization, mixed finite elements, and regularization techniques for shape optimization of plates.

Findings

Discretization and regularization errors are bounded.

Coupling of regularization parameter and grid size is analyzed.

Numerical examples demonstrate the effectiveness of the method.

Abstract

We consider an optimal shape design problem for the plate equation, where the variable thickness of the plate is the design function. This problem can be formulated as a control in the coefficient PDE-constrained optimal control problem with additional control and state constraints. The state constraints are treated with a Moreau-Yosida regularization of a dual problem. Variational discretization is employed for discrete approximation of the optimal control problem. For discretization of the state in the mixed formulation we compare the standard continuous piecewise linear ansatz with a piecewise constant one based on the lowest-order Raviart-Thomas mixed finite element. We derive bounds for the discretization and regularization errors and also address the coupling of the regularization parameter and finite element grid size. The numerical solution of the optimal control problem is realized with a semismooth Newton algorithm. Numerical examples show the performance of the method.