Quest for the control on the second order derivatives: topology optimization with functional includes the state's curvature

TL;DR

This paper explores the control of second order derivatives in PDE-based systems, introducing a numerical approach to optimize a functional involving the state's curvature, which is a novel contribution in this area.

Contribution

It presents the first numerical investigation into controlling second order derivatives in PDEs, including a regularization technique and a globalized gradient method for solving the optimality conditions.

Findings

Control on second order derivatives is feasible with the proposed method.

Numerical results demonstrate success in 2D and 3D cases.

The approach manages singularities effectively during optimization.

Abstract

Many physical phenomena, governed by partial differential equations (PDEs), are second order in nature. This makes sense to pose the control on the second order derivatives of the field solution, in addition to zero and first order ones, to consistently control the underlaying process. However, this type of control is nontrivial and to the best of our knowledge there is nigher a theoretic nor a numeric work in this regard. The present work goals to do the first quest in this regard, examining a problem of this type using a numerical simulation. A distributed parameter identification problem includes the control on the diffusion coefficient of the Poisson equation and a functional includes the state's curvature is considered. A heuristic regularization tool is introduced to manage codimension-one singularities during the functional analysis. Based on the duality principles, the approximate necessary optimality conditions is found. The system of optimality conditions is solved using a globalized projected gradient method. Numerical results, in two- and three-dimensions, implied the possibility of posing control on the second order derivatives and success of the presented numerical method.