On the Second-Order Shape Derivative of the Kohn-Vogelius Objective Functional Using the Velocity Method

Jerico B Bacani,Julius Fergy T Rabago

doi:10.1155/2015/954836

Abstract

The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the so-called Kohn-Vogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the second-order shape derivative of the objective functional using the velocity method with nonautonomous velocity fields. This work confirms the classical results of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and perturbation of identity technique.

Highlights

Shape optimization is a key research topic with many applications in various fields of pure and applied sciences, especially in biomechanics and engineering
The same functional is studied again but we focus on the application of velocity method in dealing with shape optimization problem
The study is important since it confirms a classical result of Delfour and Zolesio in relating shape derivatives of functionals using velocity method and the perturbation of identity technique

Summary

Introduction

Shape optimization is a key research topic with many applications in various fields of pure and applied sciences, especially in biomechanics and engineering (cf. [1, 2] for applications in structural mechanics, [3] for some applications in fluid mechanics or aerodynamics, and [4] for other applications). The same functional is studied again but we focus on the application of velocity method in dealing with shape optimization problem. Bacani and Peichl employed shape optimization methods to study the exterior Bernoulli FBP by. Bacani and Peichl presented two strategies in computing the first-order shape derivative of the Kohn-Vogelius objective functional. The authors computed its second-order shape derivative for general domains via the boundary differentiation scheme and via Tiihonen’s approach [24]. The study is important since it confirms a classical result of Delfour and Zolesio in relating shape derivatives of functionals using velocity method and the perturbation of identity technique (cf [26]).

Preliminaries

Analysis for the Nonautonomous Case

Conclusion