  Shape optimization based on shape calculus has received a lot of attention in
recent years, particularly regarding the development, analysis, and
modification of efficient optimization algorithms. In this paper we propose and
investigate nonlinear conjugate gradient methods based on
Steklov-Poincar\'e-type metrics for the solution of shape optimization problems
constrained by partial differential equations. We embed these methods into a
general algorithmic framework for gradient-based shape optimization methods and
discuss the numerical discretization of the algorithms. We numerically compare
the proposed nonlinear conjugate gradient methods to the already established
gradient descent and limited memory BFGS methods for shape optimization on
several benchmark problems. The results show that the proposed nonlinear
conjugate gradient methods perform well in practice and that they are an
efficient and attractive addition to already established gradient-based shape
optimization algorithms.
