Non-convex shape optimization by dissipative Hamiltonian flows

Shape optimization is important for improving the performance of mechanical components. In the past, gradient based optimization methods using the elegant adjoint formalism dominated the field. Gradient descent optimization, however, tends to get stuck in local minima if the optimization problem is non-convex. In this article, an improved approach to shape optimization is proposed that is capable of overcoming local minima. It is motivated by gradient based training methods in machine learning, which involve momentum. Here, the momentum method is derived from the formalism of dissipative Hamiltonian flows, which generalize gradient flows. While gradient descent can be seen as an Euler scheme discretization of the gradient flow, a symplectic Euler scheme is proposed in this article to discretize the Hamiltonian dynamics for the momentum method. To show the efficiency of the method, it is applied to a shape optimization problem involving ceramic components, with reliability, volume and avoidance of construction space constraints as target functions. Construction space constraints lead to manifestly non-convex shape optimization problems. Numerical experiments demonstrate that the proposed method solves these non-convex problems more efficiently than the traditional gradient descent approach.