Optimal shape for a nozzle design problem using an arbitrary Lagrangian-Eulerian finite element method

This paper is concerned with the investigation of mathematical and numerical methods to find the optimal shape of a nozzle by means of shape optimization. The non-viscous, incompressible potential field within the nozzle is assumed to satisfy the Laplace PDE with mixed boundary conditions. We try to track the geometry of the nozzle to match the resulting velocity field with a prescribed one in some given critical subdomain. The problem is reformulated as an output least-squares minimization problem with the nozzle boundary being the control variable. The shape gradient of the cost functional is derived by combining the adjoint method and the techniques in the Lie derivative framework proposed by Hiptmair and Li (2012). An arbitrary Lagrangian-Eulerian finite element method is proposed to numerically solve the problem in an efficient way. Numerical experiments are presented to demonstrate the applicability and effectiveness of the proposed method.