Some problems in constrained shape optimization are considered. The goal in our optimization process is to maximize a measure of device performance computed using CAE, with a CAD compatible representation and specified geometric constraints. This approach illustrates several issues in integration of CAD and CAE systems. We test our ideas on idealized internal flow devices where the underlying device physics is governed by either the Laplace or the Navier-Stokes equations. The geometric shape of the device is represented by Bézier curves. Analysis tools such as the grid generator and the fluid flow solver are treated as a black box. The search pattern during the optimization process is suggested by the design of experiment methodology. The proposed framework is tested with one potential flow problem and two laminar flow diffuser problems.