Numerical Solutions for Optimal Control Problems under Spde Constraints

The primary source of aircraft noise is the fan noise from the engines; natural approaches to reducing this noise involve acoustic shape optimization of the inlet and impedance optimization of the liner. This project will use optimal control to systematically determine the inlet shape and the linear material impedance factor that minimize the fan noise. A novel feature of this approach is that we automatically incorporate uncertainty and data measurement errors. Specifically we assume that the acoustic wave number is a random variable/field instead of a constant. This means that the computed answers are valid, not merely for a single configuration, but for a wide range. Our numerical results show significant noise reduction with the optimal impedance factor. Since the wave number is random, the underlying partial differential equation–Helmholtz equation in our case, is a stochastic partial differential equation. In this project, we have constructed efficient Monte Carlo methods as well as stochastic finite element methods to solve stochastic partial differential equations. Rigorous error estimates are obtained and numerical simulations are conducted to support the error analysis. Formulation of the optimal control problem In this research, we treat liner impedance optimization as an optimal control and parameter estimation problem. The parameter is the acoustic impedance factor of the acoustic liner. We define a cost function that reflects the amount of noise radiated from the engine inlet. The parameter estimation problem then is to seek the parameter that minimizes the cost function. The geometry of the domain in which the control problem is posed has the generic shape represented in Figure 1. The modal composition of the noise source is supposed to be known on the source plane Γ1. The nacelle boundary is made up of two parts, the first part being the interior boundary Γ2 to which some acoustic liner material is attached, and the second part being Γ3 that constitutes the rest of boundary of the nacelle geometry. The boundary Γ4 is assumed to be sufficiently far from the noise source so that the Sommerfeld radiation boundary condition holds. The nacelle symmetry axis is denoted by Γ5. We assume that the mean flow is zero. Then the acoustic pressure u satisfies the Helmholtz equation (1) ∆u + ku = 0 on Ω