Adjoint-based shape optimization of an annular combustion chamber in a simplified gas turbine engine

The top-right figure shows a 3D rendering of a simplified model of an annular combustion chamber in an aircraft gas turbine engine. The bottom-right shows a slice through the combustion chamber, showing the plenum on the left, the feed tubes in the middle, and the combustion chamber on the right. The figure on the left shows the acoustic natural frequencies (eigenmodes) of this system, calculated with an acoustic network model. Two modes have positive growth rate (grey region) and five have negative growth rate (white region). Adjoint methods are used to calculate the sensitivity of both unstable modes to all possible geometry changes. This gradient information is used to subtly change the shape of the plenum and combustion chamber until both modes are stable. Only small changes are required, and they are mainly in the plenum.