59866 - A Hybrid Adjoint Network Model for Thermoacoustic Optimization 

Adjoint methods allow to compute sensitivities both efficiently and in a straight-forward manner, with significant benefits for a wide variety of optimization and control problems. 
The present study is concerned with the application of adjoint methods to the prediction and control of thermoacoustic combustion instabilities.

In general, there are two approaches to calculate the adjoint system: The discrete adjoint and the continuous adjoint. 
While the discrete adjoint can be obtained directly from the discretized direct system, thus eliminating the need to derive the adjoint equations, its application to network models is limited, as the adjoint eigenmodes feature spurious oscillation at jump and boundary conditions.
On the other hand, deriving the continuous adjoint equations for each element of the network system can be cumbersome.
We introduce a method that allows the computation of the continuous adjoint network system based on the direct equation. Our approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct system.
This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. 
We show how the obtained continuous adjoint system may be utilized to optimize the thermoacoustic model by determining the optimal damper placement for an annular combustor.