Helmholtz-x: Parallelized adjoint open source solver for the thermoacoustic Helmholtz equation

We create and describe an inhomogeneous Helmholtz equation solver, helmholtz-x, written in an open-source framework. The mesh is generated with Gmsh and the solver uses DOLFINx and UFL from FEniCSx. The performance, validity, stability and extensibility of the solver are demonstrated through several examples of thermoacoustic instability, from the one-dimensional Rijke tube to the three-dimensional MICCA combustor. The implementation of Bloch-type boundary conditions is explained and tested. The adjoint capability of the solver is also shown, and used to obtain derivatives of the eigenvalue with respect to shape parameters. This is exploited to find shape changes that reduce the thermoacoustic growth rate.