Perturbation theory of nonlinear, non-self-adjoint eigenvalue problems: Simple eigenvalues

Degenerate perturbation theory in thermoacoustics: high-order sensitivities and exceptional points

“…This is also correctly captured by the presented high-order perturbation theory, which is able to accurately reconstruct the three-dimensional modeshape of the thermoacoustic modes at hand when varying τ . As was shown in Mensah et al (2020), the accuracy of the estimated eigenvectors within the radius of convergence increases at higher perturbation orders. Figure 7 shows a plenum cut view of the magnitude of the pressure modeshapes of the split eigenvalues for τ = 2.65 ms.…”

“…Using them, it is possible to map nonlinear eigenproblems, which are difficult to solve, into a series of linear non-homogeneous equations, which are easier to solve, to approximate eigensolutions to any desired order. For simple eigenvalues, general formulae based on the high-order expansion of the eigenvalue problem have been derived by the thermoacoustic community (Mensah et al 2020). The same level of generality, however, has not been reached for degenerate thermoacoustic modes, which are often found in practise due to the rotational symmetries of annular and can-annular combustors in gas turbines.…”

“…In thermoacoustics, arbitrarily high-order adjoint-based perturbation theory for non-degenerate eigenvalues has been presented in Mensah et al. (2020). We report here the key ideas and results of the method since they serve as a starting point for the discussion of the degenerate case, which is the main focus of this study.…”

“…This helps the recursive implementation of perturbation theory (Mensah et al. 2020). In appendix A, we provide a general formula for at any order, and its explicit expressions for .…”

“…This calculation may be computationally demanding if systems with millions of degrees of freedom are considered. In order to calculate the drift of eigenvalues and eigenvectors due to changes in parameters at an affordable computational cost, high-order adjoint-based perturbation theory can instead be used (Mensah, Orchini & Moeck 2020).…”