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

Abstract

We discuss high-order adjoint based degenerate perturbation theory of semisimple eigenvalues resulting from nonlinear eigenvalue problems, which are typical of acoustic problems and vibrational analysis. The theory is discussed for a general matrix operator with no further properties assumed. The unperturbed operator is thus in general non-normal, which requires the use of adjoint methods. The main challenge is to handle arbitrary order eigenvalue split for eigenvalue problems of arbitrary multiplicity: this is aided by a tree-structure formalism, which can be straightforwardly implemented numerically. We highlight the existence of a cross-branch solvability condition that needs to be satisfied when the perturbation causes the eigenvalues to split. After deriving the equations that govern the eigenproblem expansion order by order, we apply the method to three different test cases – which span eigenproblems of academic and industrial interest arising in structural mechanics and thermoacoustic stability – having different degrees of degeneracy and levels of complexity. We validate our algorithm by comparing the eigenvalues and eigenvectors reconstructed from perturbation theory up to 20th-order against exact solutions obtained using nonlinear eigenvalue solvers, and we discuss the advantages and limits of the method.