Beyond the limitations of perturbation methods for real random eigenvalue problems using Exceptional Points and analytic continuation

A numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods. The radius of convergence of eigenvalue perturbation methods, based on Taylor series, is known to dec...

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Bibliographic Details
Published in:Journal of sound and vibration Vol. 480; p. 115398
Main Authors: Ghienne, Martin, Nennig, Benoit
Format: Journal Article
Language:English
Published: Amsterdam Elsevier Ltd 18.08.2020
Elsevier Science Ltd
Elsevier
Subjects:
Algorithms
Approximation
Convergence
Defective eigenvalue
Eigenvalues
Engineering Sciences
Exceptional point
Flutter
Mathematical models
Mechanics
Numerical analysis
Numerical methods
Numerical models
Parameters
Parametric eigenvalue problem
Perturbation methods
Puiseux series
Robustness (mathematics)
Rotating machinery
Singularities
Structural mechanics
Taylor series
Uncertainty propagation
Veering
Vibration
Vibrations
Waveguides
Veering
Uncertainty propagation
Exceptional point
Defective eigenvalue
Puiseux series
Parametric eigenvalue problem
defective eigenvalue
parametric eigenvalue problem
ISSN:0022-460X, 1095-8568
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Summary:A numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods. The radius of convergence of eigenvalue perturbation methods, based on Taylor series, is known to decrease when eigenvalues are getting closer to each other. This phenomenon, known as veering in structural dynamics, is a direct consequence of the existence of branch point singularity in the complex plane of the varying parameters where some eigenvalues are defective. When this degeneracy, referred to as Exceptional Point (EP), is close to the real axis, the veering becomes stronger. The main idea of the proposed approach is to combined a pair of eigenvalues to remove this singularity. To do so, two analytic auxiliary functions are introduced and are computed through high order derivatives of the eigenvalue pair with respect to the parameter. This yields a new robust eigenvalue reconstruction scheme which is compared to Taylor and Puiseux series. In all cases, theoretical bounds are established and all approximations are compared numerically on a three degrees of freedom toy model. This system illustrates the ability of the method to handle the vibrations of a structure with a randomly varying parameter. Computationally efficient, the proposed algorithm could also be relevant for actual numerical models of large size, arising from other applications involving parametric eigenvalue problems, e.g., waveguides, rotating machinery or instability problems such as squeal or flutter. •Standard perturbation methods are extended using eigenvalues high order derivatives.•New eigenvalue representations based on analytic functions are proposed.•The radius of convergence of each representation depends on exceptional point.•Eigenvalue loci is reconstructed on a large range using a single computation point.•An application to random eigenvalue problems acceleration is considered.
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ISSN:0022-460X
1095-8568
DOI:10.1016/j.jsv.2020.115398