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

Abstract

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, knwon 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 illustrate 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.

Structural reliability analysis of automotive parts based on strength degradation and abrasion and correlation

“…At present, the methods to solve the dynamic reliability of mechanical parts mainly include random perturbation, 20 traverse, 21 orthogonal expansion, and random simulation methods. The stochastic perturbation method provides an important method for reliability analysis of static, dynamic, and nonlinear stochastic structural systems.…”

Poroelastic lamellar metamaterial for sound attenuation in a rectangular duct

“…The interested reader is referred to the seminal paper of Tester [23] for locally reacting materials and [25,26] for rigid frame porous material and metamaterial having periodic structures. In the context of structural dynamics, this phenomenon is also known as veering and linked to the existence of exceptional points [38] which leads to strong attenuation [39,40] due to the absence of beating phenomenon between both modes. This is well illustrated for the perpendicular configuration in Fig.…”

A direct solution method for generalized stochastic eigenvalue problems of structures with random parameters

“…Qiu 24 put forward a direct‐variance‐analysis method based on the matrix perturbation theory to predict the mean values and variances of the eigenvalues. Ghienne 25 focused on the perturbation method for the generalized stochastic eigenvalue problems, and utilized the non‐Hermitian framework and analytic continuation around the Exceptional Points to solve the problems.…”