A Benchmark Study on Eigenfrequencies of Fluid-Loaded Structures

In this paper, a coupled finite/infinite element method is applied for computing eigenfrequencies of structures in exterior acoustic domains. The underlying quadratic eigenvalue problem is addressed by a contour integral method based on resolvent moments. The numerical framework is applied to an academic example of a hollow sphere submerged in water. Comparisons of the computed eigenfrequencies to those obtained by boundary element discretizations as well as finite element discretizations in conjunction with perfectly matched layers verify the proposed numerical framework. Furthermore, extensive parameter studies are carried out illustrating the performance of the method with regard to both projection and discretization parameters. Finally, we point out that the proposed method achieves significantly smaller residuals of the computed eigenpairs than the Rayleigh Ritz procedure with second-order Krylov subspaces.

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Wavenumber-explicit convergence analysis for finite element discretizations of time-harmonic wave propagation problems with perfectly matched layers

Communications in Mathematical Sciences ◽

10.4310/cms.2022.v20.n1.a1 ◽

2022 ◽

Vol 20 (1) ◽

pp. 1-52

Author(s):

Théophile Chaumont-Frelet ◽

Dietmar Gallistl ◽

Serge Nicaise ◽

Jérôme Tomezyk

Keyword(s):

Finite Element ◽

Wave Propagation ◽

Convergence Analysis ◽

Harmonic Wave ◽

Perfectly Matched Layers ◽

Time Harmonic ◽

Finite Element Discretizations ◽

Matched Layers ◽

Element Discretizations

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THE HYPERBOLIC QUADRATIC EIGENVALUE PROBLEM

Forum of Mathematics Sigma ◽

10.1017/fms.2015.14 ◽

2015 ◽

Vol 3 ◽

Cited By ~ 2

Author(s):

XIN LIANG ◽

REN-CANG LI

Keyword(s):

Eigenvalue Problem ◽

Convergence Rates ◽

Natural Generalization ◽

Steepest Descent ◽

Cauchy Type ◽

Quadratic Eigenvalue Problem ◽

Nonlinear Conjugate Gradient ◽

Gradient Type ◽

Ritz Procedure ◽

Quadratic Eigenvalue

The hyperbolic quadratic eigenvalue problem (HQEP) was shown to admit Courant–Fischer type min–max principles in 1955 by Duffin and Cauchy type interlacing inequalities in 2010 by Veselić. It can be regarded as the closest analog (among all kinds of quadratic eigenvalue problem) to the standard Hermitian eigenvalue problem (among all kinds of standard eigenvalue problem). In this paper, we conduct a systematic study on the HQEP both theoretically and numerically. On the theoretical front, we generalize Wielandt–Lidskii type min–max principles and, as a special case, Fan type trace min/max principles and establish Weyl type and Wielandt–Lidskii–Mirsky type perturbation results when an HQEP is perturbed to another HQEP. On the numerical front, we justify the natural generalization of the Rayleigh–Ritz procedure with existing principles and our new optimization principles, and, as consequences of these principles, we extend various current optimization approaches—steepest descent/ascent and nonlinear conjugate gradient type methods for the Hermitian eigenvalue problem—to calculate a few extreme eigenvalues (of both positive and negative type). A detailed convergence analysis is given for the steepest descent/ascent methods. The analysis reveals the intrinsic quantities that control convergence rates and consequently yields ways of constructing effective preconditioners. Numerical examples are presented to demonstrate the proposed theory and algorithms.

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