A Sylvester equation approach for the computation of zero-group-velocity points in waveguides

Plestenjak, Bor (author), Kiefer, Daniel A. (author), Gravenkamp, Hauke (author)

Eigenvalues of parameter-dependent quadratic eigenvalue problems form eigencurves. The critical points on these curves, where the derivative vanishes, are of practical interest. A particular example is found in the dispersion curves of elastic waveguides, where such points are called zero-group-velocity (ZGV) points. Recently, it was revealed that the problem of computing ZGV points can be modeled as a multiparameter eigenvalue problem (MEP), and several numerical methods were devised. Due to their complexity, these methods are feasible only for problems involving small matrices. In this paper, we improve the efficiency of these methods by exploiting the link to the Sylvester equation. This approach enables the computation of ZGV points for problems with much larger matrices, such as multi-layered plates and three-dimensional structures of complex cross-sections.

• Language: English
• Keywords: parameter-dependent quadratic eigenvalue problem, eigencurves, zero-group-velocity point, Sylvester equation, method of fixed relative distance, two-parameter eigenvalue problem
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.11.2025
• Year: 2025
• Number of pages: Str. 1443–1456
• Numbering: Vol. 76, iss. 5
• PID: 20.500.12556/RUL-175777
• UDC: 519.6
• ISSN on article: 0178-7675
• DOI: 10.1007/s00466-025-02656-8
• COBISS.SI-ID: 245905155
• Publication date in RUL: 07.11.2025

Record is a part of a journal

• Title: Computational mechanics
• Shortened title: Comput. mech.
• Publisher: Springer
• ISSN: 0178-7675
• COBISS.SI-ID: 6668037

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: parametrični kvadratični problem lastnih vrednosti, lastne krivulje, točka ničelne skupinske hitrosti, Sylvestrova enačba, metoda fiksne relativne razdalje, dvoparametrični problem lastnih vrednosti