Deflation schemes for the generalized nonlinear eigenvalue problem

Deflation schemes for the generalized nonlinear eigenvalue problem | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Deflation schemes for the generalized nonlinear eigenvalue problem Wellington Tatagiba de Carvalho, Renan Costa Sales, Ney Augusto Dumont This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3934035/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Deflation is an artifice to remove from a search subspace of interest already evaluated eigenpairs. It is a necessary step in solving structure vibration or even the complete simulation of time-dependent problems with a large number of degrees of freedom, which in our particular case is in the frame of a generalized modal analysis related to high-order nonlinear eigenvalue problems. We start with the linear algebra outline of the complex symmetric, nonlinear eigenvalue problem for structures submitted to viscous damping. However, we soon evolve to a more general development for complex, nonsymmetric, general matrices that may be expressed as closed-form analytical functions or as a power series of the eigenvalues -- as long as the matrices are simultaneously diagonalizable. After a short literature review of deflation techniques for linear and some restricted nonlinear cases, we propose a few families of deflation schemes applicable to problems that may not have mechanical meaning, thus in terms of sheer linear algebra. We show that a general deflation scheme works as penalty functions accrued to the original problem to prevent already obtained results from being re-evaluated. We illustratively assess the issues of combining the proposed deflation schemes with a generalized, inverse-free Krylov subspace iteration technique, as applied to nonlinear symmetric problems generated randomly. The numerical application to more general, nonsymmetric eigenvalue problems is also presented. Structural dynamics nonlinear eigenvalue problems advanced modal analysis eigenvalue deflation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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