Computational Mechanics

, Volume 46, Issue 2, pp 215–228 | Cite as

Extracting the left and right critical eigenvectors from the LDU-decomposed non-symmetric Jacobian matrix in stability problems

Original Paper


In the previous publications of the authors, an eigenanalysis-free computational procedure has been proposed to extract the bifurcation buckling mode(s) from the LDL T -decomposed symmetric stiffness matrix in the vicinity of a stability point. Any eigensolver, for instance, inverse iteration or subspace method, is not necessary. The procedure has been verified in numerical examples and well works in multiple and clustered bifurcation problems too. This present paper will extend the eigenanalysis-free procedure to the LDU-decomposed non-symmetric Jacobian matrix, from which both left and right critical eigenvectors relevant to the stability point may be extracted in a similar way. The idea is mathematical and totally independent of the physical problem to be solved, so that it is applicable to any non-symmetric square matrix in stability problems including plasticity with non-associated flow rules, contact and fluid-structure interaction. The linear-algebraic background of non-symmetric eigenvalue problems is firstly described. The present paper will then mention the role play of the left and right critical eigenvectors in stability analysis and the eigenanalysis-free LDU-procedure is proposed. Numerical examples of elastoplastic bifurcation are illustrated for verification and discussion. In APPENDICES, a bench model visualizes the mechanical meaning of the left and right critical singular vectors of a rectangular matrix.


Bifurcation Eigenanalysis-free Non-symmetric matrix Left and right eigenvectors Left and right singular vectors 


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mathematical and Computational EngineeringGifu UniversityGifuJapan
  2. 2.Department of Civil and Environmental EngineeringTohoku UniversitySendaiJapan

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