Rakenteiden Mekaniikka (Journal of Structural Mechanics) Vol. 45, No 3, 2012, pp. 104 – 124 Geometric approach in numerical eigenproblem analysis Alexis Fedoroff and Reijo Kouhia Summary. Non-linear eigenproblems can be encountered in a wide range of physical systems, stability analysis being an important source of such problems. The current technique used in commercial finite element codes to solve non-linear eigenproblems consists in linearizing the criticality equation with respect to the bifurcation parameter evaluated at the origin. Since the numerical technique is approximative, it is essential to assess the error both on the eigenvalue and the eigenmode. The latter one is of particular importance, since the eigenmode is typically used as initial imperfection in imperfection sensitivity analysis. In this paper the authors propose a new geometric approach, in which the eigenvector is considered as a locally smooth function defined on the criticality manifold. Given two such eigenvectors, one for the non-linear eigenproblem and one for the linear one, it is possible to evaluate the error between the two eigenvectors considering the location of their respective arguments on the criticality manifold and the intrinsic properties of the criticality manifold itself. Simple, illustrative examples taken from structural stability will be shown for the sake of clarity as well as numerical computations on engineering problems. Key words: non-linear eigenproblems, Riemannian geometry, structural stability Introduction If we look at the role of engineers in the assessment of structural integrity for a given system it is rather clear that in order to avoid catastrophic events, emphasis should be placed on the study of qualitative changes in system behavior. From a theoretical point of view such qualitative changes in system behavior have been widely studied in the context of bifurcation theory [8], [23], [14], but as systems get more and more complex, numerical issues related to locating bifurcation points have to be taken into consideration. In the numerical study of non-linear phenomena eigenproblems play a significant role as a means of locating bifurcation points. Numerical eigenproblem solution strategies Actually there are many different numerical strategies that have been developed over the years related to non-linear eigenproblem solving, and broadly we can conceive three types of approaches: indirect, direct and polynomial approximation approaches. The indirect approach consists in following the primary equilibrium path using continuation while monitoring some test function related to the criticality condition [21], [1], [3], [20], [12],[13]. Although continuation methods and their implementations in finite element codes tend to be reliable, these approaches need some experience from the user in order to properly set step length and other continuation parameters and certainly it is a time consuming approach. The direct approach consists in solving iteratively an augmented system consisting of equilibrium, criticality and normalization condition [10], [22], [18], 104