Keywords: non-linear eigenvalue problem, equilibrium equations, preconditioned iterations.

Computation of critical points on an equilibrium path requires solution of a non-linear eigenvalue problem. When the external load is parameterized using a single parameter, the non-linear stability eigenvalue problem consists of solving the equilibrium equations at the criticality condition. Several techniques exist for solution of such a system. Their algorithmic treatment is usually focused on direct linear solvers and thus the use of the block elimination strategy. This paper places special emphasis on a strategy which can also be used with iterative linear solvers. Due to the non-uniqueness of the critical eigenmode a normalizing condition is required. In addition, for bifurcation points, the Jacobian matrix of the augmented system is singular at the critical point and additional stabilization is required in order to maintain the quadratic convergence of the Newton's method. Depending on the normalizing condition, convergence to a critical point with a negative load parameter value can happen. The form of the normalizing equation is critically discussed and an alternative form is proposed which quarantees convergence to a positive critical load value.

If properly scaled, all the examined constraint equations yield a similar type of convergence of the extended system. When starting from a random eigenvector, one step of inverse power iteration considerably improved the robustness of the Newton's iteration. In the present study, the second derivatives of the residual vector for the geometrically exact three-dimensional beam element have been computed from exact expressions. This approach guarantees the asymptotic quadratic convergence rate of the Newton's method. Numerical example computations of some spatial structures clearly demonstrates this.

The proposed simple preconditioning strategy, which requires only a standard preconditioner for the stiffness matrix, seems to work well. However, a really high quality preconditioner is needed to obtain faster execution than using the block elimination strategy with a fast direct solver. This is mainly due to the small bandwidth of the shell structures discretized using the finite element method.

Further studies will be focused on increasing the robustness and efficiency of the non-linear eigenvalue procedure as well as developing a suitable preconditioning strategy. Also procedures enlarging the convergence domain will be developed. Furthermore, utilization of the information already present in the non-linear eigenvalue algorithm to branch out to the possible secondary paths should be investigated.

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