Incremental inverse iteration for the nonlinear eigenvalue problem in structural dynamics

Abstract eng:
Calculation of modes is of primal importance in structural dynamics. Most of the time discretization of the equation of motion leads to a quadratic eigenvalue problem to solve [1]. But when one wants to include more elaborate mechanical model (as viscoelasticity, feedback control, or fluid-structure interaction, ...), general nonlinear eigenvalue problem has to be solved. Nonlinearity leads to the loss of usual properties of modes and forces to adapt the procedures commonly used to solve linear eigenvalue problems, so that it constitutes a modern challenge in computational mechanics as noted by Mehrmann and Voss [2]. It is well known that direct application of Newton’s method to the linear eigenvalue problem leads to the Rayleigh quotient iteration (which can be identified as the inverse iteration method with a shift updated at every iterations). This explains why the idea to solve the nonlinear eigenvalue problem with the same Newton procedure (often called ”inverse iteration”) is so popular. To minimize the number of factorization of the operator used in inverse iteration (due to successive shifts), Neumaier [3] proposed to perform the iterative process without updating the shift, leading to the ”residual inverse iteration”. Because based on the Newton’s method, both above methods (inverse iteration and residual inverse iteration) converge at the conditions that initial guess is close enough to the sought-after mode, and that the functional considered is regular enough. In industrial applications, what is important is to determine the m first modes of the structure, and initial guess of the resonance frequencies are seldomly available with a sufficient precision to use directly Newton’s based procedures. What is proposed in this paper is to circumvent this problem, by using an incremental approach to localize successive eigenvalues λ from a starting point. Increment are driven by a monotonic and controlled shift λshif t . This enables to avoid problems inherent to Newton’s method when problem is genuinely nonlinear (failure by oscillating around local minima or by jumping roots). Numerical examples demonstrate the good behavior of the proposed method for Hermitian nonlinear eigenvalue problem.

• Export as: BibTeX | MARC | MARCXML | DC | EndNote | NLM | RefWorks
• Add to your basket