Full-order eigenpair perturbations with mode tracking applications in aeroelasticity and optimization.

General methodology for calculating changes in eigenvalues and eigenvectors resulting from parameter perturbations in the eigenproblems is presented for self-adjoint and nonself-adjoint cases. Perturbing the eigenproblems and retaining all terms leads to coupled equations for the eigenpair perturbations which must be solved iteratively. The familiar singularity in the solution for the eigenvector change is removed using Nelson's method, which is successful despite the approximate nature of the singularity. The normalization task necessary for the eigenvector perturbation also retains all terms, leading to a quadratic equation for a weighting factor which, for the self-adjoint case, has a uniquely defined correct root. For the nonself-adjoint case, magnitude normalization is not sufficient to define unique perturbed left and right eigenvectors, and an additional phase correction is imposed. Enhancements to the basic algorithm that are discussed are the use of under-relaxation and the computation of perturbations in the presence of repeated eigenvalues. The resulting algorithms, the higher order eigenpair perturbation algorithm (HOEP) and its complex extension (C-HOEP), are robust in that they can converge to the exact eigenpairs for very large parameter perturbations. Mode tracking techniques are developed and applied to problems in structural optimization and aeroelastic analysis. The goal is to eliminate difficulties caused by mode-switching (i.e. frequency crossing). Out of several candidate methods, two methods for mode tracking are successful. The first method is the higher order eigenpair perturbation algorithm. By computing perturbations for each eigenpair individually, it maintains the correspondence between the baseline and perturbed eigenpairs. The second method is the cross-orthogonality check method which uses mass orthogonality (self-adjoint case) or mass biorthogonality (nonself-adjoint case) to reestablish correspondence after a standard reanalysis. Applications of mode tracking technology that are presented are frequency-constrained optimization, optimization with mode shape constraints, V-g aeroelastic analysis, and p-k aeroelastic analysis. Each application procedure is outlined and examples are given. Recommendations are made based on method efficiency and robustness in the example problems.PhDAerospace engineeringApplied SciencesComputer scienceMechanicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/129185/2/9409681.pd