An inner–outer nonlinear programming approach for constrained quadratic matrix model updating

(a)The constrained quadratic matrix model updating problem is tackled in this work.(b)An inner-outer nonlinear programming approach for that problem is proposed.(c)The goal is to avoid spurious modes in the frequency range of interest.(d)Constraints based on a quadratic Rayleigh quotient are dynamically included.(e)Only a few eigenpairs of the associated quadratic matrix pencil need to be computed.

The Quadratic Finite Element Model Updating Problem (QFEMUP) concerns with updating a symmetric second-order finite element model so that it remains symmetric and the updated model reproduces a given set of desired eigenvalues and eigenvectors by replacing the corresponding ones from the original model. Taking advantage of the special structure of the constraint set, it is first shown that the QFEMUP can be formulated as a suitable constrained nonlinear programming problem. Using this formulation, a method based on successive optimizations is then proposed and analyzed. To avoid that spurious modes (eigenvectors) appear in the frequency range of interest (eigenvalues) after the model has been updated, additional constraints based on a quadratic Rayleigh quotient are dynamically included in the constraint set. A distinct practical feature of the proposed method is that it can be implemented by computing only a few eigenvalues and eigenvectors of the associated quadratic matrix pencil.