Highly convergent dynamic models obtained by modal synthesis with application to short wave pulse propagation

Abstract

A general approach for obtaining the matrices of a substructure ensuring minimum modal frequency errors of the whole structure is presented. The mass and stiffness matrices of a small component domain of selected dimension are obtained by applying the modal synthesis of a limited number of close-to-exact modes such that after assembling a larger joined domain model the modal convergence rate of the latter should be as high as possible. The goal is achieved by formulating the minimization problem for the penalty-type target function representing the cumulative relative modal frequency error of the joined sample domain and by applying the gradient descent minimization method. After the optimum matrices of a component domain are obtained, they can be used in any structure as higher-order elements or super-elements. The well-known generalized mass matrices obtained as a weighted sum of lumped and consistent components can be treated as a special case of the presented approach. The obtained dynamic models are used for modelling short transient waves and wave pulses propagating in elastic or acoustic environments by using a only a few nodal points per pulse length.