Model reduction based on matrix interpolation and distorted finite element meshes for dynamic analysis of 2D nearly periodic structures

Abstract

The dynamic analysis of 2D nearly periodic structures of finite dimensions, subject to harmonic excitations, is addressed. Such structures are often made up of slightly different locally resonant layered substructures whose geometrical properties randomly vary in space and which are described here by means of distorted finite element (FE) meshes. It is well known that purely periodic structures with resonant substructures possess band gap properties, i.e., frequency bands where the vibration levels are low. The question arises whether nearly periodic structures provide additional features, e.g., the fact that the vibrational energy remains localized around the excitation points. Predicting the harmonic responses of such structures via efficient numerical approaches is the motivation behind the present paper. Usually, the Craig Bampton (CB) method is used to model the substructures in terms of reduced mass and stiffness matrices, which can be further assembled together to model a whole structure. The issue arises because the reduced mass and stiffness matrices of the substructures need to be computed several times — i.e., for several substructures whose properties differ to each other —, which is computationally cumbersome. To address this issue, a strategy is proposed which involves computing the reduced matrices of the substructures for some particular distorted FE meshes (a few number), and interpolating these matrices between these “interpolation points” for modeling substructures with random FE meshes. The relevance of the interpolation strategy, in terms of computational time saving and accuracy, is highlighted through comparisons with the FE and CB methods. Three structures are analyzed, i.e., (1) a plate with 8 × 8 substructures, (2) a plate with 15 × 15 substructures, and (3) a plate with 8 × 4 substructures embedded in a floor panel. Results show that, at high frequencies, the vibration levels of the nearly periodic structures undergo an overall reduction compared to the purely periodic cases.