Combined Approximation Method Applied To Reduced Non-Linear Vibroacoustic Problems

Abstract eng:
The vibroacoustic modeling using the finite element method (FEM) still remains costly in numerical computing. This is notably due to the formulation that can lead to a non-symmetric problem, and/or to a very large number of degrees of freedom to deal with [1]. The non-linear behavior commonly found in the vibration of thin structures, complicates the numerical solving. The reduced order method is used to reduce the numerical computation costs. The classical reduced order method is known to be unsuitable for non-linear problems due to the modal superposition principle that is not valid anymore. The combined approximation method [2] is a reduced order method which has been first developed for optimization and robustness studies via reanalysis [3]. It consists in building a reduced order basis issued from the modal basis of the initial problem enriched by a function taking into account the changes in the problem. This function is developed in a Taylor expansion to avoid the inversion of the current stiffness matrix. In this paper, this method is adapted to geometrical nonlinearities by considering the non-linear problem as a perturbation of the initial linear model. For thin structures [4] a coupling between inplane and bending movements appears for large displacements. The initial basis should then take into account this effect. By considering a light fluid as air, vibroacoustic coupling effect can modify the acoustic behavior. A structural reduced order basis is built on by using the combined approximation method. The acoustic basis that includes eigenvectors of the conservative acoustic model is enriched by the static response of the fluid due to the presence of the structure. Therefore the fluid-structure reduced-order computational model is constructed using an extended Ritz basis which allows the nonlinear dynamical response of the vibroacoustic problems to be predicted with a good accuracy. The temporal response of a thin plate supported on a closed cavity is presented to illustrate the efficiency of the method. Results show a high precision level, and the error produced by the reduced order model is explicitly related to the level of non-linearity.

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