In this paper a geometric multiscale finite element method (GMsFEM), recently developed by the authors, is applied to the analysis of wave propagation in damaged plates. The proposed methodology is based on the formulation of both two- and three-dimensional multi-node (or multiscale) elements capable of describing small defects without resorting to excessive mesh refinements. Each multiscale element is equipped with a local mesh that is used to compute the interpolation functions of the element itself and to resolve the local fluctuations of the solution near the defect. The computed shape functions guarantee the continuity of the solution between multiscale and conventional elements. This allows using an undistorted discretization in the uniform portion of the domain while limiting the use of multiscale elements only in the vicinity of the defects. In this article the method is applied to evaluate the reflection coefficients due to cracks of different size and orientation in an otherwise homogeneous plate. Also, numerical simulations of wave-damage interaction are used to compute the scattering coefficients associated to three-dimensional defects in isotropic plates.
This work illustrates the possibility to extend the field of application of the Multi-Scale Finite Element
Method (MsFEM) to structural mechanics problems that involve localized geometrical discontinuities
like cracks or notches. The main idea is to construct finite elements with an arbitrary number of
edge nodes that describe the actual geometry of the damage with shape functions that are defined as
local solutions of the differential operator of the specific problem according to the MsFEM approach.
The small scale information are then brought to the large scale model through the coupling of the
global system matrices that are assembled using classical finite element procedures. The efficiency of
the method is demonstrated through selected numerical examples that constitute classical problems of
great interest to the structural health monitoring community.
Efficient numerical models are essential for the simulation of the interaction of propagating waves with localized
defects. Classical finite elements may be computationally time consuming, especially when detailed discretizations
are needed around damage regions. A multi-scale approach is here propose to bridge a fine-scale mesh
defined on a limited region around the defect and a coarse-scale discretization of the entire domain. This "bridging"
method is formulated in the frequency domain in order to further reduce the computational cost and provide a general framework valid for different types of structures. Numerical results presented for propagating elastic waves in 1D and 2D damaged waveguides illustrate the proposed technique and its advantages.
Periodic arrays of hybrid shunted piezoelectric actuators are used to suppress vibrations in an aluminum plate.
Commonly, piezoelectric shunted networks are used for individual mode control, through tuned, resonant RLC
circuits, and for broad-band vibration attenuation, through negative impedance converters (NIC). Periodically
placed resonant shunts allow broadband reduction resulting from the attenuation of propagating waves in frequency
bands which are defined by the spatial periodicity of the array and by the shunting parameters considered
on the circuit. Such attenuation typically occurs at high frequencies, while NICs are effective in reducing the
vibration amplitudes of the first modes of the structure. The combination of an array resonant shunts and NICs
on a two-dimensional (2D) panel allows combining the advantages of the two concepts, which provide broadband
attenuation in the high frequency regimes and the reduction of the amplitudes of the low frequency modes.
Numerical results are presented to illustrate the proposed approach, and frequency response measurements on a
cantilever aluminum plate demonstrate that an attenuation region of about 1000Hz is achieved with a maximum
8 dB vibration reduction.