### 8th European Workshop On Structural Health Monitoring (EWSHM 2016)

5-8 July 2016, Spain, Bilbao

Right Now in Bilbao 12:40 Sat 14

16:50 Wednesday 6. Jul - A2

### Ultrafast Wave Finite Element Method for the computation of dispersion properties in periodic viscoelastic waveguides

Abstract »Knowledge of complex dispersive properties of waveguides, i.e. dispersion curve and attenuation, plays a pivotal role in long range guided waves based nondestructive evaluation and structural health monitoring applications. Dispersion and attenuation curves can be obtained solving k(ω) eigenvalue problems, where ω is the real wave circular frequency and k = kr + iki is the complex wavenumber, by means of Semi Analytical Finite Element (SAFE) approaches. Nevertheless, the use of these algorithms requires the implementation of ad-hoc finite element (FE) coding, limiting their adoption to some research groups only, and does not allow to account for waveguides periodicity. As such, alternative approaches based on standard/commercially available FE routines, generally known as Wave Finite Element methods (WFE), have been proposed. WFE method exploits a FE discretization of a finite length portion of the waveguide, the unit cell, and impose Floquet-Bloch boundary conditions at the edges of the cell to build the dispersive wave equation. The flexibility of the standard FE approach allows to model waveguides with arbitrarily unit cell geometry and viscoelastic rheology with no particular effort. However, the potential of WFE formulations is partially counteracted by the higher computational cost when compared to SAFE methods, given the large system of equations generated by the unit cell domain discretization. As such, in this work a fast computational algorithm based on a Model Reduced WFE approach for the extraction of complex dispersion curves of mono-dimensional waveguides is proposed. The algorithm is based on the WFE approach combined with a Component Mode Synthesis (CMS) model reduction. The complex stiffness matrix K and mass matrix M of the waveguide are in fact reduced by a CMS reduction based on the Craig Bampton modal reduction scheme. In detail the internal degree of freedoms (dofs) of the unit cell are substituted by a reduced set of fixed interface modal coordinates and boundary cross-section dofs are left untouched. As a result, a reduced description of the unit cell is obtained yielding to a smaller k(ω) eigenvalue problem. Criteria for the selection of unit cell length and number of internal modes are discussed by analyzing different conventional waveguides as well as periodic/locally resonant systems for wavefiltering applications.

#### Authors

Palermo, Antonio***Palermo, Antonio***

antonio.palermo6@unibo.it

+39 051 20 9 3374

*Biography:*

Antonio Palermo is a Ph.D student in the Department of Civil, Chemical, Environmental and Materials Engineering at the University of Bologna. He got is Bachelor and Master Degree in Civil Engineering at the University of Bologna and a MSc in Earthquake Engineering at Imperial College. His Ph.D research is focused on the development of computational algorithm for analyzing waves dispersive properties of periodic/locally resonant metamaterial for vibration control applications.

*Affiliation:*

University of Bologna

Department of Civil, Chemical, Environmental and Materials Engineering

40136 Bologna

ItalyMarzani, Alessandro

**Marzani, Alessandro**

alessandro.marzani@unibo.it

+39 051 20 9 3506

Homepage

*Affiliation:*

University of Bologna

Department of Civil, Chemical, Environmental and Materials Engineering (DICAM)

40136 Bologna

Italy

+390512093506

www.dicam.unibo.it/en

*Other Presentations:*

Tue 5. 10:50, C1

Session: Piezoelectrics as sensors/actuators

Title: A stamp size, 40mA, 5 grams sensor node for impact detection and location

Tue 5. 10:30, C1

Session: Piezoelectrics as sensors/actuators

Title: A design strategy for highly directional piezoelectric transducers for Lamb waves inspections

Wed 6. 16:30, A2

Session: Guided waves based methods for SHM-I

Title: Ultrasonic Guided waves Communications in smart materials: the case of tapered waveguides

**Contact*