Periodically arranged acoustic metamaterial in industrial applications: The need for uncertainty quantification

Abstract Structures with periodically arranged resonators have attracted attention within vibroacoustics recently. These periodic structures attenuate wave propagation in a defined frequency range. Engineers can tune the resonators to produce structures with a desired band gap behavior. The majority of the established stop band materials is produced by additive manufacturing today. While promising scientific results have been achieved with this method, it is not yet fit for industrial applications. In order to develop new manufacturing approaches and to close the gap between science and industry, the influence of uncertainties of periodic structures has to be quantified. In this study, the influence of geometrical uncertainties on periodic structures is investigated. The authors develop a finite element model of an epoxy plate with beam resonators for this purpose. In a parameter study, the influence of some parameters is identified. Afterwards, the behavior of stop band material with uncertain input parameters is studied using spectral stochastic methods. Having predefined probability density functions, the generalized polynomial chaos expansion is used to propagate the uncertainty of these parameters. The stop band behavior is accordingly represented by the generalized polynomial chaos having unknown deterministic coefficients. A collocation-based stochastic simulation is used to estimate the coefficients employing the deterministic finite element model as a black-box. The results show the necessity of regarding uncertainties in periodic structures. The performed stochastic simulations are suitable to define manufacturing tolerances for the production of stop band material.

[1]  J. Xiang,et al.  The band gap and transmission characteristics investigation of local resonant quaternary phononic crystals with periodic coating , 2015 .

[2]  K. Sepahvand.  Stochastic finite element method for random harmonic analysis of composite plates with uncertain modal damping parameters , 2017 .

[3]  Massimo Ruzzene,et al.  Multiscale finite element analysis of wave propagation in periodic solids , 2016 .

[4]  M. Petyt,et al.  Random response of periodic structures by a finite element technique , 1975 .

[5]  Experimental and calculated research on a large band gap constituting of tubes with periodic narrow slits , 2009 .

[6]  S. Marburg,et al.  Stochastic Dynamic Analysis of Structures with Spatially Uncertain Material Parameters , 2014 .

[7]  M. Ruzzene,et al.  Floquet–Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems ☆ , 2011 .

[8]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[9]  Massimo Ruzzene,et al.  A general FEM technique to model wave propagation in cellular periodic structures , 2003, SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring.

[10]  Leping Feng,et al.  Reduction of the sound transmission of a periodic sandwich plate using the stop band concept , 2015 .

[11]  Steffen Marburg,et al.  UNCERTAINTY QUANTIFICATION IN STOCHASTIC SYSTEMS USING POLYNOMIAL CHAOS EXPANSION , 2010 .

[12]  H. Matthies,et al.  Uncertainties in probabilistic numerical analysis of structures and solids-Stochastic finite elements , 1997 .

[13]  Takayuki Yamada,et al.  Topology optimization of an acoustic metamaterial with negative bulk modulus using local resonance , 2013 .

[14]  D. Duhamel,et al.  A wave-based model reduction technique for the description of the dynamic behavior of periodic structures involving arbitrary-shaped substructures and large-sized finite element models , 2015 .

[15]  Wim Desmet,et al.  On the potential of tuned resonators to obtain low-frequency vibrational stop bands in periodic panels , 2013 .

[16]  L. Brillouin.  Wave propagation in periodic structures : electric filters and crystal lattices , 1953 .

[17]  S. Marburg,et al.  Reducing mechanical cross-coupling in phased array transducers using stop band material as backing , 2018, Journal of Sound and Vibration.

[18]  Sheng,et al.  Locally resonant sonic materials , 2000, Science.

[19]  S. Marburg,et al.  On Construction of Uncertain Material Parameter using Generalized Polynomial Chaos Expansion from Experimental Data , 2013 .

[20]  Paul-Remo Wagner,et al.  Robust-to-uncertainties optimal design of seismic metamaterials , 2018 .

[21]  D. Duhamel,et al.  A wave finite element-based approach for the modeling of periodic structures with local perturbations , 2016 .

[22]  Bert Pluymers,et al.  Dynamic Metamaterials for Structural Stopband Creation , 2016 .

[23]  O. von Estorff,et al.  A membrane-type acoustic metamaterial with adjustable acoustic properties , 2016 .

[24]  W. Desmet,et al.  A Bloch wave reduction scheme for ultrafast band diagram and dynamic response computation in periodic structures , 2018, Finite Elements in Analysis and Design.

[25]  F. Bloch.  Über die Quantenmechanik der Elektronen in Kristallgittern , 1929 .

[26]  Jihong Wen,et al.  Sound transmission loss of metamaterial-based thin plates with multiple subwavelength arrays of attached resonators , 2012 .

[27]  Wim Desmet,et al.  Probability that a band-gap extremum is located on the irreducible Brillouin-zone contour for the 17 different plane crystallographic lattices , 2017 .

[28]  Yuesheng Wang,et al.  Propagation of bending waves in phononic crystal thin plates with a point defect , 2009 .

[29]  Kuo-Chih Chuang,et al.  Experimental study on slow flexural waves around the defect modes in a phononic crystal beam using fiber Bragg gratings , 2016 .

[30]  B. Mace,et al.  Modelling wave propagation in two-dimensional structures using finite element analysis , 2008 .

[31]  Arthur W. Leissa,et al.  Vibration of Plates , 2021, Solid Acoustic Waves and Vibration.

[32]  P. Sheng,et al.  Acoustic metasurface with hybrid resonances. , 2014, Nature materials.

[33]  Steffen Marburg,et al.  Uncertainty quantification in natural frequencies and radiated acoustic power of composite plates: Analytical and experimental investigation , 2015 .

[34]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[35]  Annett Wechsler,et al.  Formulas For Natural Frequency And Mode Shape , 2016 .

[36]  Tsung-Tsong Wu,et al.  Waveguiding and frequency selection of Lamb waves in a plate with a periodic stubbed surface , 2009 .

[37]  S. Marburg,et al.  Random and Stochastic Structural Acoustic Analysis , 2016 .

[38]  D. Xiu,et al.  Modeling uncertainty in flow simulations via generalized polynomial chaos , 2003 .

[39]  Steffen Marburg,et al.  More Than Six Elements Per Wavelength: The Practical Use of Structural Finite Element Models and Their Accuracy in Comparison with Experimental Results , 2017 .

[40]  Wim Desmet,et al.  Design and validation of metamaterials for multiple structural stop bands in waveguides , 2017 .

[41]  K. Sepahvand,et al.  Spectral stochastic finite element vibration analysis of fiber-reinforced composites with random fiber orientation , 2016 .

[42]  Robin S. Langley,et al.  A note on the force boundary conditions for two-dimensional periodic structures with corner freedoms , 1993 .

[43]  C. Kittel.  Introduction to solid state physics , 1954 .

[44]  Jie Yang,et al.  Wave propagation in viscoelastic phononic crystal rods with internal resonators , 2018, Applied Acoustics.

[45]  X. Wen,et al.  Flexural wave band gaps in locally resonant thin plates with periodically attached spring–mass resonators , 2012 .

[46]  Wim Desmet,et al.  On the acoustic radiation efficiency of local resonance based stop band materials , 2014 .