Analysis of wave motion in 1D waveguides with microstructure
2020-12
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Analysis of wave motion in 1D waveguides with microstructure
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2020-12
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The main objectives of this dissertation include: (I) low-frequency homogenization of the wave equation in Rd with non-uniformly oscillating coefficients, (II) low-frequency solution of boundary value problem for 1D quasi-periodic waveguides, (III) solution of boundary value problem for 1D purely periodic waveguides at arbitrary frequency, as described below. Effective field equation. The focus of this work is dispersive, second-order effective model describing the low-frequency wave motion in heterogeneous (e.g. functionally- graded) media endowed with periodic microstructure. For this class of quasi-periodic medium variations, homogenization of the scalar wave equation in R^d, d >= 1 is pursued within the framework of multiple scales expansion. When either d = 1 or d = 2, this model problem bears direct relevance to the description of (anti-plane) shear waves in elastic solids. By adopting the lengthscale of microscopic medium fluctuations as the perturbation parameter, the germane low-frequency behavior is synthesized via a fourth- order differential equation (with smoothly varying coefficients) governing the mean wave motion in the medium, where the effect of microscopic heterogeneities is upscaled by way of the so-called cell functions. Effective boundary value problem. In an effort to demonstrate the relevance of the proposed analysis, presented in objective (I), toward solving boundary value problems (deemed to be the ultimate goal of most homogenization studies), effective boundary conditions are also developed, up to the second order of asymptotic approximation, applicable to one-dimensional (1D) shear wave motion in a macroscopically heterogeneous solid with periodic microstructure. The analysis is illustrated numerically in 1D by considering (i) low-frequency wave dispersion, (ii) mean-field homogenized description of the shear waves propagating in a finite domain, and (iii) full-field homogenized description thereof. In contrast to (i) where the overall wave dispersion appears to be fairly well described by the leading-order model, the results in (ii) and (iii) demonstrate the critical role that higher-order corrections may have in approximating the actual waveforms in quasi-periodic media. From d’Alembert to Bloch and back. Regarding the third objective, a simple semi-analytical model is developed for solving boundary value problems in one- dimensional (1D) periodic waveguides at arbitrary frequency. When the boundary value problem is well-posed in that the excitation frequency does not match resonant frequency of the waveguide, any solution that satisfies the field equation and boundary conditions is the (unique) solution of the problem. This motivates this study to seek the solution in the form of Bloch waves that by design solve the wave equation with periodic coefficients. For a given excitation frequency this assumption leads to a quadratic eigenvalue problem in terms of the wavenumber k which, in turn, produces a d’Alembert - type solution in terms of the left- and right- “propagating” waves. In this vein, the solution of a boundary value problem is obtained by computing the “left” and “right” amplitudes of the Bloch waves from the prescribed boundary conditions. For completeness, the situations are considered when the driving frequency is (i) inside a passband, (ii) inside a band gap, (iii) at the edge of a band gap (the so-called exceptional points), and (iv) at a crossing point between dispersion branches (repeated eigenfrequencies). From simulations, it is found that the semi-analytical solution (expressed in terms of either propagating or evanescent waves) is computationally very effective and reproduces the numerical results with high fidelity. Consistent with related studies, it is reported that the solution of the boundary value problem undergoes sharp transition in neighborhood of exceptional points. Perhaps the most remarkable result of this study is the find- ing that the resonance of a waveguide can be induced by adjusting the “microscopic” location of the support points so they coincide with the nodes of the Bloch eigenfunctions (if any) at given frequency. Vice versa, one can minimize the vibration amplitude (for given boundary condition) by selecting the micro-location of the support points near peaks of the germane Bloch eigenfunctions. This opens the door toward design of adaptive periodic systems whose performance can be tuned by adjusting the boundary conditions as opposed to altering the properties of the periodic medium. Applications of our initial study may include adaptive energy harvesting for sensing purposes and vibration isolation. The extension of this work to nonlinear periodic systems, those with internal resonators (metamaterials), and more complex physical systems such as beams is envisioned.
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University of Minnesota Ph.D. dissertation. 2020. Major: Civil Engineering. Advisor: Bojan Guzina. 1 computer file (PDF); 90 pages.
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Panahandeh-Shahraki, Danial. (2020). Analysis of wave motion in 1D waveguides with microstructure. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/219301.
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