We consider scalar waves in periodic media through the lens of a second-order effective, i.e., macroscopic description, and we aim to compute the sensitivities of the germane effective parameters due to topological perturbations of a microscopic unit cell. Specifically, our analysis focuses on the tensorial coefficients in the governing mean field equation-including both the leading order (i.e., quasi-static) terms, and their second-order companions bearing the effects of incipient wave dispersion. The results demonstrate that the sought sensitivities are computable in terms of (i) three unit cell solutions used to formulate the unperturbed macroscopic model; (ii) two adointfield solutions driven by the mass density variation inside the unperturbed unit cell; and (iii) the usual polarization tensor, appearing in the related studies of nonperiodic media, that synthesizes the geometric and constitutive features of a point-like perturbation. The proposed developments may be useful toward (a) the design of periodic media to manipulate macroscopic waves via the microstructure-generated effects of dispersion and anisotropy, and (b) subwavelength sensing of periodic defects or perturbations.
We consider the homogenized boundary and transmission conditions governing the mean-field approximations of 1D waves in finite periodic media within the framework of two-scale analysis. We establish the homogenization ansatz (up to the second order of approximation), for both types of problems, by obtaining the relevant boundary correctors and exposing the enriched boundary and transmission conditions as those of Robin type. Rigorous asymptotic analysis is performed for boundary conditions, while the applicability to transmission conditions is demonstrated via numerical simulations. Within this framework, we also propose an optimized second-order model of the homogenized wave equation for 1D periodic media, that follows more accurately the exact dispersion relationship and generally enhances the performance of second-order approximation. The proposed analysis is applied toward the long-wavelength approximation of waves in finite periodic bilaminates, subject to both boundary and transmission conditions. A set of numerical simulations is included to support the mathematical analysis and illustrate the effectiveness of the homogenization scheme.
This article concerns an extension of the topological derivative concept for 3D elasticity problems involving elastic inhomogeneities, whereby an objective function J is expanded in powers of the characteristic size a of a single small inhomogeneity. The O(a 6) approximation of J is derived and justified for an inhomogeneity of given location, shape and elastic properties embedded in a 3D solid of arbitrary shape and elastic properties; the background and the inhomogeneity materials may both be anisotropic. The generalization to multiple small inhomogeneities is concisely described. Computational issues, and examples of objective functions commonly used in solid mechanics, are discussed.
This work presents an enriched finite element method (FEM) dedicated to the numerical resolution of Webster's equation in the time-harmonic regime, which models many physical configurations, e.g. wave propagation in acoustic waveguides or vibration of bars with varying cross-section. Building on the wavebased methods existing in the literature, we present new enriched finite element bases that account for both the frequency of the problem and the heterogeneity of the coefficients of the equation. The enriched method is compared to the classical fifth-order polynomial FEM, and we show they share the same asymptotic convergence order, present the same easiness of implementation and have similar computational costs. The main improvement brought by these enriched bases relies on the convergence threshold (the mesh size at which the convergence regime begins) and the convergence multiplicative constant, which are observed to be (i) better than the ones associated with polynomial bases and (ii) not only dependent on the resolution (the number of elements per wavelength), but also on the frequency for a fixed resolution, making the method we propose well adapted to high-frequency regimes. Moreover, taking into account the heterogeneity of the coefficient in Webster's equation by using an element-dependent enrichment leads to a significant decrease of the approximation error on the considered examples compared to a uniform enrichment, again with almost no additional cost. Several possible extensions of this work are finally discussed.
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