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AbstractA method of dynamic homogenization was recently proposed as an alternative technique for three-dimensional (3D) core calculations using a direct approach. This technique allows for producing homogenization parameters for a subdomain within the core at its actual state, and offers the advantage of avoiding off-line calculations, cross-section interpolations, and expensive 3D transport calculations. The methodology has shown a remarkable reduction in computational cost compared to 3D transport. However, the application of direct calculation schemes in multiphysics and multicycle problems still demands an extensive use of computational resources. Within the framework of traditional two-step calculations, several multilevel schemes have been developed to accelerate lattice calculations for the generation of the multiparameter cross-section libraries.With a similar objective, this work investigates how the multilevel approach can be applied within the framework of direct calculation schemes and discusses how depletion calculations may be performed. The objective is to enhance their performance and reduce memory requirements in favor of a high-fidelity model of the matter behavior in the reactor. We have then identified suitable flux solvers for each level and explored various homogenization options. We tested the methodology in a 3D pressurized water reactor core problem inspired by the TVA Watts Bar Unit 1 Multi-Physics Multi-Cycle OECD/NEA Benchmark, and performed a comparative analysis to assess the accuracy and computational efficiency of these new calculation schemes against direct single-level calculations and a two-step calculation based on pin-by-pin homogenization, as well as Monte Carlo simulations.Our analysis proves that the computational resources required to solve the full-core problem using a multilevel scheme are significantly reduced, and the best computational features are provided by multilevel dynamic homogenization. Furthermore, we observed that the multilevel approach not only allows for speedup, but can also be advantageously applied in order to improve the accuracy of the final solution of a calculation scheme when the core solver relies on an approximate transport operator. [ABSTRACT FROM AUTHOR]

We pursue a low-wavenumber, second-order homogenized solution of the time-harmonic wave equation at both low and high frequency in periodic media with a source term whose frequency resides inside a band gap. Considering the wave motion in an unbounded medium R d ( d ⩾ 1), we first use the (Floquet-)Bloch transform to formulate an equivalent variational problem in a bounded domain. By investigating the source term's projection onto certain periodic functions, the second-order model can then be derived via asymptotic expansion of the Bloch eigenfunction and the germane dispersion relationship. We establish the convergence of the second-order homogenized solution, and we include numerical examples to illustrate the convergence result. [ABSTRACT FROM AUTHOR]

The focus of our work is a 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, we pursue homogenization of the scalar wave equation in R d , d ≥ 1 , 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, we synthesize the germane low-frequency behavior 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. In an effort to demonstrate the relevance of our analysis toward solving boundary value problems (deemed to be the ultimate goal of most homogenization studies), we also develop effective boundary conditions, up to the second order of asymptotic approximation, applicable to one-dimensional (1D) shear wave motion in a macroscopically heterogeneous solid with periodic microstructure. We illustrate the analysis numerically in one dimension 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. [ABSTRACT FROM AUTHOR]

Mechanical metamaterials have recently gathered increasing attention for their uncommon mechanical responses enabling unprecedented applications for elastic wave control. Many research efforts are driven towards the conception of always new metamaterials' unit cells that, due to local resonance or Bragg-Scattering phenomena, may produce unorthodox macroscopic responses such as band-gaps, cloaking, focusing, channeling, negative refraction, etc. To model the mechanical response of large samples made up of these base unit cells, so-called homogenization or upscaling techniques come into play trying to establish an equivalent continuum model describing these macroscopic metamaterials' characteristics. A rather common approach is to assume a priori that the target continuum model is a classical linear Cauchy continuum featuring the macroscopic displacement as the only kinematical field. This implies that the parameters of such continuum models (density and/or elasticity tensors) must be considered to be frequency-dependent to capture the complex response of the considered mechanical systems in the frequency domain. These frequency-dependent models can be useful to describe some of the aforementioned macroscopic metamaterials' properties, yet, they suffer some drawbacks such as featuring negative masses and/or elastic coefficients in some frequency ranges which are close to resonance frequencies of the underlying microstructure. This implies that the considered Cauchy continuum is not positive-definite for all the considered frequencies. In this paper, we present a procedure, based on the definition of extra kinematical variables (with respect to displacement alone) and through the use of the inverse Fourier transform in time, to convert a frequency-dependent model into an enriched continuum model of the micromorphic type. All the parameters of the associated enriched model are constant (i.e., frequency-independent) and the model itself remains positive-definite for all the considered frequency ranges. The response of the frequency-dependent model and the associated micromorphic model coincide in the frequency domain, in particular when looking at the dispersion curves. Moreover, the micromorphic (frequency-independent) model results to be well defined both in time- and in the frequency-domain, while the Cauchy (frequency-dependent) model can only exist in the frequency domain This paper aims to build a bridge between the upscaling techniques usually found in the literature and our persuasion that macroscopic continua of the micromorphic type should be used to model metamaterials' response at the macroscopic scale. • Conversion of a frequency-dependent Cauchy model into an enriched continuum model of the micromorphic type. While the first can only be defined in the frequency domain, the latter is well defined both in the frequency and the time domain. • The parameters of the enriched models are constant and frequency-independent. • The frequency-dependent Cauchy continua may exhibit negative material properties (e.g., mass and/or elastic tensor) close to resonance frequencies, while the micromorphic models avoids these issues and remains well-defined and positive definite. • This paper wants to build a bridge between frequency-dependent models and macroscopic (homogenized) models of the micromorphic type for the description of mechanical metamaterials. [ABSTRACT FROM AUTHOR]

Recent years have seen considerable research success in the field of dynamic homogenization which seeks to define frequency dependent effective properties for heterogeneous composites for the purpose of studying wave propagation. There is an approximation involved in replacing a heterogeneous composite with its homogenized equivalent. In this paper we propose a quantification to this approximation. We study the problem of reflection at the interface of a layered periodic composite and its dynamic homogenized equivalent. It is shown that if the homogenized parameters are to appropriately represent the layered composite in a finite setting and at a given frequency, then reflection at this special interface must be close to zero at that frequency. We show that a comprehensive homogenization scheme proposed in an earlier paper results in negligible reflection in the low frequency regime, thereby suggesting its applicability in a finite composite setting. In this paper we explicitly study a 2-phase composite and a 3-phase composite which exhibits negative effective properties over its second branch. We show that based upon the reflected energy profile of the two cases, there exist good arguments for considering the second branch of a 3-phase composite a true negative branch with negative group velocity. Through arguments of calculated reflected energy we note that infinite-domain homogenization is much more applicable to finite cases of the 3-phase composite than it is to the 2-phase composite. In fact, the applicability of dynamic homogenization extends to most of the first branch (negligible reflection) for the 3-phase composite. This is in contrast with a periodic composite without local resonance where the approximation of homogenization worsens with increasing frequency over the first branch and is demonstrably bad on the second branch. We also study the effect of the interface location on the applicability of homogenization. The results open intriguing questions regarding the effects of replacing a semi-infinite periodic composite with its Bloch-wave (infinite domain) dynamic properties on such phenomenon as negative refraction. © 2014 Elsevier B.V.

The time-domain propagation of scalar waves across a periodic row of inclusions is considered in 2D. As the typical wavelength within the background medium is assumed to be much larger than the spacing between inclusions and the row width, the physical configuration considered is in the low-frequency homogenization regime. Furthermore, a high contrast between one of the constitutive moduli of the inclusions and of the background medium is also assumed. So the wavelength within the inclusions is of the order of their typical size, which can further induce local resonances within the microstructure. In Pham et al. (J. Mech. Phys. Solids 106:80–94, 2017), two-scale homogenization techniques and matched-asymptotic expansions have been employed to derive, in the harmonic regime, effective jump conditions on an equivalent interface. This homogenized model is frequency-dependent due to the resonant behavior of the inclusions. In this context, the present article aims at investigating, directly in the time-domain, the scattering of waves by such a periodic row of resonant scatterers. Its effective behavior is first derived in the time-domain and some energy properties of the resulting homogenized model are analyzed. Time-domain numerical simulations are then performed to illustrate the main features of the effective interface model obtained and to assess its relevance in comparison with full-field simulations. [ABSTRACT FROM AUTHOR]

The goal of this paper is to characterize the dynamic behavior of porous materials containing parallel cylindrical voids. Unlike static approaches, micro-inertia effects are accounted for in the modeling which infer a strong dependence of the dynamic response upon void geometry. Since cylindrical voids are considered, the void radius and void length both play a crucial role in the overall response of the porous material. A theoretical approach is developed, founded on the dynamic homogenization scheme proposed by Molinari and Mercier (J Mech Phys Solids 49:1497–1516, 2001) for spherical voids embedded in a viscoplastic matrix material. Considering a cylindrical unit cell, a constitutive response of porous material containing cylindrical void is developed for general homogeneous boundary conditions. For illustrative purpose, the analysis focuses on axisymmetric loadings considering a perfectly plastic matrix material. Micro-inertia effects are exemplified considering various loading conditions such as, among others, spherical loading and plane strain loading. In particular, the peculiar effect of the length of the cylindrical void is revealed. Indeed, particular attention has been paid to the response of short and elongated cylindrical voids. All predictions of the present model are verified against numerical simulations developed for various axisymmetric loading paths. Our findings can be used in several applications such as thick wall honeycomb structures or additively manufactured materials submitted to dynamic loading. [ABSTRACT FROM AUTHOR]

In this study, we establish an inclusive paradigm for the homogenization of scalar wave motion in periodic media (including the source term) at finite frequencies and wavenumbers spanning the first Brillouin zone. We take the eigenvalue problem for the unit cell of periodicity as a point of departure, and we consider the projection of germane Bloch wave function onto a suitable eigenfunction as descriptor of effective wave motion. For generality the finite wavenumber, finite frequency homogenization is pursued in Rd via second-order asymptotic expansion about the apexes of 'wavenumber quadrants' comprising the first Brillouin zone, at frequencies near given (acoustic or optical) dispersion branch. We also consider the junctures of dispersion branches and 'dense' clusters thereof, where the asymptotic analysis reveals several distinct regimes driven by the parity and symmetries of the germane eigenfunction basis. In the case of junctures, one of these asymptotic regimes is shown to describe the so-called Dirac points that are relevant to the phenomenon of topological insulation. On the other hand, the effective model for nearby solution branches is found to invariably entail a Dirac-like system of equations that describes the interacting dispersion surfaces as 'blunted cones'. For all cases considered, the effective description turns out to admit the same general framework, with differences largely being limited to (i) the eigenfunction basis, (ii) the reference cell of medium periodicity, and (iii) the wavenumber- frequency scaling law underpinning the asymptotic expansion. We illustrate the analytical developments by several examples, including Green's function near the edge of a band gap and clusters of nearby dispersion surfaces. [ABSTRACT FROM AUTHOR]

Abstract The paper deals with the differences between effective and homogeneous solution in case of dynamic of continuous media with random micro-structure. In particular, these differences, called "residuals", are considered for the dynamic linear problem of the Euler-Bernoulli's beam with random Young's modulus. The differential operator with random coefficient, that describes the eigenvalues problem, is taken into account. The convergence to the effective solution is analysed by introducing two measures: the normalized error between apparent and effective Young's moduli and between the modes shapes. The obtained results permit to highlight the dependence of the residuals from the micro-structure dimensionless length and the effect of the modes order; these aspects should be considered in the homogenization of dynamic behaviour of random heterogeneous composites. The assessment of the Rapresentative Volume Element (RVE) by convergence of the Statistical Volume Element (SVE) is also discussed. Highlights • The residuals for the dynamic behaviour of heterogeneous beams have been highlighted. • Two normalized residuals have been introduced. • The effects of the mode order have been underlined. • The convergence of SVE to RVE is discussed. [ABSTRACT FROM AUTHOR]

Highlights • A dynamic homogenization model is developed for corrugated sandwich plates. • Analytical solutions are derived for low-frequency dispersion relations. • The spectral element method is used to validate the homogenization model. Abstract In the present work, a new dynamic homogenization model is developed to investigate the long-wavelength wave propagation in a corrugated sandwich plate. With the harmonic motion assumption and using a shifting operator, the governing equations of the plate are firstly represented in a state-space form. Then, a dynamic homogenization model is developed via the two-scale homogenization method. Based on this model and considering the propagation of sinusoidal waves, the dispersion relations and corresponding wave modes can be easily obtained. In order to validate the developed homogenization model, the obtained dispersion relations are compared with those predicted by the spectral element method. It is found that the present method gives accurate results in low frequency range. Furthermore, the effects of some geometric and material parameters on the dispersion relations for the corrugated sandwich plate are also discussed. The developed homogenization model is expected to be helpful in the prediction and control of dynamic responses of corrugated or even lattice sandwich structures. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

When considering an effective, i.e. homogenized description of waves in periodic media that transcends the usual quasi-static approximation, there are generally two schools of thought: (i) the two-scale approach that is prevalent in mathematics and (ii) the Willis' homogenization framework that has been gaining popularity in engineering and physical sciences. Notwithstanding a mounting body of literature on the two competing paradigms, a clear understanding of their relationship is still lacking. In this study, we deploy an effective impedance of the scalar wave equation as a lens for comparison and establish a low-frequency, long-wavelength dispersive expansion of the Willis' effective model, including terms up to the second order. Despite the intuitive expectation that such obtained effective impedance coincides with its two-scale counterpart, we find that the two descriptions differ by a modulation factor which is, up to the second order, expressible as a polynomial in frequency and wavenumber. We track down this inconsistency to the fact that the two-scale expansion is commonly restricted to the free-wave solutions and thus fails to account for the body source term which, as it turns out, must also be homogenized--by the reciprocal of the featured modulation factor. In the analysis, we also (i) reformulate for generality the Willis' effective description in terms of the eigenfunction approach, and (ii) obtain the corresponding modulation factor for dipole body sources, which may be relevant to some recent efforts to manipulate waves in metamaterials. [ABSTRACT FROM AUTHOR]

Magneto-electro-elastic (MEE) heterogeneous materials with periodic microstructure are investigated, with special focus on the particular case of a layered MEE material. By exploiting the transfer matrix method in combination with the Floquet–Bloch boundary conditions, the frequency dispersion spectrum can be derived by solving an eigenproblem, involving a 12 × 12 transfer matrix with palindromic characteristic polynomial. In particular, by assuming the cubic symmetry of layers, the problems involving longitudinal and transverse elastic waves are uncoupled. This circumstance has the advantage of simplifying the treatment and the size of the subproblems to be solved, corresponding to a 4 × 4 transfer matrix for the longitudinal case, and a 8 × 8 transfer matrix of the transverse problem, both characterized by palindromic characteristic polynomials. Afterward, a dynamic multi-field continualization approach is proposed to investigate wave propagation in MEE layered periodic material by matching the Z -transform of the vector collecting the nodal fields to the two-sided Laplace transform of the same vector at the macroscale. The continualization technique allows identifying multi-field integral-type non-local continua as well as multi-field generalized gradient-type (higher-order) non-local continua. Finally, to verify the accuracy of the proposed approach, the dispersion curves derived from the continualization technique are compared with the curves provided by the Floquet–Bloch theory. • Magneto-Electro-Elastic periodic microstructured materials are investigated. • Transfer matrix method is applied to obtain complex frequency dispersion spectra. • A dynamic multi-field continualization scheme of layered MEE is conceived. • Multi-field generalized gradient-type (higher-order) non-local continua are identified. [ABSTRACT FROM AUTHOR]

• Perform analysis of wave propagation in crossply CFRP with Floquet wave theory; • Focus the wave energy with velocity correction for TFM imaging of crossply CFRP; • Optimize imaging with selected wave frequency, filtering, and angle restriction; Carbon fiber reinforced polymers (CFRPs) with large thickness (≥20 mm) are being extensively used for aerospace structures, and are prone to the manufacturing and in-service defects due to the complexity and difficulty of manufacturing. This study addresses an intensive interrogation of multi-parameter optimization towards a high-quality ultrasonic total focusing method (TFM) imaging for diverse and multiple defects in crossply CFRP. Firstly, material parameters of unidirectional laminate are inversely constructed based on Christoffel equation by inputting the experimentally measured wave speed, based on which the wave speed and homogenization region for the crossply CFRP are obtained with the Floquet wave theory. Then, the wave propagation at crossply CFRP with preset side drilled holes (SDHs) at depths of 4 mm and 12 mm is modelled, from which the full matrix data are extracted for TFM imaging using multiple parameter optimization, including wave velocity correction, propagation restriction angle, and wave central frequency. With the proposed signal-to-noise ratio as the criterion of TFM imaging quality, the combined parameters for optimized imaging are sorted out, which are strongly related to the feature of propagating and non-propagating waves calculated with the Floquet wave homogenization theory. A further experimental validation is performed to image multiple SDHs and delamination with size over 2 mm and depths covering 2.5 mm to 17.5 mm. This systematic analysis of parameter optimization for TFM imaging in thick CFRP sets up the foundation for the TFM towards a real engineering application of CFRP detection. [ABSTRACT FROM AUTHOR]

In this paper, we review the recent advances which have taken place in the understanding and applications of acoustic/elastic metamaterials. Metamaterials are artificially created composite materials which exhibit unusual properties that are not found in nature. We begin with presenting arguments from discrete systems which support the case for the existence of unusual material properties such as tensorial and/or negative density. The arguments are then extended to elastic continuums through coherent averaging principles. The resulting coupled and nonlocal homogenized relations, called the Willis relations, are presented as the natural description of inhomogeneous elastodynamics. They are specialized to Bloch waves propagating in periodic composites and we show that the Willis properties display the unusual behavior which is often required in metamaterial applications such as the Veselago lens. We finally present the recent advances in the area of transformation elastodynamics, charting its inspirations from transformation optics, clarifying its particular challenges, and identifying its connection with the constitutive relations of the Willis and the Cosserat types.

All metamaterial applications are based upon the idea that extreme material properties can be achieved through appropriate dynamic homogenization of composites. This homogenization is almost always done for infinite domains and the results are then applied to finite samples. This process ignores the evanescent waves which appear at the boundaries of such finite samples. In this paper, we first clarify the emergence and purpose of these evanescent waves in a model problem consisting of an interface between a layered composite and a homogeneous medium. We show that these evanescent waves form boundary layers on either side of the interface beyond which the composite can be represented by appropriate infinite domain homogenized relations. We show that if one ignores the boundary layers, then the displacement and stress fields are discontinuous across the interface. Therefore, the scattering coefficients at such an interface cannot be determined through the conventional continuity conditions involving only propagating modes. Here, we propose an approximate variational approach for sidestepping these boundary layers. The aim is to determine the scattering coefficients without the knowledge of evanescent modes. Through various numerical examples we show that our technique gives very good estimates of the actual scattering coefficients beyond the long wavelength limit. [ABSTRACT FROM AUTHOR]

In this paper, we review the recent advances which have taken place in the understanding and applications of acoustic/elastic metamaterials. Metamaterials are artificially created composite materials which exhibit unusual properties that are not found in nature. We begin with presenting arguments from discrete systems which support the case for the existence of unusual material properties such as tensorial and/or negative density. The arguments are then extended to elastic continuums through coherent averaging principles. The resulting coupled and nonlocal homogenized relations, called the Willis relations, are presented as the natural description of inhomogeneous elastodynamics. They are specialized to Bloch waves propagating in periodic composites and we show that the Willis properties display the unusual behavior which is often required in metamaterial applications such as the Veselago lens. We finally present the recent advances in the area of transformation elastodynamics, charting its inspirations from transformation optics, clarifying its particular challenges, and identifying its connection with the constitutive relations of the Willis and the Cosserat types. [ABSTRACT FROM AUTHOR]

The focus of the article is on the analysis of effective elastic properties of planar Solid Oxide Fuel Cell (SOFC) devices. An ideal periodic multi-layered composite (SOFC-like) reproducing the overall properties of multi-layer SOFC devices is defined. Adopting a non-local dynamic homogenization method, explicit expressions for overall elastic moduli and inertial terms of this material are derived in terms of micro-fluctuation functions. These micro-fluctuation functions are then obtained solving the cell problems by means of finite element techniques. The effects of the temperature variation on overall elastic and inertial properties of the fuel cells are studied. Dispersion relations for acoustic waves in SOFC-like multilayered materials are derived as functions of the overall constants, and the results obtained by the proposed computational homogenization approach are compared with those provided by rigorous Floquet–Bloch theory. Finally, the influence of the temperature and of the elastic properties variation on the Bloch spectrum is investigated. [ABSTRACT FROM AUTHOR]

Abstract: The analysis of the wave propagation in layered rocks masses with periodic fractures is tackled via a two-scale approach in order to consider shape and size of the rock inhomogeneities. To match the displacement fields at the two scales, an approximation of the micro-displacement field is assumed that depends on the first and second gradients of the macro-displacement through micro-fluctuation displacement functions obtained by the finite element solution of cell problems derived by the classical asymptotic homogenization. The resulting equations of motion of the equivalent continuum at the macro-scale result to be not local in space, thus a dispersive wave propagation is obtained from the model. The simplifying hypotheses assumed in the multi-scale kinematics limit the validity of the model to the first dispersive branch in the frequency spectrum corresponding to the lowest modes. Although the homogenization procedure is developed to study the macro-scale wave propagation in rock masses with bounded domain, the reliability of the proposed method has been evaluated in the examples by considering unbounded rock masses and by comparing the dispersion curves provided by the rigorous process of Floquet–Bloch with those obtained by the method presented. The accuracy of the method is analyzed for compressional and shear waves propagating in the intact-layered rocks along the orthotropic axes. Therefore, the influence of crack density in the layered rock mass has been analyzed. Vertical cracks have been considered, periodically located in the stiffer layer, and two different crack densities have been analyzed, which are differentiated in the crack spacing. A good agreement is obtained in case of compressional waves travelling along the layering direction and in case of both shear and compressional waves normal to the layering. The comparison between two crack systems with different spacing has shown this aspect to have a remarkable effect on waves travelling along the direction of layering, and limited in the case of waves propagating normal to the layers. The equivalent continuous model obtained through the dynamic homogenization technique here presented may be applied to the computational analysis of non-stationary wave propagation in rock masses of finite size, also consisting of sub-domains with different macro-mechanical characteristics. This avoids the use of computational models represented at the scale of the heterogeneities, which may be too burdensome or even unfeasible. [Copyright &y& Elsevier]

Abstract: This paper examines ultrasonic wave propagation through strongly heterogeneous materials such as cementitious materials, and deals meanly with the formulation of a multiphase approach of a self-consistent multiple scattering model, the so-called dynamic generalized self-consistent model (DGSCM) proposed by Yang [J. Appl. Mech. 70(2003) 575–582]. This extended model can describe the influence of the size and volume fraction of aggregates on cementitious materials, as well as the interaction, contribution, and influence of entrapped air voids together with the aggregates on frequency-dependent parameters such as the phase velocity and the attenuation coefficient. To show the performance of this approach, theoretical predictions were compared with experimental ultrasonic measurements over a wide frequency range from several mortar specimens with different features in their microstructure properties and concentrations of aggregates up to 60%. The multiphase approaches of both the DGSCM and the Waterman–Truell model (WT) were also compared. The obtained results of the multiphase DGSCM were found to be significantly better than those obtained from the N-phase WT model for ultrasonic measurements from cementitious materials at high aggregate concentrations. The feasibility of material characterization using the multiphase approach of DGSCM was also discussed. [ABSTRACT FROM AUTHOR]

Metamaterials are artificial composite structures designed for controlling waves or fields, and exhibit interaction phenomena that are unexpected on the basis of their chemical constituents. These phenomena are encoded in effective material parameters that can be electronic, magnetic, acoustic, or elastic, and must adequately represent the wave interaction behavior in the composite within desired frequency ranges. In some cases—for example, the low frequency regime—there exist various efficient ways by which effective material parameters for wave propagation in metamaterials may be found. However, the general problem of predicting frequency-dependent dynamic effective constants has remained unsolved. Here, we obtain novel mathematical expressions for the effective parameters of two-dimensional metamaterial systems valid at higher frequencies and wavelengths than previously possible. By way of an example, random configurations of cylindrical scatterers are considered, in various physical contexts: sound waves in a compressible fluid, anti-plane elastic waves, and electromagnetic waves. Our results point towards a paradigm shift in our understanding of these effective properties, and metamaterial designs with functionalities beyond the low-frequency regime are now open for innovation.

• A new dynamic homogenization approach for 3D core calculations. • Equivalence theory and Flux Discontinuity Ratio techniques have been applied to reference homogenization problems with non-zero surface leakage. • We tested the methodology on a PWR MOX/UOX heterogeneous 3D configuration. • The new method shows improved accuracy of the pin power over the classical two-step calculation scheme. Three-dimensional deterministic core calculations are typically based on the classical two-step calculation scheme. The sizable number of approximations introduced in this approach can easily reduce the accuracy of the low-order operator solution, especially in high heterogeneous configurations. Several techniques have been developed in order to overcome these limitations, including direct approaches that make extensive use of computational resources, such as the 2D/1D Fusion method and direct 3D transport calculations. In this work we propose the method of Dynamic Homogenization as an alternative technique for 3D core calculations that lies between the classical and the direct approaches in terms of accuracy and performance. The methodology consists in an iterative process between core and assembly calculations that provides a set of homogenization parameters for the low-order operator, taking into account the actual environment of each assembly in the core. This is mediated defining a set of 2D reference homogenization problems that are solved with imposed core eigenvalue, fixed incoming boundary source and a simplified axial leakage expressed as a volumetric source. The methodology eliminates the infinite lattice approximation and the critical leakage model, and offers the advantage of avoiding off-line calculations, cross-section interpolations, power reconstruction techniques and expensive 3D transport calculations. The NEA "PWR MOX/UO2 Core Transient Benchmark" was chosen for our analysis where we compare the accuracy and the performance of the two-step and the Dynamic Homogenization calculations against the direct 3D transport solution. We analyzed two different core configurations: an axially uniform core and a partially rodded core. [ABSTRACT FROM AUTHOR]

In this work, the analysis of oblique anti-plane shear waves propagation and scattering in low frequency resonant micro-structured layered media with viscoelastic constituent layers is presented. The band structure of the infinitely periodic systems and scattering off a finite thickness slab of such media are determined using the transfer matrix method. A consistent dynamic field homogenization approach is applied, in which the micro-scale field equations are integrated and the overall macro-scale quantities are defined to be compatible with these integral forms. A reduced set of constitutive tensors is presented for general asymmetric repeating unit cells (RUC), utilizing the proposed homogenized macro-scale quantities combined with Onsager's principle and presumed material form of elastodynamic reciprocity. This set can be further restricted by studying the form of the dispersion equation leading to a unique constitutive tensor. It is shown that for an asymmetric RUC, the full constitutive tensor, including off-diagonal elastic moduli and Willis coupling terms are required in order to match the scattering and band structure of the micro-structured media, but all the off-diagonal parameters vanish for a symmetric RUC. Therefore, it is possible to create an equivalent homogenized representation with a uniquely determined diagonal constitutive tensor for a symmetric RUC, though, as is the case also with asymmetric RUCs, all non-zero components will be wave-vector dependent. Numerical examples are presented to demonstrate the application and consistency of the proposed method. All the diagonal terms are converging to their appropriate Voigt or Reuss averages at the long-wavelength limit for all wave directions. The wave-vector dependent nature of the off-diagonal coupling constants can still be observed even in this limit. The conditions for lossy (passive) or lossless (fully elastic) systems are presented and are shown to impose weak requirements on overall constitutive tensors. • Dynamic homogenization is applied by integrating anti-plane shear wave equations. • The integrated macro-scale quantities reproduce the scattering of micro-structure. • Unit cell asymmetry necessitates off-diagonal coupling in constitutive tensors. • Coupling parameters are linear functions of the wave vector at long wave length limit. • Passivity and lossless necessary conditions for constitutive tensors are presented. [ABSTRACT FROM AUTHOR]

• Adopted Floquet wave-based homogenization theory to analyze wave propagation in periodic FRP; • Performed FEA analysis to validate the Floquet-wave-based homogenization approach; • Proposed the FS-TFM method to image a 2 mm diameter side-drilled hole in FRP; • Selected different wave frequencies to optimize defect imaging quality. Fiber reinforced polymers (FRPs) are increasingly used in thick primary load-bearing structures. Nevertheless, manufacturing and in-service defects occur with a higher chance as the FRP thickness increases and thus the potential structure defects should be detected and evaluated. To image defects in thick FRP over 10 mm thick, this study proposes a frequency-and-structure dependent time-corrected ultrasonic total focusing method (FS-TFM) based on Floquet wave theory, which differs from conventional TFM that the wave velocity correction along different propagation direction bases on not only the structural anisotropy and inhomogeneity of FRP, but also on the probing frequency. First an analytical Floquet-wave-based dynamic homogenization for a crossply FRP laminate is performed, to obtain the homogenization region and then the wave anisotropy and dispersion therein. Then numerical finite element analysis follows, to further interrogate the time domain feature of wave signals. With the understanding of wave propagation, by accurately correcting the wave velocity anisotropy related to both the inspected crossply FPR and wave frequency, the FS-TFM imaging technique is proposed to focus the defect-scattered wave energy on the defect location. In addition, the signal-to-noise ratio (SNR) analysis of side-drilled-hole (Ø2 mm) imaging via different imaging algorithms shows that the proposed FS-TFM achieves the highest SNR at the frequency approaching the upper limit of the homogenization region. Finally, the experimental validation further indicates the potential of the proposed FS-TFM for accurate defect imaging in thick FRP. [ABSTRACT FROM AUTHOR]

• Asymptotic homogenization of thermoelasticity theory with one relaxation time. • Elastic, thermal dilatation and heat conduction overall constitutive tensors. • Multiphase materials with a periodic microstructure. • Dispersion functions benchmark between the homogenized and the heterogeneous one. [Display omitted] A multi-scale variational-asymptotic homogenization method for periodic microstructured materials in presence of thermoelasticity with periodic spatially dependent one relaxation time is introduced. The asymptotic expansions of the micro-displacement and the micro-temperature fields are rewritten on the transformed Laplace space and expressed as power series of the microstructural length scale, leading to a set of recursive differential problems over the periodic unit cell. The solution of such cell problems leads to the perturbation functions. Up-scaling and down-scaling relations are then defined, and the latter allow expressing the microscopic fields in terms of the macroscopic ones and their gradients. The variational-asymptotic scheme to establish an equivalence between the equations at macro-scale and micro-scale is developed. Average field equations of infinite order are also derived. The efficiency of the proposed technique was tested in relation to a bi-dimensional orthotropic layered body with orthotropy axis parallel to the direction of the layers, where the mechanical and temperature constitutive properties were well established. The dispersion curves of the homogenized medium, truncated at the first order are compared with the dispersion curves of the heterogeneous continuum obtained by the Floquet-Bloch theory. The results obtained with the two different approaches show a very good agreement. [ABSTRACT FROM AUTHOR]

• Development of a new dynamic homogenization approach for core calculations.. • Equivalence and Flux Discontinuity Ratios applied to RHPs with surface leakage. • We tested the methodology on a PWR MOX/UOX heterogeneous 2D configuration. Three-dimensional deterministic core calculations are typically based on the classical two-step approach, where the homogenized cross sections of an assembly type are precalculated and then interpolated to the actual state in the core. On the other hand, 3D direct transport calculations and the 2D/1D Fusion method have recently been applied showing excellent agreement with reference Monte-Carlo, but still remaining computationally expensive for multiphysics applications and core depletion calculations. In the present work, we propose a method of Dynamic Homogenization (DH) as an alternative technique that lies between the classical and the direct approaches in terms of accuracy and performance, with no need of off-line calculations and avoiding expensive 3D transport calculations. The method consists in an iterative process between core and assembly calculations, whose main feature is to homogenize the cross sections taking into account the environment of each assembly. To do that, we define a reference homogenization problem that is solved with imposed core eigenvalue and fixed incoming boundary source and that is normalized to the core macro-currents. We have also analyzed the equivalence theory and flux discontinuity ratios techniques in the framework of non-conservative boundary conditions because particular care has to be taken in this case in order to preserve the total net assembly leakage. The current work presents analysis and tests of the Dynamic Homogenization technique for two-dimensional configurations. The NEA "PWR MOX/UO2 Core Transient Benchmark" was chosen to perform the calculations and compare the different methods. [ABSTRACT FROM AUTHOR]