Multiscale Asymptotic Method Research Articles

Three-dimensional cellular structures for viscous and thermal energy control in acoustic and thermoacoustic applications

The multi-scale asymptotic method provides a separate description of the viscous and thermal response functions governing acoustic waves propagation through porous materials. However, these response functions are inherently interdependent for simple porous structures such as slits or tubes – their determination being directly influenced by the microstructural features of the geometry. This study aims to identify, characterize, and realize a microgeometry providing the ability to independently modify the viscous and thermal behaviours. A relative autonomy in the control of these phenomena would make it possible to regulate energy conversion processes within the structure, which can be either dissipative (e.g., sound absorption) or generative (e.g., thermoacoustic gain). Among the various microgeometries, cellular solids made by an assembly of cells with solid edges or faces, packed together so that they fill space, emerge as promising candidates for achieving such a goal. Within the interconnected cells (pores), the viscous losses are governed by the aperture sizes of the faces (throat section). On the other hand, the thermal exchanges are closely related to the interface between the fluid and the solid and consequently to the dimensions of the cells. The identified microstructure is a typical Kelvin cell-based geometry, modified to account for manufacturing constraints, and characterized by faces with well-defined opening ratio and thickness. This work presents a model, experimentally validated, to predict the transport parameters of such cellular solids. It provides a valuable tool for designing microstructures having the attributes to independently tune thermal and viscous effects for specific application requirements.

Cellular communication among smooth muscle cells: The role of membrane potential via connexins

Communication via action potentials among neurons has been extensively studied. However, effective communication without action potentials is ubiquitous in biological systems, yet it has received much less attention in comparison. Multi-cellular communication among smooth muscles is crucial for regulating blood flow, for example. Understanding the mechanism of this non-action potential communication is critical in many cases, like synchronization of cellular activity, under normal and pathological conditions. In this paper, we employ a multi-scale asymptotic method to derive a macroscopic homogenized bidomain model from the microscopic electro-neutral (EN) model. This is achieved by considering different diffusion coefficients and incorporating nonlinear interface conditions. Subsequently, the homogenized macroscopic model is used to investigate communication in multi-cellular tissues. Our computational simulations reveal that the membrane potential of syncytia, formed by interconnected cells via connexins, plays a crucial role in propagating oscillations from one region to another, providing an effective means for fast cellular communication.Statement of Significance: In this study, we investigated cellular communication and ion transport in vascular smooth muscle cells, shedding light on their mechanisms under normal and abnormal conditions. Our research highlights the potential of mathematical models in understanding complex biological systems. We developed effective macroscale electro-neutral bi-domain ion transport models and examined their behavior in response to different stimuli. Our findings revealed the crucial role of connexinmediated membrane potential changes and demonstrated the effectiveness of cellular communication through syncytium membranes. Despite some limitations, our study provides valuable insights into these processes and emphasizes the importance of mathematical modeling in unraveling the complexities of cellular communication and ion transport.

An Asymptotic-Homogenization Explicit Time-Domain Method for Random Multiscale Vibration Analysis of Porous Material Structures

Porous material structures have been widely used in civil engineering, mechanical engineering, aerospace engineering and other fields due to their high specific strength and specific stiffness. The stochastic response analysis of porous material structures under random excitations deserves more attention. The multiscale differential governing equations of porous material structures are derived based on the multiscale asymptotic homogenization method (AHM), and the macroscale and microscale explicit time-domain expressions of structural responses are further established. On this basis, the statistical moments of dynamic responses of porous material structures under non-stationary random excitations can be achieved with the explicit time-domain method (ETDM). The purposed method combines the advantages of AHM for high-efficient explicit formulation of macroscale and microscale dynamic responses of porous material structures and the advantages of ETDM for fast analysis of non-stationary random vibration problems. A numerical example is presented to validate the computational accuracy and efficiency of the present approach for non-stationary random vibration analysis of porous material structures.

Sound absorption of two-dimensional rough tube porous materials

In this paper, a theoretical model for predicting the sound absorption performance of two-dimensional rough tube porous materials is established based on the Johnson–Champoux–Allard–Lafarge equivalent fluid model. The shape of the two-dimensional rough tube is approximated by trigonometric functions, and the theoretical expressions of its fluid transport parameters are given, including viscous permeability, thermal permeability, tortuosity, viscous characteristic length, and thermal characteristic length. In addition, the influence of shape factor is considered when calculating the thermal permeability and the viscous characteristic length, and its theoretical expression is given. The theoretical model is verified by a numerical simulation model based on the multi-scale asymptotic method, and good agreement is achieved. Compared with smooth tubes, circumferential rough tubes and axial rough tubes, the two-dimensional rough tubes not only enhance the viscous dissipation effect but also enhance the thermal dissipation effect during the propagation of sound waves, thus, realizing the high-efficiency sound absorption at lower frequencies. This work further develops the sound absorption theory of porous materials considering the roughness effect and enriches the research and design ideas of porous materials.

Axisymmetric longitudinal waves in a cylindrical shell interacting with a nonlinear elastic medium

The evolution of axisymmetric longitudinal waves in a cylindrical Kirchhoff–Love shell interacting with a nonlinear elastic medium is investigated. A generalized model of a nonlinear elastic medium is introduced into consideration, which is reduced in particular cases to the well-known two-parameter Pasternak and Hetenyi models. Using the asymptotic multiscale method for the component of the longitudinal displacement, a non-integrable quasi-hyperbolic equation of the sixth order is derived, generalizing the Ostrovsky equation, and classes of its exact solutions are constructed in terms of the Jacobi elliptic function. It is demonstrated that exact periodic solutions are physically realizable, while solitary wave solutions are nonphysical. The further asymptotic analysis of the derived equation made it possible to obtain the nonlinear Schrödinger equation and use the Lighthill criterion. Relationships are established between the parameters of a nonlinear elastic medium, which are necessary for the development of the modulation instability of small periodic waves.

Modeling the acoustic behavior of stepwise gradient porous structures

Spatial property gradients can significantly enhance the noise reduction potential of porous structures. However, predicting the acoustic behavior of structures with gradient microstructures continues to remain a challenge. For porous structures with controlled periodic microstructures, researchers have recently demonstrated the use of a multiscale asymptotic method to extract the acoustic transport properties necessary for use in semi-empirical predictive models. Here, we propose the adaptation of this method to enable the modeling of porous absorbers with stepwise property gradients. The unit cell of the chosen microstructure is modeled in COMSOL Multiphysics and the acoustic transport properties are extracted using the multiscale method. The extracted properties are then used to formulate the transfer matrix of each unit cell. The global sound absorption and transmission loss behavior of the stepwise gradient structures are further predicted by combining the appropriate local transfer matrices. Our results show that the integration of the unit cell and transfer matrix methods provides a robust way of predicting the acoustic behavior of stepwise gradient porous structures with various combinations of layer thicknesses and geometries. The method provides a computationally efficient method to model such structures and can further the development of porous structures with application-specific acoustical properties.

Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility

<p style='text-indent:20px;'>In this paper, we propose a model to price vulnerable European options where the dynamics of the underlying asset value and the counter-party's asset value follow two jump-diffusion processes with fast mean-reverting stochastic volatility. First, we derive an equivalent risk-neutral measure and transfer the pricing problem into solving a partial differential equation (PDE) by the Feynman-Kac formula. We then approximate the solution of the PDE by pricing formulas with constant volatility via multi-scale asymptotic method. The pricing formula for vulnerable European options is obtained by applying a two-dimensional Laplace transform when the dynamics of the underlying asset value and the counter-party's asset value follow two correlated jump-diffusion processes with constant volatilities. Thus, an analytic approximation formula for the vulnerable European options is derived in our setting. Numerical experiments are given to demonstrate our method by using Laplace inversion.</p>

Modeling of the Turbulent Poiseuille–Couette Flow in a Flat Channel by Asymptotic Methods

Developed turbulent flow of a viscous incompressible fluid in a channel of small width at high Reynolds numbers is considered. The instantaneous flow velocity is represented as the sum of a stationary component and small perturbations, which are generally different from the traditional averaged velocity and fluctuations. The study is restricted to the search for and consideration of stationary solution components. To analyze the problem, an asymptotic multiscale method is applied to the Navier–Stokes equations, rather than to the RANS equations. As a result, a steady flow in the channel is found and investigated without using any closure hypotheses. The basic phenomenon in the Poiseuille flow turns out to be a self-induced fluid flow from the channel center to the walls, which ensures that kinetic energy is transferred from the maximum-velocity zone to the turbulence generation zone near the walls, although the total averaged normal velocity is, of course, zero. The stationary solutions for the normal and streamwise velocities turn out to be viscous over the entire width of the channel, which confirms the well-known physical concept of large-scale “turbulent viscosity.”

Dimensional reduction analyzing the thermoelastic behavior of wind turbine blades based on the variational asymptotic multiscale method

Accurate and effective prediction of the thermoelastic behavior of composite wind blades is critical to the optimal design and performance evaluation. In this paper, a high-fidelity dimensional reduction model of composite wind blade is established based on the variational asymptotic multiscale method. First, the thermoelastic variational expression of 3-D composite wind blade was derived. Then, the thermoelastic energy functional was extended asymptotically to a series of 2-D energies by using the inherent small parameters of the structure, so that the original 3-D thermoelastic problem of composite wind blades was strictly divided into 2-D airfoil section analysis and 1-D beam analysis. The local displacement and stress fields were recovered based on the global response without a priori assumptions and unnecessary scale separation assumptions. Finally, the numerical examples of composite wind blades with NACA2412 airfoil and MH 104 airfoil are provided to verify the accuracy and effectiveness of the model by comparison with the three-dimensional finite element model (3D-FEM). The results show that the predicted thermoelastic behavior and local 3-D stress field of the composite wind blades are in good agreement with 3D-FEM, but the calculation efficiency is greatly improved.

Hierarchical cellular ferroelectric metamaterials: A design motif to enhance multifunctional figures of merit

The engineering of microstructures has been adopted as an effective approach to tune the overall performance of advanced ferroelectric materials. In this article, inspired by hierarchical porous biomaterials, the ferroelectric properties of hierarchical cellular ferroelectric metamaterials constructed by six commonly used primitive cubic unit cells are investigated. Both multiscale asymptotic homogenization method and scaling relationship are proposed to predict the effective ferroelectric properties of hierarchical cellular metamaterials. Analysis on the influence of design parameters, e.g. hierarchical order, cell topologies, and relative density on their effective ferroelectric figures of merit is conducted. The 2nd-order hierarchical cellular ferroelectric metamaterial exhibits remarkable improvement compared to the corresponding 2nd order unit cell. For example, the normalized FOM33 of hybrid 2nd-order T3-2/L3-1 (ρ1 = 0.2, ρ2 = 0.25) is 117.49, while 19.95 for 1st-order L3-1 (ρ = 0.05). Increasing structural hierarchical order can further improve the effective ferroelectric properties of hierarchical cellular ferroelectric metamaterials. This work highlights the potential of ultralight hierarchical ferroelectric metamaterials as the next generation of hydrophone, IR detector, flexible self-powered sensors, and thermal energy harvesting devices.

A multiscale parallel algorithm for dual-phase-lagging heat conduction equation in composite materials

In this paper, a new numerical scheme which combines the multiscale asymptotic method and the Laplace transformation, is presented for solving the 3-D dual-phase-lagging equation in composite materials. The convergence results of the truncated first-order and second-order multiscale approximate solutions are given rigorously. The numerical experiments are carried out to validate the theoretical results of this paper. It is pointed out that the proposed method allows us to choose a relative coarse grid and solve problems in parallel, and therefore it can greatly save computer memory storage and CPU time.

Estimation of effective properties of composite sandwich panels with negative Poisson's ratio by using variational asymptotic multiscale method

According to the anisotropy and non-homogeneity of composite sandwich panels with negative Poisson's ratio (NPR), the variational asymptotic method combined with multiscale technology (VAMM) is used to analyze its effective properties. Firstly, in the microscale, the effective properties of the unidirectional fiber reinforced composite (UDFRC) forming the facesheets are obtained by means of asymptotic homogenization analysis, which can be used as lamina properties. Then, in the mesoscale, an asymptotic analysis of the typical unit cell is carried out to deduce the effective plate properties (A, B, and D matrices) of composite sandwich panels with NPR according to the heterogeneity of the structure. Finally, the influence of the layup configurations and fiber volume fractions of UDFRC on the equivalent plate properties of composite sandwich panels with NPR is analyzed. Numerical results show that the 3D responses of composite sandwich panels with NPR using VAMM are in agreement with those of 3D FEM model, while in a much more efficient manner. The layup configurations of facesheets have great influence on the equivalent plate properties of the structure. Furthermore, the degree of anisotropy and negative Poisson's ratio of the whole structure decrease with increasing fiber volume fraction of UDFRC.

The generalized Schamel equation in nonlinear wave dynamics of cylindrical shells

The axisymmetric propagation of longitudinal waves in an integrally stiffened physically and geometrically nonlinear cylindrical shell is studied. The case is considered when the parameters characterizing nonlinearity, dispersion and thickness–radius ratio are the same order. Using the multiscale asymptotic method, the generalized Schamel equation is derived from the equations of motion of the shell. This equation contains an additional term with the fifth derivative with respect to the spatial coordinate which characterizes the high-frequency dispersion. It is shown that the derived equation does not pass the Painleve test and therefore is not integrated by the inverse scattering transform. The geometric series method using Pade approximants is applied to construct exact solitary-wave solution of the equation. It has been established that for the existence of an exact bounded amplitude solution a “soft” type of physical nonlinearity is necessary. The Schamel–Kawahara equation containing the combined nonlinearity is considered; its exact soliton-like solution is given. Numerical simulation confirmed that the initial perturbation in the form of the exact solution generates a persistently propagating solitary wave.

Wave dispersion curves in discrete lattices derived through asymptotic multi-scale method

This paper falls within the study of dispersion feature of elastic periodic media. In most cases, no analytic description is reachable and the problem is solved via numerical computations of the dispersion curves.We propose in this paper an analytic method dedicated to lattice systems that enables to reconstruct part by part the dispersion curves via an asymptotic multi-scale method. This method is illustrated on periodic reticulated beams. At low frequency, when there is a large scale separation between the length of the cell and the characteristic size of the vibrations, the classical homogenization method allows efficiently to establish the continuous equivalent dynamic description and the associated wave propagation properties. This scale separation is lost for frequencies of the order or higher than the diffraction frequency. However, instead of considering the amplitude of the mean displacement in a unit cell, the concept of scale separation can still be used by considering the amplitude of periodic eigenmodes defined on (multi-)cells. Thus, similar principles of asymptotic multi-scale method enables to describe the large scale modulations around the eigenfrequencies of the mono-and/or multi-cells period. Finally, the properties of the modulation are straightforwardly related to the dispersion curves at the considered frequencies.

A multiscale model of epigenetic heterogeneity-driven cell fate decision-making

The inherent capacity of somatic cells to switch their phenotypic status in response to damage stimuli in vivo might have a pivotal role in ageing and cancer. However, how the entry-exit mechanisms of phenotype reprogramming are established remains poorly understood. In an attempt to elucidate such mechanisms, we herein introduce a stochastic model of combined epigenetic regulation (ER)-gene regulatory network (GRN) to study the plastic phenotypic behaviours driven by ER heterogeneity. To deal with such complex system, we additionally formulate a multiscale asymptotic method for stochastic model reduction, from which we derive an efficient hybrid simulation scheme. Our analysis of the coupled system reveals a regime of tristability in which pluripotent stem-like and differentiated steady-states coexist with a third indecisive state, with ER driving transitions between these states. Crucially, ER heterogeneity of differentiation genes is for the most part responsible for conferring abnormal robustness to pluripotent stem-like states. We formulate epigenetic heterogeneity-based strategies capable of unlocking and facilitating the transit from differentiation-refractory (stem-like) to differentiation-primed epistates. The application of the hybrid numerical method validates the likelihood of such switching involving solely kinetic changes in epigenetic factors. Our results suggest that epigenetic heterogeneity regulates the mechanisms and kinetics of phenotypic robustness of cell fate reprogramming. The occurrence of tunable switches capable of modifying the nature of cell fate reprogramming might pave the way for new therapeutic strategies to regulate reparative reprogramming in ageing and cancer.

Regimes of two-dimensional energy channeling in the inertially coupled unit-cell model subjected to an asymmetric potential

We analyze the stationary and non-stationary states emerging in a two-dimensional (2D) model consisting of an outer element that incorporates an internal rotator and is subjected to a 2D anharmonic and non-symmetric local potential. We focus on the effect of the asymmetry of the potential on the mechanism of formation and bifurcations of various stationary and non-stationary regimes in the system. The non-stationary regimes are characterized by an intense bi-directional energy transfer between the axial and lateral vibrations of the main element as well as the regimes of unidirectional energy locking. We show that the transitions between the different states of the system are controlled by the motion of the internal rotator. Our analysis is based on the complexification-averaging procedure and the singular multiscale asymptotic method. Investigation of the system's dynamics in the vicinity of the 1:1:1 resonance enables us to reproduce the bifurcation structure of the stationary regimes and describe the formation mechanisms of highly non-stationary states of intense energy transfer. Our analysis shows that the most important effect of the asymmetric potential is the breakdown of a unique complete bi-directional energy channeling mechanism into two separate mechanisms that correspond to incomplete horizontal-to-vertical and vertical-to-horizontal energy transfer, respectively, and are enabled at two different critical values of the main bifurcation parameter. Results of the analysis are in good agreement with the numerical simulations of the full model.

Multiscale Algorithms and Computations for the Time-Dependent Maxwell--Schrödinger System in Heterogeneous Nanostructures

In this paper, we discuss the multiscale computations of the time-dependent Maxwell--Schrodinger system with rapidly oscillating discontinuous coefficients. The multiscale asymptotic method for the...

Dynamics of periodic ribbed plates with inner resonance: Analytical homogenized model and dispersion features

The dynamic behavior of a periodic ribbed plate with local resonance is investigated. The behavior of the cell made of a beam clamped along a plate edge analyzed through multi-scale asymptotic method enables to derive the governing equations of the effective mechanical behavior. This approach allows obtaining a full homogenized analytical model that provides a relevant representation of the flexural and torsional mechanisms at both global and local scales. The complex dynamic behavior is shown to encompass several mechanisms associated with enriched kinematics. Two types of flexural and torsional waves are evidenced and governed by two distinct differential equations that describes (i) waves where both beam and plate moves, and (ii) guided waves where the plate only is set in motion. The inner resonance of the plate induces unconventional dispersion features, with singularities associated either with the symmetric eigenmodes for the bending waves or the antisymmetric modes for the torsional waves. The guided waves are alternatively related to the symmetric and antisymmetric modes of the bended plate and are propagative above the corresponding eigenfrequencies. The predictions of the homogenized model are successfully compared to numerical calculations conducted using Wave Finite Element based methods, for two realistic examples of ribbed plates. The study provides design rules to tailor ribbed plate panels having specific atypical features in a given frequency range.

A semi-analytical model for the influence of temperature on sound propagation in sintered metal fiber materials

A semi-analytical model is developed to predict the influence of temperature on the sound absorbing performance of sintered metal fiber materials (SMFMs) by extending the Johnson-Champoux-Allard-Lafarge (JCAL) model. In this model, three micro-structure factors of the SMFMs – the Kozeny number and the thermal and viscous shape factors – are calculated by applying the multi-scale asymptotic method (MAM), and the temperature effect is taken into account by considering the variations of the thermo-physical parameters of saturated air with temperature. Key transport parameters (e.g., viscous and thermal permeability, tortuosity, viscous and thermal characteristic lengths) are then directly determined by fiber diameter and porosity of the material. High-efficiency of the semi-analytical model is demonstrated for estimating the influence of topological parameters upon sound absorption under different temperatures. The transport parameters obtained by the proposed model agree well with those calculated using fully numerical simulations. Also, the model predictions are in accordance with existing experimental measurements at different temperatures. The model reveals not only the underlying mechanisms of temperature effect on sound propagation in porous metals, but also provides a theoretical guideline for the sound absorption application of SMFMs in high temperature environments.

Multi-Scale Homogenization of Transversal Waves in Periodic Composite Beams

This paper deals with the deduction of new homogenized models for the flexural wave in bi-periodic beams. According to the homogenization theory, the long-wave assumption is used and the valid frequency range of homogenized models is limited to the first Bragg band gap. However, the classical homogenization method, whose idea is taking the component’s mean values as effective material properties, has limitations in mimicking the dispersive behavior and the real valid frequency range is far less than the limit. Thus, enriched homogenized models, derived by the multi-scale asymptotic homogenization method, are proposed to provide more accurate homogenization models with larger real valid frequency range. The new homogenized models are validated by investigating the dispersion relation in the infinite case and the frequency response function in the finite case. Wave finite element method (WFEM) are used to provide associated references. A parametric study is carried out in the infinite case while two different boundary conditions are considered in the finite case.