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Sprays generated by atomization processes have been used in a wide range of fields. The size and spatial distribution of volumetric droplets in a spray are critical parameters in industrial applications. Astigmatic interferometric particle imaging (AIPI), an extension of the traditional interferometric particle imaging (IPI), is developed to simultaneously measure the size and 3D position of volumetric droplets in a sparse spray. Based on the generalized Huygens–Fresnel integral and the transfer matrix, the size and depth position of droplet can be respectively extracted from the fringe spacing and orientation of interferogram in AIPI. An AIPI setup is established to characterize droplets in a sparse spray generated by a nozzle with the AIPI calibration procedure adopted. The measured parameters by AIPI are compared with those obtained synchronously by digital inline holography, which is regarded as a standard measurement technique. Results show that the average deviation values of droplet size and depth position are respectively 3.8% and 6.8%. AIPI has been demonstrated with high accuracy in simultaneous 3D positions and size measurements.

The analytical solutions are presented for the distributions of temperature, displacements, and stresses in simply-supported layered rectangular plates under thermo-mechanical loads. The material properties of every layer are temperature-dependent. A slice model is presented for discretization: dividing every layer into finite thin slices, the material properties within each slice are assumed to be uniform. The temperature varies along the thickness and follows the one-dimensional (1-D) heat conduction equations. An iterative algorithm is proposed for the prediction of the 1-D temperature field. Then, the exact three-dimensional (3-D) thermoelasticity theory and the transfer matrix technique are applied for the prediction of the displacement and stress fields in the plate. The influences of temperature can be divided into two parts: producing thermal deformations and thermal stresses and affecting the deformations and stresses induced by mechanical loads. Finally, the influences of the surface temperature and the layer thickness ratio on the thermo-mechanical responses of a sandwich plate are studied in detail.

The dynamic behavior of bi-dimensional functionally graded beams is investigated by using the continuous elements method. The material properties are assumed to vary exponentially along the beam thickness and length. First, the transfer matrix is developed. Second, the characteristic frequency equation of simply supported beams is obtained. The proposed method is assessed and validated through comparison with available results. Then, parametric studies are carried out to analyze the influences of the gradient indexes and the beam slenderness ratio on the natural frequencies of bi-dimensional functionally graded beams.

The principle of the ultrasonic technique consists of exciting the elastic plate immersed in water by an incident plane wave, the scattering acoustic phenomena are observed and then the backscattering waves are detected. In this study, we provide a theoretical model that is developed to study and understand the acoustic scattering. This model is based on the transfer matrix, taking into account the matrix components, we obtain the reflection and transmission coefficient by appropriate relationship. Furthermore, we will present a study an acoustic signal backscattering, under a normal incident wave, for different plates (Glass and Plexiglas). The time–frequency representations such as spectrogram (SP) are used. The images obtained by these representations are compared to the frequency spectral method. Consequently, the results obtained from the time–frequency analysis are interesting.

The aim of this paper is to present a theoretical study of the transmission coefficient for a periodic CdMnTe / C d T e system containing a geometric and material defect layer. The dynamic tuning of this defect layer is investigated using the transfer matrix for the theoretical calculations. It is shown that the thickness and the concentration of the defect layer have a strong impact on the electronic states (called also defect modes) localized in the band gaps. Moreover, the transmission coefficients of these defect states reach maximal values when the defect layer is placed in the middle of the structure; this is due to the constructive interferences caused by the symmetry of the system. Further, the defect states shift to lower energies with the increase of the concentration or the thickness of the defect layer as well as the decrease of the quality factor. These findings are useful to design a nano-energy filtering and sensoring devices.

Combining the ABCD transfer matrix and the generalised Huygens-Fresnel integral equation method, the closed-form expression of a finite Airy Gaussian array beam (FAGAB) threading through a paraxial optical system was established in a space domain. We systematically investigated the evolution of the FAGAB's properties travelling through a curved slab system based on previously derived formula. The wave's characteristics through the flat slab system were the special cases of the main findings. We also assess the influence of the truncation factor and adjustment parameter on the FAGAB's intensity distribution on the slab system model's input and output cross-sections.

This paper proposes a novel realization approach to generate a low-order Fornasini–Marchesini (F-M) (second) model for multidimensional (n-D) systems. First, based on two developed matrix relation properties, the F-M model realization problem is converted to establishing a desired standard matrix by performing elementary operations on an associated matrix initialized from the given transfer matrix. Then, a general constructive procedure and two corresponding techniques to compute the F-M models are established for both the left and right matrix fractional descriptions (MFDs) of the given transfer matrix. It turns out that compared with the existing methods the proposed approach can produce an F-M model with lower order, and furthermore, this is the first method that can treat both the left and right MFDs for obtaining an F-M model realization in a unified manner. Non-trivial examples are given to illustrate the details and effectiveness of the proposed approach.

In this work we investigate the effects of a Fermi velocity modulation in a valley filter in graphene created by a combination of a magnetic and electric barrier. With the effective Dirac equation of the system, we use the transfer matrix formalism to obtain the transmittance. We verify that the valley transport in graphene is very sensitive to a Fermi velocity modulation, which is able to choose which valley will be filtered with perfect filtering. Also, it is possible to use a Fermi velocity modulation to filter both valleys or to make the valley filter transparent. It reveals that the Fermi velocity is a powerful tool that can be used to tune a graphene valley filter, since it has a total control in its transport properties.

Vibration characteristics play a fundamental role in structural analysis and dynamics design. The determination of vibration characteristics requires the solution of an eigenproblem. In this work, a novel parallel hybrid finite element (FE) transfer matrix method (TMM) is studied for a large-scale structural eigenproblem. The full structure is divided into some substructures, and the vibration characteristics of each substructure are calculated with their proper solution method (FE or TMM). Then by using the order-reduction transformation of the full structure based on an improved fixed interface modal synthesis, the reduced global equation of the full structure is obtained easily, and it can be used to solve the vibration characteristics of the full structure efficiently. To overcome the numerical instability of the ordinary transfer matrix technology in a large-scale structural analysis, the Riccati transform is also adopted. This hybrid method has good parallelism, and the corresponding parallel algorithms are presented. Finally, this method is applied to a vacuum vessel structure of the China fusion engineering test reactor, which has cyclic symmetry conditions. The simulation results demonstrate that this method has low computational cost and satisfactory accuracy.

Due to the curse of dimensionality caused by the increasing number of objectives, it is very challenging to tackle many-objective optimization problems (MaOPs). Aiming at this issue, this paper proposes a novel many-objective evolutionary algorithm , called Tk-MaOEA, based on transfer matrix assisted by Kriging model. In this approach, for the global space optimization, a transfer matrix is used as a map tool to reduce the number of objectives, which can simplify the optimization process. For the objective optimization, the Kriging model is incorporated to further reduce the computation cost. In addition, the fast non-dominated sorting and farthest-candidate selection (FCS) methods are used to guarantee the diversity of solutions. Comprehensive experiments on a set of benchmark functions have been conducted. Experimental results show that Tk-MaOEA is effective for solving complex MaOPs.

The transfer matrix method has been proposed to analyze the acoustic black hole effect in duct terminations. The latter is achieved by placing a retarding waveguide structure inside the duct, which consists in a set of rings whose inner radii decrease to zero following a power law. The rings are separated by thin air cavities. If the number of ring-cavity ensembles is large enough, wave propagation inside the waveguide can be treated as a continuous problem. A governing differential equation can be derived for the acoustic black hole which intrinsically relies on assumptions from transfer matrix theory. However, no formal demonstration exists showing that the transfer matrix solution is consistent and formally tends to the solution of the continuous problem. Proving such consistency is the main goal of the paper and an original option has been adopted to this purpose. First, we prove the differential equation for the acoustic black hole is identical to the wave equation for a metafluid with a power-law varying density. Transfer matrices are then applied to solve wave propagation in a discretization of the metafluid into layers of constant density. It is shown that when the number of layers tends to infinity and their thicknesses to zero, the transfer matrix solution satisfies the metafluid equation and therefore the acoustic black hole one. The transfer matrices for the metafluid discrete layers take a particularly simple form, which is a great advantage. This work allows one to interpret the retarding waveguide structure as a particular realization of the metafluid.

In this study, a new simulation method is proposed and verified for fiber Bragg grating patterned on polarization maintaining fiber(PM-FBG) using the transfer matrix approach. The method is designed to solve the twisted PM-FBG simulation. The four mode matrix element of a period grating is derived from the S matrix element and the method only involves the torsion-induced shear strain entry e x y which can be considered as the rotation of the principal axis of the polarization maintaining fiber and ignores the torsion-induced shear strain entries e x z and e y z . The validity of the method is confirmed by numerical simulation of the twisted PM-FBG. The novel method provides the ability to analyze any type of PM-FBG.

The transfer matrix method (TMM) is a common method for the modelling of acoustical systems. Traditionally, this method requires each unique layer within a system to be defined by a transfer matrix and then for each matrix to be multiplied together in the sequential order of the system. Whilst the resulting matrix can be used to find the effective material properties of the modelled system, the resulting analytical expressions of these properties are often unwieldy for use. Here, a simplified approach is proposed to obtain simple analytical expressions in the low frequency regime for the effective properties of acoustical systems based on the components of the TMM and inspired by the Champoux and Stinson model. It was shown that the proposed approximation of TMM matches the effective fluid properties of a cylindrical rigid tortuous pore derived with the Champoux and Stinson model. Using this approach, analytical expressions for the effective fluid properties of a waveguide of constant cross section, side-loaded by an arbitrary number of Helmholtz resonators, were derived. These expressions were validated against the traditional transfer matrix method and with numerical computation. The result of this work offers a validated general approach that provides simple analytical low frequency approximations of the acoustical properties of media which consist of complicated networks of pores or side-branches.

In this paper, a theoretical analysis of a flat-top microring resonator (FTMRR) is undertaken. The intensity and group delay responses of the FTMRR are derived from a transfer matrix formalism, which shows that flat-top resonance is possible in this single-ring cavity by stimulating and manipulating slow light in the counterclockwise (CCW) mode and fast light in the clockwise (CW) mode. The analytical solutions required for achieving the flat-top responses are obtained by assuming that the slope around the resonant wavelength equals zero. In addition, an improvement in the bandwidth of the FTMRR (in comparison with the MRR), the tunable and reconfigurable optical responses for filtering/switching, and the small variation in intensity during the tuning of the flat-top group delay observed in the previous experiments are all proved through theoretical analysis.

Huazhong University of Science and Technology has been developing a proton therapy facility based on a superconducting cyclotron. As an important section of the facility, the gantry beamline has been designed using TRANSPORT code with simplified fringing field model and linear transfer matrix. However, realistic magnetic fringing field and second-order terms of particle motion cannot be precisely described in this approach, which may deviate the actual result from the design value and thereby increase uncertainties for radiation therapy during beam delivery. For this consideration, the specific effects of fringing field and second-order aberrations have been investigated using OPERA code and ODE solvers. For the potential problems of the designed gantry beamline, COSY INFINITY code, which can calculate transfer matrix of any order under the realistic fringing field represented by Enge function, was used to optimize the optics of present gantry beamline. The final results suggest the beam analysis and optimization methods mentioned in this paper are effective and valuable for relevant applications.

We propose a theoretical model to study the spin transport, particularly the interfacial resistance in a noncollinear system with a diffuse interface. The realistic interface region arising from the sputtering process is first modeled by using Fick’s law. Subsequently, the spin transport behavior in magnetic system is determined via a generalized spin accumulation model taking into account spin dephasing term. Our model is able to deal with magnetic systems with arbitrary orientation of magnetization by employing the transfer matrix approach. The model is first used to investigate spin accumulation in a bilayer system consisting of a ferromagnet (FM) and a non-magnet (NM). The influence of important transport parameters: spin dephasing, interdiffusion, injected current density and spin polarization on magnetoresistance (MR) are studied. Next we investigate the transport properties of a FM/NM/FM system where the interface disorder giving rise to enhanced spin dephasing is modeled. We found that the interdiffusion of the FM/NM/FM interface gives rise to large spin scattering, which enhances the GMR. The results also show that including the disorder arising from interface diffusion is an important contribution to the transport properties.

An effective method for the calculations of stress and residual strength of complex spatial pipelines at static and dynamic loading with taking into account the vibration of internal liquid and Puasson coupling is presented. It based on the well-known transfer matrix approach. The method utilizes the exact mechanical equations for the stress-strained state of straight beam with distributed mass, which connect unknown parameters of forces and displacements at the end of beam calculation element with the parameters at the beginning of the element. At a general level, the solution scheme and calculation models are the same for static and dynamic calculations, the only difference – are the equations. Two different methods are used to find natural vibration frequencies of the distributed mass system: straightforward (“brute force”) and quick, based on Williams-Wittrick frequencies counter. Within such counter a special procedure is developed for taking into account fluid vibration with consideration of volumetric balance condition at T-nozzles. The excellent accuracy of the method is demonstrated by the modeling of thin-walled, water filled pipe behavior after the rod impact. Two seismic analysis procedures are implemented on the base of developed method: standard response spectrum procedure and normal-mode summation procedure (“exact” dynamical), which uses accelerograms. These methods and procedures have been already implemented in calculation software for pipeline strength assessment with the possibility of building of arbitrary complexity models by the user-friendly, visual way. Their effectiveness is demonstrated by the seismic calculation for primary circuit pipings of Zaporizhia NPP.

Vibroacoustic performance of the doubly curved thick shell is explored based on the three dimensional sound propagation approach as well as state space solution. In fact, the main aim is particularly focused on inspecting the influence of using three-dimensional theory through sound transmission loss (STL) of the structure which includes more reliable and accurate results especially for relatively thick and thick shells even in high frequency domain in comparison with other theories. In order to achieve this end, firstly stress and strain components are developed to present the governing equations of thick shell. This procedure is carried out by dividing the shell into s layers. Then, a solution technique is provided on the basis of state vector methodology wherein approximate layer model along with local transfer matrix are performed. Moreover, this method is followed by global transfer matrix method for the all layers of structure. As an outcome, in results section, not only the accuracy of the offered results is proved but also the importance of employing the current theory in high frequencies is revealed. Another remarkable achievement of this work is related to nominate the dip points of STL diagram. On contrary to panels, doubly curved shells contain two dips, because of their both radii of curvatures. In this work, the first dip is nominated as curvature frequency. The second dip is similar to that of panels at high frequency zone. Thus, it is called as coincidence frequency. Finally, the behavior of the transmitted pressure and the effect of curvatures on the position of curvature frequency are discussed.

This paper presents a state-vector equation method (SVEM)-based spectral element method (SEM) to formulate frequency domain spectral element models for non-uniform 1-D structures. In the SVEM-based SEM, the frequency domain homogeneous governing differential equations for non-uniform 1-D structures are first transformed into state-vector equations. The system matrix and state-vector are represented with power series. These power series representations are then substituted into the state-vector equation, and the differential transformation method is used to efficiently derive a recurrence formula for the coefficients of the power series solution of the state-vector equation. By computing these coefficients from the recurrence formula, the transfer function that relates the state-vector at an arbitrary position to that at the initial position of a finite element is derived. The transfer matrix is then obtained from the transfer function. Finally, the spectral element matrix or exact dynamic stiffness matrix is obtained from the transfer matrix. The accuracy and computational efficiency of the proposed SVEM-based SEM are verified in due course through comparisons with other solution methods for non-uniform axial rods, Bernoulli−Euler beams, and Timoshenko beams with different spatial variations.

"Ternary photonic crystals have shown a superior performance over the binary ones. In this work, a ternary photonic crystal which has the structure dielectric/plasma/ dielectric is considered. Transmission through the proposed photonic crystal at oblique incidence on a specific PC structure is analyzed using the transfer matrix technique. The effect of the plasma layer of the photonic band gap is investigated. The use of the proposed photonic crystal as an optical sensor for detection of small changes within the plasma electron density or the index of refraction of the dielectric layers is investigated." "Ternary photonic crystals have shown a superior performance over the binary ones. In this work, a ternary photonic crystal which has the structure dielectric/plasma/ dielectric is considered. Transmission through the proposed photonic crystal at oblique incidence on a specific PC structure is analyzed using the transfer matrix technique. The effect of the plasma layer of the photonic band gap is investigated. The use of the proposed photonic crystal as an optical sensor for detection of small changes within the plasma electron density or the index of refraction of the dielectric layers is investigated."

Boasting the advantages of low maximum ground pressure and low locomotion resistance of tracked mechanisms, a traveling wave sandwich piezoelectric transducer with a beam-ring combined structure was proposed and developed to directly drive a metal track, and preliminary experimental investigations have confirmed the feasibility of the transducer design and its operating principle. However, the dynamic behavior of the sandwich piezoelectric transducer has not yet been studied theoretically. To enable practical applications, the sandwich piezoelectric transducer needs to be optimized for better output performances. Therefore, a semi-analytical model is of significance to describe the dynamic behavior of the sandwich piezoelectric transducer, to predict its driving effect, and to improve its output performance. In this study, an electromechanical coupling model is developed for the sandwich piezoelectric transducer by employing the transfer matrix method, in which a novel longitudinal-bending coupling vibration transfer matrix is created for the combined PZT elements and an in-plane bending vibration transfer matrix is developed for the closed curved beam element. To validate the proposed model, experimental investigations are carried out to measure the dynamic behavior of the transducer prototype and compare to calculation results. The differences between the measured and calculated resonant frequencies for two operating vibration modes are, respectively, 74 Hz and 63 Hz. Also the vibration shapes at the respective resonant frequencies match well. Comparisons demonstrate the feasibility of the developed transfer matrix model. To improve the output performance of the transducer, model-based optimization is conducted. The optimized geometrical sizes are obtained to meet optimization goals of the transducer.

The authors developed the modified power method (MPM) in previous publications to obtain multiple eigenmodes of an eigenvalue problem at the same time by employing a generalized eigenvalue problem (GEP) of the form of WX = VXK. Special attention has been paid to the Monte Carlo (MC) implementation of the MPM because it always suffers from the inherent statistical noises. In this paper, a preconditioning method for the GEP has been developed for the MC MPM, so that the performance is more stable and robust to the MC statistical noises. This preconditioning method is crucial for MC solving of problems with degenerated eigenmodes, which requires the accumulation of a so-called transfer matrix and the division of the system space into multiple sub-regions, the number of sub-regions being greater than the number of eigenmodes to be solved. The preconditioning method solves the issues arising from the mismatch between the number of sub-regions and the target number of eigenmodes to be calculated. The numerical results for a model cube problem and BEAVRS whole core neutron transport eigenvalue problem successfully demonstrate the validity of the preconditioning method and the extended applicability of the MC MPM for practical problems.

A 2-dimensional ( 1 , ∞ ) -runlength limited constraint S h s is the set of ( 0 , 1 ) -matrices/arrays that do not have adjacent horizontal or vertical 1’s. Let f ( m , n ) be the number of ( m + 1 ) × ( n + 1 ) ( m , n ≥ 0 ) matrices that satisfy the constraint S h s . It is known (Calkin and Wilf, 1998) that f ( m , n ) is equal to the number of independent sets in an ( m + 1 ) × ( n + 1 ) grid graph and f ( m , n ) = 1 t T m n 1 , where T m is an associated transfer matrix and 1 is the all 1’s vector. The capacity of the constraint S h s , c a p ( S h s ) = lim m , n → + ∞ log 2 f ( m , n ) m n , has been bounded actively in the last decades (e.g., Calkin and Wilf, 1998; Engel, 1990; Nagy and Zeger, 2000; Weber, 1988), with the most recent bounds being 0 . 5878911617 . . . ≤ c a p ( S h s ) ≤ 0 . 5878911618 . . . (Nagy and Zeger, 2000). In this paper we derive various algebraic and combinatorial properties of the transfer matrix T m and by which we show that lim m , n → + ∞ f ( m , n ) 1 m n = lim m → + ∞ λ m λ m − 1 , and c a p ( S h s ) ≈ 0 . 587891161775 . . . , where λ m is the largest eigenvalue of T m . Interestingly, our approach can be applied to study other related 2-dimensional or multi-dimensional constraints (for example, when applied to the 3-dimensional dimer problem, it generates better bounds than the ones given in Lundow, 2001).

The propagation mechanism of vibration in a media such as a non-uniform rotating shaft is an important topic since it can help to control the vibration of the media, or to reduce the noise. Vibration propagates in a non-uniform rotating shaft in the form of elastic waves and the propagation characteristics of the waves are affected by the non-uniform factors, the transverse cracks and the rotating speed, etc. This study deals with the propagation characteristics of the elastic waves in a non-uniform shaft with transverse cracks by adopting the Transfer Matrix method. First, the transfer matrix for a single non-uniform shaft is derived by constructing the motion equations derived with Equilibrium Equations. Second, the transfer matrix for a transverse crack is studied. Crack Mode III is adopted, and the transverse crack is modeled as a local spring, the local flexibility coefficient of which is deduced. The transfer matrix for the non-uniform shaft with transverse cracks is then derived by combining the individual matrixes. Finally, several numerical examples are used to illustrate the influence of different crack depths and alternative rotating speeds on the propagation characteristics of the elastic waves, which are compared with the results from the experimental analyses. It's shown that a new stop band will come out at lower frequency region when a transverse crack occurs and the bandwidth of which will be wider and wider as the increase of the depth of the crack, and the rotating speed mainly affects the first stop band, the central frequency of which will decrease as the increase of the rotating speed.

The orientation of a transparent spheroidal particle can be obtained from the speckle orientation of a given out-of-focus image. In this paper, we present a method for obtaining the speckle orientation of the out-of-focus image. Based on optical transfer matrix theory, we study the effect the number, phase difference, internal distribution, and dimension of the emitting points and the relative position of the outermost emitting points have on the speckle orientation of out-of-focus images. We acquired out-of-focus images with different speckle orientations and applied both a Fourier transform and autocorrelation to process the simulative and experimental out-of-focus images. We found that the orientation of the Fourier transform of an out-of-focus image is perpendicular to the speckle orientation of the image. Further, the orientation of the autocorrelation of an out-of-focus image is parallel to the speckle orientation of the image. We therefore propose a method for obtaining the speckle orientation of an out-of-focus image using either a Fourier transform or autocorrelation. The method can be used to acquire the orientation of a transparent spheroidal particle.

Spin-dependent resonant tunneling in CdTe/CdMnTe double-barrier heterostructures was theoretically investigated. The transfer-matrix method was employed to study the spin transport properties under zero applied field. The effect of Dresselhaus spin- orbit coupling was considered in the symmetrical heterostructure. The influences of the barrier width and height on the polarization efficiency, barrier transparency and dwell time of light and heavy holes were observed and discussed.

We proposed a numerical method, which is based on the RCWA method combined with T matrix. It is applied as an effective method to design controllable-reflection metallic microstructures from wide incidence angles. The specific structures were fabricated by laser processing, and were analyzed and compared with simulation. Their optical characters showed low-reflection features and was verified in experiments.

This paper proposes a new approach that can generate a low-order Fornasini–Marchesini (F–M) model realization for a given n -D system by taking into account the column structural properties of its transfer matrix. Specifically, a new necessary and sufficient realization condition is developed for the F–M model realization based on a resolvent invariant space associated with the Gleason’s problem specified by the given n -D transfer matrix. Then, a new constructive procedure with respect to the columns of a given transfer matrix is proposed for constructing a low-order F–M model realization. In order to apply this procedure to the multiple-input multiple-output case more effectively, an improved realization procedure based on a polynomial description is also proposed. It will be shown, by both algorithmic analysis and illustrative examples, that for a transfer matrix having more rows than columns the proposed realization approach may generate a much lower realization order than the existing one with respect to the rows of a given n -D transfer matrix recently developed by Cheng et al. Particularly, for a given scalar transfer function, our new approach always generates an F–M model realization with lower order than the existing methods.

Two forms of dynamic transfer matrix are derived for a one-dimensional (beam-like) repetitive pin-jointed structure with point masses located at nodal cross-sections, the displacement-force transfer matrix G, and the displacement-displacement transfer matrix, H. Similarity matrices are introduced to relate G and H, together with their respective metrics. Symplectic orthogonality relationships for the eigenvectors of both G and H are derived, together with relationships between their respective sets of eigenvectors. New expressions for the group velocity are derived. For repetitive structures of finite length, natural frequency equations are derived employing both G and H, including phase-closure and the direct application of boundary (end) conditions. Besides an exposition of the theory, some familiar but much new, the focus of the present paper is on the relationships between the two forms of transfer matrix, including their respective (dis)advantages. Numerical results, together with further theory necessary for interpretation, are presented in companion papers.

Birman--Murakami--Wenzl (BMW) algebra was introduced in connection with knot theory. We treat here interaction round the face solvable (IRF) lattice models. We assume that the face transfer matrix obeys a cubic polynomial equation, which is called the three block case. We prove that the three block theories all obey the BMW algebra. We exemplify this result by treating in detail the $SU(2)$ $2\times 2$ fused models, and showing explicitly the BMW structure. We use the connection between the construction of solvable lattice models and conformal field theory. This result is important to the solution of IRF lattice models and the development of new models, as well as to knot theory., Comment: 16 pages, no figure

An analytical treatment of an earth and rockfill dam in a radially inhomogeneous multi-layered semi-cylindrical canyon in a half-space under obliquely incident plane SH waves is presented. In terms of a radial wave function expansion and a transfer matrix approach, a rigorous approach is formulated to consider inhomogeneity of the deformable canyon and associated phenomena of wave redirection, reflection, transmission, scattering and concentration. Upon confirmation of its accuracy with a past exact solution for the degenerated case of a dam in a homogeneous canyon, the multi-layer solution is extended to the case of a dam in a canyon covered by a surficial zone with a power-law shear-wave velocity model as a generalized class of smooth in-situ variations. A comprehensive set of numerical examples are presented to illustrate the sensitivity of the dam response to the inhomogeneity profile of the top canyon zone, the frequency content and the angle of the seismic wave incidence. As illustrated in both frequency and time domains, wave concentration is found to be the main reason for the high amplifications of the displacement within the dam.

We introduce a new robust stability measure for retarded type systems, which is called smoothed spectral abscissa. The main characteristics of the smoothed spectral abscissa are that it is smooth with respect to the system parameters, and it provides a trade-off between the spectral abscissa and the H2 norm of a transfer matrix related with the system. It is given by an implicit relation with the H2 norm of an auxiliary system, and its computation is based on the delay Lyapunov matrix. We show that these features make the smoothed spectral abscissa suitable for the design of robust controllers by using standard gradient-based optimization techniques.

In two and three dimensions, the standard treatment of the scattering problem for a multi-delta-function potential, $v(\mathbf{r})=\sum_{n=1}^N\mathfrak{z}_n\delta(\mathbf{r}-\mathbf{a}_n)$, leads to divergent terms. Regularization of these terms and renormalization of the coupling constants $\mathfrak{z}_n$ give rise to a finite expression for the scattering amplitude of this potential, but this expression has an important short-coming; in the limit where the centers $\mathbf{a}_n$ of the delta functions coincide, it does not reproduce the formula for the scattering amplitude of a single-delta-function potential, i.e., it seems to have a wrong coincidence limit. We provide a critical assessment of the standard treatment of these potentials and offer a resolution of its coincidence-limit problem. This reveals some previously unnoticed features of this treatment. For example, it turns out that the standard treatment is incapable of determining the dependence of the scattering amplitude on the distances between the centers of the delta functions. This is in sharp contrast to the treatment of this problem offered by a recently proposed dynamical formulation of stationary scattering. For cases where the centers of the delta functions lie on a straight line, this formulation avoids singularities of the standard approach and yields an expression for the scattering amplitude which has the correct coincidence limit., Comment: 18 pages, version accepted for publication in Annals of Physics

This paper proposes a method of H 2 -optimal vibration control using regional eigenvalue assignment based only on measured receptances, typically from a modal test. The sum of the squared weighted H 2 -norm of closed-loop receptance matrix and the transfer matrix from the external input to the control input is minimized by exploiting the flexibility from assigning the closed-loop eigenvalues to prescribed stable regions with single- or multiple-input control. It is found that regional eigenvalue assignment and the optimization of a free parameter matrix (in the multiple-input case) provide extra flexibility for improving the closed-loop performance over and above the selection of weighting matrices, as in standard H 2 control. Therefore, the internal stability, satisfactory response, and minimum control energy are readily guaranteed by the proposed method, whereas the standard approach is dependent upon the arbitrarily chosen weighting matrices. The method inherits the advantages of both optimal control and eigenvalue assignment by the receptance method. The working of the method and its advantages over the standard H 2 control are demonstrated numerically.

We consider output processes which are realizable by stochastic linear time-invariant (LTI) systems. Such processes can always be realized by LTI systems in forward innovation form, and we study the transfer matrices of such LTI realizations. We show that such a transfer matrix is consistent with an acyclic directed graph if and only if the edges of this graph represent Granger-causality relations among the components of the output process. By consistency we mean that if there is no edge between two vertices of the graph, then the corresponding block of the transfer matrix is zero. Under this assumption, conditional Granger non-causality between the components of the process is equivalent with a zero block in the transfer matrix.

In this paper, a high-accuracy manifold element modelling scheme for dynamic fracture analysis of two-dimensional elastic cracked bodies is presented. The displacement discontinuity across crack surfaces is naturally simulated attributing to the dual cover systems in the numerical manifold method. The singularity of field gradients at crack tips is appropriately captured through the use of asymptotic basis functions. The discrete equations for dynamic analysis are solved by a new fully explicit direct time integration algorithm based on precise time step integration method, in which Taylor expansion theorem and downscaling time step integration method are used to calculate the exponential transfer matrix with high accuracy. To eliminate the sensitivity of accuracy of results to time-step size, Fourier expansion with trigonometric basis functions is utilized to obtain the approximate expression for different time-dependent loadings, the exponential transfer matrix is also further utilized to calculate the response matrix of inhomogeneous term. A matrix-based spectral analysis is used to prove the stability and convergence of proposed scheme, the results demonstrate that the present scheme is unconditionally stable and convergent. Finally, the applicability and accuracy of the proposed scheme for calculating dynamic stress intensity factors are examined by two numerical examples with different crack configurations and time-dependent loadings. By comparing with the analytical/reference solutions, it is proved that the present scheme has significant advantages in accuracy and efficiency than Newmark scheme for dynamic fracture analysis.

A theoretical framework is presented to compute the dispersive properties of a locally-resonant Euler–Bernoulli beam hosting a periodic array of beam-like resonators, the bending-torsion response of which may be either uncoupled or coupled depending on the cross-section geometry. It is shown that, in the frequency domain, the reaction of every beam-like resonator can be expressed in an exact analytical form depending only on deflection and rotation of the host beam at the connection points. In this way, the beam-like resonators can be substituted by equivalent fictitious external supports with frequency dependent stiffness. On this basis: (a) band gaps of the infinite locally-resonant beam are easily computed employing a transfer matrix approach; (b) the frequency response, as well as the transmittance of the finite locally-resonant beam, are computed in closed analytical forms extending an efficient formulation recently proposed by the authors, based on generalized functions. In addition to presenting the novel theoretical framework, the main purpose of the paper is promoting coupled bending-torsion beam-like resonators as a novel concept for locally-resonant structures, which may prove even more efficient than conventional uncoupled bending-torsion beam-like resonators. Indeed, it is shown the former offer more and better opportunities to open band gaps, taking advantage of the inherent coupling between bending and torsional vibrations.

An investigation is presented in this paper into the applicability of empirical wavelet transform, combined with mode transfer matrix and/or modal superposition, to reconstruction of responses of structures subject to typical excitations. Empirical wavelet transform, of which the parameters are determined from scale-space representation, is used to decompose each measured response signal into a number of mono-components. Structural responses at unmeasured locations are reconstructed from those mono-components, either through superposition of normal modes, or through mode transfer matrices, both derived from a finite element model of the structure in question. The method is applied to a high-speed train-used extruded aluminium panel subject to typical excitations. Considered excitations include impact excitations, stochastic excitations, periodic excitations and periodic impact excitations. The accuracy of the method is demonstrated both numerically and experimentally. The robustness of the method to measurement noise is also investigated for the panel. Results show that the proposed method is not only able to overcome difficulties in existing methods caused by shortages in measurement point, mode mixing and/or periodic excitations, but also robust to measurement noise.

In this article we report a novel characteristic in the dispersion relation for dielectric–graphene photonic crystal, resulting from the modulation of the chemical potential ( μ g , j ) of the graphene sheets in the unit cell that form the entire optical structure. The photonic dispersion relation of the system is obtained employing the transfer matrix formalism taking into account the intra and interband contributions of the optical conductivity of graphene. The chemical potential of each graphene sheet is distributed with discrete values following envelope functions such as Gaussian, linear, Lorentzian and Poschl–Teller. We found that this kind of dielectric–graphene photonic crystal shows the formation of spikes in the dispersion relation due to the discrete values of chemical potential in its unit cell. The spikes are located in the frequency positions which correspond to 2 μ g , j for each distribution and can be modulated by optical and geometric parameters.

A transfer matrix approach that combines with finite element calculations is proposed to characterize homogeneous and non-homogeneous acoustic absorbing materials with in-plane periodicity. The characterized acoustic materials are mainly metamaterials made of multiple layers, where at least one layer consists of a non-homogeneous material (NHM). The NHM is a porous or solid matrix with embedded periodic simple or complex inclusions and Helmholtz resonators. The equivalent transfer matrix of the NHM is determined by the proposed approach which is similar to the three-microphone two-load method. By using the equivalent transfer matrix of the NHM coupled analytically in series with other transfer matrices, complex multilayer systems can be modeled easily and quickly in configurations wherein the use of finite element calculation alone will be more expensive and time consuming. The results of the normal incidence sound transmission loss predicted by the present method are compared with finite element results for different configurations of metamaterials and good agreements are obtained. The proposed method offers an efficient way to characterize and design at normal incidence complex multilayer acoustic metamaterials with periodic inclusions. It can be used to optimize multilayer system containing non-homogeneous materials.

Analytical model for Pulsed Eddy Current Testing (PECT) of a multi-layered metallic pipe has not been solved, which limits the development of PECT technology. In this paper, the analytical model for vertical coils near a multi-layered metallic pipe is proposed. However, the analytical solution is hard to solve due to the large number of undetermined coefficients. Combined with the global matrix method, an improved transfer matrix across a conductor is established, which overcomes unsolvable problem in the traditional transfer matrix method. The induced voltage of pickup coil calculated based on the analytical model is verified by Finite Element Method (FEM) and experiments. Furthermore, the influence of several working conditions of pipelines on the PECT signal is studied based on the analytical model, providing guidance for engineering applications.

Fast and accurate prediction of indoor airborne contaminant distribution is of great significance to the safety and health of occupants. Several Markov chain models have been developed and proved to be the potential solutions. However, there is a lack of comparison in terms of accuracy, computing cost, and robustness among these models, which limits their practical application. To this end, this study compared the performance of three Markov chain models, in which the state transfer matrix was calculated based on different principles, i.e., Markov chain model with flux-based method, with Lagrangian tracking, and with set theory approach. The investigation was conducted in a 2D ventilated cavity and a two-zone ventilated chamber. The simulation by Eulerian model for contaminant and experimental data were used as the benchmarks for the 2D and 3D cases, respectively. It is revealed that all three Markov chain models can provide a correct prediction. In the 2D case, the Markov chain model with set theory approach is the most accurate, followed by Lagrangian tracking. In the 3D case, the accuracy of Markov chain models with flux-based method and Lagrangian tracking is comparable. The Markov chain model with Lagrangian tracking is the fastest, and the time step size in this model can be relatively large. Finally, the selection guideline is given for the application of Markov chain models in different scenarios.

We establish transfer matrices in explicit form of pre-stressed elastic layers for compressible and incompressible materials. Because these transfer matrices are totally explicit, they are a powerful tool for solving various problems related to the wave propagation in layered pre-stressed elastic media. To prove this point, we investigate the reflection of a qP wave from a pre-stressed compressible elastic layer overlying a pre-stressed compressible elastic half-space. By using the transfer matrix of a pre-stressed compressible elastic layer, we derive the formulas in closed form for the reflection coefficients. Based on the obtained formulas, we carry out some numerical examples to investigate the dependence of the reflection coefficients on the incident angle, the wave frequency, the prestresses and the ratio of mass densities.

Ground and track vibrations are important environmental and safety issues which must be properly addressed when planning and operating a high-speed railway. In the past most of the research is directed to ground and track vibrations induced by a single train. To save land and construction cost, many double-track railways are built in such a way that the two tracks are narrowly separated, e.g. only by 5 m. Thus, this is a common operational situation for two high-speed trains of the same design to pass each other closely. The relative speed of the two trains can be as high as 600 km/h if the trains run at 300 km/h. This may cause many new safety, ride comfort and environmental issues including ground and track vibrations. In this paper, a method is established by combining the wavenumber finite method and transfer matrix method, verified by comparison with the two-and-half dimensional finite and boundary element methods (2.5D FEM-BEM), and applied to investigate track and ground vibrations generated by two sets of moving axle loads passing each other. The track-ground system is a typical non-ballasted and slab double track-embankment-layered ground system. The investigation shows that, the environmental vibration due to two passing trains has spectial features which should taken into account when planning a railway line. When evaluating ground vibration by measurement, the measurement points must be chosen carefully, since the location of the maximum response along a given line parallel with the track depends on the relative position of the two trains.

Vibration-based energy harvesting using piezoelectric stacks has received increasing attention in recent years for its higher mechanical-to-electrical energy conversion capability in the d33 mode. However, most research on this type of harvesters is based on either the simplified one-degree-of-freedom model or transfer matrix model. This paper presents a distributed-parameter model for the multilayer piezoelectric stack transducer based on the axial vibration theory of a continuous bar. The presented analytical model is free from the assumption that the generated current is identically distributed over all the piezoelectric layers, which has been widely used by the existing models in literature. A first-order numerical model is also introduced to validate the performance of the analytical model on the prediction of voltage, current and power outputs of the harvester under different types of external excitations. Both the analytical and numerical models are firstly validated by experiments and then are used to predict the electrical responses of the stack harvester under general periodic and random excitations with different intensities. Experiment and simulation results demonstrate that the proposed distributed-parameter model has a good accuracy and reliable performance in the prediction of the electrical responses.

We devise an algorithm to calculate upper bounds on the growth rates of the number of independent sets on a variety of regular two-dimensional lattices, using an amalgamation of techniques from linear algebra, combinatorics, and statistical mechanics. Our method uses Calkin and Wilf's transfer matrix eigenvalue upper bound together with the Collatz–Wielandt formula from linear algebra. To obtain a good bound, we need an approximate eigenvector, which we find using Baxter's corner transfer matrix ansatz and Nishino and Okunishi's corner transfer matrix renormalisation group method. This results in an algorithm for computing upper bounds which is far faster in practice than all other known methods. It is also the first algorithm for this problem with a polynomial, rather than exponential, memory requirement, and it is extremely parallelisable. This allows us to make dramatic improvements to the previous best known upper bounds. We apply our algorithm to five models, including independent sets on the square lattice (also known as the hard squares lattice gas from statistical mechanics). In all cases we extend the number of rigorously known digits of the growth rate by 4–6 digits.

Wave-based method for dynamic responses of net structures can provide analytic out-of-plane responses in full frequency range, helping to avoid the problem of control spillover, comparing with traditional modal analysis. However, as the scale of the net structure increases, more computing resources are required. Here, a wave-based method inspired by transfer matrix method is proposed for both orthogonal and non-orthogonal symmetric net structures. According to the symmetric characteristics of the net structure, we first divide the whole structure into several periodic elements and calculate the transfer matrix of the periodic element by using the basic wave-based method. Then by multiplying transfer matrixes of the periodic element and using all the boundary condition of net structure, the out-of-plane displacments of the net structure can be finally obtained. Our proposed wave-based transfer matrix method is proved to be more accurate and efficient than the finite element analysis for large net structures.

We report the experimental observation of doorway states phenomenon for acoustic waves in porous silicon structures. We designed and fabricated a structure capable to exhibit the acoustic doorway states phenomenon around 1 GHZ. The structure consists of two porous silicon layers, one with low porosity and another with higher porosity and smaller thickness, followed by a much more thicker layer of crystalline silicon. We study the doorway state phenomenon in the frequency and in the time domain. To do this, we measured the acoustic transmission spectra in the frequency domain, and by using the Fourier transform, we obtain the transmission in the time domain. The transmission spectrum of the fabricated sample shows acoustic doorway states in frequencies around 1 GHz. The experimental results are in good agreement with the transfer matrix calculations.

This work presents a fiber Bragg grating (FBG) displacement sensing system to experimentally investigate the dynamic behaviors of phononic crystal (PC) beams through impact-induced early short time or long time transient responses. Based on the couple-mode theory and the optical transfer matrix (T-matrix) formulation, we first show that it is feasible to achieve linear displacement sensing using a single FBG without extra demodulators such as matching gratings. To validate its effectiveness, the proposed self-demodulated FBG system is applied to measure the point-wise transient displacement responses of a cantilever PC beam subjected to steel ball impacts. To demonstrate the transient sensing performance, the measured early short time transient responses are compared with the corresponding ones predicted by the method of reverberation-ray matrix (MRRM) theoretically and the finite element method (FEM) numerically. The excellent agreements verify together the experimental, theoretical and numerical results. Finally, conducting Fast-Fourier transform (FFT) to the early short time or long time transient responses gives the frequency responses that clearly show the existence of the band gaps. In addition to providing a new displacement sensing method using the self-demodulated FBGs, this work also offers another route to investigate the band structures of PC beams through the frequency responses transformed from the early short time transient responses, long before the structural damping becomes dominant.

The reflection and transmission coefficients through multilayered structure of finite thickness and the dispersive relations of Bloch waves in one-dimension infinite periodic structure (phononic crystals) consisting of two repetitive different micropolar slabs are studied. Due to the additional microrotational degree of freedom, the modes of elastic waves in a micropolar solid are different from that in the traditional elastic medium. The transfer matrix and the stiffness matrix of each slab and further one single cell are obtained for both in-plane and anti-plane situations. The dispersive equations are derived by using Bloch theorem . The relation between the minimum eigenvalue of transfer matrix of single cell and the stop bands are presented. The reflection and transmission coefficients through finite multilayered structure are also calculated based on the stiffness matrix method to avoid the numerical instability of the transfer matrix method. The energy fluxes carried by the reflection and transmission waves are estimated and the energy conservation is checked to validate the numerical results. Based on the numerical results, the influence of micropolar parameters on the dispersive feature of Bloch waves are discussed. It is found that the dispersive feature and the bandgap of Bloch waves in the micropolar phononic crystal are evidently different from that in the classic phononic crystal. The dispersive feature and bandgap of Bloch wave at high frequency are more sensitive to the micropolar constants than at low frequency range.