Propagator Matrix Research Articles

Reflections and transmissions from a porous layer sandwiched between seawater and solid substrate

Ocean sediments are composed of a porous layer sandwiched between seawater and a solid elastic base. The reflection and transmission of acoustic waves from ocean sediments are essential in ocean engineering, seafloor seismology, etc. The existing reflection and transmission (R/T) formulas are determined by complex matrix equations in the order of 9 × 9 or higher. To obtain more concise R/T coefficient formulas, we established the propagator matrix of the coupled water-porous-layer-solid elastic system, wherein the above-water is ideal and the middle porous marine sediment is defined by the Biot–Stoll model. Based on the propagator matrix, the displacement R/T coefficients were given by a 3 × 3 matrix equation for two P-wave incidence cases: the down-going case from the water, and the up-going case from the ocean floor. For zero-frequency or zero-thickness limitation, the proposed R/T coefficient formulas are the same as the displacement R/T coefficients at a single fluid–solid interface. The numerical results of the reflected or transmitted waves in the above-water show that the R/T coefficients exhibit regular variations at frequencies far lower than the characteristic frequency. The R/T coefficient amplitudes decreased with increasing frequency at a fixed incidence angle and ratio between the thickness and wavelength.

Manipulation of acoustic wave scattering from cylindrical shells using a coating of FG magneto-electro-elastic composite materials: Active acoustic cloaking

Acoustically cloaking an object from view of a system that uses acoustic waves for detection is an attractive and practical capability. The aim of the current investigation is to hide a steel cylinder containing axial flow acoustically, which is achieved through the application of a coating of FG electro-magnetic material. This coating is employed to minimize the sound pressure that is scattered from the structure, and it comprises piezoelectric (BaTiO3) and magnetostrictive (CoFe2O4) make up the coating. Accordingly, the method presented is based on optimal control and incorporates an improved particle swarming optimization algorithm. The decision variables are the magnetic and electrical potentials on the inner and outer surfaces of the coating, whereas the cost function is the scattered acoustic pressure. In other words, what are the values of electrical and magnetic potentials on the inner and outer surfaces of the coating to minimize the scattered pressure? Subsequently, by defining the electrical and magnetic boundary conditions in a certain way, it is feasible to achieve the complete cancelation of scattered sound pressure. The governing equation of each layer is extracted using an orthotropic laminate composite model that is based on the 3-D theory of elasticity. A state-vector method is then applied to obtain the propagator matrix that links the state variables at each layer's surfaces. The results show that the proposed acoustic cloaking technique has an excellent capability for hiding the shell immersed in the fluid.

Imperfect and multilayered magneto-electro-elastic nanoplates bending response analysis based on the nonlocal state-space approach

The present article is about the 3D static responses of the multilayered magneto-electro-elastic (MEE) nanoplates with imperfect interfaces and nonlocal effect analytically by the state-space method. To model the imperfect interfaces, which have been assumed to be electro-magnetically weakly conducting, the spring type layer model has been used. The nonlocal effect is taking into account in the modeling through to the Eringen nonlocal elasticity theory. The propagator matrix technique is utilized to derive the solution at any position in the transverse direction of the MEE nanoplates. The nonlocal length effect on the whole amount state variables has been highlighted in numerical examples. It has been carried out that, this parameter influences weakly the stresses, the electric and the magnetic displacements, respectively; but it has a strong influence on the elastic displacements, the electric and the magnetic potentials. Furthermore, the effects of the staking sequence and the interface parameters have been clearly illustrated.

A New ACD-OMP Accelerator With Clustered Computing Look-Ahead

The orthogonal matching pursuit (OMP) has been widely explored to realize real-time compressed sensing (CS) reconstruction. The matrix pseudo-inverse of the least squares (LSs) is the most computationally complex operation in the OMP. Among various algorithms to realize this complex operation, the alternative Cholesky decomposition (ACD) algorithm performs the best. However, it typically involves a very long computation time due to its iterative procedure. To accelerate the ACD-OMP algorithm, a novel method called clustered computing look-ahead (CCL) is proposed. Inspired by the famous parallel carry look-ahead adder (CLA), CCL adds a propagation matrix to decouple the data dependency in ACD and then uses a clustering operator to transform the iterative computation of ACD into a pipelined and parallelized computation. This brief also proposes an efficient hardware architecture of the CCL-based ACD-OMP algorithm for CS reconstruction. The proposed algorithm is implemented on field programmable gate array (FPGA). For sparse signals with the same sparsity and length, the proposed implementation is 1.96 times faster than state-of-the-art work.

Three-dimensional unsteady state solution of multilayered saturated anisotropic finite media subjected to point source

The three-dimensional (3D) unsteady state solution of a saturated anisotropic finite porous medium due to a point source is presented in this paper. First, Laplace transform is introduced to address the governing equations of anisotropic poroelasticity. Next, the eigen equation method and the modified pseudo-Stroh formalism are used to obtain the transient solution for homogeneous saturated finite media. Then, the propagation matrix method is used to address the corresponding multilayered porous elastic media, and the 3D unsteady state solution of the layered porous medium subjected to a point source is obtained. The solution for a finite porous foundation is presented to demonstrate the use of the closed form solution. The numerical results of a homogeneous foundation due to an internal time-dependent point source are given. Two different point sources are used to show the transient responses of the proposed method. By comparing with those obtained by the finite element method, the accuracy of the proposed method has been proved. Meanwhile, the influence of the depth of the point source on the hydromechanical responses of the homogeneous foundation is discussed. Finally, the transient response of an anisotropic sandwich foundation due to a point source at different depths is investigated. These numerical examples prove the feasibility of the transient 3D solution. The closed form solution presented in this paper can provide help for 3D transient fluid-deforming coupling analysis of laminated porous media.

Propagator and transfer matrices, Marchenko focusing functions and their mutual relations

SUMMARY Many seismic imaging methods use wavefield extrapolation operators to redatum sources and receivers from the surface into the subsurface. We discuss wavefield extrapolation operators that account for internal multiple reflections, in particular propagator matrices, transfer matrices and Marchenko focusing functions. A propagator matrix is a square matrix that ‘propagates’ a wavefield vector from one depth level to another. It accounts for primaries and multiples and holds for propagating and evanescent waves. A Marchenko focusing function is a wavefield that focuses at a designated point in space at zero time. Marchenko focusing functions are useful for retrieving the wavefield inside a heterogeneous medium from the reflection response at its surface. By expressing these focusing functions in terms of the propagator matrix, the usual approximations (such as ignoring evanescent waves) are avoided. While a propagator matrix acts on the full wavefield vector, a transfer matrix (according to the definition used in this paper) ‘transfers’ a decomposed wavefield vector (containing downgoing and upgoing waves) from one depth level to another. It can be expressed in terms of decomposed Marchenko focusing functions. We present propagator matrices, transfer matrices and Marchenko focusing functions in a consistent way and discuss their mutual relations. In the main text we consider the acoustic situation and in the appendices we discuss other wave phenomena. Understanding these mutual connections may lead to new developments of Marchenko theory and its applications in wavefield focusing, Green’s function retrieval and imaging.

Three-dimensional thermoelastic analysis of orthotropic laminated cylindrical shells subjected to uniformly localized thermal boundary conditions

Accurate evaluation of three-dimensional (3D) thermal-elastic analysis of laminated cylindrical shells under different loads has a significant impact on their safety design. In this study, we present a 3D thermoelastic solution of laminated orthotropic cylindrical shells subjected to uniformly localized thermal boundary conditions. For a homogeneous orthotropic layer, a thinning layer approach is introduced to deal with the heat conduction equation and the elastic equation. Based on the eigenequation method and pseudo-Stroh formalism, general solutions for physical quantities of a homogeneous layer are derived. The propagator matrix method is used to deal with multilayered media. A closed form solution for the laminated cylindrical shells is obtained. A numerical example of a 3-layered cylindrical shell subjected to convective heat transfer boundary conditions is presented to verify the solutions’ correctness. An excellent convergence of the current approach is found and the physical quantities predicted by the current method agree well with the numerical results by the finite element method. Finally, the numerical solutions of a 3-layered orthotropic cylindrical shell subjected to uniformly localized thermal boundary conditions are obtained by the current method. The closed form solution presented in this article can be used for 3D thermoelastic analysis of laminated cylindrical shells under complex boundary conditions.

Calculating InN/GaN Transmission Coefficient from Single Barrier to Five Barriers with Propagation Matrix and Transfer Matrix Methods

In this study, the value of transmission coefficient on InN/GaN semiconductor from a single barrier to five barriers was determined by using the propagation matrix method and the transfer matrix method. This study aims to see the effect of adding a barrier to the number of resonance tunneling that occurs, to see the difference in transmission coefficient values which was obtained with the two methods, and to determine the effectiveness of the program execution process time from the propagation matrix and transfer matrix methods using Matlab programming. The results obtained indicated that the value of the transmission coefficient obtained from the two methods was the same. As the number of barriers increases, the number of resonance tunneling that occurs will increase. These two matrix methods had differences in terms of the effectiveness of the program execution process time and calculation process. The propagation matrix method was considered more effective than the transfer matrix method.

Implementation and propagation of prestress forces in pin-jointed and tensegrity structures

The implementation of targeted prestress forces in pin-jointed structures, such as tensegrity structures, is often elusive since the activation of a prestressing device in a single cable or strut propagates forces into all other structural elements. Simultaneously controlling the internal forces and the shape of the structure through an appropriate set of length variations imposed by the devices, referred to as prestress scenario herein, has been a focus of research for over three decades. Overall, this article proposes a pedagogical state-of-the-art on the topic of prestressing and presents an alternative to the force sensitivity matrix, a commonly used method to anticipate the forces in all elements due to the unit length variation imposed on each element. Contrary to this element-to-element approach, a modal paradigm is presented here to simplify the mathematical complexity of a prestressing problem and provide a more intuitive understanding of the propagation phenomenon. Key concepts introduced in the article include the modal scenario, stiffness matrix of the self-stress modes, and propagation matrix. The propagation matrix quantifies the propagation of self-stress levels from one mode to another and offers a valuable design tool to plan the prestressing stages at each construction step or to correct errors observed in the self-stress levels. The article also introduces the self-stress scenario, which anticipates the elastic deformations caused by prestressing to achieve exact targeted forces and geometry while avoiding the propagation of prestress forces. Two examples of the application of the proposed concepts are given.

1-D and 2-D Direction of Arrival Estimation in a Conical Conformal Array: Design and Implementation.

Direction of arrival (DOA) estimation for conformal arrays is challenging due to non-omnidirectional element patterns and shadow effects. Conical conformal array (CCA) can avoid the shadow effect at small elevation angles. So CCA is suitable for DOA estimation on both azimuth and elevation angles at small elevation angles. However, the element pattern in CCA cannot be obtained by conventional directional element coordinate transformation. Its local element pattern also has connection with the cone angle. The paper establishes the CCA radiation pattern in local coordinate system using 2-D coordinate transformation. In addition, in the case of large elevation angle, only half elements of the CCA can receive signal due to the shadow effect. The array degrees of freedom (DOF) are reduced by halves. We introduce the difference coarray method, which increases the DOF. Moreover, we propose a more accurate propagator method for 2-D cases. This method constructs a new propagation matrix and reduces the estimation error. In addition, this method reduces computational complexity by using linear computations instead of eigenvalue decomposition (EVD) and avoids spectral search. Simulation and experiment verify the estimation performance of the CCA. Both demonstrate the CCA model established in this paper is corresponding to the designed CCA antenna, and the proposed algorithms meet the needs of CCA angle detection. When the number of array elements is 12, the estimation accuracy is about 5 degrees.

A novel method for calculating dislocation Green's functions and deformation in a transversely isotropic and layered elastic half-space

A novel and comprehensive method is proposed for calculating the dislocation Love numbers (DLNs), Green's functions (GFs), and the corresponding deformation in a transversely isotropic and layered elastic half-space. It is based on the newly introduced Fourier-Bessel series system of vector functions, along with the dual variable and position method. Two important features associated with this new system are: (1) it is much faster than the conventional cylindrical system of vector functions; (2) we can even pre-calculate the DLNs which are only possible in terms of this new system. This is due to the fact that the variables to be solved in the new system are functions of the simple discrete zero points of the Bessel functions, instead of the numerical integration of continuous Bessel functions between the neighboring zero points as in the conventional system. The introduced dual variable and position method is unconditionally stable as compared to the traditional propagator matrix method in dealing with layering. Exact asymptotic expressions of the DLNs for large wavenumber are further derived, which makes the Kummer's transformation applicable in accelerating the convergence of the corresponding GFs. For the reduced case of homogeneous and isotropic half-space, the present solutions amazingly reduce to the existing exact closed-form solutions. These new features are further seamlessly combined for calculating the deformation due to a finite dislocation (or a finite fault in geophysics) in the layered structure, which are demonstrated to be accurate and efficient.

An Efficient Method for Predicting the Shielding Effectiveness of an Apertured Enclosure with an Interior Enclosure based on Electromagnetic Topology

A fast analytical method has been proposed for predicting the shielding effectiveness (SE) and resonances of an apertured enclosure with an interior enclosure. Under the concept of electromagnetic topology, the monitor point and the walls are treated as nodes, and the space between them is treated as tubes. The propagation relationships at tube level and reflection relationships at node level are derived as the propagation matrix. After modeling the front wall of the interior enclosure as a junction between two waveguides, an equivalent circuital model of the enclosures is derived. The front wall and the window structure in front of the adjacent space of the interior enclosure are considered as a three-port scattering matrix. Then we can use the extended BLT equations to calculate the voltage response at each node. Results from the proposed method are compared with those from the numerical method, and the results have a good agreement while it can dramatically save calculation time.

The propagator matrix reloaded

The standard way to perform calculations for quantum field theories involves the S-matrix and the assumption that the theory is free at past and future infinity. However, this assumption may not hold for field theories in non-trivial backgrounds such as curved spacetimes or finite temperature. In fact, even in the simple case of finite temperature Minkowski spacetime, there are a lot of misconceptions and confusion in the literature surrounding how to correctly take interactions into account when setting up the initial conditions. The objective of this work is to clear up these misconceptions and provide a clean and simple derivation of a formalism which includes interactions in the initial conditions and assesses whether or not it is legitimate to ignore them. The ultimate conclusion is that we cannot ignore them: quantum field theories at finite temperature are not free in the infinite past.

Three-dimensional semi-analytical solutions of arbitrary functionally graded piezoelectric doubly curved shell panel under thermo-electric load

In this article, a methodological approach for the three-dimensional (3D) static behavior of functionally graded (FG) thermo-electro-elastic doubly curved shells subjected to a thermal and electrical load is presented. A coupling procedure based on the state space, a multilayered Runge-Kutta and transfer matrix methods is elaborated for numerical solution of the 3D thermo-electro-elastic governing equations. In the thickness direction, recursive relationships using a propagator matrix are elaborated allowing to handle arbitrary functionally graded effects. A global transfer matrix is formulated permitting to investigate the effects of various types of bottom and top surface temperature loads and heat flux. Some benchmark tests are presented to validate the presented approach and to analyze the influence of the material-property gradient index, the curvature radius ratio, the span-to-thickness ratio, the temperature, normal heat flux, and the electric voltage on the mechanic, electric, and thermal variables of doubly curved shells.

Orbital Angular Momentum Radio-Wave-Based 2-D Target Angle Estimation With Mutual Coupling

Vortex electromagnetic waves are conveniently produced by a uniform circular array (UCA). The mutual coupling matrix (MCM) of UCAs presents rotational symmetry, which can be employed to cut down the parameters in the circuit model. The impedance matrix also does not rely on a specific port. This work uses rotational symmetry to expand the MCM into the orbital angular momentum (OAM) mode domain. The OAM-domain phase sequence is derived by the discrete Fourier transform (DFT) of the first row of the MCM. The connection between the number of OAM modes and the excitation of elements is described. Furthermore, an improved propagator method (PM) for OAM-based decoupled 2-D angle estimation is proposed. This method employs the original steering vector of OAM waves to process the receiving signal, which avoids errors caused by the Bessel function approximation. The estimation error is also reduced by redefining the propagator matrix. In terms of computational complexity, the proposed method avoids spectral search-and-element level eigenvalue decomposition (EVD). The results demonstrate that the coupling is compensated effectively by the OAM-mode domain MCM. The error of the proposed method is less than 1° under the condition of SNR = 20 dB and 1000 snapshots. The computational complexity is reduced by 1–2 orders of magnitude compared to the origin PM. The method is also suitable for 1-D estimation scenarios.

Ballistic spin transport in DC-bias single top gate p-type narrow channel device with Zeeman–Rashba effects

The ballistic transport of holes in a p-type quasi-one-dimensional channel system is investigated without considering the coupling effect of the light and heavy holes. A top gate (TG) is used to form a potential barrier in the quantum channel with Zeeman–Rashba (ZR) effects. The Luttinger Hamiltonian and propagation matrix method are used to examine the relationship between the conductance and the incident hole energy. The results show that when the gate width L is large, an electron-like bound state (E-BS) structure is formed at the inflection point of the spin-down branch. When a positive gate potential energy is applied (U0>0), the light and heavy hole spins form an electron-like quasi-bound state (E-QBS) at the top of the branch under the hole spin. Conversely, when a negative gate potential energy (U0<0) is applied, the light and heavy hole spins form a hole-like quasi-bound state (H-QBS) structure at the bottom of the branch.

Beamforming Based on a SSS Angle Estimation Algorithm for 5G NR Networks

The current 5G-NR standard includes the transmission of multiple synchronization signal blocks (SSBs) in different directions to be exploited in beamforming techniques. However, choosing a pair of these beams leads to performance degradation, mainly for the cases where the transmit and receive beams are not aligned, because it considers that only few fixed directions among wide beams are established. Therefore, in this article, we design a new 3GPP-standard- compliant beam pair selection algorithm based on secondary synchronization signal (SSS) angle estimation (BSAE) that makes use of multiple synchronization signal blocks (SSBs) to maximize the reference signal received power (RSRP) value at the receiver. This optimization is performed using the SSSs present in each SSB to perform channel estimation in the digital domain. Afterwards, the combination of those estimations is used to perform equivalent channel propagation matrix estimation without the analog processing effects. Finally, through the estimated channel propagation matrix, the angle that maximizes the RSRP is determined to compute the most suitable beam. The proposed algorithm was evaluated and compared with a conventional beam pair selection algorithm. Ours has better performance results. Furthermore, the proposed algorithm achieved performance close to the optimal performance, where all channel state information (CSI) is available, emphasizing the interest of the proposed approach for practical 5G mmWave mMIMO implementations.

A new semianalytical algorithm for rapid simulation of triaxial electromagnetic logging responses in multilayered biaxial anisotropic formations

A set of compact and stable formulas are developed for the computation of triaxial electromagnetic (EM) logging responses in multilayered formations with biaxial anisotropy. These formulas are based on a new global amplitude propagator matrix (GAPM) algorithm, which derives the amplitudes of upgoing/downgoing waves at the boundaries rather than the reflection/transmission coefficients of each layer. The kernel of the GAPM algorithm is assembling the local continuity equations into a global linear system, which has addressed the notorious overflow problems commonly encountered by the traditional coefficient propagator matrix approach. To fast and accurately convert the EM fields from the spectral domain into the spatial domain, an efficient twofold inverse Fourier transform (IFT) technique is established for the computation of triaxial EM logging responses. Two main techniques are developed to significantly improve the computation accuracy and speed of IFT. On the one hand, the spectral fields at the first integral quadrant can be used to express the fields at the other three quadrants. Consequently, the original twofold infinite integral is analytically simplified to a semiinfinite one and the computation cost can be reduced by three-fourths. On the other hand, a two-level subtraction scheme is developed to reduce the integral singularity. The first-level subtraction is achieved by separating the primary fields from the total fields. The second-level process includes subtracting the spectral scattered fields of an equivalent transverse isotropic (TI) layered medium, integrating the residual fields, and computing the spatial fields of the TI model. The subtractions on the primary and scattered fields can make the integral convergence increase by more than 10 orders of magnitude in favorable conditions. Numerical experiments demonstrate that the new semianalytical formulas are well suited for any dipping angle and an arbitrary number of layers, as well as for thin and thick formations. The modeling method is efficient and robust, which can provide a fast simulator for the interpretation of triaxial EM logging data.

Mixed-field source localization based on robust matrix propagator and reduced-degree polynomial rooting

This paper develops two new algorithms based on multiple signal classification (MUSIC) and matrix propagator for near-field (NF) and far-field (FF) source localization, which substantially reduce the computational complexities and can be applied no matter whether the source number is known as a priori. In the developed algorithms, two special fourth-order cumulant (FOC) matrices are constructed. Then the noise subspace is obtained by incorporating lower-upper (LU) decomposition into the matrix propagator approach. This avoids the computationally burdensome eigenvalue decomposition (EVD) or singular value decomposition (SVD) as well as the dependence of source number. The direction-of-arrivals (DOAs) of FF and NF sources are distinguished by the oblique projection and then estimated based on the MUSIC spectrum. To further improve the computational efficiency, an alternating root-propagator scheme is proposed to determine DOAs, where the real lower-degree polynomial is formulated, which increases the computational speed by a factor about eight compared to the standard root-MUSIC. Finally, the NF ranges are calculated from the FOC and the DOA estimates via the least square (LS) criterion. The effectiveness and performance of the proposed approach are validated by using numerical simulation.

A methodology for data-driven adjustment of variation propagation models in multistage manufacturing processes

In the current paradigm of Zero Defect Manufacturing, it is essential to obtain mathematical models that express the propagation of manufacturing deviations along Multistage Manufacturing Processes (MMPs). Linear physical-based models such as the Stream of Variation (SoV) model are commonly used, but its accuracy may be limited when applied to MMPs with a large amount of stages, mainly because of the modeling errors at each stage that are accumulated downstream.In this paper we propose a methodology to calibrate the SoV model using data from the inspection stations and prior engineering-based knowledge. The data used for calibration does not contain information about the sources of variation, and they must be estimated as part of the model adjustment procedure. The proposed methodology consists of a recursive algorithm that minimizes the difference between the sample covariance of the measured Key Product Characteristic (KPC) deviations and its estimation, which is a function of a variation propagation matrix and the covariance of the deviation of the variation sources. To solve the problem with standard convex optimization tools, Schur complements and Taylor series linearizations are applied. The output of the algorithm is an adjusted model, which consists of a variation propagation matrix and an estimation of the aforementioned variation source covariance.In order to validate the performance of the algorithm, a simulated case study is analyzed. The results, based on Monte Carlo simulations, show that the estimation errors of the KPC deviation covariances are proportional to the measurement noise variance and inversely proportional to the number of processed parts that have been used to train the algorithm, similarly to other process estimators in the literature.