Reciprocity Theorem Research Articles

Unit cell of three nonlinear layers for the design of tunable nonlinear metamaterials

Tang et al. (2012) and Kube (2017–2018) discussed the harmonic scattering of elastic waves from nonlinear inclusions through a theoretical perspective modeling early-stage damages as quadratic and cubic nonlinear elastic material models. Achenbach and Wang (2017–2018) studied the harmonic scattering of a wave from a single nonlinear layer embedded in a linear elastic material using the reciprocity theorem in elastodynamics and the concept of compensatory waves. They demonstrated sensitivity of the back and forward scattered waves towards the dimension and the intensity. Based on this understanding, a new unit cell is proposed in this study that can act as a tunable nonlinear metamaterial, where we can tune the amplitudes of the back and forward-scattered nonlinear waves. The unit cell contains three nonlinear layers of equal widths; the intensity of the central layer, along with the total width of the unit cell, is varied to tune the harmonic responses of scattered waves. Computational studies are carried out to demonstrate the presence and absence of harmonically backscattered waves from a unit cell. In some case studies, parameters are tuned so that the amplitudes of 2nd harmonics (2f) of forward, backscattered waves and waves recorded at the interfaces of the three nonlinear layers.

Influence of structural disorder on plasmonic metasurfaces and their colors—a coupled point dipole approach: tutorial

The optical properties of plasmonic metasurfaces are determined not only by the shape and size of the constituting nanostructures, but also by their spatial arrangement. The fast progress in nanofabrication has facilitated the emergence of many advanced metasurface designs that enable controlling the propagation of light on the nanoscale. While simple metasurface designs can be derived from theoretical considerations, it is inevitable to employ computational approaches for complex manipulations of incident light. However, most of the currently available full-wave simulation approaches such as the finite element method (FEM) or finite difference time domain method come with drawbacks that limit the applicability to certain usually simplified or less complex geometries. Within this tutorial, different approaches are outlined for modeling light propagation in complex metasurfaces. We focus on an approach that approximates the nanostructure ensemble as a coupled set of point dipoles and determine their far-field response via the reciprocity theorem. This coupled point dipole approximation (CPDA) model is used to examine randomly distributed, oriented, and scaled nanostructure ensembles. A disorder formalism to introduce the randomness is developed that allows one to progressively perturb periodic arrangements of identical nanostructures and thereby investigate the effects of disorder and correlation. Several disorder metrics are provided that allow one to quantify the disorder, and the relation with the far-field scattering properties is discussed. Spatially and angle resolved hyperspectral datasets are computed for various disordered metasurfaces to assess the capabilities of the CPDA model for different polarization states and incidence angles, among others. The hyperspectral datasets are converted into sRGB color space to deduce the appearances in the image and Fourier planes. Very good agreement of the simulation results with Mie theory, FEM results, and experiments is observed, and possible reasons for the present differences are discussed. The presented CPDA model establishes a highly efficient approach that provides the possibility to rapidly compute the hyperspectral scattering characteristics of metasurfaces with more than 10,000 structures with moderate computational resources, such as state-of-the-art desktop computers with sufficient memory; 16 GB allow for the simulations in this paper, whereas scaling to up to more memory by the factor of N2 allows for the simulation of N times more dipoles. For that reason, the CPDA is a suitable approach for tailoring the bidirectional reflectance distribution function of metasurfaces under consideration of structural perturbations and experimental parameters.

Convolved action principles for couple stress elastodynamics

A novel set of variational principles and integral theorems is developed for consistent couple stress elastodynamics, including the principle of virtual convolved action, the principle of stationary convolved action, the theorem of dynamic reciprocity and the integral representation theorem. Both time-domain and Laplace-domain versions are presented. Most notably, all of these are based on the concept of convolved action, involving temporal convolutions, and are applicable to the most general case of bi-anisotropic couple stress materials. Furthermore, the principle of virtual convolved action extends beyond elastodynamics to viscoelastic and plastic material response. In addition, the two-dimensional infinite space fundamental solutions and kernel functions are derived in the Laplace-domain and used to formulate a new boundary element method (BEM) for transient response, which also originates from a convolved action. The resulting BEM is applied to four prototype examples to demonstrate the validity and robustness of the method, and to explore interesting behavior of consistent couple stress elastodynamics in the isotropic case. Although the focus here is solely on couple stress elastodynamics, the convolved action framework extends to a broad range of time-dependent problems, including those of a multiphysical nature.

Dissipation-induced acoustic nonreciprocity

Linear acoustic transmission through a stationary elastic medium possesses Rayleigh reciprocal symmetry with respect to pressure produced by a source at point A(B) and measured at point B(A), pA(rB) = pB(rA). In ideal (inviscid) fluids, pressure plays the role of the scalar potential for velocity, v∼∇p. The Rayleigh reciprocity theorem is formulated for velocity potentials but not for their gradients. Any function of the absolute value of velocity, v(r), lacks reciprocal symmetry. In particular, acoustic intencity I = pv is not symmetric, IA(rB) ≠ IB(rA). Dynamics of ideal fluid is time-reversible, and lack of reciprocal symmetry is attributed to asymmetry in scattering. However, in a viscous environment dissipation breaks T symmetry making fluid dynamics irreversible. We report that acoustic transmission is not only irreversible but also becomes nonreciprocal due to different amount of dissipated energy for forward and backward propagation, provided that mirror symmetry (P symmetry) is broken. Since dissipation occurs within a narrow viscous layer at solid–fluid interface, the amount of nonreciprocity in transmission strongly depends on the quality of the surface of scatterers. Experiments performed with scatterers of different surface quality show how surface roughness affects the level of nonreciprocity. In a series of experiments with asymmetric phononic crystal, Tesla valve, and acoustic cavity, we demonstrate dissipation-induced nonreciprocity in transmission. [This work is supported by the NSF under EFRI Grant No. 1741677.]

Intelligent element: Coupling Green function approach and artificial intelligence to reduce discretization effort

This research work presents a method that modifies a classical numerical method using artificial intelligence (AI) and takes advantage of an analytical method to minimize the usual need for increasing discretization. Its formulation is based on the integration of two main concepts: the reciprocity theorem and the generalization capability of artificial neural networks (ANNs). The reciprocity theorem is used to formulate the mathematical expression governing the geomechanical problem, which is then discretized in space into intelligent elements. The behavior of the strain field inside these new elements is predicted using an ANN. To make these predictions, the neural network uses displacement boundary conditions, material properties, and the geometric shape of the element as input data. The comparison was performed for two examples, in which the first had a uniform depletion of the reservoir, while the second had a non-uniform variation of the pore pressure. For the same level of accuracy, the proposed method was 10 times faster than the traditional method for the first example and five times faster for the second example on a computer with 12 threads of 2.6 GHz and 32 GB RAM.

ELF-EM fields in the multi-layer spherical ‘Earth-ionosphere’ model based on WKB

Abstract With a high signal-to-noise ratio and a great depth of exploration, the wireless electromagnetic method (WEM) has wide applications in the exploration of deep mineral resources and oil and gas reservoirs. Extremely low-frequency electromagnetic (ELF) waves emitted from a horizontal antenna are used to achieve synchronous acquisition for different receivers of multi-coverage information in a global region. However, previous research based on a planar model ignored the curvature of the Earth. This work focuses on the electromagnetic fields (EM fields) in the model of a spherical ‘Earth ionosphere’ to extend the coverage of WEM. By transferring the EM fields from a vertical electric dipole (VED) as well as a vertical magnetic dipole (VMD) in the multi-layered medium of the Earth, we obtain the formulae for the EM fields emitted by a horizontal electric dipole (HED) by using a reciprocity theorem. The correctness of the proposed method is verified by comparing it with the approximate analytical formula and previous work. Based on the above results, we have studied the propagation and frequency characteristics of electromagnetic fields in a spherical waveguide consisting of the ionosphere and earth. The results show that the electromagnetic fields under the spherical model produce interference effects that are different from those of the planar model. The electromagnetic response of the layered Earth was then discussed, and its potential as an electromagnetic technique for exploring the deep Earth was demonstrated.

A novel mathematical model on generalized thermoelastic diffusion theory

In this work, a new linear model of thermo-diffusion in elastic solids has been proposed in the context of generalized thermo-elasticity. The theories of coupled thermoelastic diffusion and of generalized thermoelastic diffusion problem with one relaxation time and also of generalized thermoelastic diffusion in GN models as well as Lord-Shulman, Green-Nagdhi, two-phase-lag and three-phase-lag models on thermo-elasticity follow as limit cases. A uniqueness theorem of the proposed model followed by the reciprocity theorem for diffusion is derived. The variational theorem for the governing equations has also been derived. Also, some applications of the model in dynamic and quasi-static problems are considered.

Thermotactic navigation of an artificial microswimmer near a plane wall

Despite significant advances in the field of man-made micro- and nanomotors, it remains a challenge to precisely control their motion in bounded environments. Here, we present a theoretical analysis of a thermally activated micromotor near a plane wall under the action of a background linear temperature field. The coupling between the autonomous and field-directed motions has been resolved using a combined analytical–numerical framework comprising general solutions in bispherical coordinates and the reciprocal theorem for creeping flows. Results reveal giant augmentation in swimming speeds, the controlling parameter zones for positive and negative thermotaxes and the flexibility of steering perpendicular to the field gradient for an isolated micromotor. Boundary-instigated thermo-fluidic modulations at different levels of confinements and preferential orientations cause directional switching of both the vertical translation and rotation parallel to the wall, thereby drastically altering the phase portraits of the swimmer dynamics. Contrasting trajectory characteristics, e.g. escape, attraction, are partitioned by unstable separatrices in the phase portraits, while competitive repulsion (attraction) after attraction (repulsion) characteristics emerge for different relative field strengths$\mathcal {S}$and gradient orientations$\theta _T$. Below$\mathcal {S}=0.25$, highly counter-intuitive trajectories result when the micromotor is initially launched from an overlapping escape zone. Moreover, external-field-assisted microswimming can uniquely tune the directionality of wall-parallel translation, broadening the scope of dynamic regulation of self-propulsion. Thus, providing insights into a precisely controlled fuel-free actuation of micromotors near a physical obstacle, the present study stands as a step toward addressing the increasing demand for successful implementation of micromotors in futuristic clinical and environmental applications.

Achievable Rate of Near-Field Communications Based on Physically Consistent Models

This paper introduces a novel information-theoretic approach for studying the effects of mutual coupling (MC), between the transmit and receive antennas, on the overall performance of single-input-single-output (SISO) near-field communications (NFC). By incorporating the finite antenna size constraint using Chu’s theory and under the assumption of canonical-minimum scattering (CMS), we derive the MC between two radiating volumes of fixed sizes. Expressions for the self and mutual impedances are obtained by the use of the reciprocity theorem. Based on a circuit-theoretic two-port model for SISO radio communication systems, we first establish its input-output relationship where the noise depends on the self/mutual impedances of the antennas, unlike the conventional assumption of independent additive white Gaussian noise. We then characterise the achievable data rate for a given pair of transmit and receive antenna sizes, thereby providing an upper bound on the system performance under physical size constraints. Through the lens of these findings, we shed new light on the influence of MC on the information-theoretic limits of near-field communications using compact antennas.

Imaging of thrust structures under Colombian foothills landscape through Marchenko approach

The Marchenko imaging approach successfully deals with multiple scattering during seismic migration, obtaining clear images in structurally complex geological zones below at a specific depth. Through the Marchenko equation, the method retrieves Green's function of a virtual source in profundity using reflections recorded on the surface and a smoothed velocity model. According to the theorem of reciprocity, Green's function represents the medium's response to a source on the surface recorded by a virtual receiver in-depth and decomposes it into a down-going wavefield and an up-going wavefield. Both wave fields' deconvolution provides the reflection response of virtual sources and virtual receivers repositioned in the subsurface, mimicking redatumming. The result is an image free of multiple internal reflections from the area below virtual both sources and receivers. After successfully testing with synthetic seismograms generated in a plane-horizontal layers model, we applied the Marchenko algorithm to seismic data of the Upper Magdalena Valley project using an available velocity model. Compared to an image supplied by a Kirchhoff pre-stack depth migration algorithm, the new image correctly positions the seismic events avoiding misleading interpretations in locating complex structures.

What are recent observations telling us in light of improved tests of distance duality relation?

As an exact result required by the Etherington reciprocity theorem, the cosmic distance duality relation (CDDR), η(z)=DL(z)(1+z)−2/DA(z)=1 plays an essential part in modern cosmology. In this paper, we present a new method (η(zi)/η(zj)) to use the measurements of ultra-compact structure in radio quasars (QSO) and the latest observations of type Ia supernova (SN Ia) to test CDDR. By taking the observations directly from SN Ia and QSOs, one can completely eliminate the uncertainty caused by the calibration of the absolute magnitudes of standard candles (MB) and the linear sizes of standard rulers (lm). Benefit from the absence of nuisance parameters involved in other currently available methods, our analysis demonstrates no evidence for the deviation and redshift evolution of CDDR up to z=2.3. The combination of our methodology and the machine learning Artificial Neural Network (ANN) would produce 10−3 level constraints on the violation parameter at high redshifts. Our results indicate perfect agreement between observations and predictions, supporting the persisting claims that the Etherington reciprocity theorem could still be the best description of our universe.

Electrophoretic trajectories of non-uniformly charged particles in viscoelastic fluids: the weak surface charge limit

Electrophoretic motion of a particle carrying a weak but arbitrary non-uniform surface charge density in an Oldroyd-B fluid is analysed here in the thin electrical double layer limit. A semi-analytical generic framework, based on regular perturbation, the Lamb's general solutions and the generalized reciprocal theorem, assuming the viscoelastic effects to remain subdominant, is developed for tracing the particle's trajectory using its instantaneous translational velocity and accounting for the temporal evolution of its surface charge driven by rotation. Our results reveal that in a viscoelastic medium, non-uniformly charged particles may generally follow distinct trajectories depending on their sizes, which is in stark contrast to Newtonian fluids. By considering the multipole moments of the surface charge, we show that the particle may initially rotate until its dipole moment becomes collinear with the imposed electric field, and the nature of the surrounding medium does not alter this fundamental behaviour. However, during the course of rotation, the excess polymeric stresses within the electrical double layer and the bulk may cause the particle to migrate perpendicular to the applied field, by forcing the multipole moments of the surface charge to interact with each other. The final steady-state trajectory of the particle and its possible migration normal to the applied electric field are also largely governed by these interactions and more specifically, presence of non-zero quadrupole moments. The present framework may be helpful towards designing tools for particle separation and sorting, relevant in many biological applications.

On a variational principle for the Upper Convected Maxwell model

A variational principle for the Upper Convected Maxwell model is presented. The stationary value of the appropriate functional is the drag on an immersed object. From the principle, a formula is derived for the derivative of the drag with respect to the Deborah number for an arbitrarily shaped particle in a circular duct under creeping flow conditions. The formalism is compared with the conventional reciprocal theorem. Whereas the reciprocal theorem gives the drag as a volume integral involving the Stokesian stress tensor, the variational principle involves the stress from the adjoint equation. For low Deborah numbers both approaches provide the correction to the Stokes drag as a volume integral involving only the Stokesian rate-of-strain tensor, in line with second-order fluid theory. • A variational principle for the Upper Convected Maxwell model is presented. • At stationarity the functional equals the drag on a partcle in a duct. • For low Deborah numbers the drag correction involves only the Stokesian strain rate. • This is in correspondence with the reciprocal theorem and second-order fluid theory.

Fluctuations of the ground surface at blasting operations on tunnel structures

The paper considers the fluctuations of the ground surface during blasting operations on tunnel structures. To estimate the vibration levels of the soil surface, under the explosive action inside a cylindrical cavity, we will consider the plane problem of elasticity theory as a first approximation. The main purpose of the work is to determine the stress-strain state of pipelines under the influence of seismic waves. As a result of this impact, longitudinal and transverse waves will propagate in the ground. To solve the problem, the reciprocity theorem is applied, for dynamic problems of elasticity theory and the results obtained are compared by wave dynamics methods. The obtained solution makes it possible to determine the vibration levels of the soil surface that occur during explosive impacts inside the cylindrical cavity of the elastic half-space. The results can be used to assess the vibration levels of the soil surface that occur in the structures of shallow tunnels laid by drilling and blasting.

Delamination thickness measurement based on Stoneley wave in bilayered composite structure

Introduction: For composite structures, delamination usually occurs at the interface. Ultrasonic guided waves have been widely used to detect the delamination. However, most researches focus on the measurement of the delamination length along the interface, and the measurement of the delamination thickness is less studied. Method: In this paper, based on the characteristic that the Stoneley wave only propagates along the interface, the reflection coefficient of interaction between Stoneley wave and delamination is used to measure the delamination thickness. The effects of delamination thickness and frequency on the reflection coefficient are investigated via dividing integral region and reciprocity theorem. Some experimental validations are carried out on two aluminum-steel bilayered composite plates with different delamination thickness. Results: It is found the reflection coefficient increases linearly at first, then its rate of increase slows down gradually, and finally becomes stable, in theory. And the experimental results can verify the theoretical relationship between the reflection coefficients and the delamination thickness. Conclusion: The variation of reflection coefficient provides a reference for the measurement of delamination thickness in Stoneley-wave-based non-destructive testing.

Nonreciprocal transmission characteristics in double-cavity double-optomechanical system

<sec>Optical non-reciprocal devices such as the isolators are quite important components in optical systems. To realize the non-reciprocal transmission of the light, the Lorenz reciprocity theorem must be broken first and the main method is that Faraday magnetic rotation effect is used to change the polarization state of the signal through magneto-optical materials. However, this method is difficult to achieve on-chip integration. So using optomechanical system is presented to overcome the difficulty.</sec><sec>In order to improve the isolation characteristics of the device, a double-cavity double-optomechanical system, which is coupled to two optical modes by two mechanical oscillators with two different optomechanical coupling strengths, is proposed. Driven by the red detuning field in such a system, the non-reciprocal phenomenon can be realized by regulating the phase difference, and the direction of light transmission and isolation can be determined as well. This property is determined by the quantum interference effect between the optomechanical coupling strengths and the couplings of the optical cavity modes. The method is that the relative operators are represented by their average value plus their relative fluctuations, and then according to the input-output relationship the transmission amplitude and the isolation rate are obtained.</sec><sec>We mainly discuss the distribution of the isolation rate as a function of the optomechanical coupling strength. The results are that the combined action of two mechanical modes can make the system have higher fault tolerance rate. The other mechanical mode can make the system achieve a large isolation rate at two specific frequencies and the reverse transmission in the resonant frequency signals at the same time.</sec>

Comparison of Various Voltage Metrics for the Evaluation of the Nonradiative Voltage Loss in Quantum-Structure Solar Cells

There are several methods for the evaluation of the performance of solar cells through the voltage loss. However, the advantages and disadvantages of each method have not been thoroughly discussed nor elucidated. This study compares these various methods to clarify their advantages and disadvantages. The bandgap offset W <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">OC</sub> has been used for the evaluation of the voltage loss. However, it was found that such approach gives inaccurate assessment of the voltage loss in thin-bulk and quantum-structure solar cells. Our study discusses the reasons for such inaccuracy and proves that we can accurately evaluate the nonradiative voltage loss in quantum-structure solar cells using the optoelectronic reciprocity theorem and detailed-balance theory. Our findings enable a fair, accurate, and accessible evaluation of the nonradiative voltage loss in solar cells which facilitates the development of high-efficiency quantum-structure solar cell.

Reciprocal and variational theorems on the two-temperature Green–Lindsay thermoelasticity theory

Reciprocal and variational theorems on the two-temperature Green–Lindsay thermoelasticity theory

Electromechanical Reciprocity Applied to the Sensing Properties of Guided Elastic Wave Transducers

Guided elastic wave (GEW) transducers for structural health monitoring (SHM) can act as transmitters (senders) and receivers (sensors). Their performance in both cases depends on the structure to which they are coupled. Therefore, they must be characterized as system transducer- structure. The characterization of the transducer-structure as transmitter using a Scanning Laser Doppler Vibrometer (SLDV) is straightforward, whereas its characterization as receiver is non-trivial. We propose to exploit electromechanical reciprocity, which is an identity between the transfer functions of electrical-to-mechanical and mechanical-to-electrical conversions. For this purpose, the well-known electromechanical reciprocity theorem was adapted to the following situation: The two reciprocal states are "electrical excitation and detection of the surface velocity at point P" and "mechanical excitation at P and measurement of the electrical quantities". According to the derived formulas, the quantities on the mechanical and electrical sides must be chosen appropriately to ensure reciprocity as well as that the corresponding transfer functions are equal. We demonstrate the reciprocity with experimental data for correctly chosen transfer functions and show the deviation in reciprocity for a different choice. Furthermore, we propose further applications of electromechanical reciprocity.

"NUMERICAL SIMULATION OF ELASTIC-PLASTIC CONTACT WITH ISOTROPIC HARDENING "

The contacts of mechanical components transmit loads that lead to subsurface stresses developing in the contacting bodies. In an efficient tribological design, these stresses are expected to remain under the yield strength of the softer contacting material. When this condition is not met, plastic flow occurs in the softer body. Under the assumption of isotropic hardening, the yield strength increases with the development of additional plastic strains. As plastic flow processes are dissipative and therefore path dependent, the elastic-plastic problem is unsolvable through analytical endeavours, but can be approached with a numerical algorithm capable of simulating the loading history. The Betti’s reciprocal theorem provides the theoretical framework for the application of superposition principle to elastic-plastic stresses and displacement. An algorithm consisting in three nested loops is assembled from the solutions of simpler problems: (1) the purely elastic rough contact problem, (2) the inclusion problem and (3) the problem of the plastic strain increment. The numerical simulations suggest that the residual stresses decrease the intensity of the total stresses, thus delaying additional plastic flow. With increasing load, the heart-shaped plastic strain volume advances toward the surface, enveloping a plastic core near the initial point of contact. Compared to the purely elastic case, the elastic-plastic pressure shows a flatter distribution, while the contact radius is increased.