Reciprocity Theorem Research Articles

Analysis and Optimization of Truss Structures, Constrained Handling using Genetic Algorithm

In this study, an attempt is made to minimize the weight of Howe roof and ten member-6 Node trusses, separately. Two constraints, maximum allowable deflection and maximum allowable member stresses have been considered. For the first truss, permissible deflection is not known from the literature; therefore, it is determined using the exhaustive search method. Once magnitudes of the constraints are identified, member cross-sectional areas are varied to get the optimal weight. Both the exhaustive search method and the genetic algorithm have been implemented for this purpose. During the optimization, members tending to form a string may be eliminated from the structure. Doing this, we could further reduce the weights of the trusses and even less than the minimum available in the literature. The second truss is an indeterminate structure, and Maxwell Betti reciprocal theorem is applied to calculate the member forces. Also, further reduction of members is made for this truss, keeping in mind that the truss becomes determinate with the decrease in the member(s).

The radiation from ultrafast point dipoles, moving uniformly near chiral media

The article discusses the features of the radiation of ultrafast point clusters of charged particles moving uniformly near a gyrotropic medium interface. It is shown that some types of electromagnetic radiation – transient and/or Cherenkov radiation – have the characteristics of superchiral fields. Therefore, they can be effectively used to study chiral structures (for example, to detect circular dichroism, the frequency characteristics of refractive indices), various materials, including biomaterials. Ultrafast (relativistic) particles can serve as a «tool» not only for studying the structure of various materials. They can also be used as «generators» of quasiparticles that determine the «dynamic» properties of the materials under study, as well as the features of their interaction with radiation of various nature and response to external influences. In this paper, some types of circularly polarized EM waves propagating in optically active (magnetoactive, naturally active, gyrotropic, and chiral) media are considered. Using the generalized reciprocity theorem for media characterized by the Hermitian permittivity tensor, we consider the transient and Cherenkov radiation excited by a uniformly moving bunch of charged particles when it crosses (or moves along) the interface of media, one of which is an optically active gyrotropic medium. It is shown that the superchiral electromagnetic fields of the transition and Cherenkov radiation of dipoles can serve as a source of chiral collective excitations in magnetoactive and naturally active media. The investigated mechanisms of interaction of electromagnetic radiation with chiral materials (structures and media) are one of the possible physical approaches to solving the problem of the chiral purity of the biosphere and to elucidate the factor of deracemization of the organic primeval environment. A new hypothesis is presented, suggesting that ultra-high-speed clumps of charged particles of cosmic origin can cause deracemization of the prebiosphere.

Viscous propulsion of a two-dimensional Marangoni boat driven by reaction and diffusion of insoluble surfactant

A simple model of viscous propulsion of a two-dimensional Marangoni boat caused by surface diffusion and sublimation (reaction) of insoluble surfactant is formulated and studied analytically. The speed of the boat is determined explicitly as a function of a Damkohler number by fully resolving the Marangoni-induced bulk flow caused by the release of surfactants onto the surface of a semi-infinite viscous fluid. A complementary general integral formula for the boat speed is also derived using the reciprocal theorem.

Generalized electromagnetic theorems for nonlocal plasmonics

The ultraconfined light of plasmonic modes put their effective wavelength close to the mean free path of electrons inside the metal electron gas. The Drude model, which can not take the repulsive interactions of electrons into account, then clearly begins to show its limits. In an intermediate length scale where a full quantum treatment is computationally prohibitive, the semiclassical hydrodynamic model, instrinsically non-local, has proven successful. Here we generalize the expression for the absorption volume density and the reciprocity theorem in the framework of this hydrodynamic model. We validate numerically these generalized theorems and show that using classical expressions instead leads to large discrepancies.

Digital beam forming methods for variable polarimetric monopulse radar

In variable polarimetric monopulse radars, due to the number and types of data channels, the polarisation and angle information of the target in the receiving signals are non-linearly coupled, which results in incorrect polarisation measurement and erroneous angle estimation through a linear digital beam forming (DBF) method. Based on a variable polarimetric monopulse radar with four channels, two non-linear DBF methods are proposed to improve radar ability. Through the angle monopulse estimation principle and the reciprocity theorem, new pairs of summation and difference signals in all polarisation channels are formed to generate the correct polarimetric scattering and angle estimation results. The simulation and experiment results show that the proposed methods achieve a higher precision of polarisation measurement and more consistent angle estimation results compared to the linear DBF method.

An elastodynamic reciprocity theorem-based closed-form solution to second harmonic generation of lamb waves by a fatigue crack: Theory & experimental validation

Characterization of small-scale damage (with dimensions smaller than 1/10 of the probing wavelength) using nonlinear guided ultrasonic waves (GUWs) has been practiced over years, numerically and experimentally. To compensate for the insufficiency of analytical solutions that are able to interpret the underlying physical aspects of nonlinearity in GUWs induced by the small-scale damage and in particular fatigue damage, a new theoretical model based on the elastodynamic reciprocity theorem is developed. The model yields a closed-form solution to the modulation mechanism of a fatigue crack with ‘breathing’ attributes on Lamb wave propagation, gaining insight into the generation of second harmonics in Lamb waves. The model depicts the ‘breathing’ crack as an additional wave source imposing extra forces on crack surfaces that is equivalent to the integral of the stress tensor, and the source interferes with the wavefield of the original probing wave. In a time-frequency domain, this additional wave source is linked to the second harmonic generation in spectra. By virtue of the model, a nonlinear damage indicator, governed by the quantified second harmonic generated by the crack, is defined, to calibrate crack severity quantitatively. Finite element simulation is performed to verify the analytical model and demonstrate its accuracy when used for evaluating damage onset. Proof-of-concept experimental validation is conducted to verify the proportional trend of the damage indicator with respect to damage severity. This elastodynamic reciprocity-driven model and the closed-form solution shed light, from an analytical perspective, on the nonlinear interaction of GUWs with damage of small scale featuring ‘breathing’ attributes.

Analytical Electromagnetic Field Coupling to Transmission Line Model Exploiting the Reciprocity Theorem

In this work, an analytical electromagnetic field coupling to transmission line model is proposed to efficiently evaluate the induced voltage/current responses of loads connected at the terminals. Specifically, an equivalent circuit is developed by combing the Lorentz reciprocity theorem with the classic passive transmission line model, it is found that active contribution by impinging field could be cast in analytic form, as a function of the far-field radiation pattern and input impedance of transmission line system in transmitting mode. Comparing to Agrawal’s model, the proposed method reports concise and physical formulations without degrading the accuracy. Moreover, it is demonstrated that defining impinging field in spherical coordinates aligning parallelly with line is advantageous in revealing the wave propagation process.

Spherical surface and volume averages in electrostatics using Green’s reciprocity theorem

In the present work we apply Green’s reciprocity theorem for the determination of surface and volume averages of the potential and electric field over an imaginary sphere in an electrostatic field. To the best of our knowledge this approach has not been used so far for such averaging. The derived results have importance when studying the properties and behaviour of dielectric media.

Fast computation of scattering by isolated defects in periodic dielectric media

Scattering by an isolated defect embedded in a dielectric medium of two-dimensional periodicity is of interest in many sub-fields of electrodynamics. Present approaches to compute this scattering rely on either the Born approximation and its quasi-analytic extensions or ab initio computation that requires large domain sizes to reduce the effects of boundary conditions. The Born approximation and its extensions are limited in scope, while the ab initio approach suffers from its high numerical cost. In this paper, I introduce a hybrid scheme in which an effective local electric susceptibility tensor of a defect is estimated by solving an inverse problem efficiently. The estimated tensor is embedded into an S-matrix formula based on the reciprocity theorem. With this embedding, the computation of the S-matrix of the defect requires field solutions only in the unit cell of the background. In practice, this scheme reduces the computational cost by almost two orders of magnitude, while sacrificing little in accuracy. The scheme demonstrates that statistical estimation can capture sufficient information from cheap calculations to compute quantities in the far field. I outline the fundamental theory and algorithms to carry out the computations in high dielectric contrast materials, including metals. I demonstrate the capabilities of this approach with examples from optical inspection of nano-electronic circuitry where the Born approximation fails and the existing methods for its extension are also inapplicable.

Post-seismic crustal internal deformation in a layered earth model

SUMMARY This study introduces a novel method for computing post-seismic crustal internal deformation in a layered earth model. The surface dislocation Love number (DLN) calculated by the reciprocity theorem was implemented as the initial value. Furthermore, numerical integration of the value from the Earth's surface to the interior was undertaken to obtain the internal DLN. This method does not require a combination of the general solution and particular solution for the calculation of internal deformation above the seismic source, thus avoiding the loss of precision. When the post-seismic deformation within a certain period is calculated, the particular solutions at the beginning and end of the considered period cancel each other. This simplifies the calculation of post-seismic internal deformation. The numerical results depict that as the degrees increase, the post-seismic DLN reaches stability in a shorter interval of time. Thus, for improved efficiency of the post-seismic internal deformation calculation, the post-seismic DLNs should be calculated within 2000 degree and integrated with the coseismic results. As an application, the post-seismic Coulomb failure stress changes (∆CFS) induced by the 2011 Tohoku-Oki earthquake in the near field around the Japanese archipelagos and two major faults in Northeast China were simulated. The results exhibit that the ∆CFS values in the near field agree well with those simulated by the method in a half-space layered earth model, thus verifying the present method. The coseismic ∆CFS on the Mishan-Dunhua fault in Northeast China, as an example, is only 0.094–0.668 KPa. However, the ∆CFS caused by the viscoelastic relaxation of the mantle within 5 yr following the 2011 Tohoku-Oki event on the same fault exceeds the coseismic results. Therefore, the cumulative effect of the viscoelastic relaxation of the mantle is deserving of attention.

Mathematical Modelling of a Novel Hetero-junction Dual SIS ZnO-Si-SnO Solar Cell

For the last few decades scientists across the world have achieved significant improvement in performance of conventional silicon p-n junction solar cell. Sophisticated high temperature doping technology is unavoidable in the fabrication of these conventional solar cells. Back in 1970s scientists proposed an alternative solar cell technology with Schottky barrier which can cut down the burden on thermal budget of manufacturing process. Later the metal-semiconductor Schottky barrier further modified with hetero junction semiconductor-semiconductor solar cells. A thin intrinsic layer sandwiched between semiconductor-semiconductor junctions can repair the junction defect efficiently. These SIS solar cells became popular for its low thermal budget and considerable efficiency. In this paper we have tried to propose a mathematical model of a novel dual side SIS solar cell which is basically a multi junction solar cell. We have introduced a third semiconductor layer (SnO) at the back side of the cell which can provide an inversion layer much similar to PERT solar cell. This structure is first of its kind and thus a theoretical analysis is required before implementation. We have studied the effect of this back field on the performance of the cell and propose a mathematical model based on reciprocity theorem of charge collection. The efficiency of conventional ZnO-pSi SIS solar cell was computed ~5.2% while the back SnO p+ layer is expected to enhance the efficiency up to ~7.9% according to our mathematical model. We have concluded with significant mathematical justification to implement this structure with proposed electro-chemical experiments.

Nonlinear thermodynamics of biological signal transduction for predicting conservation of entropy production rate

Cell signal transduction is an example of a nonequilibrium phenomenon. In this study, a nonequilibrium nonlinear thermodynamic model was formulated. First, we obtained a condition in which the Onsager’s reciprocity theorem holds in the signal transduction. Second, it was found that the entropy production rate per signal molecule is conserved through signal transduction. Finally, it was determined that when Onsager’s reciprocity theorem does not hold, fluctuation circulation is given by the phosphorylation rate of signal molecules. The simple relation implies that the fluctuation circulation can be an essential quantity of the signal transduction amount. These results expand the limit of nonequilibrium thermodynamics and can be used to provide ideas for signal transduction quantification.

Uniqueness, continuous dependence and reciprocity theorems in thermoelastic relaxed micromorphic continuum

A mixed initial-boundary value problem is formulated for linear thermo-elastic relaxed micromorphic continuum with asymmetric micro-distortion tensor, micro-dislocation tensor and temperature field. First, Lagrange identity involving two admissible processes at different instants for the considered mixed initial-boundary value problem is established. Using this identity, four theorems are proved: (i) uniqueness of solution without recourse either to an energy conservation law or to any boundedness assumptions on the thermoelastic coefficients, (ii) continuous dependence of solution of the problem on loads, (iii) continuous dependence of solution of the problem on the initial data, and (iv) reciprocity relation. The use of positive definiteness conditions on the thermoelastic coefficients is also avoided.

SIMPLIFIED METHOD FOR CALCULATING THE PASSIVE EARTH PRESSURE OF COHESIVE BACKFILL AGAINST RIGID RETAINING WALLS

The horizontal slice element method for calculating the passive earth pressure against rigid retaining walls is extended to cohesive backfill. The formula of the nonlinear distribution of passive earth pressure is derived to satisfy the reciprocal theorem of shear stress. According to the Mohr-Coulomb theory, the lateral passive earth pressure coefficient can be determined, and a simplified expression of the total passive force is obtained. The influence of the shear stress increment between soil strips on the passive earth pressure is discussed by two calculation models. The results are compared with the existing classical theory. The comparison shows that the formula in this paper degenerates to the Rankine formula when there is no uniformly distributed load. If the backfill is cohesionless soil, the results are consistent with those in the literature that considered the soil arching effect. The calculated results are also consistent with the existing experimental results.

New approach to interface crack problems in transversely isotropic materials

The novelty of the approach contains several aspects: the method of derivation of the governing equations, the form and number of equations, the optimal choice of elastic constants. While majority of published articles derives the governing equations from the Green’s function for the compound space and the use of the reciprocal theorem, we use the combination of Green’s functions for 2 different half-spaces and Fourier transform. The usual approach leads to 3 hypersingular integral equations, we arrive at 2 integro-differential equations (one of them being complex). While the coefficients of the governing equations in existing publications are presented in terms of the results of the solution of a set of linear algebraic equations and are too cumbersome to be written explicitly in terms of the basic elastic constants, our choice of the basic constants leads to quite elegant explicit expressions for these coefficients and reveals certain symmetry, which was noticed only numerically in previous publications. The new form of the governing equations allows us to obtain an exact closed form solution for an axisymmetric interface crack problem by elementary means. The same method can be applied to obtain the exact solution for a non-axisymmetric problem. A comparison is made with the existing exact solutions and some are shown to be incorrect.

Programmable nonreciprocal meta-prism

Optical prisms are made of glass and map temporal frequencies into spatial frequencies by decomposing incident white light into its constituent colors and refract them into different directions. Conventional prisms suffer from their volumetric bulky and heavy structure and their material parameters are dictated by the Lorentz reciprocity theorem. Considering various applications of prisms in wave engineering and their growing applications in the invisible spectrum and antenna applications, there is a demand for compact apparatuses that are capable of providing prism functionality in a reconfigurable manner, with a nonreciprocal/reciprocal response. Here, we propose a nonreciprocal metasurface-based prism constituted of an array of phase- and amplitude-gradient frequency-dependent spatially variant radiating super-cells. In conventional optical prisms, nonreciprocal devices and metamaterials, the spatial decomposition and nonreciprocity functions are fixed and noneditable. Here, we present a programmable metasurface integrated with amplifiers to realize controllable nonreciprocal spatial decomposition, where each frequency component of the incident polychromatic wave can be transmitted under an arbitrary and programmable angle of transmission with a desired transmission gain. Such a polychromatic metasurface prism is constituted of frequency-dependent spatially variant transistor-based phase shifters and amplifiers for the spatial decomposition of the wave components. Interesting features include three-dimensional prism functionality with programmable angles of refraction, power amplification, and directive and diverse radiation beams. Furthermore, the metasurface prism can be digitally controlled via a field- programmable gate array (FPGA), making the metasurface a suitable solution for radars, holography applications, and wireless telecommunication systems.

On the application of the beam model for linear dynamic analysis of pile and suction caisson foundations for offshore wind turbines

Piles and suction caissons are the most common foundation solutions for fixed Offshore Wind Turbines at intermediate water depths. They are generally used as a single element, presenting large diameters and short aspect ratios. These specific dimensions drastically differ from the ones of classical applications (offshore platforms, bridges, tall buildings etc.). Thus, in this paper the validity of their modelling as beam elements for the particular problem of OWT is revised. The results of a soil-beam model, based on the integral Reciprocity Theorem in Elastodynamics and specific Green’s functions for the layered half-space for the soil behaviour coupled with Timoshenko’s beam Finite Elements, are benchmarked against the ones of a soil-shell model, based on Boundary Elements for the soil coupled with shell Finite Elements. The comparative study is conducted in terms of foundation characterization variables (impedance functions and kinematic interaction factors). Their influence on the OWT seismic response is also studied through a substructuring procedure. From the results, some expressions for determining the applicability range of the beam simplification are proposed as functions of the relative foundation-soil stiffness ratio. It is observed that this applicability range goes beyond that the one commonly considered. .

Constraints of Pore-Bulk Strain Ratio and Interference Time on the Evolution of Coal Permeability during CO2 Injection

CO2 injection into coal seam triggers a series of processes that are coupled all together through a permeability model. Previous studies have shown that current permeability models cannot explain experimental data as reported in the literature. This knowledge gap defines the goal of this study. We hypothesize that this failure originates from the assumption that the pore strain is the same as the bulk strain in order to satisfy the Betti-Maxwell reciprocal theorem. This assumption is valid only for the initial state without gas sorption and deformation and for the ultimate state with uniform gas sorption and uniform deformation within the REV (representative elementary volume). In this study, we introduce the pore-bulk strain ratio and interference time to characterize the process of gas sorption and its associated nonuniform deformation from the initial state to the ultimate state. This leads to a new nonequilibrium permeability model. We use the model to fully couple the coal deformation and gas flow. This new coupled model captures the impact of coal local transient behaviors on gas flow. Results of this study clearly show that coal permeability is constrained by the magnitudes of initial and ultimate pore-bulk strain ratios and interference time, that current permeability data in the literature are within these bounds, and that the evolutions of coal permeability all experience similar stages from the initial value to the ultimate one.

Oriented beamformig of a multi-mode antenna based on characteristic mode analysis

In this paper, a method is presented for antenna beamforming based on characteristic mode theory. A multi-mode antenna with two elements and four significant orthogonal modes is presented in 5 GHz. Feed positions of the proposed antenna to excite its four characteristic modes are determined; Then with the help of characteristic mode analysis and the reciprocity theorem, amplitude and phase of the excitation coefficients of the proposed antenna are found to change the antenna beam direction in whole azimuth plane. Moreover, changing the weights of the antenna feeds enables elevation beamforming of the proposed antenna.

Influence of Considering the Sorption Effect in the Betti-Maxwell Reciprocal Theorem on Gas Transport Capacity in Unconventional Reservoirs

Pore bulk modulus in continuum apparent permeability models was generally treated as an independent parameter or deduced from the Betti-Maxwell reciprocal theorem neglecting the sorption effect, which is not appropriate for the unconventional reservoirs with strong gas sorption. In this work, pore bulk modulus was firstly derived from the Betti-Maxwell theorem with the consideration of the gas sorption strain and was incorporated into the solid–fluid coupling apparent permeability model. Numerical simulation was then conducted under the uniaxial strain condition and the stress confined condition to investigate the gas transport characteristics. The results showed that when taking into account for the sorption effect in the Betti-Maxwell theorem, the pore bulk modulus dramatically decreased with the falling pore pressure during the gas transport process under two loading conditions. Moreover, gas transport ability under the stress confined condition was greatly degraded while it was hardly changed under the uniaxial strain condition. Both the pore bulk modulus and the stress distribution in the reservoirs were found responsible for the variation of gas flow capacity. Finally, the effects of the parameters associated with the pore bulk modulus were examined to further discuss the impact of involving the sorption effect in the Betti-Maxwell theorem.