Superposition Laws Research Articles

Stochastic Dynamical Modeling of Wind Farm Turbulence

Low-fidelity engineering wake models are often combined with linear superposition laws to predict wake velocities across wind farms under steady atmospheric conditions. While convenient for wind farm planning and long-term performance evaluation, such models are unable to capture the time-varying nature of the waked velocity field, as they are agnostic to the complex aerodynamic interactions among wind turbines and the effects of atmospheric boundary layer turbulence. To account for such effects while remaining amenable to conventional system-theoretic tools for flow estimation and control, we propose a new class of data-enhanced physics-based models for the dynamics of wind farm flow fluctuations. Our approach relies on the predictive capability of the stochastically forced linearized Navier–Stokes equations around static base flow profiles provided by conventional engineering wake models. We identify the stochastic forcing into the linearized dynamics via convex optimization to ensure statistical consistency with higher-fidelity models or experimental measurements while preserving model parsimony. We demonstrate the utility of our approach in completing the statistical signature of wake turbulence in accordance with large-eddy simulations of turbulent flow over a cascade of yawed wind turbines. Our numerical experiments provide insight into the significance of spatially distributed field measurements in recovering the statistical signature of wind farm turbulence and training stochastic linear models for short-term wind forecasting.

Physics-driven mode decomposition based on a training-free shallow neural network

A mode decomposition method based on a training-free shallow neural network (TFSNN-MD) is proposed. The TFSNN-MD underlies the interaction between the eigenmode superposition laws and the neural network (NN), eliminating the massive beam pattern datasets and training process. It has obvious advantages with high accuracy, fast operation and great versatility. In the case of using only a shallow NN with only two convolutional layers, the errors of modal weights and modal relative phases are lower than 0.50% and 0.60%, theoretically. Furthermore, it performs well in generalization regardless of the variation of the FMF’s core radius and NA. In experiment, the correlation between the real and reconstructed beam patterns is higher than 95%.

Behaviors of type-I instabilities in natural circulation with a heating channel under ocean conditions

Characteristics of two-phase natural circulation and density wave instability with a single-heating channel under ocean conditions were studied by using PNCMC PROGRAM (Program for Natural Circulation under Motion Condition). The superposition laws of oscillations of both mass flow rate and density wave caused by ocean conditions are found, and the corresponding stability boundaries of the natural circulation system are also proposed here. The use of a simplified mathematical description of fluid flow to describe the additional motions with waves reduces the difficulty of analysis. Specifically speaking, based on the study of stability boundary in vertical condition, first of all, it was found that inclined ocean condition can weaken the ability of natural circulation and can move down stability boundary of natural circulation, and the greater the inclination angle, the greater the amount of going-down. Secondly, the stability boundary of the system moves up in heaving circumstances, and the smaller the period, the greater the amount of going-up. Thirdly, the amplitude of rolling plays an important role in the rolling situation. In the case of rolling with large amplitude, the period of flow complex oscillation is clear, which is half the rolling period. However, in the case of rolling with a short period, the period of flow complex oscillation is ambiguous, the oscillation frequency of density wave under vertical condition is preserved in the flow complex oscillation. The stability boundary of natural circulation is moved down by rolling with a short period, where the frequency of flow complex oscillation was equal to that of density wave oscillation under vertical conditions. In contrast, the stability boundary of natural circulation is moved up by rolling with a long period, where the short-period local oscillation appears in the process of flow complex oscillation. Finally, the oscillation superpositions of rolling and heaving motion will result in a complex long-period oscillation with motion amplitudes being important.

Использование композиционных и суперпозиционных законов распределения вероятностей отказов при оценке надежности электротехнических изделий военного назначения

An implementation of the developed computational structural method for assessing the reliability of complex electrical products. The basis of this method is the use of combinations of composition and superposition laws of distribution of probabilities of failures. The estimation of reliability of a rotating transformer type ВT-5. The calculated reliability indices are compared with results of standard test ВT-5 for reliability. Received high convergence of the results.

Edge detection algorithm for omnidirectional images, based on superposition laws on Blach’s sphere and quantum entropy

This paper presents an edge detection algorithm for omnidirectional images based on superposition law on Bloch’s sphere and quantum local entropy. Omnidirectional vision system has become an essential tool in computer vision, duo to its large field of view. However, classical image processing algorithms are not suitable to be applied directly in this type of images without taking into account the spatial information around each pixel. To show the performance of the proposed method, a set of experimentation was done on synthetic and real images devoted to agriculture applications. Later, Fram & Deutsh criterion has been adopted to evaluate its performance against three algorithms proposed on the literature and developed for omnidirectional images. The results show a good performance of the proposed method in term of edge quality, edge community and sensibility to noise.

Long-term properties of carbon fiber-reinforced shape memory epoxy/polymer composites exposed to vacuum and ultraviolet radiation

Shape memory polymer composites (SMPCs) are being evaluated for aerospace applications due to their lightweight and ease of self-deployment. For such applications, the mechanical properties of SMPCs, including long-term behavior, need to be determined in the particularly harsh environments that are likely to be encountered. In this study, the storage modulus of carbon fiber-reinforced SMPCs (CF-SMPCs) was investigated under high vacuum condition and ultraviolet (UV) irradiation. The storage modulus of CF-SMPCs generally degrades as a function of time and temperature. However, the opposite behavior was observed under vacuum and UV exposure due to induced crosslinking. These two phenomena were characterized separately using acceleration tests at various temperatures under UV irradiation in a vacuum chamber and were modeled as two time-temperature superposition laws. The long-term mechanical behavior of CF-SMPCs in vacuum and UV environment is predicted by a linear product of shift factors obtained from the two acceleration tests.

Topology-Driven Single-Molecule Conductance of Carbon Nanothreads.

Highly conductive single-molecule junctions typically involve π-conjugated molecular bridges, whose frontier molecular orbital energy levels can be fine-tuned to best match the Fermi level of the leads. Fully saturated wires, e.g., alkanes, are typically thought of as insulating rather than highly conductive. However, in this work, we demonstrate in silico that significant zero-bias conductance can be achieved in such systems by means of topology. Specifically, caged saturated hydrocarbons offering multiple σ-conductance channels afford transmission far beyond what could be expected based upon conventional superposition laws, particularly if these pathways are composed entirely from quaternary carbon atoms. Computed conductance of molecular bridges based on carbon nanothreads, e.g., polytwistane, is not only of appreciable magnitude; it also shows a very slow decay with increasing nanogap, similarly to the case of π-conjugated wires. These findings offer a way to manipulate the transport properties of molecular systems by means of their topology, alternatively to the traditionally invoked electronic structure.

Hypercomplex analysis and integration of systems of ordinary differential equations

We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of hypercomplexification, which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati‐type systems, we obtain invariants of Abel‐type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs. Copyright © 2016 John Wiley & Sons, Ltd.

Prediction of the yield strength of a secondary-hardening steel

Based on detailed three-dimensional (3-D) local-electrode atom-probe (LEAP) tomographic measurements of the properties of Cu and M2C precipitates, the yield strength of a high-toughness secondary-hardening steel, BA160, as a function of aging time is predicted using a newly developed 3-D yield strength model. Contributions from each strengthening constituent are evaluated with the model, and superposition laws are applied to add each contribution. Prediction of the yield strength based entirely on 3-D microstructural information is thus achieved. The accuracy of the prediction depends on the superposition laws and the LEAP tomographic measurements, especially the mean radius and volume fraction of M2C precipitates.

Lie's reduction method and differential Galois theory in the complex analytic context

This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generalization of linear systems, and the more general class of differential equations adimitting superposition laws, as recently stated in [5]. A Lie-Vessiot system is automatically translated into a equation in a Lie group that we call automorphic system. Reciprocally an automorphic system induces a hierarchy of Lie-Vessiot systems. In this work we study the global analytic aspects of a classical method of reduction of differential equations, due to S. Lie. We propose an differential Galois theory for automorphic systems, and explore the relationship between integrability in terms of Galois theory and the Lie's reduction method. Finally we explore the algebra of Lie symmetries of a general automorphic system.

Effect of heat treatment on the strain hardening behaviour of an Al–Zn–Mg alloy

Abstract The strain hardening of the Al-5.2Zn-1.2Mg alloy AA7030 in different conditions has been investigated. The dislocation related strengthening was described by the Voce equation and the quantitative effects of different superposition laws have been studied. All tempers, except the over-aged, suffer from plastic instability before the geometric Considère condition is reached. This was attributed primarily to the occurrence of shear failure due to dynamic strain ageing effects. For plastic strain levels below approximately 8 %, preceding the plastic instability, the results show that the saturation stress in the under and peak-aged tempers is independent of heat treatment, whereas the kinetics of structure formation differs. Using a linear superposition law in the under and peak aged tempers has negligible effects, whereas the use of Pythagorean addition for the over-aged temper significantly decreases the goodness-of-fit, although not to unacceptable levels. Furthermore, the naturally aged temper shows strong influence of dynamic precipitation.

High temperature creep of modified α + β brasses

Good machinability and formability of technical brasses α+β have to be preserved when alternative modifications are developed instead of alloying with toxic lead. Several model brasses of this type, which exhibit satisfactory machinability, are creep tested at 780°C to compare their high temperature formability. The observed stationary creep rate and local ductility may highly differ for various alloy types. To characterize the impact of individual additions, superposition laws for the respective quantities are proposed and applied to the measured values. The strong effect of even small amounts of some additions on the creep rate and the boundary component of ductility is suggested to be due to segregation phenomenon. Rigid particles, such as phosphides, enhance significantly the creep resistance of the material, though restrict the macroscopic stability of plastic flow. On the other hand, the contribution of liquid particles to the creep damage may be effectively obstructed by phosphides.

Nonlinear gravitational waves and their polarization

Vacuum gravitational fields invariant for a non-Abelian Lie algebra generated by two Killing fields whose commutator is light-like are analyzed. It is shown that they represent non-linear gravitational waves obeying to two non-linear superposition laws. The energy and the polarization of this family of waves are explicitly evaluated.

An integrable model in general relativity

It is shown that gravitational fields invariant for a non Abelian 2-dimensional Lie algebra of Killing fields are parameterized either by solutions of a transcendental equation (the tortoise equation) or by solutions of a linear second order partial differential equation (the Laplace equation or the Darboux equation) on the plane. Those determined via Laplace or Darboux equations are exact nonlinear gravitational waves obeying to two nonlinear superposition laws.

The isotropic shear modulus of multicomponent Fe-base solid solutions

A critical analysis of the available experimental data for the effect of alloying elements on the isotropic shear modulus of bcc (body-centered cube) Fe– X ( X=Al, Be, C, Co, Cr, Ge, Ir, Mn, Ni, Pt, Re, Rh, Ru, Si and V) solid solutions is carried out. The total effect of a solute on the shear modulus is decomposed into two contributions: the electronic (or chemical) and the volumetric. A systematic trend of the electronic contribution is demonstrated as a function of electron-to-atom ( e/a) ratio and the ground-state electronic configuration of the solute atom. Based on the demonstrated trend, we predict the chemical contribution of the shear modulus of Cu, Mo, N, Nb, Ti and W in ferromagnetic α-Fe (bcc), and that of Ti and V in paramagnetic γ-Fe [face-centered cube (fcc)]. These along with the corresponding volumetric contributions enable us to predict the total effect of a solute on the shear modulus in α-Fe and γ-Fe. In the case of γ-Fe, we derive the chemical and volumetric contributions of Ni and Pt from the experimental shear modulus data of paramagnetic Fe–Ni and Fe–Pt alloys while those of C, Co, Cr, Mn, Mo, N and Si are derived from the shear modulus of paramagnetic Fe–Ni– X alloys. In the case of Al, Be, Cu, Ge, Ir, Nb, Re, Rh , Ru and W, the total effect on the shear modulus is calculated by assuming that the electronic contribution to the shear modulus in γ-Fe is the same as in α-Fe. To calculate the isotropic shear modulus of multicomponent bcc and fcc solid solutions, we propose linear superposition laws. The proposed relationships are validated using the experimental data of a large number of multicomponent alloys having austenitic, ferritic, and lath martensitic microstructures. It is demonstrated that for all three microstructures, in most cases the shear modulus can be predicted with an accuracy of ±3% in multicomponent solid solutions. It is also found that the high dislocation density in lath martensite accounts for a decrease in shear modulus by about 5% compared to the ferritic counterpart. We also demonstrate that the temperature dependence of shear modulus in multicomponent bcc and fcc solid solutions is similar to that of pure α- and γ-Fe, respectively, for up to about 800 K.

Determination of Martensite Start Temperature for Engineering Steels Part II. Correlation between Critical Driving Force and <I>M</I>s Temperature

Various physical models are discussed based on different correlations established between the is temperature, the critical driving force, and the steel chemistry. These correlations were derived for a group of Fe-(0.2-0.5)C-(0.5-2.0)Mn-(0.5-2.0)Si-(0.5-2.0)Cr-(0.1-0.7)Mo test alloys, which serve as a good representative for most low alloy engineering steels. The chemical driving force was calculated by thermodynamic software and the is temperature was predicted by a validated artificial neural network model. Two basic physical models are discussed: the is-dependent model and the chemistry-dependent model. In the is-dependent model, the critical chemical driving force is linearly related to the Ms temperature; the effect of the steel chemistry is indirect. The standard error of the simple is-dependent model is 51.9 J/mol when the spontaneous Zener ordering of carbon atoms is taken into account. The chemistry-dependent model is based on the hypothesis that the critical driving force can be fully represented in terms of the steel chemistry. The critical driving force has been estimated using either linear, exponential, Pythagorean or mixed superposition laws. Comparisons of the critical driving force predicted by these addition methods with the thermodynamic result indicate that an exponential addition method, with the optimum exponent index value of 2.07 (approximately square) gives the best predictive result. The quality of the linear relation between the critical driving force and the is temperature is improved slightly if the critical driving force is corrected for the elastic strain energy, estimated by assuming that the elastic moduli, lattice constants, and molar volumes of ferrite and austenite are both temperature- and chemistry-dependent, is removed from the critical driving force. Analysis indicates that the simple is-dependent model will be improved after considering the extra minor effect of steel chemistry. In contrast, the error of the chemistry-dependent model can be hardly related to the is temperature. Based on the above analysis, an accurate mixed is-dependent plus chemistry-influence model, in which the elastic strain energy is considered, is developed, and which yields for the chemical driving force -ΔG * r =3247 -4.8446 Ms (°C), with a standard error of only 40.7 J/mol.

Kinetics of F.C.C. → B.C.C. heterogeneous martensitic nucleation—I. The critical driving force for athermal nucleation

Employing available experimental data for athermal f.c.c. → b.c.c. martensitic transformation in binary, ternary and multicomponent Fe-base alloys, a model is developed and tested for the critical driving force at the M s temperature. Incorporating the theory of solid solution hardening, we describe the composition dependence of the athermal frictional work for martensitic interface motion governing the kinetics of barrierless heterogeneous nucleation. The available data suggests that the composition dependence of the athermal frictional work is of the same form as that for slip deformation. We have evaluated the athermal strengths of 14 alloying elements Al, C, Co, Cr, Cu, Mn, Mo, N, Nb, Ni, Si, Ti, V and W from the experimental data. Except for Al, Ni and Co, the athermal strengths of the common substitutional alloying elements are similar in magnitude, while the interstitial solutes C and N exert a stronger influence. Previously proposed superposition laws are used to account for the presence of multiple solutes having different athermal strengths. With an improved assessment of the magnetic parameters of alloy systems, the model predicts M s temperatures within ±40 K for M s > 300 K where thermal contributions to the frictional work can be neglected.

Plastic Deformation of Ionic Crystals: Statistical Analysis of Yield Stress Measurements

Statistical methods of data analysis were used to study yield stress measurements of ionic crystals. First, the effect of specimen geometry on the reproductibility of yield stress measurements were studied in order to determine the optimum dimensions and shape of the specimen so as to obtain the most reproductibile measurements. Secondly, the causes of inconsistencies between values of n (the exponent of the power law which expresses the variation of the yield stress gainst impurity concentration) were investigated. A set of values of n obtained with various regression methods and various superposition laws to calculate the impurity-induced-hardening were analyzed by using the variance analysis method. Results of the analysis show that the choice of a superposition law influences the determination of n. The choice of a superposition law together with the lack of a characterization of the aggregation state of impurities are thought to be the main causes of discrepancies between the values of n published over the last ten years.

Superposition laws for solutions of differential matrix Riccati equations arising in control theory

Representation formulas are given for the general solution of the N \times N matrix Riccati equation \dot{W} = A + WB + CW + WDW using n known solutions, with n = 1, ..., 5 ( n -representations). The 5- representation is a superposition formula, in that it expresses the general solution explicitly as a function of five particular solutions and N2arbitrary constants ( N \geq 2 ), using no further information. The representation formulas can be used in numerical calculations. The 4- and 5- representations are specially useful when a solution W(t) has a singularity for some finite t = t_{0} . They also clarify the properties of the solution space: the matrix elements of W(t) are meromorphic functions of t having simple poles as the only possible singularities. The relation between the representation formulas and previously known results is discussed.

Homogeneous quadratic invariants for one-dimensional classical homogeneous quadratic Hamiltonians: Some properties of the auxiliary equation

Using a previously derived characterization of the homogeneous quadratic invariants for an arbitrary one-dimensional classical homogeneous quadratic Hamiltonian we obtain various ‘‘nonlinear superposition laws.’’ We establish a close connection between one of these (the quadratic superposition law) and the original characterization.