Manifold-informed state vector subset for reduced-order modeling

Reduced-order models (ROMs) for turbulent combustion rely on identifying a small number of parameters that can effectively describe the complexity of reacting flows. With the advent of data-driven approaches, ROMs can be trained on datasets representing the thermo-chemical state-space in simple reacting systems. For low-Mach flows, the full state vector that serves as a training dataset is typically composed of temperature and chemical composition. The dataset is projected onto a lower-dimensional basis and the evolution of the complex system is tracked on a lower-dimensional manifold. This approach allows for substantial reduction of the number of transport equations to solve in combustion simulations, but the quality of the manifold topology is a decisive aspect in successful modeling. To mitigate manifold challenges, several authors advocate reducing the state vector to only a subset of major variables when training ROMs. However, this reduction is often done ad hoc and without giving detailed insights into the effect of removing certain variables on the resulting low-dimensional data projection. In this work, we present a quantitative manifold-informed method for selecting the subset of state variables that minimizes unwanted behaviors in manifold topologies. While many authors in the past have focused on selecting major species, we show that a mixture of major and minor species can be beneficial to improving the quality of low-dimensional data representations. The desired effects include reducing non-uniqueness and spatial gradients in the dependent variable space. Finally, we demonstrate improvements in regressibility of manifolds built from the optimal state vector subset as opposed to the full state vector.     

A hybrid approach for the modal analysis of continuous systems with discrete piecewise-linear constraints

The analysis of continuous systems with piecewise-linear constraints in their domains have previously been limited to either numerical approaches, or analytical methods that are constrained in the parameter space, boundary conditions, or order of the system. The present analysis develops a robust method for studying continuous systems with arbitrary boundary conditions and discrete piecewise-linear constraints. A superposition method is used to generate homogeneous boundary conditions, and modal analysis is used to find the displacement of the system in each state of the piecewise-linear constraint. In order to develop a mapping across each slope discontinuity in the piecewise-linear force-deflection profile, a variational calculus approach is taken that minimizes the L₂ energy norm between the previous and current states. An approach for calculating the finite-time Lyapunov exponents is presented in order to determine chaotic regimes. To illustrate this method, two examples are presented: a pinned-pinned beam with a deadband constraint, and a leaf spring coupled with a connector pin immersed in a viscous fluid. The pinned-pinned beam example illustrates the method for a non-operator based analysis. Results are used to show that the present method does not necessitate the need of a large number of basis functions to adequately map the displacement and velocity of the system across states. In the second example, the leaf spring is modeled as a clamped-free beam. The interaction between the beam and the connector pin is modeled with a preload and a penalty stiffness. Several experiments are conducted in order to validate aspects of the leaf spring model. From the results of the convergence and parameter studies, a high correlation between the finite-time Lyapunov exponents and the contact time per period of the excitation is observed. The parameter studies also indicate that when the system's parameters are changed in order to reduce the magnitude of the impact velocity between the leaf spring and the connector pin, the extent of the regions over which a chaotic response is observed increases.     

Fourier modal methods for modeling optical dielectric waveguides

This work contains new materials relative to the use of Fourier expansion techniques, also called plane-wave expansion techniques, for modelling normal modes of optical waveguides. Two rigorous fully vectorial methods are presented and benchmarked for a classical rib waveguide geometry well studied in the literature. The first method relies on a pole extraction and on a one-dimensional expansion scheme. A four-digit accuracy for the normalized propagation constant B is obtained for the dominant TE and TM modes of the benchmark problem and for a small number of retained Fourier harmonics. Better accuracy is anticipated for larger truncation ranks. The second method relies on a two-dimensional expansion scheme in Fourier space and provides a three-digit accuracy for the normalized propagation constant.     

Stability change of intrinsic localized mode in finite nonlinear coupled oscillators

Intrinsic localized mode (ILM) is spatially localized and temporally periodic oscillation in nonlinear coupled oscillators. We numerically investigate the dynamical stability of ILMs in a microcantilever array, in which ILMs were experimentally observed by Sato et al. It is found that the stability change of ILMs is due to the ratio in nonlinear potentials. This phenomenon also occurs in the array without harmonic potentials. Consequently, the stability of ILMs substantially depends on the ratio in nonlinear on-site and inter-site potentials.     

A hybrid SEA/modal technique for modeling structural-acoustic interior noise in rotorcraft

This paper describes a hybrid technique that combines Statistical Energy Analysis (SEA) predictions for structural vibration with acoustic modal summation techniques to predict interior noise levels in rotorcraft. The method was applied for predicting the sound field inside a mock-up of the interior panel system of the Sikorsky S-92 helicopter. The vibration amplitudes of the frame and panel systems were predicted using a detailed SEA model and these were used as inputs to the model of the interior acoustic space. The spatial distribution of the vibration field on individual panels, and their coupling to the acoustic space were modeled using stochastic techniques. Leakage and nonresonant transmission components were accounted for using space-averaged values obtained from a SEA model of the complete structural-acoustic system. Since the cabin geometry was quite simple, the modeling of the interior acoustic space was performed using a standard modal summation technique. Sound pressure levels predicted by this approach at specific microphone locations were compared with measured data. Agreement within 3 dB in one-third octave bands above 40 Hz was observed. A large discrepancy in the one-third octave band in which the first acoustic mode is resonant (31.5 Hz) was observed. Reasons for such a discrepancy are discussed in the paper. The developed technique provides a method for modeling helicopter cabin interior noise in the frequency mid-range where neither FEA nor SEA is individually effective or accurate.     

Two-dimensional modeling of an aircraft engine structural bladed disk-casing modal interaction

In modern turbo machines such as aircraft jet engines, structural contacts between the casing and bladed disk may occur through a variety of mechanisms: coincidence of vibration modes, thermal deformation of the casing, rotor imbalance due to design uncertainties to name a few. These nonlinear interactions may result in severe damage to both structures and it is important to understand the physical circumstances under which they occur. In this study, we focus on a modal coincidence during which the vibrations of each structure take the form of a k-nodal diameter traveling wave characteristic of axi-symmetric geometries. A realistic two-dimensional model of the casing and bladed disk is introduced in order to predict the occurrence of this very specific interaction phenomenon versus the rotation speed of the engine. The equations of motion are solved using an explicit time integration scheme in conjunction with the Lagrange multiplier method where friction is accounted for. This model is validated from the comparison with an analytical solution. The numerical results show that the structures may experience different kinds of behaviors (namely damped, sustained and divergent motions) mainly depending on the rotational velocity of the bladed disk.     

利用表象变换精确求解最一般双耦合谐振子的能量本征值

将变换后的哈密顿量进一步变换到占有数表象,使哈密顿量对角化,精确求解出最一般双耦合谐振子,即哈密顿量中含非对角项(-λx1x2 vp1p2)的玻色谐振子的能量本征值．讨论了由耦合所引起的能级分裂。     

Numerical modeling of thermal loading of diamond crystal in X-ray FEL oscillators

Due to high reflectivity and high resolution of X-ray pulses, diamond is one of the most popular Bragg crystals serving as the reflecting mirror and mono-chromator in the next generation of free electron lasers(FELs).The energy deposition of X-rays will result in thermal heating, and thus lattice expansion of the diamond crystal,which may degrade the performance of X-ray FELs. In this paper, the thermal loading effect of diamond crystal for X-ray FEL oscillators has been systematically studied by combined simulation with Geant4 and ANSYS, and its dependence on the environmental temperature, crystal size, X-ray pulse repetition rate and pulse energy are presented. Our results show that taking the thermal loading effects into account, X-ray FEL oscillators are still robust and promising with an optimized design.     

基于高增益降维观测器的一类混沌同步

针对一类混沌系统, 当不满足观测器匹配条件时, 讨论了基于观测器的混沌同步及保密通讯问题.通过设计辅助驱动信号, 使得观测器匹配条件得以满足.为了处理辅助驱动信号中的未知变量, 使用高增益观测器作为近似微分器, 不仅给出了辅助驱动信号的估计值, 还给出了辅助驱动信号的微分估计值.基于辅助驱动信号的估计值, 提出了一种能与非线性项和干扰影响解耦的降维观测器作为响应系统, 达到了与驱动系统的同步.在同步的基础上, 再基于辅助驱动信号及其微分的估计值, 提出了一种保密信息还原的方法.最后, 针对Rössler系统进行仿真, 仿真结果表明所提出的方法是有效的. 关键词： 辅助驱动信号 混沌同步 降维观测器 保密通讯     

Properties of r-centroids for equal-frequency harmonic oscillators

Basic properties of r-centroids are discussed in the model of harmonic oscillators with equal frequencies and different oscillation centers (r_e values). In this approximation, r-centroids are found to be given by a very simple formula. They are characteristic of each Δυ=constant sequence and change regularly from one sequence to the next. The corresponding change in r-centroid is inversely proportional to the distance between the oscillation centers. The important special case of two abutting levels is studied and it is found that the r-centroid of the two levels is their common turning point. The conditions under which r-centroids can be considered as mean internuclear distances are specified. Applications of these simple results to molecular vibronic interactions are discussed. It is shown that the r-centroid of degenerate perturbing vibronic levels is located at the crossing of the corresponding electronic potential curves, so that, contrary to transition moments, the variation of electronic perturbation parameters with the internuclear separation R cannot be accurately deduced from only experimental data.     

Application of magnetomechanical sensors for modal testing

A new modal testing technique using magnetomechanical sensors is proposed in this paper. To list some advantages of this technique, sensors are cost-effective and require no direct physical contact with a structure. The specific application made in this paper is the modal testing for the bending vibration of a solid circular beam. The theoretical analysis explaining the principle of the magnetostrictive sensor-based modal testing is presented for beam bending. The present results are compared with those obtained by the use of standard accelerometers. Although the application of this technique is made to relatively simple problems, the potential of magnetomechanical sensors for modal testing has been revealed.     

Gelfand-Tsetlin basis for representations of Yangians

We construct an analog of the Gelfand-Tsetlin basis in the finite-dimensional irreducible representations of Yangians and find the matrix elements of the Drinfeld generators in this basis. As a special case of this construction, we obtain the well-known Gelfand-Tsetlin basis in the representations of the Lie algebra gl(N).     

Solvable model for chimera states of coupled oscillators

Networks of identical, symmetrically coupled oscillators can spontaneously split into synchronized and desynchronized subpopulations. Such chimera states were discovered in 2002, but are not well understood theoretically. Here we obtain the first exact results about the stability, dynamics, and bifurcations of chimera states by analyzing a minimal model consisting of two interacting populations of oscillators. Along with a completely synchronous state, the system displays stable chimeras, breathing chimeras, and saddle-node, Hopf, and homoclinic bifurcations of chimeras.     

Determination of limit cycles for strongly nonlinear oscillators

An innovative approach to finding limit cycles is proposed and illustrated on the van der Pol equation. The technique developed in this Letter is similar to the Ritz's method in variational theory. The present theory can be applied to not only weakly nonlinear equations, but also strongly nonlinear ones, and the obtained results are valid for the whole solution domain.     

Liapunov spectra for infinite chains of nonlinear oscillators

We argue that the spectrum of Liapunov exponents for long chains of nonlinear oscillators, at large energy per mode, may be well approximated by the Liapunov exponents of products of independent random matrices. If, in addition, statistical mechanics applies to the system, the elements of these random matrices have a distribution which may be calculated from the potential and the energy alone. Under a certain isotropy hypothesis (which is not always satisfied), we argue that the Liapunov exponents of these random matrix products can be obtained from the density of states of a typical random matrix. This construction uses an integral equation first derived by Newman. We then derive and discuss a method to compute the spectrum of a typical random matrix. Putting the pieces together, we see that the Liapunov spectrum can be computed from the potential between the oscillators.     

Obtaining certificates for complete synchronisation of coupled oscillators

In this paper, we provide a novel reformulation of sufficient conditions that guarantee global complete synchronisation of coupled identical oscillators to make them computationally implementable. To this end, we use semidefinite programming techniques. For the first time, we can efficiently search for and obtain certificates for synchronisability and, additionally, also optimise associated cost functions. In this paper, a Lyapunov-like function (certificate) is used to certify that all trajectories of a networked system consisting of coupled dynamical systems will eventually converge towards a common one, which implies synchronisation. Moreover, we establish new conditions for complete synchronisation, which are based on the so called Bendixson’s Criterion for higher dimensional systems. This leads to major improvements on the lower bound of the coupling constant that guarantees global complete synchronisation. Importantly, the certificates are obtained by analysing the connection network and the model representing an individual system only. In order to illustrate the strength of our method we apply it to a system of coupled identical Lorenz oscillators and to coupled van der Pol oscillators.     

Revealing direction of coupling between neuronal oscillators from time series: phase dynamics modeling versus partial directed coherence

The problem of determining directional coupling between neuronal oscillators from their time series is addressed. We compare performance of the two well-established approaches: partial directed coherence and phase dynamics modeling. They represent linear and nonlinear time series analysis techniques, respectively. In numerical experiments, we found each of them to be applicable and superior under appropriate conditions: The latter technique is superior if the observed behavior is "closer" to limit-cycle dynamics, the former is better in cases that are closer to linear stochastic processes.     

振动系统的模态分析实验设计

利用模态分析原理和简单试验仪器，得到了直观的悬臂梁和圆盘前4阶模态振型，使学生熟悉了模态分析原理和测试方法，有利于学生对振动规律的理解．     

Mathematical modeling of localized melting around graphite nodules during laser surface hardening of austempered ductile iron

An attempt has been made to mathematically predict the optimum conditions of laser surface hardening (LSH) of austempered ductile iron (ADI) that can ensure a predominantly martensitic microstructure and preclude partial/complete dissolution of graphite nodules in the laser hardened zone during laser irradiation. The exercise involves prediction of the thermal profile (using the Ashby and Easterling model), and consequently, the carbon diffusion profile around the graphite nodules at different depths from the surface for the given conditions of LSH. Microstructural investigations have been carried out by optical and scanning electron microscopy to study the morphology, shape and width of the partially/completely melted graphite nodules as a function of the LSH parameters. Finally, the predicted maximum width of the melted zone around the graphite nodules is compared with the relevant experimental data to validate the proposed model.     

An asymptotic expansion for energy eigenvalues of anharmonic oscillators

In the present contribution, we derive an asymptotic expansion for the energy eigenvalues of anharmonic oscillators for potentials of the form V(x)=κx^2q+ωx²,q=2,3,…$V\left(x\right)=\kappa x^{2q}+\omega x^2,q=2,3,\dots$ as the energy level n$n$ approaches infinity. The asymptotic expansion is obtained using the WKB theory and series reversion. Furthermore, we construct an algorithm for computing the coefficients of the asymptotic expansion for quartic anharmonic oscillators, leading to an efficient and accurate computation of the energy values for n≥6$n\ge 6$.