Set Of Eigenmodes Research Articles

A finite element framework for fluid–structure interaction of turbulent cavitating flows with flexible structures

We present a finite element framework for the numerical prediction of cavitating turbulent flows interacting with flexible structures. The vapor–fluid phases are captured through a homogeneous mixture model, with a scalar transport equation governing the spatio-temporal evolution of cavitation dynamics. High-density gradients in the two-phase cavitating flow motivate the use of a positivity-preserving Petrov–Galerkin stabilization method in the variational framework. A mass transfer source term introduces local compressibility effects arising as a consequence of phase change. The turbulent fluid flow is modeled through a dynamic subgrid-scale method for large eddy simulations. The flexible structure is represented by a set of eigenmodes, obtained through a modal decomposition of the linear elasticity equations. While a partitioned iterative approach is adopted to couple the structural dynamics and cavitating fluid flow, the deforming flow domain is described by an arbitrary Lagrangian–Eulerian frame of reference. We establish the fidelity of the proposed framework by comparing it against experimental and numerical studies for both rigid and flexible hydrofoils in cavitating flows. Under unstable partial cavitating conditions, we identify specific vortical structures leading to cloud cavity collapse. We further explore features of cavitating flow past a rigid body such as re-entrant jet and turbulence-cavity interactions during cloud cavity collapse. Based on the validation study conducted over a flexible NACA66 rectangular hydrofoil, we elucidate the role of cavity and vortex shedding in governing the structural dynamics. Subsequently, we identify a broad spectrum frequency band whose central peak does not correlate to the frequency content of the cavitation dynamics or the natural frequencies of the structure, indicating the induction of unsteady flow patterns around the hydrofoil. Finally, we discuss the coupled fluid–structure dynamics during a cavitation cycle and the underlying mechanism associated with the promotion and mitigation of cavitation.

Singularity Reconfiguration of Non-Diffraction Structured Light

It's a basic characteristic that there exists a phase singularity located at the center for pure vortex light. If combining several vortex lights belonging to the same eigenmode set, the central higher-order singularity may split into several singularities distributed around the space. In this paper, we propose a mechanism to reconfigure such splitting singularities. The splitting singularities can be manipulated to be close to or away from each other. Especially, they form a higher-order singularity or annihilate when two singularities coincide. Because all the combined lights are non-diffraction, the synthetic field is also non-diffraction. We realize the reconfiguration of splitting singularities in the experiment and verify the non-diffraction property at a 1m distance.

Structured Beamforming Based on Orbital Angular Momentum Mode-Group

The structured beams, which can manipulate arbitrary intensity and phase of electromagnetic rays, are significant in communication and sensing. The vortex electromagnetic wave carrying orbital angular momentum (OAM) exhibits the phase distribution of the helical wavefront, which can be viewed as a complete set of eigenmodes for electromagnetic waves. It provides a flexible beamforming method to achieve the azimuthal pattern diversity by the spatial superposition of multiple specific OAM modes. We have proposed a novel form of OAM wave with a two-dimensional structure called plane spiral OAM (PS-OAM), which is conducive to overlapping different modes. Based on this, the PS-OAM mode-group can be employed to modify the spatial beam structure and form structured electromagnetic beams. This kind of beam not only retains the inherent vorticity and orthogonality of the OAM waves but also tackles the difficulties in applications stemming from the singularity-caused dark zone and beam divergence of conventional OAM waves. In this paper, the construction and manipulation of the structured beams are investigated, and the properties are analyzed. The potential applications in multiple-input multiple-output (MIMO) communication, orthogonal multiplexing communication, and spatial field digital modulation (SFDM) communication are envisioned.

Micromagnetic frequency-domain simulation methods for magnonic systems

We present efficient numerical methods for the simulation of small magnetization oscillations in three-dimensional micromagnetic systems. Magnetization dynamics is described by the Landau–Lifshitz–Gilbert equation, linearized in the frequency domain around a generic equilibrium configuration, and formulated in a special operator form that allows leveraging large-scale techniques commonly used to evaluate the effective field in time-domain micromagnetic simulations. By using this formulation, we derive numerical algorithms to compute the free magnetization oscillations (i.e., spin wave eigenmodes) as well as magnetization oscillations driven by ac radio-frequency fields for arbitrarily shaped nanomagnets. Moreover, semi-analytical perturbation techniques based on the computation of a reduced set of eigenmodes are provided for fast evaluation of magnetization frequency response and absorption spectra as a function of damping and ac field. We present both finite-difference and finite-element implementations and demonstrate their effectiveness on a test case. These techniques open the possibility to study generic magnonic systems discretized with several hundred thousands (or even millions) of computational cells in a reasonably short time.

Recoil experiments determine the eigenmodes of viscoelastic fluids

We experimentally investigate the transient recoil dynamics of a colloidal probe particle in a viscoelastic fluid after the driving force acting on the probe is suddenly removed. The corresponding recoil displays two distinct timescales which are in excellent agreement with a microscopic model which considers the probe particle to be coupled to two bath particles via harmonic springs. Notably, this model exhibits two sets of eigenmodes which correspond to reciprocal and non-reciprocal force conditions and which can be experimentally confirmed in our experiments. We expect our findings to be relevant under conditions where particles are exposed to non-steady shear forces as this is encountered e.g. in microfluidic sorting devices or the intermittent motion of motile bacteria within their natural viscoelastic surrounding.

Branching High‐Order Exceptional Points in Non‐Hermitian Optical Systems

AbstractExceptional points are complex‐valued spectral singularities that lead to a host of intriguing features such as loss‐induced transparency—a counterintuitive process in which an increase in the system's overall loss can lead to enhanced transmission. In general, the associated enhancements scale with the order of the exceptional points. Consequently, it is of great interest to devise new strategies to implement realistic devices capable of exhibiting high‐order exceptional points. Here, it is shown that high‐order ‐photon exceptional points can be generated by exciting non‐Hermitian waveguide arrangements with coherent light states. Using photon‐number resolving detectors it then becomes possible to observe ‐photon enhanced loss‐induced transparency in the quantum realm. Further, it is analytically shown that the number‐resolved dynamics occurring in the same nonconservative waveguide arrays will exhibit eigenspectral ramifications having several exceptional points associated to different sets of eigenmodes and dissipation rates.

Eigenmode Decomposition of the Diffusion Equation: Applications to Pressure-Rate Deconvolution and Reservoir Pore Volume Estimation

SummaryThis paper proposes the application of the eigenmode decomposition to the problem of pressure-rate deconvolution. The method enabled us to write the solution of the pressure-rate deconvolution as an infinite series of eigenmodes (eigenfunctions). Each eigenmode is associated with a hydraulic diffusion time scale (eigenvalue) that depends on the intrinsic reservoir properties. The set of eigenmodes forms a basis of the solution space and the analytical solution shows some desirable properties: It is smooth, physically constrained for a closed system, and some parameters have a physical interpretation. As a result, in both single and multiwell problems, the late-time unit slope appears naturally and the curvature penalty does not have to be imposed, thus addressing current knowledge gaps. As an additional contribution, we also propose a mathematically well-defined expression in terms of eigenfunctions to the concept of investigated volume during a well test and a procedure for untangling the single-well deconvolved response from interference effects of neighboring wells under certain conditions. Applications to the analysis of drillstem tests (DST) and reservoir characterization are presented.

Fiber Vector Eigenmode Multiplexing Based High Capacity Transmission Over 5-km FMF With Kramers-Kronig Receiver

Vector modes (VMs) with spatially nonhomogeneous polarization patterns, combining polarization and spatial-mode degrees of freedom together, are regarded as the eigenmodes of optical fiber. Such eigenmodes have potential to be used to implement the high capacity spatial division multiplexing (SDM)/mode division multiplexing (MDM) based optical fiber communication systems. Here, we respectively demonstrate 2 VMs of the same order (l = +2, EH11o and EH11e modes) and 2 VMs of different orders (l = 0, HE11o mode and l = +2, EH11e mode) multiplexing transmissions over 5-km few-mode fiber (FMF). The mode crosstalk after FMF transmission has been minimized to prepare high capacity signal transmission without multiple-input-multiple-output (MIMO) digital signal processing (DSP). To increase the transmission capacity and decrease the system cost, the Kramers-Kronig (KK) receiver has been employed and the single-wavelength two sets of eigenmodes multiplexed 360 Gbit/s and 400 Gbit/s signal transmissions over 5-km FMF link are realized, respectively. Then, to further burst the transmission capacity, 2 vector-mode-division-multiplexing channels (HE11o and EH11e modes) in combination of 5 wavelength-division-multiplexing (WDM) channels with a total of 10 parallel channels have been demonstrated. A total data rate of 1.12 Tbit/s with 28 GBaud 16-state quadrature amplitude modulation (QAM) signal has been realized over the 5-km FMF. In all data-carrying experimental demonstrations, the bit-error rate (BER) performance of each VM channel is below the 7% hard decision forward error correction threshold (FEC) at BER of 3.8 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> . The experimental results highlight that the VMDM-based optical transmission technology could find enormous potential in huge-capacity short-reach optical interconnects.

Lagrangian drift in a permeable bottom layer induced by internal gravity waves

The mean drift in a permeable homogeneous bottom layer caused by internal gravity waves in an overlying stratified fluid, is investigated theoretically. We apply a Lagrangian formulation, together with the long-wave approximation, for the motion in the fluid and the fluid-saturated permeable layer. In the stratified fluid, we assume a constant Brunt-Väisälä frequency and inviscid flow. In the permeable bed, the density is constant. Here we apply a simple macroscopic version of Darcy’s law. For internal waves with frequency a fundamental small parameter in the permeable bed is Here is the effective permeability, and is the overall eddy viscosity, representing the small-scale turbulence of the interstitial fluid. The Lagrangian mean flow in each layer is weakly damped, with a damping rate proportional to and composed of contributions from an infinite, but discrete set of eigen-modes. For each mode the mean drift in the permeable bed is an order smaller than the Stokes drift in internal waves at the bottom of the fluid layer. For spatially damped waves, it is particularly interesting that the wave-induced Eulerian mean current in the permeable bed may exceed the Stokes drift if the bed thickness is smaller than the upper layer thickness. It is suggested that the explored wave-induced particle drift in the permeable bottom layer could provide a physical model for the slow net transport of bio aerosols and smoke particles in the tropical rainforest.

Flight dynamics modeling of a flexible wing unmanned aerial vehicle

This paper shows the results of traditional aerodynamic analysis and the flexible flight dynamics modeling including the effects of structural motion for a flexible wing unmanned aerial vehicle. The influence of some design parameters such as wing flexibility, horizontal/vertical tail aerodynamics is investigated for aeroelasticity and flight dynamics of flexible aircraft. The research platform is an Unmanned Aerial Vehicle (UAV), made of composite material. Its wing span is 4 m and reaches a high aspect ratio, whose value is 18.9. For a traditional analysis, the Vortex Lattice Method (VLM) was used to obtain conventional aerodynamic and control derivatives. The flexible flight dynamics model is based on the work of Waszack and Schmidt. In this approach, it is used the mean-axes reference system and it is assumed that structural deformations is small and described by a set of eigenmodes. The dynamics model incorporates the first normal modes obtained by Ground Vibration Test (GVT) campaigns. A focus of the paper lies on providing a useful model for dynamics system identification and in-flight aeroelastic testing. A developed platform for in-flight system identification and the acquisition system is described. A flight path reconstruction process from a flight test campaign results is shown.

The Atiyah–Patodi–Singer index on a lattice

Abstract We propose a nonperturbative formulation of the Atiyah–Patodi–Singer (APS) index in lattice gauge theory in four dimensions, in which the index is given by the $\eta$ invariant of the domain-wall Dirac operator. Our definition of the index is always an integer with a finite lattice spacing. To verify this proposal, using the eigenmode set of the free domain-wall fermion we perturbatively show in the continuum limit that the curvature term in the APS theorem appears as the contribution from the massive bulk extended modes, while the boundary $\eta$ invariant comes entirely from the massless edge-localized modes.

Modal Analysis of Internal Wave Propagation and Scattering over Large-Amplitude Topography

AbstractCoupled-mode equations describing the propagation and scattering of internal waves over large-amplitude arbitrary topography in a two-dimensional stratified fluid are derived. They consist of a simple set of ordinary differential equations describing the evolution of modal amplitudes, based on an orthogonality condition that allows one to distinguish leftward- and rightward-propagating modes. The coupling terms expressing exchange of energy between modes are given in an analytical form using perturbation theory. This allows the derivation of a weak-topography approximate solution, generalizing previous linear solutions for a barotropic forcing that were described in 2002 by Llewellyn Smith and Young . In addition, the orthogonality condition derived is valid for a different set of eigenmodes defined on a sloping bottom, which shows a better convergence rate when compared with the standard set of modes. The work presented here provides a useful and simple framework for the investigation of internal wave propagation in an inhomogeneous ocean, along with theoretical insight.

How proteins' negative cooperativity emerges from entropic optimisation of versatile collective fluctuations.

The fact that allostery, a nonlocal signaling between distant binding sites, can arise mainly from the entropy balance of collective thermal modes, without conformational changes, is by now well known. However, the propensity to generate negative cooperativity is still unclear. Starting from an elastic-network picture of small protein complexes, in which effector binding is modeled by locally altering interaction strengths in lieu of adding a node-spring pair, we elucidate mechanisms particularly for such negative cooperativity. The approach via a few coupled harmonic oscillators with internal elastic strengths allows us to trace individual eigenmodes, their frequencies, and their statistical weights through successive bindings. We find that the alteration of the oscillators' couplings is paramount to covering both signs of allostery. Binding-modified couplings create a rich set of eigenmodes individually for each binding state, modes inaccessible to an ensemble of noninteracting units. The associated shifts of collective-mode frequencies, nonuniform with respect to modes and binding states, result in an enhanced optimizability, reflected by a subtle phase map of allosteric behaviors.

Power steered modes conversions and rotations in nonlocal nonlinear media

Propagations of optical beams in nonlocal nonlinear media can be described by the nonlocal nonlinear Schrödinger equation (NNLSE). Hermite–Gaussian (HG) modes and Laguerre–Gaussian (LG) modes are two different sets of eigenmodes to the NNLSE. The inter-conversions can be achieved between the HG and the LG modes if they both carry a kind of cross phase. During the modes conversion processes, mode rotations appear. Furthermore, both the inter-conversions and the rotations between the two modes can be controlled exactly by the input optical power. These power induced mode conversions and rotations are discussed in the nematic liquid crystal as an exemplary nonlocal nonlinear media. The theoretical results provide potential applications in the beam shaping in optics.

Azimuthal capillary waves on a hollow filament – the discrete and the continuous spectrum

We study the temporal spectrum of linearised, azimuthal, interfacial perturbations imposed on a cylindrical gaseous filament surrounded by immiscible, viscous, quiescent fluid in radially unbounded geometry. Linear stability analysis shows that the base state is stable to azimuthal perturbations of standing wave form. Normal mode analysis leads to a viscous dispersion relation and shows that in addition to the discrete spectrum, the problem also admits a continuous spectrum. For a given azimuthal Fourier mode and Laplace number, the discrete spectrum yields two eigenfunctions which decay exponentially to zero at large radii and thus cannot represent far field perturbations. In addition to these discrete modes, we find an uncountably infinite set of eigenmodes which decay algebraically to zero. The completeness theorem for perturbation vorticity may be expressed as a sum over the discrete modes and an integral over the continuous ones. We validate our normal mode results by solving the linearised, initial value problem (IVP). The initial perturbation is taken to be an interfacial, azimuthal Fourier mode with zero perturbation vorticity. It is shown that the expression for the time dependent amplitude of a capillary standing wave (in the Laplace domain, ) has poles and branch points on the complex plane. We show that the residue at the poles yields the discrete spectrum, while the contribution from either side of the branch cut provides the continuous spectrum contribution. The particular initial condition treated here in the IVP, has projections on the discrete as well as the continuous spectrum eigenmodes and thus both sets are excited initially. Consequently the time evolution of the standing wave amplitude and the perturbation vorticity field have the form of a sum over discrete exponential contributions and an integral over a continuous range of exponential terms. The solution to the IVP leads to explicit analytical expressions for the standing wave amplitude and the vorticity field in the fluid outside the filament. Linearised analytical results are validated using direct numerical simulations (DNS) conducted using a code developed in-house for solving the incompressible, Navier–Stokes equations with an interface. For small perturbation amplitude, analytical predictions show excellent agreement with DNS. Our analysis complements and extends earlier results on the discrete and the continuous spectrum for interfacial viscous, capillary waves on unbounded domain.

Rigorous space-time coupled-wave analysis for patterned surfaces with temporal permittivity modulation [Invited

Material patterns with temporal modulation of permittivity provide an unprecedented platform for active light manipulation and multifunctional devices. The ongoing discovery of new classes of tunable optical materials combined with various multiphysics paradigms demands an appropriate computational tool to characterize and design the wave manipulation patterns with the additional degree of freedom (time). Here we present, for the first time, a fast and accurate numerical technique to compute light propagation through patterned layers with spatial and temporal modulation of permittivity by extending the well-established rigorous coupled-wave analysis framework. We expand the electric and magnetic field components of an arbitrarily shaped space-time unit cell into a basis of spatio-temporal harmonics and compute their modal field profiles as a set of eigenmodes of the unit cell. The relation between the wave amplitudes of the incident, reflected, and transmitted spatio-temporal harmonics are then computed by enforcing the boundary conditions at each interface of the pattern. The validity of the proposed method is rigorously verified against the results of a time-variant finite-difference time-domain technique. The applicability of the method to study dielectric and plasmonic time-modulated metasurfaces is exemplified by considering two metasurfaces with spatially-variant phase and frequency of temporal modulation, based on electro-optical modulation of materials such as silicon and graphene. Utilizing the developed technique, we demonstrate some recently reported phenomena associated with time-modulated metasurfaces, such as anomalous steering of higher-order frequency harmonics, and dynamic optical beam generation. Even though we restrict ourselves to one-dimensional patterns for simplicity, the developed method can be flexibly employed for the design, characterization and optimization of two-dimensional spatio-temporal modulation patterns of arbitrary composition and shape.

How do spatially distinct frequency specific MEG networks emerge from one underlying structural connectome? The role of the structural eigenmodes

Functional networks obtained from magnetoencephalography (MEG) from different frequency bands show distinct spatial patterns. It remains to be elucidated how distinct spatial patterns in MEG networks emerge given a single underlying structural network. Recent work has suggested that the eigenmodes of the structural network might serve as a basis set for functional network patterns in the case of functional MRI. Here, we take this notion further in the context of frequency band specific MEG networks. We show that a selected set of eigenmodes of the structural network can predict different frequency band specific networks in the resting state, ranging from delta (1–4 Hz) to the high gamma band (40–70 Hz). These predictions outperform predictions based from surrogate data, suggesting a genuine relationship between eigenmodes of the structural network and frequency specific MEG networks. We then show that the relevant set of eigenmodes can be excited in a network of neural mass models using linear stability analysis only by including delays. Excitation of an eigenmode in this context refers to a dynamic instability of a network steady state to a spatial pattern with a corresponding coherent temporal oscillation. Simulations verify the results from linear stability analysis and suggest that theta, alpha and beta band networks emerge very near to the bifurcation. The delta and gamma bands in the resting state emerges further away from the bifurcation. These results show for the first time how delayed interactions can excite the relevant set of eigenmodes that give rise to frequency specific functional connectivity patterns.

Localised streak solutions for a Blasius boundary layer

Streaks are a common feature of disturbed boundary-layer flows. They play a central role in transient growth mechanisms and are a building block of self-sustained structures. Most theoretical work has focused on streaks that are periodic in the spanwise direction, but in this work we consider a single spatially localised streak embedded into a Blasius boundary layer. For small streak amplitudes, we show the perturbation can be described in terms of a set of eigenmodes that correspond to an isolated streak/roll structure. These modes are new, and arise from a bi-global eigenvalue calculation; they decay algebraically downstream and may be viewed as the natural three-dimensional extension of the classical two-dimensional Libby &amp; Fox (J. Fluid Mech., vol. 17 (3), 1963, pp. 433–449) solutions. Despite their bi-global nature, we show that a subset of these eigenmodes (including the slowest decaying) is fundamentally related to the solutions first presented by Luchini (J. Fluid Mech., vol. 327, 1996, pp. 101–116), as derived for spanwise-periodic disturbances (at small spanwise wavenumber). This surprising connection is made by an analysis of the far-field decay of the bi-global state. We also address the fully non-parallel downstream development of nonlinear streaks, confirming that the aforementioned eigenmodes are recovered as the streak/roll decays downstream. Encouraging comparisons are made with available experimental data.

Realization of Beam Steering Based on Plane Spiral Orbital Angular Momentum Wave

A new form of plane spiral orbital angular momentum (PS-OAM) carrying wave with the spiral harmonics of exp(− $jl\varphi$ ) as a complete set of eigenmodes is proposed to realize beam steering. Such a wave possesses the fundamental characteristics of an OAM-carrying wave, i.e., a spiral phase distribution, yet propagates along transverse directions. Hence, it owes unique advantages for use in realizing desired radiation beams, while avoids the puzzles encountered by traditional OAM-carrying beams. Various OAM mode grouping using the superposition of PS-OAM waves are demonstrated here via theoretical calculations as well as experimental verifications. The generation methods for PS-OAM waves are also proposed and experimentally validated to authenticate the feasibility. Both the calculated and experimental results show that the PS-OAM grouping is capable of providing a practical realization of eigenmode beamforming.

How to: Using Mode Analysis to Quantify, Analyze, and Interpret the Mechanisms of High-Density Collective Motion

While methods from statistical mechanics were some of the earliest analytical tools used to understand collective motion, the field has expanded in scope beyond phase transitions and fluctuating order parameters. In part, this expansion is driven by the increasing variety of systems being studied, which in turn, has increased the need for innovative approaches to quantify, analyze, and interpret a growing zoology of collective behaviors. For example, concepts from material science become particularly relevant when considering the collective motion that emerges at high densities. Here, we describe methods originally developed to study inert jammed granular materials that have been borrowed and adapted to study dense aggregates of active particles. This analysis is particularly useful because it projects difficult-to-analyze patterns of collective motion onto an easier-to-interpret set of eigenmodes. Carefully viewed in the context of non-equilibrium systems, mode analysis identifies hidden long-range motions and localized particle rearrangements based solely on the knowledge of particle trajectories. In this work, we take a ``how-to'' approach and outline essential steps, methods, diagnostics, and know-how used to apply this analysis to study densely-packed active systems.