Finite Number Of Modes Research Articles

Hydrodynamic force interaction of two fixed spheres in a wall-bounded linear shear flow

We examine the modulations of the drag coefficient Cd and lift coefficient Cl on a spherical particle caused by the presence of a neighboring sphere when positioned in close proximity to a wall. Data are generated through the utilization of a fast and accurate Particle-Resolved Direct Numerical Simulation (PR-DNS) method that relies on the Direction-Splitting algorithm and that was extensively validated in our prior study (Goyal and Wachs, 2023a, 2023b; Morente et al., 2023). We further validate the method in the flow configuration of the present study by comparing them with data published in the literature. We consider three Reynolds numbers Re=20, Re=50 and Re=100, which describe the flow conditions in the vicinity of the wall. The primary sphere is positioned at a distance δ from the wall, while the neighboring sphere is positioned at a constant distance from the primary sphere encompassing all possible angular positions θ. We report the modulations of Cd and Cl compared to a situation without any neighboring sphere as a function of δ, θ and Re. The inclusion of a wall intensifies the influence of the neighboring sphere on the primary sphere and significantly modifies the values of Cd and Cl. We attempt to relate the values of the modulations of Cd and Cl to the modulations in the contours of the pressure and of the spanwise vorticity on the surface of the primary sphere. We take advantage of the periodic nature of Cd and Cl as a function of the relative angular position θ of the neighboring sphere and propose a Fourier Predictive Model (FPM) to effectively represent the data using a finite number of modes in the cosine Fourier series. We show that our FPM is able to estimate the modulations of Cd and Cl excellently with a coefficient of determination R2≳0.99 in the parameter space investigated in our study.

Mitigating modal noise in multimode circular fibres by optical agitation using a galvanometer

Abstract Modal noise appears due to the non-uniform and unstable distribution of light intensity among the finite number of modes in multimode fibres. It is an important limiting factor in measuring radial velocity precisely by fibre-fed high-resolution spectrographs. The problem can become particularly severe as the fibre’s core become smaller and the number of modes that can propagate reduces. Thus, mitigating modal noise in relatively small core fibres still remains a challenge. We present here a novel technique to suppress modal noise. Two movable mirrors in the form of a galvanometer re-image the mode-pattern of an input fibre to an output fibre. The mixing of modes coupled to the output fibre can be controlled by the movement of mirrors applying two sinusoidal signals through a voltage generator. We test the technique for four multimode circular fibres: 10 and 50 $\mu$m step-index, 50 $\mu$m graded-index, and a combination of 50 $\mu$m graded-index and 5:1 tapered fibres (GI50t). We present the results of mode suppression both in terms of the direct image of the output fibre and spectrum of white light obtained with the high-resolution spectrograph. We found that the galvanometer mitigated modal noise in all the tested fibres, but was most useful for smaller core fibres. However, there is a trade-off between the modal noise reduction and light-loss. The GI50t provides the best result with about 60 per cent mitigation of modal noise at a cost of about 5 per cent output light-loss. Our solution is easy to use and can be implemented in fibre-fed spectrographs.

Modal approximation for time-domain elastic scattering from metamaterial quasiparticles

This paper aims at quantitatively understanding the elastic wave scattering due to negative metamaterial structures under wide-band signals in the time domain. Specifically, we establish the modal expansion for the time-dependent field scattered by metamaterial quasiparticles in elastodynamics. By Fourier transform, we first analyze the modal expansion in the time-harmonic regime. With the presence of quasiparticles, we validate such an expansion in the static regime via quantitatively analyzing the spectral properties of the Neumann-Poincaré operator associated with the elastostatic system. We then approximate the incident field with a finite number of modes and apply perturbation theory to obtain such an expansion in the perturbative regime. In addition, we give polariton resonances as simple poles for the elastostatic system. Finally, we show that the low-frequency part of the scattered field in the time domain can be well approximated by using the resonant modal expansion with sharp error estimates.

On the Discrete Normal Modes of Quasigeostrophic Theory

AbstractThe discrete baroclinic modes of quasigeostrophic theory are incomplete, and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal-mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple modes with no internal zeros, a finite number of modes with negative norms, and the fact that their vertical structures form a basis capable of representing any quasigeostrophic state with a differentiable series expansion. These properties are a consequence of the Pontryagin-space setting of the Rossby wave problem in the presence of boundary buoyancy gradients (as opposed to the usual Hilbert-space setting). We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. A natural application of these modes is the development of a weakly nonlinear wave-interaction theory of geostrophic turbulence that takes topography into account.

Reliability analysis of a class of stochastically excited nonlinear Markovian jump systems

It is inevitable for the nonlinear systems to suffer from external stochastic disturbance. Meanwhile, the components failure will bring abrupt changes in its substructures, which can be considered as the internal stochastic disturbance. It is demonstrated that the components failure performs random jumpy factors switching between a finite number of modes. This salient feature allows us to identify this type of dynamic behaviors as response of nonlinear hybrid systems undergoing Markovian jumps. In this paper a novel method is presented to analyze the dynamical reliability of multi-degree-of-freedom (DOF) nonlinear stochastic systems undergoing Markovian jumps. Firstly, the Markovian jump process is introduced to formulate the aforementioned systems as continuous-discrete hybrid systems. Secondly, a two-step approximate method is applied to convert the hybrid systems to one governed by a set of averaged Itô stochastic differential equations (SDEs) of mechanical energies. Then the associated averaged backward Kolmogorov equation and the Pontryagin equation are constructed and solved to yield the conditional reliability function and the mean first-passage time of the original system respectively. Finally, an example of two coupled Duffing oscillators is illustrated to compare the analytical results and those from Monte Carlo simulation to verify the effectiveness of the proposed method.

Optimal modes for spatially multiplexed free-space communication in atmospheric turbulence

In near-field free-space optical (FSO) communication, spatial-mode multiplexing (SMM) increases transmission capacity by transmitting independent information streams in orthogonal modes. Propagation through atmospheric turbulence causes phase and amplitude distortions that can degrade SMM performance. In this paper, we show there exist optimal modes for transmission through turbulence with minimum degradation, under a realistic assumption that a transmitter knows the turbulence statistics but not the instantaneous state of the atmosphere. These modes are determined by performing a Karhunen-Loève expansion of the optical electric field in the receiver aperture. We show that these modes are Laguerre-Gauss (LG) modes whose beam waist is chosen depending on the field coherence length in the receiver plane. These adaptive-waist LG modes, when ordered by decreasing eigenvalue, can approximate a received signal field by a finite number of modes with lowest mean-square error among all orthonormal mode sets. Hence, they represent optimal transmit and receive bases for SMM FSO. Using numerical simulation, we study SMM FSO transmission at various turbulence strengths and signal-to-noise ratios. We compare the performance using the adaptive-waist LG modes to that using fixed-waist LG modes (which assume no knowledge of turbulence statistics) and instantaneous eigenmodes (which assume knowledge of the instantaneous state of the turbulence). We also study the performance using the orbital angular momentum subsets of the adaptive-waist LG mode and fixed-waist LG mode sets.

Modal approximation for strictly convex plasmonic resonators in the time domain: The Maxwell's equations

We study the possible expansion of the electromagnetic field scattered by a strictly convex metallic nanoparticle with dispersive material parameters placed in a homogeneous medium in a low-frequency regime as a sum of modes oscillating at complex frequencies (diverging at infinity), known in the physics literature as the quasi-normal modes expansion. We show that such an expansion is valid in the static regime and that we can approximate the electric field with a finite number of modes. We then use perturbative spectral theory to show the existence, in a certain regime, of plasmonic resonances as poles of the resolvent for Maxwell's equations with non-zero frequency. We show that, in the time domain, the electric field can be written as a sum of modes oscillating at complex frequencies. We introduce renormalised quantities that do not diverge exponentially at infinity.

Buckling between soft walls: sequential stabilization through contact

Motivated by applications of soft-contact problems such as guidewires used in medical and engineering applications, we consider a compressed rod deforming between two parallel elastic walls. Free elastica buckling modes other than the first are known to be unstable. We find the soft constraining walls to have the effect of sequentially stabilizing higher modes in multiple contacts by a series of bifurcations, in each of which the degree of instability (the index) is decreased by one. Further symmetry-breaking bifurcations in the stabilization process generate solutions with different contact patterns that allow for a classification in terms of binary symbol sequences. In the hard-contact limit, all these bifurcations collapse into highly degenerate ‘contact bifurcations’. For any given wall separation at most a finite number of modes can be stabilized and eventually, under large enough compression, the rod jumps into the inverted straight state. We chart the sequence of events, under increasing compression, leading from the initial straight state in compression to the final straight state in tension, in effect the process of pushing a rod through a cavity. Our results also give new insight into universal features of symmetry-breaking in higher mode elastic deformations. We present this study also as a showcase for a practical approach to stability analysis based on numerical bifurcation theory and without the intimidating mathematical technicalities often accompanying stability analysis in the literature. The method delivers the stability index and can be straightforwardly applied to other elastic stability problems.

Energy-Constrained Discrimination of Unitaries, Quantum Speed Limits, and a Gaussian Solovay-Kitaev Theorem.

We investigate the energy-constrained (EC) diamond norm distance between unitary channels acting on possibly infinite-dimensional quantum systems, and establish a number of results. First, we prove that optimal EC discrimination between two unitary channels does not require the use of any entanglement. Extending a result by Acín, we also show that a finite number of parallel queries suffices to achieve zero error discrimination even in this EC setting. Second, we employ EC diamond norms to study a novel type of quantum speed limits, which apply to pairs of quantum dynamical semigroups. We expect these results to be relevant for benchmarking internal dynamics of quantum devices. Third, we establish a version of the Solovay-Kitaev theorem that applies to the group of Gaussian unitaries over a finite number of modes, with the approximation error being measured with respect to the EC diamond norm relative to the photon number Hamiltonian.

Designing locally maximally entangled quantum states with arbitrary local symmetries

One of the key ingredients of many LOCC protocols in quantum information is a multiparticle (locally) maximally entangled quantum state, aka a critical state, that possesses local symmetries. We show how to design critical states with arbitrarily large local unitary symmetry. We explain that such states can be realised in a quantum system of distinguishable traps with bosons or fermions occupying a finite number of modes. Then, local symmetries of the designed quantum state are equal to the unitary group of local mode operations acting diagonally on all traps. Therefore, such a group of symmetries is naturally protected against errors that occur in a physical realisation of mode operators. We also link our results with the existence of so-called strictly semistable states with particular asymptotic diagonal symmetries. Our main technical result states that the Nth tensor power of any irreducible representation of SU(N) contains a copy of the trivial representation. This is established via a direct combinatorial analysis of Littlewood-Richardson rules utilising certain combinatorial objects which we call telescopes.

Discretized Conical Waves in Multimode Optical Fibers.

Multimode optical fibers are essential in bridging the gap between nonlinear optics in bulk media and single-mode fibers. The understanding of the transition between the two fields remains complex due to intermodal nonlinear processes and spatiotemporal couplings, e.g., some striking phenomena observed in bulk media with ultrashort pulses have not yet been unveiled in such waveguides. Here we generalize the concept of conical waves described in bulk media towards structured media, such as multimode optical fibers, in which only a discrete and finite number of modes can propagate. Such propagation-invariant optical wave packets can be linearly generated, in the limit of superposed monochromatic fields, by shaping their spatiotemporal spectrum, whatever the dispersion regime and waveguide geometry. Moreover, they can also spontaneously emerge when a rather intense short pulse propagates nonlinearly in a multimode waveguide, their finite energy is also associated with temporal dispersion. The modal distribution of optical fibers then provides a discretization of conical emission (e.g., discretized X waves). Future experiments in multimode fibers could reveal different forms of dispersion-engineered conical emission and supercontinuum light bullets.

Stochastic response analysis of multi-degree-of-freedom vibro-impact system undergoing Markovian jump

The paper treats stationary response of a stochastically excited multi-degree-of-freedom (multi-DOF) vibro-impact system undergoing Markovian jump. The vibro-impact system with sudden abrupt changes in substructures or external excitations is modeled as a continuous-discrete Markovian jump system, which is essentially different from the traditional vibro-impact model. It is demonstrated that the random jump factors switch between a finite number of modes. This salient feature allows us to identify this type of dynamic behaviors as response of hybrid vibro-impact systems undergoing Markovian jump. Utilizing a two-step approximate technique, we can reduce the considered multi-DOF hybrid system to one-dimensional averaged Ito equation of the form of system’s total energy. The approximate analytical solution of the associated Fokker–Planck–Kolmogorov (FPK) equation of system’s energy is derived to predict the stationary response of original hybrid systems.

Longitudinal vibration of double nanorod systemsusing doublet mechanics theory

This paper investigates the free and forced longitudinal vibration of a double nanorod system using doublet mechanics theory. The doublet mechanics theory is a multiscale theory spanning between lattice dynamics and continuum mechanics. Equations of motion and boundary conditions for the double nanorod system are obtained using Hamilton\'s principle. Clamped-clamped and clamped-free boundary conditions are considered. Frequencies and dynamic displacements are determined to demonstrate the effects of length scale parameter of considered material and geometry of the nanorods. It is shown that frequencies obtained by the doublet mechanics theory are bounded from above (van Hove singularity) and unlike classical elasticity theory doublet mechanics theory predicts finite number of modes depending on the length of the nanotube. The present doublet mechanics results have been compared to molecular dynamics, experimental and nonlocal theory results and good agreement is observed between the present and other mentioned results. The difference between wave frequencies of graphite is less than 10% between doublet mechanics and experimental results near to the end of the first Brillouin zone.

Semiclassical asymptotics of the solution of the Helmholtz equation in a three-dimensional layer of variable thickness with a localized right-hand side

We construct a semiclassical asymptotics of the solution of the Helmholtz equation in a three-dimensional layer of variable thickness with a localized right-hand side, assuming the absence of “trap” states and the fulfillment of radiation conditions at infinity (such as the Sommerfeld condition). Wave part of the asymptotic solution can be represented as an decomposition into a finite number of modes, each mode is connected with a pair of Lagrangian manifolds.

Asymptotic Solution of the Helmholtz Equation in a Three-Dimensional Layer of Variable Thickness with a Localized Right-Hand Side

The asymptotics of the solution to the Helmholtz equation in a three-dimensional layer of variable thickness with a localized right-hand side in the absence of “trap” states and under the asymptotic radiation conditions at infinity is constructed. The wave part of the solution has a finite number of modes. The resulting formula makes sufficiently clear the influence of the shape of the source on the wave part of the solution.

Assessing Smoothing Effects of Wind Power around Trondheim via Koopman Mode Decomposition

To cope with a large-scale introduction of renewables into power systems, it is important to understand the reduction of variability in the aggregated generation or the so-called smoothing effect. Knowledge of the degree of smoothing is used for assessing the potential impact of intermittent generation on the power system operation. Here, smoothing effects of aggregated wind power are assessed around Trondheim, Norway, by applying a recently proposed smoothing index based on the so-called Koopman Mode Decomposition (KMD). The method is shown to effectively decompose complex time-series of wind power outputs into a finite number of modes, each of which oscillates with a single frequency for all locations (or hypothetical wind farms). It is shown that the method is able to reconstruct the original power outputs well by only a small number of modes that retain the variability of the original time-series, and is able to provide a relevant quantification of the smoothing effects for each individual frequency (or mode).

On the analytical solutions for water waves generated by a prescribed landslide

This paper presents a suite of analytical solutions, for both the free-surface elevation and the flow velocity, for landslide-generated water waves. The one-dimensional (horizontal, 1DH) analytical solutions for water waves generated by a solid landslide moving at a constant speed in constant water depth were obtained for the linear and weakly dispersive wave model as well as the linear and fully dispersive wave model. The area enclosed by the landslide was shown to have stronger lasting effects on the generated water waves than the exact landslide shape. In addition, the resonance solution based on the fully dispersive wave model was examined, and the growth rate was derived. For the 1DH linear shallow water equations (LSWEs) on a constant slope, a closed-form analytical solution, which could serve as a useful benchmark for numerical models, was found for a special landslide forcing function. For the two-dimensional (horizontal, 2DH) LSWEs on a plane beach, we rederived the solutions using the quiescent water initial conditions. The difference between the initial conditions used in the new solutions and those used in previous studies was found to have a permanent effect on the generated waves. We further noted that convergence rate of the 2DH LSWE analytical solutions varies greatly, and advised that case-by-case convergence tests be conducted whenever the modal analytical solutions are numerically evaluated using a finite number of modes.

Modelo mecánico acoplado para la simulación 2D del tránsito peatonal por estructuras esbeltas

This work focuses on evaluating the vertical movements that appear when a pedestrian is walking on a slender beam-like structure. For that, the pedestrian is modelled by a dynamic system consisting on a concentrated mass, located in its center of masses, supported by two springs which simulate the stiffness of the legs. The success of this simple mechanical model is that the resulting contact forces are well adjusted to those observed experimentally for human gait. In this way, the structure is undergoing the moving mass and is loaded with realistic forces, resulting in a proper assembly to study interaction effects. The dynamic response of the beam is obtained at any coordinate by the modal superposition method, considering a finite number of modes. Differential equations of the resulting coupled problem are solved using proper numerical techniques. Solutions obtained for different scenarios (depending on the mass of the pedestrian with respect to the mass of the beam and on the pace) are presented.

Joint probability distributions for projection probabilities of random orthonormal states

The quantum chaos conjecture applied to a finite dimensional quantum system implies that such a system has eigenstates that show similar statistical properties as the column vectors of random orthogonal or unitary matrices. Here, we consider the different probabilities for obtaining a specific outcome in a projective measurement, provided the system is in one of its eigenstates. We then give analytic expressions for the joint probability density for these probabilities, with respect to the ensemble of random matrices. In the case of the unitary group, our results can be applied, also, to the phenomenon of universal conductance fluctuations, where the same mathematical quantities describe partial conductances in a two-terminal mesoscopic scattering problem with a finite number of modes in each terminal.

Existence of a Steady Flow of Stokes Fluid Past a Linear Elastic Structure Using Fictitious Domain

We use fictitious domain method with penalization for the Stokes equation in order to obtain approximate solutions in a fixed larger domain including the domain occupied by the structure. The coefficients of the fluid problem, excepting the penalizing term, are independent of the deformation of the structure. It is easy to check the inf-sup condition and the coercivity of the Stokes problem in the fixed domain. Subtracting the structure equations from the fictitious fluid equations in the structure domain, we obtain a weak formulation in a fixed domain, where the continuity of the stress at the interface does not appear explicitly. Existence of a solution is proved when the structure displacement is generated by a finite number of modes.