Fourier Space Research Articles

Predictive modeling of Alfvén eigenmode stability in inductive scenarios in JT-60SA

The JT-60SA device offers unique conditions before ITER for the study of the interaction of energetic particles with plasma waves. With similar dimensions to JET, e.g., a major radius but with a slightly more elongated plasma volume, JT-60SA is used as a high-power device where additional heating power (including 10 MW of the 500 keV Neutral Beam Injection) of up to 41 MW and the potential for high non-inductive plasma current operation pave the path for numerous challenges in physics on MHD stability, in particular, when considering the effects of energetic particles. Several operational scenarios with ITER and DEMO-relevant plasma regimes, in terms of non-dimensional plasma parameters, are anticipated. In this work, the stability of Alfvén eigenmodes (AEs) in variants of two of the most relevant operational scenarios with single null is analyzed: a full Ip inductive scenario at high density (1.1 × 1020 m−3 on-axis electron density) and 5.48MA/2.05T toroidal plasma current and magnetic field, and an advanced (hybrid) scenario with an ion energy transport barrier (ITB) and 3.5MA/2.28T toroidal plasma current and magnetic field. The workflow included the CRONOS code to establish the scenario, the ASCOT code to calculate the slowing-down energetic particle distributions for a positive/negative ion source-based neutral beam, and the MISHKA/CASTOR-K suite to calculate the MHD spectra of AEs and the associated drive/damping contributions from the NBI energetic ions, as well as the thermal ion landau damping. The systematic analysis, over a large Fourier space of the toroidal mode number/mode frequency, provides evidence that although a significant fraction of supra-Alfvénic particles stemming from the negative ion source-based neutral beam (500 keV) can, in some cases, drive to AEs in both scenarios, it is not enough to overcome the thermal ion landau damping. In addition, the advanced scenario with ITB is shown to be stable against AEs localized in the vicinity of the barrier as well, offering good prospects of sustainability of the plasma performance and of ITB. Finally, some sensitivity scan results are shown on the influence of fast ion density and q-profile on the AE mode spectra and stability.

600-GHz Fourier imaging based on heterodyne detection at the 2nd sub-harmonic.

Fourier imaging is an indirect imaging method which records the diffraction pattern of the object scene coherently in the focal plane of the imaging system and reconstructs the image using computational resources. The spatial resolution, which can be reached, depends on one hand on the wavelength of the radiation, but also on the capability to measure - in the focal plane - Fourier components with high spatial wave-vectors. This leads to a conflicting situation at THz frequencies, because choosing a shorter wavelength for better resolution usually comes at the cost of less radiation power, concomitant with a loss of dynamic range, which limits the detection of higher Fourier components. Here, aiming at maintaining a high dynamic range and limiting the system costs, we adopt heterodyne detection at the 2nd sub-harmonic, working with continuous-wave (CW) radiation for object illumination at 600 GHz and local-oscillator (LO) radiation at 300 GHz. The detector is a single-pixel broad-band Si CMOS TeraFET equipped with substrate lenses on both the front- and backside for separate in-coupling of the waves. The entire scene is illuminated by the object wave, and the Fourier spectrum is recorded by raster scanning of the single-detector unit through the focal plane. With only 56 µW of power of the 600-GHz radiation, a dynamic range of 60 dB is reached, sufficient to detect the entire accessible Fourier space spectrum in the test measurements. We present a detailed comparison between plane-to-plane imaging and Fourier imaging, and show that, with both, a lateral spatial resolution of better than 0.5 mm, at the diffraction limit, is reached.

Fourier Transform of the Lippmann-Schwinger Equation: Solving Vectorial Electromagnetic Scattering by Arbitrary Shapes

In Electromagnetics, the field scattered by an ensemble of particles—of arbitrary size, shape, and material—can be obtained by solving the Lippmann–Schwinger equation. This singular vectorial integral equation is generally formulated in the direct space Rn (typically n=2 or n=3). In the article, we rigorously computed the Fourier transform of the vectorial Lippmann–Schwinger equation in the space of tempered distributions, S′(R3), splitting it in a singular and a regular contribution. One eventually obtains a simple equation for the scattered field in the Fourier space. This permits to draw an explicit link between the shape of the scatterer and the field through the Fourier Transform of the body indicator function. We compare our results with accurate calculations based on the T-matrix method and find a good agreement.

Beyond superoscillation: general theory of approximation with bandlimited functions

We give a general strategy to construct superoscillating/supergrowing functions using an orthogonal polynomial expansion of a bandlimited function. The degree of superoscillation/supergrowth is controlled by an anomalous expectation value of a pseudodistribution that exceeds the band limit. The function is specified via the rest of its cumulants of the pseudodistribution. We give an explicit construction using Legendre polynomials in the Fourier space, which leads to an expansion in terms of spherical Bessel functions in the real space. The other expansion coefficients may be chosen to optimize other desirable features, such as the range of super behavior. We provide a prescription to generate bandlimited functions that mimic an arbitrary behavior in a finite interval. As target behaviors, we give examples of a superoscillating function, a supergrowing function, and even a discontinuous step function. We also look at the energy content in a superoscillating/supergrowing region and provide a bound that depends on the minimum value of the logarithmic derivative in that interval. Our work offers a new approach to analyzing superoscillations/supergrowth and is relevant to the optical field spot generation endeavors for far-field superresolution imaging.

Multi–materials topology optimization using deep neural network for coupled thermo–mechanical problems

In this paper, a density-based topology optimization method using neural networks is used to design domains composed of multi-materials under combined thermo–mechanical loading. The neural network produces the topology density field by minimizing a loss function defined for the problem. Length scale control is conducted using a Fourier space projection. JAX, a high-performance automatic differentiation (AD) Python library is used to build an end-to-end differentiable neural network. The model can handle the sensitivity analysis automatically using the backpropagation process (automatic differentiation of loss function) and the need for solving adjoint equations manual sensitivity analysis is removed. The optimization problem is solved by minimizing the loss function of the network and other optimization algorithms such as the Method of Moving Asymptotes (MMA) are not required. The performance of the method is demonstrated through several examples and compared with those obtained from the popular Solid Isotropic Material with Penalization (SIMP) method. The network is able to handle high-resolution re-sampling, resulting in a more refined and smooth variation of optimal topologies.

GEO-FPT: a model of the galaxy bispectrum at mildly non-linear scales

We present GEO-FPT (Geometric Fitted Perturbation Theory), a new model for the galaxy bispectrum anisotropic signal in redshift space, with functional form rooted in perturbation theory. It also models the dependence of the bispectrum with the geometric properties of the triangles in Fourier space, and has a broader regime of validity thanstate-of-the-art theoretical models based on perturbation theory. We calibrate the free parameters of this model using high-resolution dark matter simulations and perform stringent tests to show that GEO-FPTdescribes the galaxy bispectrum accurately up to scales of k ≃ 0.12hMpc-1 for different cosmological models, as well as for biased tracers of the dark matter field, considering a survey volume of 100 (Gpc h-1)3. In particular, a joint analysis ofthe power spectrum and bispectrum anisotropic signals, taking into account their full covariance matrix, reveals that therelevant physical quantities —the BAO peak position (along and across the line-of-sight), and the growth of structure parameters times the amplitude of dark matter fluctuations, fσ 8 — are recovered in an unbiased way, with an accuracy better than 0.4% and 2% respectively (which is our 2σ statistical limit of the systematic error estimate).In addition, the bispectrum signal breaks the fσ 8 degeneracy without detectable bias: f and σ 8 are recovered with better than 2.7% and 3.8% accuracy respectively (which is our 2σ statistical limit of the systematic error estimate).GEO-FPT boosts the applicability of the bispectrum signal of galaxy surveys beyond the current limitation of k ≲ 0.08hMpc-1 and makes the bispectrum a key statistic to unlock the information content from the mildly non-linear regime in the on-going and forthcoming galaxy redshift surveys.

Entanglement and quantum teleportation conditions from non-local quantum computing theory

Entanglement and quantum teleportation are two inter-related subjects and can be explained through four fundamental non-local quantum computing diagonal operator-state relations, which model the conservative phase-interaction between two adjacent atoms of an entangled atomic chain. Each model atom possesses four eigen-states. There exists a minimum focus distance between the two atoms for the entanglement to exist. Consequently, there exist perpetual elastic phase-changing events on the entire entangled chain at every discrete time interval that is evaluated and shown here. It is analogous to Newton’s first law of motion in discrete time. Quantum teleportation means every extra $2\pi$ phase between the atoms can be translated in Fourier space into an additional separation distance, a quantized value.

Circular cumulant reductions for macroscopic dynamics of oscillator populations with non-Gaussian noise.

We employ the circular cumulant approach to construct a low dimensional description of the macroscopic dynamics of populations of phase oscillators (elements) subject to non-Gaussian white noise. Two-cumulant reduction equations for α-stable noises are derived. The implementation of the approach is demonstrated for the case of the Kuramoto ensemble with non-Gaussian noise. The results of direct numerical simulation of the ensemble of N=1500 oscillators and the "exact" numerical solution for the fractional Fokker-Planck equation in the Fourier space are found to be in good agreement with the analytical solutions for two feasible circular cumulant model reductions. We also illustrate that the two-cumulant model reduction is useful for studying the bifurcations of chimera states in hierarchical populations of coupled noisy phase oscillators.

Neutrino winds on the sky

We develop a first-principles formalism to compute the distortion to the relic neutrino density field caused by the peculiar motions of large-scale structures. This distortion slows halos down due to dynamical friction, causes a local anisotropy in the neutrino-CDM cross-correlation, and reduces the global cross-correlation between neutrinos and CDM. The local anisotropy in the neutrino-CDM cross-spectrum is imprinted in the three point cross-correlations of matter and galaxies, or the bispectrum in Fourier space, producing a signal peaking at squeezed triangle configurations. This bispectrum signature of neutrino masses is not limited by cosmic variance or potential inaccuracies in the modeling of complicated nonlinear and galaxy formation physics, and it is not degenerate with the optical depth to reionization. We show that future surveys have the potential to detect the distortion bispectrum.

Machine learning for phase-resolved reconstruction of nonlinear ocean wave surface elevations from sparse remote sensing data

Accurate short-term predictions of phase-resolved water wave conditions are crucial for decision-making in ocean engineering. However, the initialization of remote-sensing-based wave prediction models first requires a reconstruction of wave surfaces from sparse measurements like radar. Existing reconstruction methods either rely on computationally intensive optimization procedures or simplistic modelling assumptions that compromise the real-time capability or accuracy of the subsequent prediction process. We therefore address these issues by proposing a novel approach for phase-resolved wave surface reconstruction using neural networks based on the U-Net and Fourier neural operator (FNO) architectures. Our approach utilizes synthetic yet highly realistic training data on uniform one-dimensional grids, that is generated by the high-order spectral method for wave simulation and a geometric radar modelling approach. The investigation reveals that both models deliver accurate wave reconstruction results and show good generalization for different sea states when trained with spatio-temporal radar data containing multiple historic radar snapshots in each input. Notably, the FNO demonstrates superior performance in handling the data structure imposed by wave physics due to its global approach to learn the mapping between input and output in Fourier space.

Fast construction of cluster interaction matrix for data-driven cluster-based reduced-order model and prediction of elastoplastic stress-strain curves and yield surface

The data-driven cluster-based reduced-order model (CROM) methods[47–50] are very effective for predicting the nonlinear effective properties of heterogeneous materials. The Cluster Interaction Matrix (CIM) in these methods can be regarded as a clustered discrete Green's function and is the critical part of CROM methods, which more and more scholars have realized. However, the explicit expression of Green's function is available only for homogeneous material. The CIM for heterogeneous materials needs to be calculated through numerical approaches, which face the challenge of computational efficiency. Therefore, this paper proposes an efficient method to construct the CIM of heterogeneous materials. By deriving the Green's function in Fourier space under the governing equation of FCA, the CIM of homogeneous materials that can provide the self-equilibrium stress field is calculated efficiently. Then, to avoid the effects of stress averaging on clusters, we augment the proposed cluster-element interaction matrix with the self-equilibrium stress basis vector of heterogeneous material without introducing additional computational cost. Based on the augmented cluster-element interaction matrix, the self-equilibrium stress space is improved, and the CIM of highly heterogeneous materials is calculated using the minimum complementary energy principle. The proposed algorithm significantly improves the offline computational accuracy of FCA, enabling efficient prediction of nonlinear effective properties and yield surface under complex load paths. Numerical examples demonstrate the efficiency and accuracy of the proposed method. In addition, we provide the relationship between the CIMs of a homogeneous material of FCA and SCA. Based on the proposed algorithm in this paper, it can be helpful for other CROM methods.

A fast Fourier transform-based solver for elastic micropolar composites

This work presents a spectral micromechanical formulation for obtaining the full-field and homogenized response of elastic micropolar composites. The algorithm relies on a coupled set of convolution integral equations for the micropolar strains, where periodic Green’s operators associated with a linear homogeneous reference medium are convolved with functions of the Cauchy and couple stress fields that encode the material’s heterogeneity, as well as any potential material nonlinearity. Such convolution integral equations take an algebraic form in the reciprocal Fourier space that can be solved iteratively. In this vein, the fast Fourier transform (FFT) algorithm is leveraged to accelerate the numerical solution, resulting in a mesh-free formulation in which the periodic unit cell representing the heterogeneous material can be discretized by a regular grid of pixels in two dimensions (or voxels in three dimensions). For verification, the numerical solutions obtained with the micropolar FFT solver are compared with analytical solutions for a matrix with a dilute circular inclusion subjected to plane strain loading. The developed computational framework is then used to study length-scale effects and effective (micropolar) moduli of composites with various topological configurations.

Fast multi-scale galaxy cluster detection with weak lensing: Towards a mass-selected sample

The sensitivity and wide area reached by ongoing and future wide-field optical surveys allows for the detection of an increasing number of galaxy clusters uniquely through their weak lensing signal. This motivates the development of new methods for analysing the unprecedented volume of data faster and more efficiently. Here we introduce a new multi-scale weak lensing detection method based on the application of wavelet filters to convergence maps. We compare our results to those obtained from four commonly used single-scale approaches based on the application of aperture mass filters to the shear in real and Fourier space. The method was tested and validated on Euclid-like mocks from the DUSTGRAIN-pathfinder simulations. We introduce a new matching procedure that takes into account the theoretical signal-to-noise of detection by weak lensing and the size of the filter under consideration. We performed a qualitative and quantitative analysis of the methods, including a complete analysis of the filters and a comparison of the purity and the completeness of the resulting catalogues. We show explicitly that equivalent results are obtained when the detection is undertaken in real and Fourier space, and when the algorithms are applied to the shear and the convergence. We show that the multi-scale method applied to the convergence is faster and more efficient at detecting clusters than single-scale methods applied to the shear. In particular, we obtain an increase of 25% in the number of detections while maintaining the same purity compared to the most up-to-date aperture mass filter. We analysed the detected catalogues and quantified the efficiency of the matching procedure, showing in particular that fewer than 5% of the detections from the multi-scale method can be ascribed to line-of-sight alignments. The method is well adapted to the more sensitive, wider-area optical surveys that will become available in the near future, and paves the way to cluster samples that are as near as possible to being selected by total matter content.

Energy transfer and vortex structures: visualizing the incompressible turbulent energy cascade

The transfer of kinetic energy from large to small scales is a hallmark of turbulent flows. Yet, a precise mechanistic description of this transfer, which is expected to occur via an energy cascade, is still missing. Several conceptually simple configurations with vortex tubes have been proposed as a testing ground to understand the energy cascade. Here, we focus on incompressible flows and compare the energy transfer occurring in a statistically steady homogeneous isotropic turbulent (HIT) flow with the generation of fine-scale motions in configurations involving vortex tubes. We start by filtering the velocity field in bands of wavenumbers distributed logarithmically, which allows us to study energy transfer in Fourier space and also visualize the energy cascade in real space. In the case of a statistically steady HIT flow at a moderate Reynolds number, our numerical results do not reveal any significant correlation between regions of intense energy transfers and vorticity or strain, filtered in corresponding wavenumber bands, nor any simple self-similar process. In comparison, in the transient turbulent flow obtained from the interaction between two antiparallel vortex tubes, we observe a qualitatively simpler organization of the intense structures, as well as of the energy transfer. However, the correlations between energy transfer and strain are small, and point to complicated dynamics of energy transfer. By imposing a structure at large scales consisting of antiparallel vortex tubes in a statistically steady flow, we observed a picture qualitatively similar to what was observed for the transient flow, but the energy transfer statistics do not reproduce the type of triadic interactions seen in HIT. These results indicate that the specific properties of the large-scale vortical structures affect the way energy is transferred, and may not be fully representative of HIT.

Decomposition of Displacement Field into the Irrotational and Solenoidal Component Using Fast Fourier Transform

Purpose: A new code based on Helmholtz decomposition is presented to separate longitudinal (pressure) and transverse (shear) components of a mixed wave field. This algorithm will help isolate shear or pressure components of an elastic wave to further concentrate on each specific wave and its physical characteristics, particularly in medical imaging instrument development and image processing techniques.
 Materials and Methods: Using the combination of Fourier transform and Helmholtz decomposition, first, the mathematical basis of the work is prepared. After reaching a usable formula, this basis is embedded in the Code written in MATLAB program. Then, various test data containing shear and pressure waves were created and fed to the Code to evaluate its ability to decompose the displacements into the shear and pressure waves.
 Results: This new algorithm successfully isolated the transverse and longitudinal wavefront of the mixed wavefield. The Code demonstrated 100% accuracy for separating the shear wave and more than 99% for the pressure wave. Moreover, the background noise was kept under 0.03% in every step.
 Conclusion: The results show that using Helmholtz decomposition in Fourier space on 3D data can help decompose a displacement field into its irrotational and solenoidal components with high accuracy. A weak dependency on wave thickness and contrast was observed, but the algorithm's accuracy never fell below 99%.

Multi-center atrial fibrillation electrocardiogram (ECG) classification using Fourier space convolutional neural networks (FD-CNN) and transfer learning

There has been a proliferation of machine learning (ML) electrocardiogram (ECG) classification algorithms reaching >85% accuracy for various cardiac pathologies. Despite the high accuracy at individual institutions, challenges remain when it comes to multi-center deployment. Transfer learning (TL) is a technique in which a model trained for a specific task is repurposed for another related task, in this case ECG ML model trained at one institution is fine-tuned to be utilized to classify ECGs at another institution. Models trained at one institution, however, might not be generalizable for accurate classification when deployed broadly due to differences in type, time, and sampling rate of traditional ECG acquisition. In this study, we evaluate the performance of time domain (TD) and frequency domain (FD) convolutional neural network (CNN) classification models in an inter-institutional scenario leveraging three different publicly available datasets. The larger PTB-XL ECG dataset was used to initially train TD and FD CNN models for atrial fibrillation (AFIB) classification. The models were then tested on two different data sets, Lobachevsky University Electrocardiography Database (LUDB) and Korea University Medical Center database (KURIAS). The FD model was able to retain most of its performance (>0.81 F1-score), whereas TD was highly affected (<0.53 F1-score) by the dataset variations, even with TL applied. The FD CNN showed superior robustness to cross-institutional variability and has potential for widespread application with no compromise to ECG classification performance.

Three- and four-wave resonances in the nonlinear quadratic Kelvin lattice

In this paper we investigate analytically and numerically the nonlinear Kelvin lattice, namely a chain of masses and nonlinear springs, as in the α-Fermi-Pasta–Ulam-Tsingou (FPUT) chain, where, in addition, each mass is connected to a nonlinear resonator, i.e., a second mass free to oscillate. Both nonlinearities are quadratic in the equations of motion. This setup represents the simplest prototype of nonlinear wave propagation on a nonlinear metamaterial. In the linear case, we diagonalize the system, and the two branches of the dispersion relation can be found. Using this result, we derive in the nonlinear case the equations of motion for the normal variables in Fourier space, obtaining a system governed by triad interactions among the two branches of the dispersion relation. We find that the transfer of energy between these two branches is ruled by three- and four-wave resonant interactions. We perform numerical simulations of the primitive equations of motion and highlight the role of resonances as an efficient mechanism for transferring energy. Moreover, as predicted by the theory, we provide direct evidence that four-wave resonances appear on a time scale that is longer than the time scale for three-wave resonances. We also assess the recurrence behaviour (usual in the FPUT system) for the nonlinear Kelvin lattice, and we show that, while recurrence is observed if all the energy is placed, at time t=0, in the lowest mode of the acoustical branch, a non-recurrent behaviour is observed if the initial energy is located in the optical branch.

Self-similar, spatially localized structures in turbulent pipe flow from a data-driven wavelet decomposition

This study aims to extract and characterize structures in fully developed pipe flow at a friction Reynolds number of $\textit {Re}_\tau = 12\,400$ . To do so, we employ data-driven wavelet decomposition (DDWD) (Floryan &amp; Graham, Proc. Natl Acad. Sci. USA, vol. 118, 2021, e2021299118), a method that combines features of proper orthogonal decomposition and wavelet analysis in order to extract energetic and spatially localized structures from data. We apply DDWD to streamwise velocity signals measured separately via a thermal anemometer at 40 wall-normal positions. The resulting localized velocity structures, which we interpret as being reflective of underlying eddies, are self-similar across streamwise extents of 40 wall units to one pipe radius, and across wall-normal positions from $y^+=350$ to $y/R=1$ . Notably, the structures are similar in shape to Meyer wavelets. Projections of the data onto the DDWD wavelet subspaces are found to be self-similar as well, but in Fourier space; the bounds of self-similarity are the same as before, except streamwise self-similarity starts at a larger length scale of $450$ wall units. The evidence of self-similarity provided in this study lends further support to Townsend's attached eddy hypothesis, although we note that the self-similar structures are detected beyond the log layer and extend to large length scales.

Analytical method for a tunnel with periodically jointed lining in the poroelastic soil under seismic waves

In this study, a longitudinal equivalent periodic shell (LEPS) model and corresponding continuous joint model are proposed for the periodically jointed lining (PJL) tunnel. Based on the LEPS model for the PJL tunnel, an analytical method for the PJL tunnel under seismic waves is established. The surrounding soil of the PJL tunnel is assumed to be the poroelastic medium, and described by the Biot's theory. The lining of the PJL tunnel is assumed to be composed of a series of lining rings linked periodically by ring joints along the longitudinal direction, which are simplified as a LEPS and described by the cylindrical shell theory. Due to the periodicity of the LEPS lining, the tunnel-soil system is treated as a periodic structure in this study. Based on the periodicity condition for the tunnel-soil system as well as the wave function expansion method, the representation for the scattering wave field in the soil is established. By using the aforementioned cylindrical shell theory and Fourier series expansion method, the Fourier space equations of motion and convolution type constitutive relations for the LEPS lining are derived. By using the lining-soil continuity conditions, the Fourier space equations of motion and convolution type constitutive relations for the LEPS lining as well as the representation for the wave field in the soil, the coupled equations for the tunnel-soil system are established, with which the Fourier components of the displacements of the LEPS lining and wave function coefficients for the scattering waves in soil can be solved for. Numerical results indicate that the presence of the ring joint in the LEPS lining of the tunnel will affect the distribution of the internal forces in the lining considerably, and it will also enhance the maximum internal forces occurring in the lining.

Fast Ewald summation for Stokes flow with arbitrary periodicity

A fast and spectrally accurate Ewald summation method for the evaluation of stokeslet, stresslet and rotlet potentials of three-dimensional Stokes flow is presented. This work extends the previously developed Spectral Ewald method for Stokes flow to periodic boundary conditions in any number (three, two, one, or none) of the spatial directions, in a unified framework. The periodic potential is split into a short-range and a long-range part, where the latter is treated in Fourier space using the fast Fourier transform. A crucial component of the method is the modified kernels used to treat singular integration. We derive new modified kernels, and new improved truncation error estimates for the stokeslet and stresslet. An automated procedure for selecting parameters based on a given error tolerance is designed and tested. Analytical formulas for validation in the doubly and singly periodic cases are presented. We show that the computational time of the method scales like O(Nlog⁡N) for N sources and targets, and investigate how the time depends on the error tolerance and window function, i.e. the function used to smoothly spread irregular point data to a uniform grid. The method is fastest in the fully periodic case, while the run time in the free-space case is around three times as large. Furthermore, the highest efficiency is reached when applying the method to a uniform source distribution in a primary cell with low aspect ratio. The work presented in this paper enables efficient and accurate simulations of three-dimensional Stokes flow with arbitrary periodicity using e.g. boundary integral and potential methods.