FFT-based Approach Research Articles

FFT-based homogenisation for efficient concurrent multiscale modelling of thin plate structures

AbstractThis paper presents a novel approach to concurrent multiscale analysis, where structures are formulated at both microscopic and macro levels for simulation purposes. The proposed method employs a plate model to formulate structures at both scales, and homogenisation is performed using the FFT-based approach, offering higher efficiency compared to conventional methods. Additionally, the macroscopic tangent operator of the microscopic model is derived through an algorithmically consistent process within the FFT-based framework, incorporating the application of Lippman–Schwinger equations as outlined in this work. The effectiveness of the proposed method is demonstrated through case studies in real simulations, revealing comparable results to traditional multiscale schemes in addressing multiscale thin plate structures. Importantly, the method significantly reduces computing time and memory usage, attributed to the efficiency of plate modelling and the FFT-based homogenisation strategy.

FFT-Based Approach for the Mechanical Analysis of Superconducting Rutherford-Type Cables

Low-temperature superconductors are widely used in high field accelerator magnets, mostly within Rutherford type cables. We have proposed the “CoCaSCOPE” approach in order to build a numerical model of impregnated low-temperature Nb <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> Sn superconducting Rutherford-type cables dedicated to mechanical computation. Here the strand structure is described as a bi-metallic material: a copper core, a superconducting bundle region and an outer copper ring. In this paper, we explore the use of the distributed solver AMITEX_FFTP based on Fast Fourier Transform (FFT). The main interest of this method is to reduce the simulation time. This approach is investigated for structures with linear material behavior in this paper. This work compares the results from the FFT approach (using AMITEX) and from the 3D Finite Elements approach (using Cast3M). In this way, we can demonstrate the higher performances of FFT-based solving method compared to the finite elements method for multi-scale structure such as Rutherford-type cables in terms of computation time. With this approach, we can reduce the computation time or, alternatively, increase the size of our models in order to analyze the behavior of a macroscale model under mechanical solicitation.

A wavelet-enhanced adaptive hierarchical FFT-based approach for the efficient solution of microscale boundary value problems

This contribution focuses on the development of an adaptive hierarchical FFT-based approach for the efficient solution of microscale boundary value problems. To this end, the classic Moulinec–Suquet scheme is revisited and enhanced by making use of wavelet analysis. Governing fields are represented in a wavelet basis and higher level stress approximations in a nested set of approximation spaces are successively derived by making use of wavelet transforms. By adaptively refining the computational grid based on the solution profile, localised features can be resolved accurately while the overall number of material model evaluations is significantly reduced. The performance is demonstrated by a detailed study of representative boundary value problems in one- and two-dimensional domains, whereby a reduction in the number of material model evaluations of up to 95% has been achieved.

PMUVis : A Large-Scale Platform to Assist Power System Operators in a Smart Grid.

Electric transmission power grids are being revamped with the widespread deployment of GPS-enabled phasor measurement units (PMUs) for real-time wide-area monitoring and control via precise, time-synchronized measurements of voltage and current. Large, concurrently produced volumes of noisy data hinder PMU usability, particularly for the analysis of power oscillation and load fluctuation events in the grid. We examine visualization challenges for events in the electric power grid and develop PMUVis, a visualization platform that supports scalable analysis of grid network topology and anomalous events in near time. PMUVis incorporates a novel FFT-based approach over raw and temporally aggregated data to examine oscillation event propagation through the grid network. We validate PMUVis with expert reviews and a case study and discuss how visualization can be leveraged to enhance real-time spatiotemporal grid analysis by advancing operator capabilities.

FFT-based Inverse Homogenization for Cellular Material Design

• An FFT-based inverse homogenization approach is proposed for efficiently designing cellular materials. • Rotating architected materials with negative Poisson's ratio are generated. • Architected cellular materials with maximum stiffness and isotropy are sought out. • The inverse homogenization method for scaffold design with specified mechanical performance is validated. Besides constituting components, the properties of composites are highly relevant to their microstructures. The work proposed a fast Fourier transform (FFT)-based inverse homogenization method implemented by the bi-directional evolutionary structural optimization (BESO) technique to explore the vast potential of cellular materials. The periodic boundary condition of self-repeated representative volume elements can be naturally satisfied in the FFT-based homogenization scheme. The objective function of the optimization problem is the specific moduli or the quadratic difference between the effective value and the target, which are obtained in terms of mutual strain energies. Its sensitivity to the design variable, namely the elemental density, is derived from the adjoint variable method and used as the criterion to remove or add material in local elements. Numerical examples show that the proposed method generates a series of architected cellular materials with maximum modulus, negative Poisson's ratio, and specific elasticity tensor. FFT-based homogenization in the method demands less memory usage but has high efficiency. Thus, it can achieve topology optimization of unit cell with one million hexahedral elements. This approach contributes to the extended application of FFT-based homogenization and can guide the microstructure design of mechanical metamaterials. :

A unified FFT-based approach to maximum assignment problems related to transitive finite group actions

This paper studies the cross-correlation of two real-valued functions whose common domain is a set on which a finite group acts transitively. Such a group action generalizes, e.g., circular time shifts for discrete periodic time signals or spatial translations in the context of image processing. Cross-correlations can be reformulated as convolutions in group algebras and are thus transformable into the spectral domain via Fast Fourier Transforms. The resulting general discrete cross-correlation theorem will serve as the basis for a unified FFT-based approach to maximum assignment problems related to transitive finite group actions. Our main focus is on those assignment problems arising from the actions of symmetric groups on the cosets of Young subgroups. The “simplest” nontrivial representatives of the latter class of problems are the NP-hard symmetric and asymmetric maximum quadratic assignment problem. We present a systematic spectral approach in terms of the representation theory of symmetric groups to solve such assignment problems. This generalizes and algorithmically improves Kondor's spectral branch-and-bound approach (SODA, 2010) to the exact solution of the asymmetric maximum quadratic assignment problem.

An FFT-based approach for Bloch wave analysis: application to polycrystals

A method based on the Fast Fourier Transform is proposed to obtain the dispersion relation of acoustic waves in heterogeneous periodic media with arbitrary microstructures. The microstructure is explicitly considered using a voxelized Representative Volume Element (RVE). The dispersion diagram is obtained solving an eigenvalue problem for Bloch waves in Fourier space. To this aim, two linear operators representing stiffness and mass are defined through the use of differential operators in Fourier space. The smallest eigenvalues are obtained using the implicitly restarted Lanczos and the subspace iteration methods, and the required inverse of the stiffness operator is done using the conjugate gradient with a preconditioner. The method is used to study the propagation of acoustic waves in elastic polycrystals, showing the strong effect of crystal anistropy and polycrystaline texture on the propagation. It is shown that the method combines the simplicity of classical Fourier series analysis with the versatility of Finite Elements to account for complex geometries proving an efficient and general approach which allows the use of large RVEs in 3D.

An order N log N parallel solver for time-spectral problems

The time-spectral method is a fast and efficient scheme for computing the solution to temporal periodic problems. Compared to traditional backward difference implicit time-stepping methods, time-spectral methods incur significant computational savings by using a temporal Fourier representation of the time discretization and solving the periodic problem directly. In the time-spectral discretization, all time instances are fully coupled to each other, resulting in a dense temporal discretization matrix, the evaluation of which scales as O(N2), where N denotes the number of time instances. However, by implementing the time-spectral method based on the fast Fourier transform (FFT) the computational cost can be reduced to O(Nlog⁡N). Furthermore, in parallel implementations, where each time instance is assigned to an individual processor, the wall-clock time necessary to solve time-spectral problems is reduced to O(log⁡N) using the FFT-based approach, as opposed to the O(N) weak scaling incurred by previous dense matrix or discrete Fourier transform (DFT) parallel time-spectral solver implementations. In this work, first an FFT-based approximate factorization (AF) scheme is used to solve time-spectral problems with large numbers of time instances. Subsequently, this solution strategy is reformulated as a preconditioner to be used in the context of a Newton-Krylov method applied directly to the complete non-linear space-time time-spectral residual. The use of the Generalized Minimal Residual Method (GMRES) Krylov method enables additional coupling between the various time instances running on different processors resulting in faster overall convergence. The GMRES/AF scheme is shown to produce significantly faster convergence than the AF scheme used as a solver alone, and achieves orders of magnitude gains in efficiency compared to previous DFT-based implementations of similar solution strategies for large numbers of time instances.

CoDockPP: A Multistage Approach for Global and Site-Specific Protein-Protein Docking.

Protein-protein docking technology is an effective approach to study the molecular mechanism of essential biological processes mediated by complex protein-protein interactions. The fast Fourier transform (FFT) correlation approach makes a good balance between the exhaustive global sampling and the computational efficiency for protein-protein docking. However, it is difficult to integrate the precise knowledge-based scoring function and site constraint information into the FFT-based approach. New docking strategies with the capability of combining both global sampling and precise scoring are strongly needed. We propose a multistage protein-protein docking strategy called CoDockPP. This program takes full advantage of the sampling efficiency of the FFT-based method to choose the valid ligand protein poses with good surface complementarity. The retained poses are transformed to the real Cartesian space for the implementation of site constraints and atomic scoring. Site constraints and a rapid table lookup scoring are applied to gradually reduce the candidate poses to a tractable number. To enhance the accuracy of docking prediction, the best fast-scoring states are expanded the local sampling points and then these neighbor poses are further evaluated by the precise knowledge-based scoring function. By testing on protein-protein docking benchmark 5.0, CoDockPP remarkably improves the success rate and hit count in both ab initio docking and site-specific docking, especially in difficult cases. The server is free and open to all users with no login requirement at http://codockpp.schanglab.org.cn .

Subthalamic and pallidal oscillatory activity in patients with Neurodegeneration with Brain Iron Accumulation type I (NBIA-I)

ObjectivesNeurodegeneration with Brain Iron Accumulation type I (NBIA-I) is a rare hereditary neurodegenerative disorder with pallidal degeneration leading to disabling generalized dystonia and parkinsonism. Pallidal or subthalamic deep brain stimulation can partially alleviate motor symptoms. Disease-specific patterns of abnormally enhanced oscillatory neuronal activity recorded from the basal ganglia have been described in patients with movement disorders undergoing deep brain stimulation (DBS). Here we studied oscillatory activity recorded from the internal globus pallidus (GPi) and the subthalamic nucleus (STN) to characterize neuronal activity patterns in NBIA-I. MethodsWe recorded local field potentials (LFP) from DBS electrodes in 6 juvenile patients with NBIA-I who underwent functional neurosurgery. Four patients were implanted in the STN and two patients in the GPi. Recordings were performed during wakeful rest. An FFT-based approach was used to analyze the power spectrum in the target area. ResultsIn all patients we found distinct peaks in the low frequency (7–12 Hz) and in 5 out 6 also in the beta frequency range (15–30 Hz) with the largest beta peak in the patient that presented with the most prominent bradykinesia. No distinct peaks occurred in the gamma frequency range (35–100 Hz). The oscillatory pattern did not differ between STN and GPi. ConclusionsHere we show for the first time the oscillatory activity pattern in the STN and the GPi in juvenile patients with dystonia plus syndrome due to NBIA-I. The low frequency peak we found is in line with previous studies in patients with isolated idiopathic dystonia. In our cohort, the pallidal beta band activity may be related to more severe motor slowing in dystonia plus syndrome such as NBIA-I. SignificanceOur results further support the link between hyperkinetic motor symptoms such as dystonia and enhanced basal ganglia low frequency activity irrespective of the underlying etiology of dystonia.

Calculating numerical derivatives using Fourier transform: some pitfalls and how to avoid them

Numerical differentiation is commonly used by a number of science students and researchers for data analysis. The differentiation of vectors of data points representing discrete samples of some underlying signal can be implemented in a computer using the central differencing scheme or the fast Fourier transform (FFT)-based approach. We point out that a simple extension of the continuous Fourier transform derivative identity to the discrete case, however, gives rise to results that are inconsistent with the central differencing scheme. In particular, for functions with step-like discontinuities, the numerical derivative computed with the FFT-based approach is corrupted by ringing artifacts. We describe the idea of a modified wave number for FFT-based numerical differentiation which leads to results that are consistent with the central differencing scheme. Modified wave number identities for numerical computation of both first and second order derivatives are described. The pitfalls of using a simplistic extension of the derivative identity for continuous Fourier transform to the discrete case and the methodology to avoid them are illustrated with numerical examples.

Novel algorithms for 3D surface point cloud boundary detection and edge reconstruction

Abstract Tessellated surfaces generated from point clouds typically show inaccurate and jagged boundaries. This can lead to tolerance errors and problems such as machine judder if the model is used for ongoing manufacturing applications. This paper introduces a novel boundary point detection algorithm and spatial FFT-based filtering approach, which together allow for direct generation of low noise tessellated surfaces from point cloud data, which are not based on pre-defined threshold values. Existing detection techniques are optimized to detect points belonging to sharp edges and creases. The new algorithm is targeted at the detection of boundary points and it is able to do this better than the existing methods. The FFT-based edge reconstruction eliminates the problem of defining a specific polynomial function order for optimum polynomial curve fitting. The algorithms were tested to analyse the results and measure the execution time for point clouds generated from laser scanned measurements on a turbofan engine turbine blade with varying numbers of member points. The reconstructed edges fit the boundary points with an improvement factor of 4.7 over a standard polynomial fitting approach. Furthermore, through adding artificial noise it has been demonstrated that the detection algorithm is very robust for out-of-plane noise lower than 25% of the cloud resolution and it can produce satisfactory results when the noise is lower than 75%. Highlights Introducing novel boundary point detection algorithm and spatial FFT-based filtering approach. Enabling reliable detection of boundary points and smooth boundary reconstruction. FFT-based edge reconstruction works better than polynomial curve fitting. Optimally smoothed edges facilitate the direct use of tessellated models for CAM tasks. The new detection algorithm tolerates out-of-plane noise. The reconstructed edges fit well the detected boundary points.

Anisotropic Elastic Field of 3D Prismatic Dislocation Loops in Bounded and Voided Single Crystal Films

Dislocations in a finite medium bring about image stresses. These image stresses play important roles in the dislocation behavior in finite sized systems such as thin films. Since the ratio of surface to volume is higher for thin films than for bulk materials, dislocation behaviors in thin films are greatly different from those in a corresponding infinite medium, which make it necessary to take into account the effects of free surfaces on the evolution of dislocations in thin films. In the investigations[4, 5], image stresses in an elastic cylinder and thin films are calculated by employing a Fourier transform (FFT) approach and isotropically elastic fields due to dislocations are adopted in their formulation. However, most crystals are anisotropic, and the anisotropic ratio changes with environment physical parameters, such as the temperature, moisture, electron field, magnetic field. A theorem based on anisotropic Stroh’s formula for calculating the image stress of infinite straight dislocations in anisotropic bicrystals has been developed by Barnett and Lothe[6]. Wu et al.[3] recently also make use of the FFT technique to investigate the general dislocation image stresses of cubic thin films, thus extending the formalism by Weinberger et al.[4,5] from isotropic to anisotropic thin films. It is clear that for the assumed in-plane elastic fields to be periodically defined within an unbounded region is an essential and indispensable prerequisite for the above FFT-based approach to be effectively implemented, thus ruling out the possibility of its being employed to analyse image stresses in bounded and/or voided thin cubic films. Our motivation here is then to make an further extension by first calculating the anisotropic elastic fields of dislocation loops in an unbounded thin film with cavities and then invoking FEM and the principle of superposition to seek the image stress solution.

Three-dimensional dominant frequency mapping using autoregressive spectral analysis of atrial electrograms of patients in persistent atrial fibrillation.

BackgroundAreas with high frequency activity within the atrium are thought to be ‘drivers’ of the rhythm in patients with atrial fibrillation (AF) and ablation of these areas seems to be an effective therapy in eliminating DF gradient and restoring sinus rhythm. Clinical groups have applied the traditional FFT-based approach to generate the three-dimensional dominant frequency (3D DF) maps during electrophysiology (EP) procedures but literature is restricted on using alternative spectral estimation techniques that can have a better frequency resolution that FFT-based spectral estimation.MethodsAutoregressive (AR) model-based spectral estimation techniques, with emphasis on selection of appropriate sampling rate and AR model order, were implemented to generate high-density 3D DF maps of atrial electrograms (AEGs) in persistent atrial fibrillation (persAF). For each patient, 2048 simultaneous AEGs were recorded for 20.478 s-long segments in the left atrium (LA) and exported for analysis, together with their anatomical locations. After the DFs were identified using AR-based spectral estimation, they were colour coded to produce sequential 3D DF maps. These maps were systematically compared with maps found using the Fourier-based approach.Results3D DF maps can be obtained using AR-based spectral estimation after AEGs downsampling (DS) and the resulting maps are very similar to those obtained using FFT-based spectral estimation (mean 90.23 %). There were no significant differences between AR techniques (p = 0.62). The processing time for AR-based approach was considerably shorter (from 5.44 to 5.05 s) when lower sampling frequencies and model order values were used. Higher levels of DS presented higher rates of DF agreement (sampling frequency of 37.5 Hz).ConclusionWe have demonstrated the feasibility of using AR spectral estimation methods for producing 3D DF maps and characterised their differences to the maps produced using the FFT technique, offering an alternative approach for 3D DF computation in human persAF studies.

Modeling solution X-ray scattering of biomacromolecules using an explicit solvent model and the fast Fourier transform

A novel computational method based on atomic form factors and the fast Fourier transform (FFT) is developed to compute small- and near-wide-angle X-ray scattering profiles of biomacromolecules from explicit solvent modeling. The method is validated by comparing the results with those from non-FFT approaches and experiments, and good agreement with experimental data is observed for both small and near-wide angles. In terms of computational efficiency, the FFT-based method is advantageous for protein solution systems of more than 3000 atoms. Furthermore, the computational cost remains nearly constant for a wide range of system sizes. The FFT-based approach can potentially handle much larger molecular systems compared with popular existing methods.

Guaranteed upper–lower bounds on homogenized properties by FFT-based Galerkin method

Guaranteed upper-lower bounds on homogenized coefficients, arising from the periodic cell problem, are calculated in a scalar elliptic setting. Our approach builds on the recent variational reformulation of the Moulinec-Suquet (1994) Fast Fourier Transform (FFT) homogenization scheme by Vond\v{r}ejc et al. (2014), which is based on the conforming Galerkin approximation with trigonometric polynomials. Upper-lower bounds are obtained by adjusting the primal-dual finite element framework developed independently by Dvo\v{r}\'{a}k (1993) and Wieckowski (1995) to the FFT-based Galerkin setting. We show that the discretization procedure differs for odd and non-odd number of grid points. Thanks to the Helmholtz decomposition inherited from the continuous formulation, the duality structure is fully preserved for the odd discretizations. In the latter case, a more complex primal-dual structure is observed due to presence of the trigonometric polynomials associated with the Nyquist frequencies. These theoretical findings are confirmed with numerical examples. To conclude, the main advantage of the FFT-based approach over conventional finite-element schemes is that the primal and the dual problems are treated on the same basis, and this property can be extended beyond the scalar elliptic setting.

A FFT-based formulation for efficient mechanical fields computation in isotropic and anisotropic periodic discrete dislocation dynamics

In this paper, we propose a novel full-field approach based on the fast Fourier transform (FFT) technique to compute mechanical fields in periodic discrete dislocation dynamics (DDD) simulations for anisotropic materials: the DDD-FFT approach. By coupling the FFT-based approach to the discrete continuous model, the present approach benefits from the high computational efficiency of the FFT algorithm, while allowing for a discrete representation of dislocation lines. It is demonstrated that the computational time associated with the new DDD-FFT approach is significantly lower than that of current DDD approaches when large number of dislocation segments are involved for isotropic and anisotropic elasticity, respectively. Furthermore, for fine Fourier grids, the treatment of anisotropic elasticity comes at a similar computational cost to that of isotropic simulation. Thus, the proposed approach paves the way towards achieving scale transition from DDD to mesoscale plasticity, especially due to the method’s ability to incorporate inhomogeneous elasticity.

Condenser-free contrast methods for transmitted-light microscopy.

Phase contrast microscopy allows the study of highly transparent yet detail-rich specimens by producing intensity contrast from phase objects within the sample. Presented here is a generalized phase contrast illumination schema in which condenser optics are entirely abrogated, yielding a condenser-free yet highly effective method of obtaining phase contrast in transmitted-light microscopy. A ring of light emitting diodes (LEDs) is positioned within the light-path such that observation of the objective back focal plane places the illuminating ring in appropriate conjunction with the phase ring. It is demonstrated that true Zernike phase contrast is obtained, whose geometry can be flexibly manipulated to provide an arbitrary working distance between illuminator and sample. Condenser-free phase contrast is demonstrated across a range of magnifications (4–100×), numerical apertures (0.13–1.65NA) and conventional phase positions. Also demonstrated is condenser-free darkfield microscopy as well as combinatorial contrast including Rheinberg illumination and simultaneous, colour-contrasted, brightfield, darkfield and Zernike phase contrast. By providing enhanced and arbitrary working space above the preparation, a range of concurrent imaging and electrophysiological techniques will be technically facilitated. Condenser-free phase contrast is demonstrated in conjunction with scanning ion conductance microscopy (SICM), using a notched ring to admit the scanned probe. The compact, versatile LED illumination schema will further lend itself to novel next-generation transmitted-light microscopy designs. The condenser-free illumination method, using rings of independent or radially-scanned emitters, may be exploited in future in other electromagnetic wavebands, including X-rays or the infrared.

Frequency Estimation of Distorted and Noisy Signals in Power Systems by FFT-Based Approach

This paper focuses on the accurate frequency estimation of power signals corrupted by a stationary white noise. The noneven item interpolation FFT based on the triangular self-convolution window is described. A simple analytical expression for the variance of noise contribution on the frequency estimation is derived, which shows the variances of frequency estimation are proportional to the energy of the adopted window. Based on the proposed method, the noise level of the measurement channel can be estimated, and optimal parameters (e.g., sampling frequency and window length) of the interpolation FFT algorithm that minimize the variances of frequency estimation can thus be determined. The application in a power quality analyzer verified the usefulness of the proposed method.

GPU-Accelerated Forward and Back-Projections With Spatially Varying Kernels for 3D DIRECT TOF PET Reconstruction

We describe a GPU-accelerated framework that efficiently models spatially (shift) variant system response kernels and performs forward- and back-projection operations with these kernels for the DIRECT (Direct Image Reconstruction for TOF) iterative reconstruction approach. Inherent challenges arise from the poor memory cache performance at non-axis aligned TOF directions. Focusing on the GPU memory access patterns, we utilize different kinds of GPU memory according to these patterns in order to maximize the memory cache performance. We also exploit the GPU instruction-level parallelism to efficiently hide long latencies from the memory operations. Our experiments indicate that our GPU implementation of the projection operators has slightly faster or approximately comparable time performance than FFT-based approaches using state-of-the-art FFTW routines. However, most importantly, our GPU framework can also efficiently handle any generic system response kernels, such as spatially symmetric and shift-variant as well as spatially asymmetric and shift-variant, both of which an FFT-based approach cannot cope with.