Number Of Fourier Modes Research Articles

A spectral projection method for the analysis of autocorrelation functions and projection errors in discrete particle simulation

AbstractDiscrete particle simulation is a well‐established tool for the simulation of particles and droplets suspended in turbulent flows of academic and industrial applications. The study of some properties such as the preferential concentration of inertial particles in regions of high shear and low vorticity requires the computation of autocorrelation functions. This can be a tedious task as the discrete point particles need to be projected in some manner to obtain the continuous autocorrelation functions.Projection of particle properties on to a computational grid, for instance, the grid of the carrier phase, is furthermore an issue when quantities such as particle concentrations are to be computed or source terms between the carrier phase and the particles are exchanged. The errors committed by commonly used projection methods are often unknown and are difficult to analyse. Grid and sampling size limit the possibilities in terms of precision per computational cost.Here, we present a spectral projection method that is not affected by sampling issues and addresses all of the above issues. The technique is only limited by computational resources and is easy to parallelize. The only visible drawback is the limitation to simple geometries and therefore limited to academic applications.The spectral projection method consists of a discrete Fourier‐transform of the particle locations. The Fourier‐transformed particle number density and momentum fields can then be used to compute the autocorrelation functions and the continuous physical space fields for the evaluation of the projection methods error. The number of Fourier components used to discretize the projector kernel can be chosen such that the corresponding characteristic length scale is as small as needed. This allows to study the phenomena of particle motion, for example, in a region of preferential concentration that may be smaller than the cell size of the carrier phase grid.The precision of the spectral projection method depends, therefore, only on the number of Fourier modes considered. Copyright © 2008 John Wiley & Sons, Ltd.

Fourier series expansion in a non-orthogonal system of coordinates for the simulation of 3D alternating current borehole resistivity measurements

We describe a method to simulate 3D resistivity borehole measurements acquired with alternating current (AC) logging instruments. It combines the use of a Fourier series expansion in a non-orthogonal system of coordinates with an existing 2D goal-oriented higher-order self-adaptive hp-finite element (FE) algorithm. The method is used to simulate measurements acquired with both wireline and logging-while-drilling (LWD) borehole logging instruments in deviated wells. It enables a considerable reduction of the computational complexity with respect to available 3D simulators, since the number of Fourier modes (basis functions) needed to solve practical applications is limited (typically, below 13). The fast convergence of the method is studied via numerical experimentation by simulating two wireline and one LWD instruments in a deviated well across a possibly invaded formation. Numerical results confirm the efficiency and reliability of the method for simulating challenging 3D AC resistivity borehole problems in deviated wells, while avoiding the expensive construction of optimal 3D grids. We accurately simulate challenging electrodynamic problems within a few seconds (or minutes) of CPU time per logging position. The method is especially well-suited for inversion of triaxial electromagnetic (EM) measurements, since we demonstrate that the number of Fourier modes needed for the exact representation of the material function is limited to only one (the central mode) for the case of borehole measurements acquired in deviated wells.

Fourier series expansion in a non-orthogonal system of coordinates for the simulation of 3D-DC borehole resistivity measurements

We describe a new method to simulate 3D borehole resistivity measurements at zero frequency (DC). The method combines the use of a Fourier series expansion in a non-orthogonal system of coordinates with an existing 2D goal-oriented higher-order self-adaptive hp-finite element algorithm. The new method is suitable for simulating measurements acquired with borehole logging instruments in deviated wells. It delivers high-accuracy simulations and it enables a considerable reduction of the computational complexity with respect to available 3D simulators, since the number of Fourier modes (basis functions) needed to solve practical applications is limited (typically, below 10). Furthermore, numerical results indicate that the optimal 2D grid based on the 0th Fourier mode (also called central Fourier mode) can be employed to efficiently solve the final 3D problem, thereby, avoiding the expensive construction of optimal 3D grids. Specifically, for a challenging through-casing resistivity application, we reduce the computational time from several days (using a 3D simulator) to just 2 h (with the new method), while gaining accuracy. The new simulation method can be easily extended to different physical phenomena with similar geometries, as those arising in the simulation of 3D borehole electrodynamics and sonic (acoustics coupled with elasticity) measurements. In addition, the method is especially suited for inversion, since we demonstrate that the number of Fourier modes needed for the exact representation of the materials is limited to only one (the central mode) for the case of borehole measurements acquired in deviated wells.

Fractal Tomlinson model for mesoscopic friction: From microscopic velocity-dependent damping to macroscopic Coulomb friction

A modified Tomlinson equation with fractal potential is studied. The effective potential is numerically generated and its mesoscopic structure is gradually adjusted to different scales by a number of Fourier modes. It is shown that with the change of scale the intensity of velocity-dependent damping in an effective Langevin equation can be gradually substituted by an equivalent constant "dry friction." For smooth macrosopic surfaces the effective equation completely reduces to the well known Coulomb law.

Dynamical transition for a particle in a squared Gaussian potential

We study the problem of a Brownian particle diffusing in finite dimensions in a potential given by ψ = ϕ2/2 where ϕ is Gaussian random field. Exact results for the diffusion constant in the high temperature phase are given in one and two dimensions and it is shown to vanish in a power-law fashion at the dynamical transition temperature. Our results are confronted with numerical simulations where the Gaussian field is constructed, in a standard way, as a sum over random Fourier modes. We show that when the number of Fourier modes is finite the low temperature diffusion constant becomes non-zero and has an Arrhenius form. Thus we have a simple model with a fully understood finite size scaling theory for the dynamical transition. In addition we analyse the nature of the anomalous diffusion in the low temperature regime and show that the anomalous exponent agrees with that predicted by a trap model.

The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains

This paper is the second part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In the first part of this series, the Fourier Singular Complement Method was introduced and analysed, in prismatic domains. In this second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges or vertices of the domain, nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used.

The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains

This is the first part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In this first part, the Fourier Singular Complement Method is introduced and analysed, in prismatic domains. In the second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges of the domain nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used.

A Nonstiff Euler Discretization of the Complex Ginzburg--Landau Equation in One Space Dimension

A nonstiff discretization method is applied to the complex Ginzburg--Landau (CGL) equation with periodic boundary conditions, \begin{eqnarray*} &\partial_t u = (1+i\nu) \partial_{xx}^2 u + Ru - (1+i\mu)|u|^2 u ; \\ &u(x + 2\pi,t) = u(x,t) \quad\mbox{ for } -\infty < x < \infty,\; t\geq 0. \end{eqnarray*} This parabolic equation is discretized first in space by a truncated Fourier series (that is, a finite Fourier sum) and then in time by an explicit Euler method. The exponential decay of Fourier modes in both, the original CGL equation, and the resulting equation for the truncated Fourier series (independently from the number of Fourier modes in the truncation) is used in an essential way to prove the nonstiffness. Also dissipativity of the discretized equation is established and numerical results are discussed.

Quantifying Age‐Related Differences in Human Reaching while Interacting with a Rehabilitation Robotic Device

New movement assessment and data analysis methods are developed to quantify human arm motion patterns during physical interaction with robotic devices for rehabilitation. These methods provide metrics for future use in diagnosis, assessment and rehabilitation of subjects with affected arm movements. Specifically, the current study uses existing pattern recognition methods to evaluate the effect of age on performance of a specific motion, reaching to a target by moving the end‐effector of a robot (an X‐Y table). Differences in the arm motion patterns of younger and older subjects are evaluated using two measures: the principal component analysis similarity factor (SPCA) to compare path shape and the number of Fourier modes representing 98% of the path ‘energy’ to compare the smoothness of movement, a particularly important variable for assessment of pathologic movement. Both measures are less sensitive to noise than others previously reported in the literature and preserve information that is often lost through other analysis techniques. Data from the SPCA analysis indicate that age is a significant factor affecting the shapes of target reaching paths, followed by reaching movement type (crossing body midline/not crossing) and reaching side (left/right); hand dominance and trial repetition are not significant factors. Data from the Fourier‐based analysis likewise indicate that age is a significant factor affecting smoothness of movement, and movements become smoother with increasing trial number in both younger and older subjects, although more rapidly so in younger subjects. These results using the proposed data analysis methods confirm current practice that age‐matched subjects should be used for comparison to quantify recovery of arm movement during rehabilitation. The results also highlight the advantages that these methods offer relative to other reported measures.

The attractor dimension of a quasi-geostrophic two-layer system

Abstract The change of the attractor dimension (Kaplan-Yorke dimension) with an increase in the number of modes included in the Fourier expansion is investigated for a quasi-geostrophic two-layer system with topographic forcing. A particular case has been chosen such that the attractor is represented by a single “weather regime”, consisting of weak fluctuations around a fixed flow pattern. In this respect, the large-scale flow structure isleast “chaotic”. Anticipation is that this kind of organized flow may be described by a low dimensional attractor. No convergence of the dimension is found by increasing the total number of Fourier modes up to 210. The result suggests that the quasi-geostrophic system cannot be reduced to a low-dimensional dynamical system in general. The crucial importance of a large number of small scale modes to predict the planetary scale dynamics, is suggested.

Fourier Methods with Extended Stability Intervals for the Korteweg–de Vries Equation

A full leap frog Fourier method for integrating the Korteweg–de Vries (KdV) equation $u_t + uu_x - \varepsilon u_{xxx} = 0$ results in an $O(N^{ - 3} )$ stability constraint on the time step, where N is the number of Fourier modes used. This stability limit is much more restrictive than the accuracy limit for many applications. In this paper we propose a method for which the stability limit is extended by treating the linear dispersive $u_{xxx} $ term implicitly. Thus timesteps can be taken up to an accuracy limit larger than the explicit stability limit. The implicit method is implemented without solving linear systems by integrating in time in the Fourier space and discretizing the nonlinear $uu_x $ term by leap frog. A second method we propose uses basis functions which solve the linear part of the KdV equation and leap frog for time integration. A linearized stability analysis of the proposed schemes proves that a version of the first scheme possesses a certain kind of unconditional stability and that the second scheme has an $O(N^{ - 1} )$ stability limit. The accuracy of the schemes for soliton propagation is analyzed by examining the truncation error. In addition, we analyze a linearization of a nonlinear finite element scheme proposed by Winther that treats the $u_{xxx} $ term by Crank–Nicolson. Numerical experiments on soliton solutions show that the linearized stability analysis gives accurate predictions for all the nonlinear schemes, that the truncation errors give an estimate for the accuracies and that the Fourier methods are more accurate than the finite element method.

The linear growth of localized disturbances in unstable compressible flows

SynopsisAccurate computation of the evolution of initially localized disturbances in compressible parallel flows is a tedious task requiring superposition of a large number of Fourier modes with differing temporal growth rates. An alternative approximate method, similar to that developed by Craik (1981, 1982) for viscous incompressible flows, is presented here. This involves asymptotic evaluation, by the saddle point method, of a double Fourier integral representation of the disturbance, with the actual dispersion relation replaced by a simpler analytic expression containing several parameters which may be adjusted to approximate the flow under investigation. Limiting cases yield informative results in simple closed form: these exemplify the possible shapes into which the disturbance may evolve. In particular, ‘splitting’ of the disturbance into two dominant regions is demonstrated.