Spectral Difference Method Research Articles

An Optimal Weighted Spectral Difference Method for Faulty Vibration Separation in Flexible Joint Robot

An Optimal Weighted Spectral Difference Method for Faulty Vibration Separation in Flexible Joint Robot

Influences of Variability in Attenuation Compensation on the Estimation of Backscatter Coefficient of Median Nerves in Vivo.

Peripheral nerves remain a challenging target for medical imaging, given their size, anatomical complexity, and structural heterogeneity. Quantitative ultrasound (QUS) applies a set of techniques to estimate tissue acoustic parameters independent of the imaging platform. Many useful medical and laboratory applications for QUS have been reported, but challenges remain for deployment in vivo, especially for heterogeneous tissues. Several phenomena introduce variability in attenuation estimates, which may influence the estimation of other QUS parameters. For example, estimating the backscatter coefficient (BSC) requires compensation for the attenuation of overlying tissues between the transducer and the underlying tissue of interest. The purpose of this study is to extend prior studies by investigating the efficacy of several analytical methods of estimating attenuation compensation on QUS outcomes in the human median nerve. Median nerves were imaged at the volar wrist in vivo and beam-formed radiofrequency (RF) data were acquired. Six analytical approaches for attenuation compensation were compared: 1-2) attenuation estimated by applying spectral difference method (SDM) and spectral log difference method (SLDM) independently to regions of interest (ROIs) overlying the nerve and to the nerve ROI itself; 3-4) attenuation estimation by applying SDM and SLDM to ROIs overlying the nerve, and transferring these properties to the nerve ROI; and 5-6) methods that apply previously published values of tissue attenuation to the measured thickness of each overlying tissue. Mean between-subject estimates of BSC-related outcomes as well as within-subject variability of these outcomes were compared among the 6 methods. Compensating for attenuation using SLDM and values from the literature reduced variability in BSC-based outcomes, compared to SDM. Variability in attenuation coefficients contributes substantially to variability in backscatter measurements. This work has implications for the application of QUS to in vivo diagnostic assessments in peripheral nerves and possibly other heterogeneous tissues.

Enhancing ultrasonic attenuation images through multi-frequency coupling with total nuclear variation.

Quantitative ultrasound is a non-invasive image modality that numerically characterizes tissues for medical diagnosis using acoustical parameters, such as the attenuation coefficient slope. A previous study introduced the total variation spectral log difference (TVSLD) method, which denoises spectral log ratios on a single-channel basis without inter-channel coupling. Therefore, this work proposes a multi-frequency joint framework by coupling information across frequency channels exploiting structural similarities among the spectral ratios to increase the quality of the attenuation images. A modification based on the total nuclear variation (TNV) was considered. Metrics were compared to the TVSLD method with simulated and experimental phantoms and two samples of fibroadenoma in vivo breast tissue. The TNV demonstrated superior performance, yielding enhanced attenuation coefficient slope maps with fewer artifacts at boundaries and a stable error. In terms of the contrast-to-noise ratio enhancement, the TNV approach obtained an average percentage improvement of 34% in simulation, 38% in the experimental phantom, and 89% in two in vivo breast tissue samples compared to TVSLD, showing potential to enhance visual clarity and depiction of attenuation images.

Galerkin spectral and finite difference methods for the solution of fourth-order time fractional partial integro-differential equation with a weakly singular kernel

Galerkin spectral and finite difference methods for the solution of fourth-order time fractional partial integro-differential equation with a weakly singular kernel

Efficient dimension-by-dimension adaptive TENO-based limiter and troubled-cell indicator for nodal-based high-order spectral difference method

A new targeted essentially non-oscillatory (TENO) limiter with adaptive dissipation has been developed for the 3rd- and 4th-order spectral difference method. This limiter can potentially be useful for other nodal-based discontinuous high-order methods as well. Unlike traditional WENO limiters for these methods which limit low-order moments one by one, the new limiter is based on a direct convex combination of reconstruction polynomial candidates. This strategy ensures the good computational efficiency for high order reconstructions. The reconstruction stencils only involve nodal values from the target cell and its neighbors sharing faces with it. The reconstruction employs Hermite polynomials, making the new limiter compact. It can be implemented in a dimension-by-dimension manner for multi-dimensional problems, which is consistent with the spectral difference method. To further enhance efficiency, a TENO-based troubled-cell indicator is also developed to only activate limiters in troubled cells. Extensive one-dimensional and two-dimensional numerical experiments validate the performance of the new TENO-based limiter and indicator. In summary, the limiter is much less dissipative than WENO limiters and can resolve richer small-scale flow structures on coarse meshes. The indicator precisely marks out troubled cells, slightly improving over the KXRCF indicator.

GPU Implementation and Optimization of a High-Order Spectral Difference Method for Aeroacoustic Problems

GPU Implementation and Optimization of a High-Order Spectral Difference Method for Aeroacoustic Problems

Distributed ultra large strain measurement range sensing based on spectral shift adjacent point difference method in OFDR

Compared with the time-domain distributed optical fiber sensing technology, optical frequency domain reflectometry (OFDR) has the advantages of high spatial resolution and low detection optical power requirements. However, in the large strain measurement, OFDR encountered such a problem: the correlation of the Rayleigh scattering spectra between strain signals and reference signals is relatively low if the loading strain is too large, resulting in incorrect demodulation. To address this issue, we propose a new method called the spectral shift adjacent point difference method. In our approach, the cross-correlation operations using two sets of non-strain signals are performed, and the average value of the cross-correlation’s spectral shift is selected as a reference standard. Then the spectral shift of the entire sensing fiber is obtained by re-sweeping frequency when the strain is loaded along the sensing fiber. The data points with a spectral shift adjacent point difference less than the reference standard will be retained, while other data points with spectral shift adjacent point difference greater than the reference standard will be considered as noisy fake spikes and removed. Finally, interpolation fitting is performed on the retained data points along the entire sensing fiber to obtain the demodulated spectral shifts realize large strain measurement. The experimental results show that the maximum measurable strain by the proposed method can reach 10,800 µε with a spatial resolution of 7.84 mm, which is an order of magnitude higher than the conventional method. The experiment demonstrates the high stability and excellent repeatability of the proposed processing method. The proposed method has the advantage of simple algorithm and easy implementation, and making it more competent in the field of large strain measurement.

Artificial viscosity-based shock capturing scheme for the Spectral Difference method on simplicial elements

An artificial viscosity-based shock capturing scheme is extended to the context of high-order triangular Spectral Difference method for solving gas dynamics problems featuring discontinuities and shock waves. The equations are regularised thanks to an artificial diffusivity method combined with a shock sensor based on dilatation and vorticity fields valid for any polynomial order of approximation. The methodological novelty of this paper is an L2 projection based iterative Restriction Prolongation filtering operation introduced for high-order triangular elements. This filtering operation applied to the artificial bulk viscosity and thermal conductivity stabilises the simulations and improves the quality of the shock capturing approach. The methodology is validated on various canonical relevant compressible test cases. Numerical results demonstrate the performance and the flexibility of the triangular high-order Spectral Difference method for solving simultaneously compressible flows and shock waves.

Unsteady oblique stagnation point flow with improved pressure field and fractional Cattaneo–Christov model by finite difference-spectral method

Unsteady oblique stagnation point flow with improved pressure field and fractional Cattaneo–Christov model by finite difference-spectral method

Quantitative ultrasound techniques for assessing thermal ablation: Measurement of the backscatter coefficient from ex vivo human liver.

Understanding the changes occurring in biological tissue during thermal ablation is at the heart of many current challenges in both therapy and medical imagingresearch. The objective of this work is to quantitatively interpret the scattering response of human liver samples, before and after thermal ablation. We report acoustic measurements performed involving n = 21 human liver samples. Thermal ablation is achieved at temperatures between 45 and 80°C and quantification of the irreversible changes in acoustic attenuation and Backscattering Coefficient (BSC) is reported, with a particular attention to thelatter. Both attenuation coefficient and BSCs were measured in the frequency range from 10 to 52 MHz. Scans were performed before heating and after cooling down. Attenuation coefficients were calculated using spectral difference method and BSC estimated using the reference phantommethod. Strong increases of attenuation coefficients and BSCs with heating temperature were observed. Quantitative ultrasonic parameters obtained with the polydisperse structure factor model (poly-SFM)are compared to histological observations and seen to be close to hepatocyte mean diameter (HMD). The results presented in this study provide a description of the impact of thermal ablation in human liver tissue on acoustic attenuation and the BSC. For the first time, quantitative agreement between the Effective Scatterer Diameter (ESD) estimated from BSC and HMD was shown, highlighting the important role of cellular network in the scattering response of the medium. This core result is an important step toward the determination of the nature of scattering sources in biologicaltissues.

Fully-discrete spatial eigenanalysis of discontinuous spectral element methods: Insights into well-resolved and under-resolved vortical flows

This study presents a comprehensive spatial eigenanalysis of fully-discrete discontinuous spectral element methods, now generalising previous spatial eigenanalysis that did not include time integration errors. The influence of discrete time integration is discussed in detail for different explicit Runge–Kutta (1st to 4th order accurate) schemes combined with either Discontinuous Galerkin (DG) or Spectral Difference (SD) methods, both here recovered from the Flux Reconstruction (FR) scheme. Selected numerical experiments using the improved SD method by Liang et al. (2009) [53,54] and Jameson (2010) [55] are performed to quantify the influence of time integration errors on actual simulations. These involve test cases of varied complexity, from one-dimensional linear advection equation studies to well-resolved and under-resolved inviscid vortical flows. When simulations are well-resolved, the overall order of accuracy of the (fully-discrete) method of choice is limited to that of the time integration scheme. Moreover, it is shown that, while both well-resolved and under-resolved simulations of linear problems correlate well with the eigenanalysis prediction of time integration errors, the correlation can be much worse for under-resolved nonlinear problems as observed via numerical experiments. In fact, in the numerical simulation of under-resolved vortical flows, the predominance of spatial errors made it practically impossible for time integration errors to be distinctly identified. As a result, the eigenanalysis predictions are expected to hold (even if partially) in direct numerical simulations of turbulence. This highlights that the interaction between space and time discretisation errors is more complex than otherwise anticipated, contributing to the current understanding about when eigenanalysis can effectively predict the behaviour of numerical errors in practical under-resolved nonlinear problems, including under-resolved turbulence computations.

Arbitrarily high-order accurate simulations of compressible rotationally constrained convection using a transfinite mapping on cubed-sphere grids

We present two major improvements over the Compressible High-ORder Unstructured Spectral difference (CHORUS) code published in Wang et al., “A compressible high-order unstructured spectral difference code for stratified convection in rotating spherical shells,” J. Comput. Phys. 290, 90–111 (2015). The new code is named CHORUS++ in this paper. Subsequently, we perform a series of efficient simulations for rotationally constrained convection (RCC) in spherical shells. The first improvement lies in the integration of the high-order spectral difference method with a boundary-conforming transfinite mapping on cubed-sphere grids, thus ensuring exact geometric representations of spherical surfaces on arbitrary sparse grids. The second improvement is on the adoption of higher-order elements (sixth-order) in CHORUS++ vs third-order elements for the original CHORUS code. CHORUS++ enables high-fidelity RCC simulations using sixth-order elements on very coarse grids. To test the accuracy and efficiency of using elements of different orders, CHORUS++ is applied to a laminar solar benchmark, which is characterized by columnar banana-shaped convective cells. By fixing the total number of solution degrees of freedom, the computational cost per time step remains unchanged. Nevertheless, using higher-order elements in CHORUS++ resolves components of the radial energy flux much better than using third-order elements. To obtain converged predictions, using sixth-order elements is 8.7 times faster than using third-order elements. This significant speedup allows global-scale fully compressible RCC simulations to reach equilibration of the energy fluxes on a small cluster of just 40 cores. In contrast, CHORUS simulations were performed by Wang et al. on supercomputers using approximately 10 000 cores. Using sixth-order elements in CHORUS++, we further carry out global-scale solar convection simulations with decreased rotational velocities. Interconnected networks of downflow lanes emerge and surround broader and weaker regions of upflow fields. A strong inward kinetic energy flux compensated by an enhanced outward enthalpy flux appears. These observations are all consistent with those published in the literature. Furthermore, CHORUS++ can be extended to magnetohydrodynamic simulations with potential applications to the hydromagnetic dynamo processes in the interiors of stars and planets.

An Extended Range of Energy Stable Flux Reconstruction Methods on Triangles

We present an extended range of stable flux reconstruction (FR) methods on triangles through the development and application of the summation-by-parts framework in two-dimensions. This extended range of stable schemes is then shown to contain the single parameter schemes of Castonguay et al. (J Sci Comput 51:224–256, 2011) on triangles, and our definition enables wider stability bounds to be developed for those single parameter families. Stable upwinded spectral difference (SD) schemes on triangular elements have previously been found using Fourier analysis. We used our extended range of FR schemes to investigate the linear stability of SD methods on triangles, and it was found that a only first order SD scheme could be recovered within this set of FR methods.

Extending the Spectral Difference Method with Divergence Cleaning (SDDC) to the Hall MHD Equations

The Hall Magnetohydrodynamic (MHD) equations are an extension of the standard MHD equations that include the “Hall” term from the general Ohm’s law. The Hall term decouples ion and electron motion physically on the ion inertial length scales. Implementing the Hall MHD equations in a numerical solver allows more physical simulations for plasma dynamics on length scales less than the ion inertial scale length but greater than the electron inertial length. The present effort is an important step towards producing physically correct results to important problems, such as the Geospace Environmental Modeling (GEM) Magnetic Reconnection problem. The solver that is being modified is currently capable of solving the resistive MHD equations on unstructured grids using the spectral difference scheme which is an arbitrarily high-order method that is relatively simple to parallelize. The GEM Magnetic Reconnection problem is used to evaluate whether the Hall MHD equations have been correctly implemented in the solver using the spectral difference method with divergence cleaning (SDDC) algorithm by comparing against the reconnection rates reported in the literature.

Extension of the Spectral Difference Method to Premixed Laminar and Turbulent Combustion

Extension of the Spectral Difference Method to Premixed Laminar and Turbulent Combustion

On construction of shock-capturing boundary closures for high-order finite difference method

High-order finite difference method is widely used in the field of computational fluid dynamics. Since it can be implemented in a dimension-by-dimension fashion, it is much more efficient than some other methods in higher dimensions. However, although interior shock-capturing schemes are mature, shock-capturing boundary closures are only in their infancy due to the issue of insufficient stencils near boundary. To address this issue, we focus on a fifth-order multi-resolution weighted compact nonlinear scheme. Firstly, to enhance the shock-capturing capability of the interior scheme, we propose a new technique by adding a free parameter to the formulae of smoothness indicators. Then we follow the idea of the nonlinear spectral difference method to construct shock-capturing boundary closures when the interior scheme cannot be used. To validate the proposed method, several benchmark numerical examples are presented. The results show that the shock-capturing capability is effectively improved and the proposed nonlinear boundary closures work well for the cases with shock transmission or shock near boundary.

Spectral difference method with a posteriori limiting: application to the Euler equations in one and two space dimensions

ABSTRACTWe present a new numerical scheme which combines the spectral difference (SD) method up to arbitrary high order with a-posteriori limiting using the classical MUSCL-Hancock scheme as fallback scheme. It delivers very accurate solutions in smooth regions of the flow while capturing sharp discontinuities without spurious oscillations. We exploit the strict equivalence between the SD scheme and a finite-volume scheme based on the SD control volumes to enable a straightforward limiting strategy. At the end of each stage of our high-order time-integration ADER (Arbitrary high order using Derivatives) scheme, we check if the high-order solution is admissible under a number of numerical and physical criteria. If not, we replace the high-order fluxes of the troubled cells by fluxes from our robust second-order MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws) fallback scheme. We apply our method to a suite of test problems for the one-dimensional and two-dimensional Euler equations. We demonstrate that this combination of SD and ADER provides a virtually arbitrary high order of accuracy while at the same time preserving good sub-element shock capturing capabilities.

Monitoring and Evaluation of 2015 Devrek Zonguldak Landslide within the scope of Flood Risk Assessment by Landsat-8 Satellite Data

The mass movement towards stream beds and their accumulation there may cause flood events as a result of landslide events. In this study, Devrek Landslide, which occurred in 2015, was investigated by using a remote sensing technique. The Çomaklar stream is located at the lower elevations of this landslide region. Çomaklar stream and surrounding settlements are at risk of flooding due to the probability of preventing the water flow in the stream because of the effects of the landslide. Devrek district is located in Zonguldak in the northern part of Turkey. LANDSAT-8 satellite images between the dates 27 January 2015 and 22 July 2015 were used to investigate the location and size of the Devrek district landslide. Satellite-derived remote sensing data have been widely used in recent years to monitor changes on the earth's surface and to provide information. In this study, the Spectral Angle Difference method was used to compare the images before and after the landslide event. A change detection analysis was conducted between pre-landslide and post-landslide images to investigate the area affected by the disaster. As a result, the landslide was determined as 1050 meters in length and 110 meters in width, and the landslide area was calculated as 10.87 ha.

Magnetofluid unsteady electroosmotic flow of Jeffrey fluid at high zeta potential in parallel microchannels

Abstract In this article, the magnetofluid unsteady electroosmotic flow (EOF) of Jeffrey fluid with high zeta potential is studied by using the Chebyshev spectral method and the finite difference method. By comparing the potential distribution and velocity distribution obtained by the Chebyshev spectral method and finite difference method, it is concluded that the Chebyshev spectral method has higher precision and less computation. Then the numerical solution obtained by the Chebyshev spectral method is used to analyze the flow characteristics of Jeffrey fluid at high zeta potential. The results show that the velocity of Jeffrey fluid increases with the increase of the wall zeta potential and electric field intensity. The oscillation amplitude of velocity distribution increases with the increase of relaxation time, but decreases with the increase of retardation time. With the increase of Hartmann number, the velocity first increases and then decreases. The positive pressure gradient promotes the flow of fluid, and the reverse pressure gradient impedes the flow of fluid.

An Arbitrarily High-order Spectral Difference Method with Divergence Cleaning (SDDC) for Compressible Magnetohydrodynamic Simulations on Unstructured Grids

This paper reports a recent development of the high-order spectral difference method with divergence cleaning (SDDC) for accurate simulations of both ideal and resistive magnetohydrodynamics (MHD) on curved unstructured grids consisting of high-order isoparametric quadrilateral elements. The divergence cleaning approach is based on the improved generalized Lagrange multiplier, which is thermodynamically consistent. The SDDC method can achieve an arbitrarily high order of accuracy in spatial discretization, as demonstrated in the test problems with smooth solutions. The high-order SDDC method combined with the artificial dissipation method can sharply capture shock interfaces with the oscillation-free property and resolve small-scale vortex structures and density fluctuations on relatively sparse grids. The robustness of the codes is demonstrated through long time simulations of ideal MHD problems with progressively interacting shock structures, resistive MHD problems with high Lundquist numbers, and viscous resistive MHD problems on complex curved domains.