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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Tuning Multigrid Methods with Robust Optimization and Local Fourier Analysis

Local Fourier analysis is a useful tool for predicting and analyzing the performance of many efficient algorithms for the solution of discretized PDEs, such as multigrid and domain decomposition methods. The crucial aspect of local Fourier analysis is that it can be used to minimize an estimate of the spectral radius of a stationary iteration, or the condition number of a preconditioned system, in terms of a symbol representation of the algorithm. In practice, this is a “minimax” problem, minimizing with respect to solver parameters the appropriate measure of work, which involves maximizing over the Fourier frequency. Often, several algorithmic parameters may be determined by local Fourier analysis in order to obtain efficient algorithms. Analytical solutions to minimax problems are rarely possible beyond simple problems; the status quo in local Fourier analysis involves grid sampling, which is prohibitively expensive in high dimensions. In this paper, we propose and explore optimization algorithms to solve these problems efficiently. Finally, several examples, with known and unknown analytical solutions, are presented to show the effectiveness of these approaches.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Monolithic Multigrid for a Reduced-Quadrature Discretization of Poroelasticity

Advanced finite-element discretizations and preconditioners for models of poroelasticity have attracted significant attention in recent years. The equations of poroelasticity offer significant challenges in both areas, due to the potentially strong coupling between unknowns in the system, saddle-point structure, and the need to account for wide ranges of parameter values, including limiting behavior such as incompressible elasticity. This paper was motivated by an attempt to develop monolithic multigrid preconditioners for the discretization developed in [C. Rodrigo et al., Comput. Methods App. Mech. Engrg, 341 (2018), pp. 467--484]; we show here why this is a difficult task and, as a result, we modify the discretization in [Rodrigo et al.] through the use of a reduced-quadrature approximation, yielding a more “solver-friendly” discretization. Local Fourier analysis is used to optimize parameters in the resulting monolithic multigrid method, allowing a fair comparison between the performance and costs of methods based on Vanka and Braess--Sarazin relaxation. Further, numerical results are presented to validate the local Fourier analysis predictions and demonstrate efficiency of the algorithms. Finally, a comparison to existing block-factorization preconditioners is also given.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Nonoverlapping block smoothers for the Stokes equations

Overlapping block smoothers efficiently damp the error contributions from highly oscillatory components within multigrid methods for the Stokes equations but they are computationally expensive. This paper is concentrated on the development and analysis of new block smoothers for the Stokes equations that are discretized on staggered grids. These smoothers are nonoverlapping and therefore desirable due to reduced computational costs. Traditional geometric multigrid methods are based on simple pointwise smoothers. However, using multigrid methods to efficiently solve more difficult problems such as the Stokes equations leads to computationally more expensive smoothers, for example, overlapping block smoothers. Nonoverlapping smoothers are less expensive, but have been considered less efficient in the literature. In this paper, we develop new nonoverlapping smoothers, the so-called triad-wise smoothers, and show their efficiency within multigrid methods to solve the Stokes equations. In addition, we compare overlapping and nonoverlapping smoothers by measuring their computational costs and analyzing their behavior by the use of local Fourier analysis.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Low-order preconditioning of the Stokes equations

A well-known strategy for building effective preconditioners for higher-order discretizations of some PDEs, such as Poisson's equation, is to leverage effective preconditioners for their low-order analogs. In this work, we show that high-quality preconditioners can also be derived for the Taylor–Hood discretization of the Stokes equations in much the same manner. In particular, we investigate the use of geometric multigrid based on the Q 1 iso Q 2 /Q 1 discretization of the Stokes operator as a preconditioner for the Q 2 /Q 1 discretization of the Stokes system. We utilize local Fourier analysis to optimize the damping parameters for Vanka and Braess–Sarazin relaxation schemes and to achieve robust convergence. Furthermore, these results are then verified and compared against the measured multigrid performance. While geometric multigrid can be applied directly to the Q 2 /Q 1 system, our ultimate motivation is to apply algebraic multigrid within solvers for Q 2 /Q 1 systems via the Q 1 iso Q 2 /Q 1 discretization, which will be considered in a companion paper.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Overlapping Schwarz Methods Are Not Anisotropy‐Robust Multigrid Smoothers

We analyze overlapping multiplicative Schwarz methods as smoothers in the geometric multigrid solution of two-dimensional anisotropic diffusion problems. For diffusion equations, it is well known that the smoothing properties of point-wise smoothers, such as Gauss Seidel, rapidly deteriorate as the strength of anisotropy increases. On the other hand, global smoothers based on line smoothing are known to generally provide good smoothing for diffusion problems, independent of the anisotropy strength. Here, a natural question is whether global methods are really necessary to achieve good smoothing in such problems, or whether it can be obtained with locally overlapping block smoothers using sufficiently large blocks and overlap. Through local Fourier analysis and careful numerical experimentation, we show that global methods are indeed necessary to achieve anisotropy-robust smoothing. Specifically, for any fixed block size bounded sufficiently far away from the global domain size, we find that the smoothing properties of overlapping multiplicative Schwarz rapidly deteriorate with increasing anisotropy, irrespective of the amount of overlap between blocks. Moreover, our results indicate that anisotropy-robust smoothing requires blocks of diameter 𝒪⁡(𝜖 −1/2 ) for anisotropy ratio 𝜖 ∈(0,1] .

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1996

Wavelet Techniques Applied to Modeling Transitional/Turbulent Flows in Turbomachinery

Computer simulation is an essential part of the design and development of jet engines for the aeropropulsion industry. Engineers concerned with calculating the flow in jet engine components, such as compressors and turbines, need simple engineering models that accurately describe the complex flow of air and gases and that allow them to quickly estimate loads, losses, temperatures, and other design parameters. In this ongoing collaborative project, advanced wavelet analysis techniques are being used to gain insight into the complex flow phenomena. These insights, which cannot be achieved by commonly used methods, are being used to develop innovative new flow models and to improve existing ones. Wavelet techniques are very suitable for analyzing the complex turbulent and transitional flows pervasive in jet engines. These flows are characterized by intermittency and a multitude of scales. Wavelet analysis results in information about these scales and their locations. The distribution of scales is equivalent to the frequency spectrum provided by commonly used Fourier analysis techniques; however, no localization information is provided by Fourier analysis. In addition, wavelet techniques allow conditional sampling analyses of the individual scales, which is not possible by Fourier methods.

Source record↗
NASA NTRS1994

Comparison of low-latitude ion and neutral zonal drifts using DE 2 data

We have used data from the ion drift meter and the wind and temperature spectrometer on the DE 2 spacecraft to make statistical comparisons of the zonal ion and neutral drifts at dip latitudes (DLAT) in the +/- 35 deg range over all local times. Fourier analysis indicates that the superrotation and the diurnal components of both flows are strongly peaked at the dip equator, with the superrotation term becoming negative for the absolute value DLAT greater than or equal 20 deg. One interesting feature is the presence of a period (2200-0500 solar local time) in the 300-400 km altitude region near the dip equator where the ion drift is more strongly eastward than the neutral flow. This would seem to indicate the presence of an electric field source of greater strength than the F region dynamo elsewhere along the geomagnetic field line. Model calculations indicate that a possible mechanism for this source lies in the vertical shear in the zonal neutral wind in the 100-200 km altitude region.

Coley, W. R.↗
NASA NTRS2012

Local Guided Wavefield Analysis for Characterization of Delaminations in Composites

Delaminations in composite laminates resulting from impact events may be accompanied by minimal indication of damage at the surface. As such, inspection techniques are required to ensure defects are within allowable limits. Conventional ultrasonic scanning techniques have been shown to effectively characterize the size and depth of delaminations but require physical contact with the structure. Alternatively, a noncontact scanning laser vibrometer may be used to measure guided wave propagation in the laminate structure. A local Fourier domain analysis method is presented for processing guided wavefield data to estimate spatially-dependent wavenumber values, which can be used to determine delamination depth. The technique is applied to simulated wavefields and results are analyzed to determine limitations of the technique with regards to determining defect size and depth. Finally, experimental wavefield data obtained in quasi-isotropic carbon fiber reinforced polymer (CFRP) laminates with impact damage is analyzed and wavenumber is measured to an accuracy of 8.5% in the region of shallow delaminations. Keywords: Ultrasonic wavefield imaging, Windowed Fourier transforms, Guided waves, Structural health monitoring, Nondestructive evaluation

Rogge, Matthew D.↗
NASA NTRS1978

On several aspects and applications of the multigrid method for solving partial differential equations

Several aspects of multigrid methods are briefly described. The main subjects include the development of very efficient multigrid algorithms for systems of elliptic equations (Cauchy-Riemann, Stokes, Navier-Stokes), as well as the development of control and prediction tools (based on local mode Fourier analysis), used to analyze, check and improve these algorithms. Preliminary research on multigrid algorithms for time dependent parabolic equations is also described. Improvements in existing multigrid processes and algorithms for elliptic equations were studied.

Dinar, N.↗
NASA NTRS1986

Solution of elliptic PDEs by fast Poisson solvers using a local relaxation factor

A large class of two- and three-dimensional, nonseparable elliptic partial differential equations (PDEs) is presently solved by means of novel one-step (D'Yakanov-Gunn) and two-step (accelerated one-step) iterative procedures, using a local, discrete Fourier analysis. In addition to being easily implemented and applicable to a variety of boundary conditions, these procedures are found to be computationally efficient on the basis of the results of numerical comparison with other established methods, which lack the present one's: (1) insensitivity to grid cell size and aspect ratio, and (2) ease of convergence rate estimation by means of the coefficient of the PDE being solved. The two-step procedure is numerically demonstrated to outperform the one-step procedure in the case of PDEs with variable coefficients.

Chang, Sin-Chung↗
DOE OSTI2022

Mass Transport in Membrane Systems: Flow Regime Identification by Fourier Analysis

The numerical calculation of local mass distributions in membrane systems by computational fluid dynamics (CFD) offers indispensable benefits. However, the concept to calculate such distributions in response to separate variations of operation conditions (OCs) makes it difficult to address overall, flow-physics-related questions, which require the consideration of the collective interaction of OCs. It is shown that such understanding-related relationships can be obtained by the analytical solution of the advection–diffusion equation considered. A Fourier series model (FSM) is presented, which provides exact solutions of an advection–diffusion equation for a wide range of OCs. On this basis, a new zeroth-order model is developed, which is very simple and as accurate as the complete FSM for all conditions of practical relevance. Advection-dominated blocked and diffusion-dominated unblocked flow regimes are identified (depending on a Péclet number which compares the flow geometry with a length scale imposed by the flow), which implies relevant requirements for the use of lab results for pilot- and full-scale applications. Analyses reveal the equivalence of variations of OCs, which offers a variety of options to accomplish desired flow regime changes.

Heinz, Stefan (ORCID:0000000248712416)↗
DOE OSTI2020

Windowed Fourier transform and cross-correlation algorithms for molecular tagging velocimetry

In this study, simulated and experimental molecular tagging velocimetry (MTV) images have been analyzed with a technique commonly used to process grid images on surfaces, the windowed Fourier transform with local spectrum analysis (WFT-LSA). A systematic synthetic image study of the modulation transfer function (MTF) and error tendencies of the WFT-LSA was performed and compared with a PIV-style cross-correlation algorithm to see if advanced strategies such as iterative image deformation can improve analysis of gridded images with high noise levels. Testing of single-pass algorithms showed that in typical MTV images, the WFT-LSA yields significantly lower bias errors than cross-correlation (CC) at displacements greater than 1 pixel but slightly higher random error at all displacements and image conditions. Analysis of the MTF shows that CC provided better resolution of spatial fluctuations than the WFT-LSA in many combinations of grid size and interrogation window. Tests of image deformation algorithms showed that the gap in performance between CC and WFT-LSA is maintained even as both methods improve. Additionally, WFT-LSA and CC methods are applied to real MTV experiments in high-speed gas jet flows. A preliminary analysis of phosphorescence lifetime provided by acetone vapor excited at 266 nm is used for assessing the required gas speeds for making MTV application feasible. The application of WFT-LSA to real MTV images demonstrates the ability of the algorithm to handle further real-world effects that could not be considered in the synthetic image analysis, like reduced signal-to-noise ratio and non-uniform intensity of the tagging grid across the image introduced by the actual laser beam energy distribution. With experimental images, CC is more accurate with shear flows but less robust to high noise levels than WFT-LSA, as predicted by the synthetic image analysis.

42 ENGINEERING↗
NASA NTRS2003

Search for 1-10 Hz Modulations in Coronal Emission with SECIS During the August 11, 1999 Eclipse

Results of the search of the periodic changes of the 530.3 nm line intensity emitted by selected structures of the solar corona in the frequency range 1-10 Hz are presented. A set of 12728 images of the section of the solar corona extending from near the north pole to the south-west were taken simultaneously in the 530.3 nm ('green') line and white-light with the Solar Eclipse Coronal Imaging System (SECIS) during the 143-seconds-long totality of the 1999 August 11 solar eclipse observed in Shabla, Bulgaria. The time resolution of the collected data is better than 0.05-s and the pixel size is approximately 4 arcsecs. Using classical Fourier spectral analysis tools, we investigated temporal changes of the local 530.3 nm coronal line brightness in the frequency range 1-10 Hz of thousands of points within the field of view. The various photometric and instrumental effects have been extensively considered. We did not find any indisputable, statistically significant evidence of periodicities in any of the investigated points (at significance level alpha = 0.05 or better).

Rudawy, P.↗
NASA NTRS1973

Relaxation factors for supercritical flows.

Relaxation procedures for solution of steady supercritical transonic flows are investigated. Von Neumann (Fourier-mode) stability analysis is used to find bounds of relaxation factors. The bounds depend on local Mach number and local mesh aspect ratio. Long wave instability of Murman-Cole implicit method is indicated. Two new relaxation procedures are introduced. Both employ central differencing exclusively. Group velocities of Fourier modes are used to study signal propagation. It was found necessary to avoid or to damp out signals propagating upstream in supersonic zones in order to obtain physically meaningful transonic solutions. It appears that requirements of high rate of convergence, of stability and accuracy, are in conflict, and that a combination of relaxation methods must be used in order not to compromise the requirements.

Kentzer, C. P.↗
DOE OSTI2020

Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation

Here, we study reproducing kernel (RK) collocation method for peridynamic Navier equation. In the first part, we apply a linear RK approximation to both displacement and dilatation, and then back-substitute dilatation and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit for a fixed nonlocal interaction length and also to the local limit as nonlocal interactions vanish. The stability is shown by comparing the collocation scheme with the standard Galerkin scheme using Fourier analysis. In the second part, we apply the RK collocation to the quasi-discrete peridynamic Navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed. The analysis is carried out on a special family of rectilinear Cartesian grids for the RK collocation method with a designated kernel with finite support. We assume the Lamé parameters satisfy λ ≥ μ to avoid extra assumptions on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.

42 ENGINEERING↗
NASA NTRS2001

Performance of the Wavelet Decomposition on Massively Parallel Architectures

Traditionally, Fourier Transforms have been utilized for performing signal analysis and representation. But although it is straightforward to reconstruct a signal from its Fourier transform, no local description of the signal is included in its Fourier representation. To alleviate this problem, Windowed Fourier transforms and then wavelet transforms have been introduced, and it has been proven that wavelets give a better localization than traditional Fourier transforms, as well as a better division of the time- or space-frequency plane than Windowed Fourier transforms. Because of these properties and after the development of several fast algorithms for computing the wavelet representation of any signal, in particular the Multi-Resolution Analysis (MRA) developed by Mallat, wavelet transforms have increasingly been applied to signal analysis problems, especially real-life problems, in which speed is critical. In this paper we present and compare efficient wavelet decomposition algorithms on different parallel architectures. We report and analyze experimental measurements, using NASA remotely sensed images. Results show that our algorithms achieve significant performance gains on current high performance parallel systems, and meet scientific applications and multimedia requirements. The extensive performance measurements collected over a number of high-performance computer systems have revealed important architectural characteristics of these systems, in relation to the processing demands of the wavelet decomposition of digital images.

El-Ghazawi, Tarek A.↗