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DOE OSTI2023

An Efficient High-Order Solver for Diffusion Equations with Strong Anisotropy on Non-Anisotropy-Aligned Meshes

This paper concerns numerical solution of the diffusion equation with strong anisotropy on meshes not aligned with the anisotropic vector field. In order to resolve the numerical pollution for simulations on a non-anisotropy-aligned mesh and reduce the associated high computational cost we propose an effective preconditioner, extending our previous work. Similar to the anisotropy-aligned mesh case, we apply the auxiliary space preconditioning framework to design a preconditioner where a continuous finite element space is used as the auxiliary space for the discontinuous finite element space. The key component is an effective line smoother that can mitigate the high-frequency errors perpendicular to the magnetic field. We design a graph-based approach to find such a line smoother that is approximately perpendicular to the vector fields when the mesh does not align with the anisotropy. Finally, numerical experiments for several benchmark problems are presented, demonstrating the effectiveness and robustness of the proposed preconditioner when applied to Krylov iterative methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2016

Flat H Redundant Frangible Joint Development

Orion and Commercial Crew Program (CCP) Partners have chosen to use frangible joints for certain separation events. The joints currently available are zero failure tolerant and will be used in mission safety applications. The goal is to further develop a NASA designed redundant frangible joint that will lower flight risk and increase reliability. FY16 testing revealed a successful design in subscale straight test specimens that gained efficiency and supports Orion load requirements. Approach / Innovation A design constraint is that the redundant joint must fit within the current Orion architecture, without the need for additional vehicle modification. This limitation required a design that changed the orientation of the expanding tube assemblies (XTAs), by rotating them 90deg from the standard joint configuration. The change is not trivial and affects the fracture mechanism and structural load paths. To address these changes, the design incorporates cantilevered arms on the break plate. The shock transmission and expansion of the XTA applies force to these arms and creates a prying motion to push the plate walls outward to the point of structural failure at the notched section. The 2014 test design revealed that parts could slip during functioning wasting valuable energy needed to separate the structure with only a single XTA functioning. Dual XTA functioning fully separated the assembly showing a discrepancy can be backed up with redundancy. Work on other fully redundant systems outside NASA is limited to a few patents that have not been subjected to functionality testing Design changes to prevent unwanted slippage (with ICA funding in 2015) showed success with a single XTA. The main goal for FY 2016 was to send the new Flat H RFJ to WSTF where single XTA test failures occurred back in 2014. The plan was to gain efficiency in this design by separating the Flat H RFJ with thicker ligaments with dimensions baselined in 2014. Other modifications included geometry changes to better disperse loads paths and to minimize air gaps. The design additionally added more structural strength to enhance the structural limits in static loads testing. The design also implemented a smoother load line through the assembly. Results / Knowledge Gained The new Flat H RFJ successfully fractured at WSTF with thicker ligaments and lower cord size. Where failure to separate occurred earlier, there is now excessive energy available for structural separation. The new challenge to provide some structural support to prevent secondary fracturing of the break plate remains to be completed. This future work is being funded by the JSC Engineering Directorate in 2017 to elevate the TRL on curved Flat H RFJs that configure with the Orion Service Panel Separation. Additional funding from JSC Engineering will provide new design testing to avoid secondary fracturing.

Brown, Chris↗

NASA NTRS1991

Alternative smoothing algorithms for on-line estimation problems

In recent years, much emphasis has been placed on the Kalman filter to solve on-line estimation problems. Smoothers, which determine state estimates at a certain time t using measurements between 0 and T, have been used primarily in off-line estimation schemes. This paper presents alternative forms of smoothers that can be used for on-line estimation problems. The alternative smoothing algorithms provide state estimates which are more refined than Kalman filtered state estimates. The alternative smoothing algorithms are illustrated in two examples. The first example considers a linear discrete-time system while the second example considers a nonlinear continuous-time system with discrete-time measurements.

Baumgartner, Eric T.↗

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 NTRS1993

Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems

In this paper, we present an analysis of a multigrid method for nonsymmetric and/or indefinite elliptic problems. In this multigrid method various types of smoothers may be used. One type of smoother which we consider is defined in terms of an associated symmetric problem and includes point and line, Jacobi, and Gauss-Seidel iterations. We also study smoothers based entirely on the original operator. One is based on the normal form, that is, the product of the operator and its transpose. Other smoothers studied include point and line, Jacobi, and Gauss-Seidel. We show that the uniform estimates for symmetric positive definite problems carry over to these algorithms. More precisely, the multigrid iteration for the nonsymmetric and/or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine (but not depending on the number of multigrid levels).

Bramble, James H.↗

NASA NTRS1998

The Application of Reflected GPS Signals to Ocean Remote Sensing

The L-band broadcast signal from the Global Positioning System (GPS) which has reflected off of the sea surface is under study for use as a ocean, coastal and wetlands remote sensing tool. The reflected signal from a given GPS satellite is cross-correlated with the pseudorandom noise code uniquely identifying that satellite. The shape of this cross-correlation, ordinarily a very sharp triangle when tracking a direct line of sight signal, becomes wider and smoother as the mean square slope of the reflecting surface increases. It is proposed that the surface wind speed can be determined by matching the recorded shape of this cross-correlation to that predicted by theoretical models as a function of wind speed and direction. The significance of these effects increases with altitude of the receiver. Experimental data have been collected using a specially modified GPS receiver on aircraft and on a balloon at altitudes of up to 25 km. These data compare favorably with predictions of analytical models and demonstrate the dependence of the waveform shape on surface wind speed and receiver altitude. The advantages that this measurement technique has over conventional scatterometers is the small size, low cost and simplicity of the receiver hardware, no requirement for a transmitter, and the ability to simultaneously collect data from usually 10 or more points (from a low earth orbiting satellite). This number could if the Russian GLONASS (Global Navigation Satellite System) satellites are also considered as additional sources of radiation. Furthermore, the bistatic scattering geometry is complementary to the backscatter used by conventional scatterometers.

Garrison, James L.↗

NASA NTRS1999

Plane Smoothers for Multiblock Grids: Computational Aspects

Standard multigrid methods are not well suited for problems with anisotropic discrete operators, which can occur, for example, on grids that are stretched in order to resolve a boundary layer. One of the most efficient approaches to yield robust methods is the combination of standard coarsening with alternating-direction plane relaxation in the three dimensions. However, this approach may be difficult to implement in codes with multiblock structured grids because there may be no natural definition of global lines or planes. This inherent obstacle limits the range of an implicit smoother to only the portion of the computational domain in the current block. This report studies in detail, both numerically and analytically, the behavior of blockwise plane smoothers in order to provide guidance to engineers who use block-structured grids. The results obtained so far show alternating-direction plane smoothers to be very robust, even on multiblock grids. In common computational fluid dynamics multiblock simulations, where the number of subdomains crossed by the line of a strong anisotropy is low (up to four), textbook multigrid convergence rates can be obtained with a small overlap of cells between neighboring blocks.

Llorente, Ignacio M.↗

NASA NTRS1998

Directional Agglomeration Multigrid Techniques for High Reynolds Number Viscous Flow Solvers

A preconditioned directional-implicit agglomeration algorithm is developed for solving two- and three-dimensional viscous flows on highly anisotropic unstructured meshes of mixed-element types. The multigrid smoother consists of a pre-conditioned point- or line-implicit solver which operates on lines constructed in the unstructured mesh using a weighted graph algorithm. Directional coarsening or agglomeration is achieved using a similar weighted graph algorithm. A tight coupling of the line construction and directional agglomeration algorithms enables the use of aggressive coarsening ratios in the multigrid algorithm, which in turn reduces the cost of a multigrid cycle. Convergence rates which are independent of the degree of grid stretching are demonstrated in both two and three dimensions. Further improvement of the three-dimensional convergence rates through a GMRES technique is also demonstrated.

Source record↗

NASA NTRS1998

Directional Agglomeration Multigrid Techniques for High-Reynolds Number Viscous Flows

Mavriplis, Dimitri J.↗

NASA NTRS1993

Mapping robust parallel multigrid algorithms to scalable memory architectures

The convergence rate of standard multigrid algorithms degenerates on problems with stretched grids or anisotropic operators. The usual cure for this is the use of line or plane relaxation. However, multigrid algorithms based on line and plane relaxation have limited and awkward parallelism and are quite difficult to map effectively to highly parallel architectures. Newer multigrid algorithms that overcome anisotropy through the use of multiple coarse grids rather than line relaxation are better suited to massively parallel architectures because they require only simple point-relaxation smoothers. The parallel implementation of a V-cycle multiple semi-coarsened grid (MSG) algorithm or distributed-memory architectures such as the Intel iPSC/860 and Paragon computers is addressed. The MSG algorithms provide two levels of parallelism: parallelism within the relaxation or interpolation on each grid and across the grids on each multigrid level. Both levels of parallelism must be exploited to map these algorithms effectively to parallel architectures. A mapping of an MSG algorithm to distributed-memory architectures that demonstrate how both levels of parallelism can be exploited is described. The results is a robust and effective multigrid algorithm for distributed-memory machines.

Overman, Andrea↗

NASA NTRS1987

Improved beta-SiC heteroepitaxial films using off-axis Si substrates

All beta-SiC films grown using on-axis (001) Si substrates that have been examined with transmission electron microscopy exhibit a high density of interfacial twins, stacking faults, and antiphase disorder. The antiphase boundaries can be decorated by chemical etching, sputter etching, wet oxidation, and beta-SiC growth in the presence of diborane. All traces of antiphase disorder are eliminated when the heteroepitaxial growth is carried out on vicinal (001) Si substrates that are tilted 2 deg about a 110-line axis. In addition, growth on the off-axis Si produces beta-SiC films that are significantly smoother than on-axis films. The density of stacking faults is apparently unaffected by growth on the off-axis substrates.

Powell, J. A.↗

NASA NTRS1978

Quiet wind tunnel

Simple and inexpensive technique suppresses background noise generated by pores in wind tunnel wall lining and makes aerodynamic data more accurate and reliable. Porous walls are covered with wire-mesh screen. Screen offers smoother surface to airflow and damps vortexes and resonance caused by wall perforations; yet it provides enough open area for perforations to cancel shock waves generated by model.

Howard, P. W.↗

NASA NTRS1994

A Hubble Space Telescope ultraviolet spectrum of SN 1993J

We obtained a Hubble Space Telescope (HST) ultraviolet spectrum of the Type II supernova SN 1993J in M81 on 1993 April 15. The approximately 1650-2900 A region is smoother than observed for SN 1987A and SN 1992A and lacks strong P Cygni absorptions caused by iron peak element lines. Synthetic spectra calculated using a parameterized Local Thermodynamic Equilibrium (LTE) procedure and a simple model atmosphere do not fit the UV region. Radio observations suggest that SN 1993J is embedded in a thick circumstellar envelope. The UV spectra of other supernovae that are believed to have thick circumstellar envelopes also have approximately 1650-2900 A regions lacking in strong P Cygni absorptions. Interaction of supernova ejecta and circumstellar matter may cause the smooth UV spectrum. If so, UV observations of supernovae will provide insight into the circumstellar environment of the supernova progenitors.

Jeffery, David J.↗

NASA NTRS2001

Towards Optimal Multigrid Efficiency for the Navier-Stokes Equations

A fast multigrid solver for the steady incompressible Navier-Stokes equations is presented. Unlike time-marching schemes, this approach uses relaxation of the steady equations. Application of this method results in a discretization that correctly distinguishes between the advection and elliptic parts of the operator, allowing efficient smoothers to be constructed. Numerical solutions are shown for flow over a flat plate and a Karman-Trefftz airfoil. Using collective Gauss-Seidel line relaxation in both the vertical and horizontal directions, multigrid convergence behavior approaching that of O(N) methods is achieved. The computational efficiency of the numerical scheme is compared with that of a Runge-Kutta based multigrid method.

Swanson, R. C.↗

NASA NTRS1999

Behavior of Filters and Smoothers for Strongly Nonlinear Dynamics

The Kalman filter is the optimal filter in the presence of known gaussian error statistics and linear dynamics. Filter extension to nonlinear dynamics is non trivial in the sense of appropriately representing high order moments of the statistics. Monte Carlo, ensemble-based, methods have been advocated as the methodology for representing high order moments without any questionable closure assumptions. Investigation along these lines has been conducted for highly idealized dynamics such as the strongly nonlinear Lorenz model as well as more realistic models of the means and atmosphere. A few relevant issues in this context are related to the necessary number of ensemble members to properly represent the error statistics and, the necessary modifications in the usual filter situations to allow for correct update of the ensemble members. The ensemble technique has also been applied to the problem of smoothing for which similar questions apply. Ensemble smoother examples, however, seem to be quite puzzling in that results state estimates are worse than for their filter analogue. In this study, we use concepts in probability theory to revisit the ensemble methodology for filtering and smoothing in data assimilation. We use the Lorenz model to test and compare the behavior of a variety of implementations of ensemble filters. We also implement ensemble smoothers that are able to perform better than their filter counterparts. A discussion of feasibility of these techniques to large data assimilation problems will be given at the time of the conference.

Zhu, Yanqui↗

NASA NTRS1998

Robust Multigrid Smoothers for Three Dimensional Elliptic Equations with Strong Anisotropies

We discuss the behavior of several plane relaxation methods as multigrid smoothers for the solution of a discrete anisotropic elliptic model problem on cell-centered grids. The methods compared are plane Jacobi with damping, plane Jacobi with partial damping, plane Gauss-Seidel, plane zebra Gauss-Seidel, and line Gauss-Seidel. Based on numerical experiments and local mode analysis, we compare the smoothing factor of the different methods in the presence of strong anisotropies. A four-color Gauss-Seidel method is found to have the best numerical and architectural properties of the methods considered in the present work. Although alternating direction plane relaxation schemes are simpler and more robust than other approaches, they are not currently used in industrial and production codes because they require the solution of a two-dimensional problem for each plane in each direction. We verify the theoretical predictions of Thole and Trottenberg that an exact solution of each plane is not necessary and that a single two-dimensional multigrid cycle gives the same result as an exact solution, in much less execution time. Parallelization of the two-dimensional multigrid cycles, the kernel of the three-dimensional implicit solver, is also discussed. Alternating-plane smoothers are found to be highly efficient multigrid smoothers for anisotropic elliptic problems.

Llorente, Ignacio M.↗

NASA NTRS1991

Microwave simulations of a tropical rainfall system with a three-dimensional cloud model

A three-dinensional cloud model-microwave radiative transfer model combination is used to study the relations among the precipitation and other microphysical characteristics of a tropical oceanic squall line and the upwelling radiance at pertinent microwave frequencies. Complex brightness temperature-rain rate relations are evident at the full horizontal resolution (1.5 km) of the models, with spatial averaging producing smoother, shifter relations in most cases. Nonprecipitating cloud water is shown to be important in understanding the resulting distribution of brightness temperature. At the mature stage, convective portions of the cloud system are shown to produce different brightness temperature relations than the stratiform portion, primarily related to the distribution of cloud water. The evolution of the convective system from a small convective complex through its mature stage and the beginning of its dissipation also is shown to result in a variation of brightness temperature-rain relations, related to the distribution of cloud water and the evolution of ice in the precipitating system. The results of the study point to the need to take into account the evolution of nonprecipitating cloud water and precipitation-sized ice in the retrieval of rain from passive microwave space observations. This effect is evident for both the life cycle of individual convective elements and the life cycle of the convective system as a whole.

Adler, Robert A.↗

NASA NTRS1999

The Behavior of Filters and Smoothers for Strongly Nonlinear Dynamics

The Kalman filter is the optimal filter in the presence of known Gaussian error statistics and linear dynamics. Filter extension to nonlinear dynamics is non trivial in the sense of appropriately representing high order moments of the statistics. Monte Carlo, ensemble-based, methods have been advocated as the methodology for representing high order moments without any questionable closure assumptions (e.g., Miller 1994). Investigation along these lines has been conducted for highly idealized dynamics such as the strongly nonlinear Lorenz (1963) model as well as more realistic models of the oceans (Evensen and van Leeuwen 1996) and atmosphere (Houtekamer and Mitchell 1998). A few relevant issues in this context are related to the necessary number of ensemble members to properly represent the error statistics and, the necessary modifications in the usual filter equations to allow for correct update of the ensemble members (Burgers 1998). The ensemble technique has also been applied to the problem of smoothing for which similar questions apply. Ensemble smoother examples, however, seem to quite puzzling in that results of state estimate are worse than for their filter analogue (Evensen 1997). In this study, we use concepts in probability theory to revisit the ensemble methodology for filtering and smoothing in data assimilation. We use Lorenz (1963) model to test and compare the behavior of a variety implementations of ensemble filters. We also implement ensemble smoothers that are able to perform better than their filter counterparts. A discussion of feasibility of these techniques to large data assimilation problems will be given at the time of the conference.

Zhu, Yanqiu↗