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474 records · Page 27

Ordering Unstructured Meshes for Sparse Matrix Computations on Leading Parallel Systems

The ability of computers to solve hitherto intractable problems and simulate complex processes using mathematical models makes them an indispensable part of modern science and engineering. Computer simulations of large-scale realistic applications usually require solving a set of non-linear partial differential equations (PDES) over a finite region. For example, one thrust area in the DOE Grand Challenge projects is to design future accelerators such as the SpaHation Neutron Source (SNS). Our colleagues at SLAC need to model complex RFQ cavities with large aspect ratios. Unstructured grids are currently used to resolve the small features in a large computational domain; dynamic mesh adaptation will be added in the future for additional efficiency. The PDEs for electromagnetics are discretized by the FEM method, which leads to a generalized eigenvalue problem Kx = AMx, where K and M are the stiffness and mass matrices, and are very sparse. In a typical cavity model, the number of degrees of freedom is about one million. For such large eigenproblems, direct solution techniques quickly reach the memory limits. Instead, the most widely-used methods are Krylov subspace methods, such as Lanczos or Jacobi-Davidson. In all the Krylov-based algorithms, sparse matrix-vector multiplication (SPMV) must be performed repeatedly. Therefore, the efficiency of SPMV usually determines the eigensolver speed. SPMV is also one of the most heavily used kernels in large-scale numerical simulations.

Oliker, Leonid↗

Low-Dimensional Dynamical Models of Thermal Convection

A low-dimensional dynamic model for transitional buoyancy-driven flow in a differentially heated tall enclosure is presented. The full governing partial differential equations with the associated boundary conditions are solved by a spectral element method for a cavity of aspect ratio A=20. Proper orthogonal decomposition is applied to the oscillatory solution at Prandtl number Pr=P tau (omega) = 0.71 and Grashof number G tau (omega) = 3.2 x 10 (exp 4) to construct empirical eigenfunctions. Using the four most energetic empirical eigenfunctions for the velocity and temperature as basis functions and applying Galerkin's method, a reduced model consisting of eight nonlinear ordinary differential equations is obtained. Close to the 'design' conditions (P tau(omega) G tau(omega)), the low-order model (LOM) predictions are in excellent agreement with the predictions of the full model. In particular, the critical Grashof number at the onset of the first temporal flow instability (Hopf bifurcation) was well as the frequency and amplitude of oscillations at supercritical conditions are in excellent agreement with the predictions of the full model. Far from the 'design' conditions, the LOM predicts the existence of multiple stable steady solutions at large values of G tau, and a unique stable steady solution at small values of G tau, and exhibits hysteretic behavior that is qualitatively similar to that observed in direct numerical simulations based on the full model.

Liakopoulos, Anthony↗

Rapid Airplane Parametric Input Design (RAPID)

RAPID is a methodology and software system to define a class of airplane configurations and directly evaluate surface grids, volume grids, and grid sensitivity on and about the configurations. A distinguishing characteristic which separates RAPID from other airplane surface modellers is that the output grids and grid sensitivity are directly applicable in CFD analysis. A small set of design parameters and grid control parameters govern the process which is incorporated into interactive software for 'real time' visual analysis and into batch software for the application of optimization technology. The computed surface grids and volume grids are suitable for a wide range of Computational Fluid Dynamics (CFD) simulation. The general airplane configuration has wing, fuselage, horizontal tail, and vertical tail components. The double-delta wing and tail components are manifested by solving a fourth order partial differential equation (PDE) subject to Dirichlet and Neumann boundary conditions. The design parameters are incorporated into the boundary conditions and therefore govern the shapes of the surfaces. The PDE solution yields a smooth transition between boundaries. Surface grids suitable for CFD calculation are created by establishing an H-type topology about the configuration and incorporating grid spacing functions in the PDE equation for the lifting components and the fuselage definition equations. User specified grid parameters govern the location and degree of grid concentration. A two-block volume grid about a configuration is calculated using the Control Point Form (CPF) technique. The interactive software, which runs on Silicon Graphics IRIS workstations, allows design parameters to be continuously varied and the resulting surface grid to be observed in real time. The batch software computes both the surface and volume grids and also computes the sensitivity of the output grid with respect to the input design parameters by applying the precompiler tool ADIFOR to the grid generation program. The output of ADIFOR is a new source code containing the old code plus expressions for derivatives of specified dependent variables (grid coordinates) with respect to specified independent variables (design parameters). The RAPID methodology and software provide a means of rapidly defining numerical prototypes, grids, and grid sensitivity of a class of airplane configurations. This technology and software is highly useful for CFD research for preliminary design and optimization processes.

Smith, Robert E.↗

Influence of Stationary Crossflow Modulation on Secondary Instability

A likely scenario for swept wing transition on subsonic aircraft with natural laminar flow involves the breakdown of stationary crossflow vortices via high frequency secondary instability. A majority of the prior research on this secondary instability has focused on crossflow vortices with a single dominant spanwise wavelength. This paper investigates the effects of the spanwise modulation of stationary crossflow vortices at a specified wavelength by a subharmonic stationary mode. Secondary instability of the modulated crossflow pattern is studied using planar, partial-differential-equation based eigenvalue analysis. Computations reveal that weak modulation by the first subharmonic of the input stationary mode leads to mode splitting that is particularly obvious for Y-type secondary modes that are driven by the wall-normal shear of the basic state. Thus, for each Y mode corresponding to the fundamental wavelength of results in unmodulated train of crossflow vortices, the modulated flow supports a pair of secondary modes with somewhat different amplification rates. The mode splitting phenomenon suggests that a more complex stationary modulation such as that induced by natural surface roughness would yield a considerably richer spectrum of secondary instability modes. Even modest levels of subharmonic modulation are shown to have a strong effect on the overall amplification of secondary disturbances, particularly the Z-modes driven by the spanwise shear of the basic state. Preliminary computations related to the nonlinear breakdown of these secondary disturbances provide interesting insights into the process of crossflow transition in the presence of the first subharmonic of the dominant stationary vortex.

Vortex structure↗

Template Matching Approach to Signal Prediction

A new approach to signal prediction and prognostic assessment of spacecraft health resolves an inherent difficulty in fusing sensor data with simulated data. This technique builds upon previous work that demonstrated the importance of physics-based transient models to accurate prediction of signal dynamics and system performance. While models can greatly improve predictive accuracy, they are difficult to apply in general because of variations in model type, accuracy, or intended purpose. However, virtually any flight project will have at least some modeling capability at its disposal, whether a full-blown simulation, partial physics models, dynamic look-up tables, a brassboard analogue system, or simple hand-driven calculation by a team of experts. Many models can be used to develop a predict, or an estimate of the next day s or next cycle s behavior, which is typically used for planning purposes. The fidelity of a predict varies from one project to another, depending on the complexity of the simulation (i.e. linearized or full differential equations) and the level of detail in anticipated system operation, but typically any predict cannot be adapted to changing conditions or adjusted spacecraft command execution. Applying a predict blindly, without adapting the predict to current conditions, produces mixed results at best, primarily due to mismatches between assumed execution of spacecraft activities and actual times of execution. This results in the predict becoming useless during periods of complicated behavior, exactly when the predict would be most valuable. Each spacecraft operation tends to show up as a transient in the data, and if the transients are misaligned, using the predict can actually harm forecasting performance. To address this problem, the approach here expresses the predict in terms of a baseline function superposed with one or more transient functions. These transients serve as signal templates, which can be relocated in time and space against the signal background. One then has the ability to reconstruct a signal regardless of the precise timing of the transients. During operation, one applies the actual start times of spacecraft activities as they occur, and produces a reconstructed, accurate predict in real-time. This general approach is valid under two important conditions. First, the transients themselves must be time-invariant. Second, the transients must be reasonably consistent with respect to different operating points. Both of these assumptions are generally valid, but in the case of a complicated system with numerous types of overlapping transients, this approach may not be effective. Fortunately, there tend to be few transients in spacecraft telemetry of sensor quantities or low-level health and status information because these signals rarely reflect multiple different types of operation. Furthermore, if the predict is at least reasonably close to actual operation, the shift either in time, or in the operating point when the transient occurs - is likely to be small. The proposed approach considers three ways to recognize a transient. The first and most reliable is to use a different signal that identifies operating mode - often the transient will be correlated to a change in operating mode, and this change can usually be detected positively from discrete signals in spacecraft telemetry. If there is no useful mode signal, the second option is to identify the transient template by hand, and detect the onset of the transient using a curve-fitting approach. Finally, if one elects not to choose by hand, there is an option for automatic selection. This technique has been applied to sensor data from several JPL missions and industrial applications, demonstrating an improvement in accurate prediction of future behavior and early detection of system problems. This technique is applicable to practically any time-varying, quantitative sensor measurement.

Mackey, Ryan↗

Small Particle Response to Fluid Motion Using Tethered Particles to Simulate Microgravity

This paper reports on ground based work conducted to support the Spaceflight Definition project SHIVA (Spaceflight Holography Investigation in a Virtual Apparatus). SHIVA will advance our understanding of the movement of a particle in a fluid. Gravity usually dominates the equations of motion, but in microgravity as well as on earth other terms can become important. Through an innovative application of fractional differential equations, two members of our team produced the first analytical solution of a fundamental equation of motion, which had only been solved numerically or by approximation before. The general solution predicts that the usually neglected history term becomes important in particle response to a sinusoidal fluid movement when the characteristic viscous time is in the same order as the fluid oscillation period and peaks when the two times are equal. In this case three force terms, the Stokes drag, the added mass, and the history drag must all be included in predicting particle movement. We have developed diagnostic recording methods using holography to save all of the particle field data, allowing the experiment to essentially be transferred from space back to earth in what we call the virtual apparatus for on-earth microgravity experimentation. We can quantify precisely the three-dimensional motion of sets of particles, allowing us to test and apply the new analytical solutions. We are examining the response of particles up to 2 mm radius to fluid oscillation at frequencies up to 80 Hz with amplitudes up to 200 microns. Ground studies to support the flight development program have employed various schemes to simulate microgravity. One of the most reliable and meaningful methods uses spheres tethered to a fine hair suspended in the fluid. We have also investigated particles with nearly neutral buoyancy. Recordings are made at the peak amplitudes of vibration of the cell providing a measure of the ratio of fluid to particle amplitude. The experiment requires precise location of the particle to within microns during recording, and techniques for achieving this are one of the project challenges. Focused microscopic images and diffraction patterns are used. To make the experiment more versatile, the spaceflight system will record holograms both on film and electronically. A cross correlation procedure enables sub pixel accuracies for electronic recordings, partially accommodating the lower spatial resolution of CCDs. The electronic holograms can be down linked providing real time data. Results of the ground experiments, the flight experiment design, and data analysis procedures are reported.

Trolinger, James↗