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NASA NTRS2014

On Formulations of Discontinuous Galerkin and Related Methods for Conservation Laws

A formulation for the discontinuous Galerkin (DG) method that leads to solutions using the differential form of the equation (as opposed to the standard integral form) is presented. The formulation includes (a) a derivative calculation that involves only data within each cell with no data interaction among cells, and (b) for each cell, corrections to this derivative that deal with the jumps in fluxes at the cell boundaries and allow data across cells to interact. The derivative with no interaction is obtained by a projection, but for nodal-type methods, evaluating this derivative by interpolation at the nodal points is more economical. The corrections are derived using the approximate (Dirac) delta functions. The formulation results in a family of schemes: different approximate delta functions give rise to different methods. It is shown that the current formulation is essentially equivalent to the flux reconstruction (FR) formulation. Due to the use of approximate delta functions, an energy stability proof simpler than that of Vincent, Castonguay, and Jameson (2011) for a family of schemes is derived. Accuracy and stability of resulting schemes are discussed via Fourier analyses. Similar to FR, the current formulation provides a unifying framework for high-order methods by recovering the DG, spectral difference (SD), and spectral volume (SV) schemes. It also yields stable, accurate, and economical methods.

Huynh, H. T.↗

NASA NTRS1982

Seeding considerations for an LV system in a large transonic wind tunnel

When it was decided to use a laser velocimeter to measure the properties of propellers, seeding was a great concern since large particles fail to flow and small ones are too small to be seen. Many methods were tried and weeded out by using a Malvern particle sizer. The most promising ones were tested in the tunnel and the laser velocimeter (LV) measurements compared to theoretical values of veocity as the particle approached a blunt nose body along a stagnation streamline. Data obtained from the LV system were compared with the one dimensional particle lag calculation. This figure showed the theoretical velocity over the blunt nose and a velocity profile for 5 um particles. This indicated the particles were approximately 5 um. The seeding method is shown. The seed was atomized by 2 seeders run with all 12 available atomizer jets on. The atomizer seed traveled from these two seeders through four 1 inch tubes 20 feet long to the plenum chamber where this cluster of tubes injected the seed into the air stream. The tubes were located 60 feet from the model and could be moved only by shutting the tunnel down. Future seeding plans are shown.

Freedman, R. J.↗

NASA NTRS1985

A study of internal and distributed damping for vibrating turbomachiner blades

Internal and distributed damping as possible methods for reducing the vibration response of turbomachine blades and theoretical methods for analyzing damped vibration were studied. It is demonstrated how the Ritz-Galerkin methods may be used to straightforwardly to analyze forced vibrations with damping. This is done directly without requiring the free vibration eigenfunctions. The Galerkin method is an effective technique for these types of problems. The Ritz method has the further advantage of not needing to satisfy the force type boundary conditions, which is particularly important for plates and shells. But proper functionals representing the forcing and damping terms must be developed, and this is done. Two types of damping--viscous and material (hysteretic) are considered. Both distributed and concentrated exciting forces are treated. Numerical results are obtained for cantilevered beams and rectangular plates. Studies showing the rates of convergence of the solutions are made. In the case of the cantilever beam, approximate solutions from the present methods are compared with the exact solutions.

Leissa, A. W.↗

NASA NTRSDate not supplied

Machine Learning Approaches for Rare-Earth Silicate Environmental Barrier Coating Thermochemical and Thermomechanical Property Predictions

Environmental barrier coatings (EBCs) are a necessary enabling technology for the transition from superalloys to silicon carbide (SiC) ceramic matrix composites (CMCs) in gas turbine engines for increased efficiency and decreased fuel costs. SiC-based CMCs are prone to oxidation-based degradation in the engine hot section, and rare-earth (RE) silicates are promising candidates for EBCs due to their close thermal expansion match to the composite substrate and oxidation resistance. However, the design of EBCs is hindered by the large chemical space of candidate materials and the difficulty in obtaining material properties for engineering optimization. This is especially difficult as research continues into mixed-cation or “high-entropy” RE silicates. First-principles computational methods such as density functional theory (DFT) are highly effective at calculating material properties to guide coating design but are limited by their computational cost. Atomistic simulations have the potential to both accelerate property calculations and expand the properties able to be calculated due to their lower computational compared to DFT. However, they require interatomic potentials (IAPs) specific to the material system of interest, and, to our knowledge, there are no suitable IAPs for RE silicates. Machine learning (ML) is a promising technique to accelerate material property predictions indirectly by generating IAPs for atomistic simulations or via direct prediction. In this work, we present two ML approaches to accelerate the calculation of RE silicate properties relevant to EBC design: 1) a ML-derived interatomic potential (IAP) for atomistic simulations of yttrium disilicate (Y2Si2O7) from DFT training data, and 2) a neural network (NN) model to directly predict thermochemical properties of RE silicates and oxides directly from easily obtainable unit cell parameters. Classical MD simulations using the IAP yield lattice properties and bond lengths in good agreement with both DFT and experimental results from x-ray diffraction. Thermodynamic properties calculated using the finite-displacement phonon method and quasi-harmonic approximation were orders of magnitude faster than DFT with good agreement to the DFT results. The IAP was also used to calculate properties such as coefficient of thermal expansion (CTE) that require large simulation supercells and are therefore difficult with DFT. The IAP correctly predicted the anisotropic nature of the CTE in three different phases of Y2Si2O7. The NN model predicts constant pressure heat capacity, Cp, orders of magnitude faster than DFT calculations, which can enable its use as a surrogate model for multiscale simulations. The two methods presented in this work demonstrate the utility of ML for accelerating the prediction of RE silicate properties, which can in turn accelerate EBC design and optimization.

machine learning↗

NASA NTRS1993

Generation and exploration of aggregation abstractions for scheduling and resource allocation

This paper presents research on the abstraction of computational theories for scheduling and resource allocation. The paper describes both theory and methods for the automated generation of aggregation abstractions and approximations in which detailed resource allocation constraints are replaced by constraints between aggregate demand and capacity. The interaction of aggregation abstraction generation with the more thoroughly investigated abstractions of weakening operator preconditions is briefly discussed. The purpose of generating abstract theories for aggregated demand and resources includes: answering queries about aggregate properties, such as gross feasibility; reducing computational costs by using the solution of aggregate problems to guide the solution of detailed problems; facilitating reformulating theories to approximate problems for which there are efficient problem-solving methods; and reducing computational costs of scheduling by providing more opportunities for variable and value-ordering heuristics to be effective. Experiments are being developed to characterize the properties of aggregations that make them cost effective. Both abstract and concrete theories are represented in a variant of first-order predicate calculus, which is a parameterized multi-sorted logic that facilitates specification of large problems. A particular problem is conceptually represented as a set of ground sentences that is consistent with a quantified theory.

Lowry, Michael R.↗

NASA NTRS1988

Semi-discrete approximations to nonlinear systems of conservation laws; consistency and L(infinity)-stability imply convergence

A convergence theory for semi-discrete approximations to nonlinear systems of conservation laws is developed. It is shown, by a series of scalar counter-examples, that consistency with the conservation law alone does not guarantee convergence. Instead, a notion of consistency which takes into account both the conservation law and its augmenting entropy condition is introduced. In this context it is concluded that consistency and L(infinity)-stability guarantee for a relevant class of admissible entropy functions, that their entropy production rate belongs to a compact subset of H(loc)sup -1 (x,t). One can now use compensated compactness arguments in order to turn this conclusion into a convergence proof. The current state of the art for these arguments includes the scalar and a wide class of 2 x 2 systems of conservation laws. The general framework of the vanishing viscosity method is studied as an effective way to meet the consistency and L(infinity)-stability requirements. How this method is utilized to enforce consistency and stability for scalar conservation laws is shown. In this context we prove, under the appropriate assumptions, the convergence of finite difference approximations (e.g., the high resolution TVD and UNO methods), finite element approximations (e.g., the Streamline-Diffusion methods) and spectral and pseudospectral approximations (e.g., the Spectral Viscosity methods).

Tadmor, Eitan↗

NASA NTRS1984

Newton like: Minimal residual methods applied to transonic flow calculations

A computational technique for the solution of the full potential equation is presented. The method consists of outer and inner iterations. The outer iterate is based on a Newton like algorithm, and a preconditioned Minimal Residual method is used to seek an approximate solution of the system of linear equations arising at each inner iterate. The present iterative scheme is formulated so that the uncertainties and difficulties associated with many iterative techniques, namely the requirements of acceleration parameters and the treatment of additional boundary conditions for the intermediate variables, are eliminated. Numerical experiments based on the new method for transonic potential flows around the NACA 0012 airfoil at different Mach numbers and different angles of attack are presented, and these results are compared with those obtained by the Approximate Factorization technique. Extention to three dimensional flow calculations and application in finite element methods for fluid dynamics problems by the present method are also discussed. The Inexact Newton like method produces a smoother reduction in the residual norm, and the number of supersonic points and circulations are rapidly established as the number of iterations is increased.

Wong, Y. S.↗

NASA NTRS1985

Newton-like minimal residual methods applied to transonic flow calculations

Wong, Y. S.↗

NASA NTRS1980

Computation of infrared cooling rates in the water vapor bands

A fast and accurate method is developed for calculating the infrared radiative terms due to water vapor - specifically, the atmospheric cooling rates. The accuracy is achieved by avoiding the constraints of band models and working directly with the absorption coefficient, which is a function of temperature and pressure as well as wavenumber. The method is based on calculation of an equivalent water vapor amount between atmospheric pressure levels and a table look-up procedure. Compared to line-by-line calculations, the present method has errors up to 4% of the maximum cooling rate. The use of a scaling factor, based on the far-wing approximation, limits the applicability of the method to the troposphere and lower stratosphere, where the line wings are responsible for most of the radiative cooling associated with water vapor.

Chou, M.-D.↗

NASA NTRS1980

Analytical study of twin-jet shielding development of a 3-dimensional model

The solution for a point source impinging on a cylinder of heated flow is presented. The indefinite integral is solved approximately using a saddle of point method. Comparison of the three-dimensional model to a previously obtained two-dimensional model of twin jet noise indicate the the approximate solution of the integral is valid. The model was analyzed to differentiate among the mechanims of shielding. Zone in which diffraction and transmission dominate are identified. The model was found to compare to experimental shielding results.

Gerhold, C. H.↗

NASA NTRS1983

Measurements of the scattering of sound by a line vortex

This paper presents measurements of the phase and magnitude of the scattered field arising from the incidence of a monochromatic plane sound field as a steady vortex. The amplitude of the scattered field was found to vary linearly with the vortex strength, and with the incident wave amplitude and frequency as predicted by solutions based on the Born approximation. The scattered field was observed to be nonsingular in the incidence direction, and this was similar to predictions by the Parabolic Equation Method (PEM) rather than the Born approximation, which predicts singular behavior in the incidence direction.

Horne, W. C.↗

NASA NTRS1972

Calculation of heat transfer on shuttle type configurations including the effects of variable entropy at boundary layer edge

A relatively simple method is presented for including the effect of variable entropy at the boundary-layer edge in a heat transfer method developed previously. For each inviscid surface streamline an approximate shockwave shape is calculated using a modified form of Maslen's method for inviscid axisymmetric flows. The entropy for the streamline at the edge of the boundary layer is determined by equating the mass flux through the shock wave to that inside the boundary layer. Approximations used in this technique allow the heating rates along each inviscid surface streamline to be calculated independent of the other streamlines. The shock standoff distances computed by the present method are found to compare well with those computed by Maslen's asymmetric method. Heating rates are presented for blunted circular and elliptical cones and a typical space shuttle orbiter at angles of attack. Variable entropy effects are found to increase heating rates downstream of the nose significantly higher than those computed using normal-shock entropy, and turbulent heating rates increased more than laminar rates. Effects of Reynolds number and angles of attack are also shown.

Dejarnette, F. R.↗

NASA NTRS1984

Time dependent computation of the Euler equations for designing 2-D cascades, including the case of transonic shock free design

A numerical method to solve the inverse problem for airfoil cascades is described. The case of compressible, inviscid flow is considered. The proposed method is based on the finite difference approximation of the time-dependent Euler equations. Numerical examples show the capability of the method to design subsonic or transonic shockless blades.

Zannetti, L.↗

NASA NTRS2003

The Filtered Abel Transform and Its Application in Combustion Diagnostics

Many non-intrusive combustion diagnosis methods generate line-of-sight projections of a flame field. To reconstruct the spatial field of the measured properties, these projections need to be deconvoluted. When the spatial field is axisymmetric, commonly used deconvolution method include the Abel transforms, the onion peeling method and the two-dimensional Fourier transform method and its derivatives such as the filtered back projection methods. This paper proposes a new approach for performing the Abel transform method is developed, which possesses the exactness of the Abel transform and the flexibility of incorporating various filters in the reconstruction process. The Abel transform is an exact method and the simplest among these commonly used methods. It is evinced in this paper that all the exact reconstruction methods for axisymmetric distributions must be equivalent to the Abel transform because of its uniqueness and exactness. Detailed proof is presented to show that the two dimensional Fourier methods when applied to axisymmetric cases is identical to the Abel transform. Discrepancies among various reconstruction method stem from the different approximations made to perform numerical calculations. An equation relating the spectrum of a set of projection date to that of the corresponding spatial distribution is obtained, which shows that the spectrum of the projection is equal to the Abel transform of the spectrum of the corresponding spatial distribution. From the equation, if either the projection or the distribution is bandwidth limited, the other is also bandwidth limited, and both have the same bandwidth. If the two are not bandwidth limited, the Abel transform has a bias against low wave number components in most practical cases. This explains why the Abel transform and all exact deconvolution methods are sensitive to high wave number noises. The filtered Abel transform is based on the fact that the Abel transform of filtered projection data is equal to an integral transform of the original projection data with the kernel function being the Abel transform of the filtering function. The kernel function is independent of the projection data and can be obtained separately when the filtering function is selected. Users can select the best filtering function for a particular set of experimental data. When the kernal function is obtained, it can be used repeatedly to a number of projection data sets (rovs) from the same experiment. When an entire flame image that contains a large number of projection lines needs to be processed, the new approach significantly reduces computational effort in comparison with the conventional approach in which each projection data set is deconvoluted separately. Computer codes have been developed to perform the filter Abel transform for an entire flame field. Measured soot volume fraction data of a jet diffusion flame are processed as an example.

Simons, Stephen N.↗

NASA NTRS1997

Locating the Discontinuities of a Bounded Function by the Partial Sums of its Fourier Series I: Periodical Case

A key step for some methods dealing with the reconstruction of a function with jump discontinuities is the accurate approximation of the jumps and their locations. Various methods have been suggested in the literature to obtain this valuable information. In the present paper, we develop an algorithm based on identities which determine the jumps of a 2(pi)-periodic bounded not-too-highly oscillating function by the partial sums of its differentiated Fourier series. The algorithm enables one to approximate the locations of discontinuities and the magnitudes of jumps of a bounded function. We study the accuracy of approximation and establish asymptotic expansions for the approximations of a 27(pi)-periodic piecewise smooth function with one discontinuity. By an appropriate linear combination, obtained via derivatives of different order, we significantly improve the accuracy. Next, we use Richardson's extrapolation method to enhance the accuracy even more. For a function with multiple discontinuities we establish simple formulae which "eliminate" all discontinuities of the function but one. Then we treat the function as if it had one singularity following the method described above.

Kvernadze, George↗

NASA NTRS1987

Periodic solutions of second-order nonlinear difference equations containing a small parameter. IV - Multi-discrete time method

It is shown that a discrete multi-time method can be constructed to obtain approximations to the periodic solutions of a special class of second-order nonlinear difference equations containing a small parameter. Three examples illustrating the method are presented.

Mickens, Ronald E.↗

NASA NTRS1993

Development of programmable artificial neural networks

Conventionally programmed digital computers can process numbers with great speed and precision, but do not easily recognize patterns or imprecise or contradictory data. Instead of being programmed in the conventional sense, artificial neural networks are capable of self-learning through exposure to repeated examples. However, the training of an ANN can be a time consuming and unpredictable process. A general method is being developed to mate the adaptability of the ANN with the speed and precision of the digital computer. This method was successful in building feedforward networks that can approximate functions and their partial derivatives from examples in a single iteration. The general method also allows the formation of feedforward networks that can approximate the solution to nonlinear ordinary and partial differential equations to desired accuracy without the need of examples. It is believed that continued research will produce artificial neural networks that can be used with confidence in practical scientific computing and engineering applications.

Meade, Andrew J.↗

NASA NTRS1992

Development of a 3-D upwind PNS code for chemically reacting hypersonic flowfields

Two new parabolized Navier-Stokes (PNS) codes were developed to compute the three-dimensional, viscous, chemically reacting flow of air around hypersonic vehicles such as the National Aero-Space Plane (NASP). The first code (TONIC) solves the gas dynamic and species conservation equations in a fully coupled manner using an implicit, approximately-factored, central-difference algorithm. This code was upgraded to include shock fitting and the capability of computing the flow around complex body shapes. The revised TONIC code was validated by computing the chemically-reacting (M(sub infinity) = 25.3) flow around a 10 deg half-angle cone at various angles of attack and the Ames All-Body model at 0 deg angle of attack. The results of these calculations were in good agreement with the results from the UPS code. One of the major drawbacks of the TONIC code is that the central-differencing of fluxes across interior flowfield discontinuities tends to introduce errors into the solution in the form of local flow property oscillations. The second code (UPS), originally developed for a perfect gas, has been extended to permit either perfect gas, equilibrium air, or nonequilibrium air computations. The code solves the PNS equations using a finite-volume, upwind TVD method based on Roe's approximate Riemann solver that was modified to account for real gas effects. The dissipation term associated with this algorithm is sufficiently adaptive to flow conditions that, even when attempting to capture very strong shock waves, no additional smoothing is required. For nonequilibrium calculations, the code solves the fluid dynamic and species continuity equations in a loosely-coupled manner. This code was used to calculate the hypersonic, laminar flow of chemically reacting air over cones at various angles of attack. In addition, the flow around the McDonnel Douglas generic option blended-wing-body was computed and comparisons were made between the perfect gas, equilibrium air, and the nonequilibrium air results.

Tannehill, J. C.↗