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At least 325 records · Page 18

All sources NASA NTRS DOE OSTI

DOE OSTI2022

Comprehensive uncertainty quantification (UQ) for full engineering models by solving probability density function (PDF) equation

This report details a new method for propagating parameter uncertainty (forward uncertainty quantification) in partial differential equations (PDE) based computational mechanics applications. The method provides full-field quantities of interest by solving for the joint probability density function (PDF) equations which are implied by the PDEs with uncertain parameters. Full-field uncertainty quantification enables the design of complex systems where quantities of interest, such as failure points, are not known apriori. The method, motivated by the well-known probability density function (PDF) propagation method of turbulence modeling, uses an ensemble of solutions to provide the joint PDF of desired quantities at every point in the domain. A small subset of the ensemble is computed exactly, and the remainder of the samples are computed with approximation of the driving (dynamics) term of the PDEs based on those exact solutions. Although the proposed method has commonalities with traditional interpolatory stochastic collocation methods applied directly to quantities of interest, it is distinct and exploits the parameter dependence and smoothness of the dynamics term of the governing PDEs. The efficacy of the method is demonstrated by applying it to two target problems: solid mechanics explicit dynamics with uncertain material model parameters, and reacting hypersonic fluid mechanics with uncertain chemical kinetic rate parameters. A minimally invasive implementation of the method for representative codes SPARC (reacting hypersonics) and NimbleSM (finite- element solid mechanics) and associated software details are described. For solid mechanics demonstration problems the method shows order of magnitudes improvement in accuracy over traditional stochastic collocation. For the reacting hypersonics problem, the method is implemented as a streamline integration and results show very good accuracy for the approximate sample solutions of re-entry flow past the Apollo capsule geometry at Mach 30.

42 ENGINEERING↗

NASA NTRS1980

Steady state solutions to dynamically loaded periodic structures

The general problem of solving for the steady state (time domain) dynamic response (i.e., NASTRAN rigid format-8) of a general elastic periodic structure subject to a phase difference loading of the type encountered in traveling wave propagation problems was studied. Two types of structural configurations were considered; in the first type, the structure has a repeating pattern over a span that is long enough to be considered, for all practical purposes, as infinite; in the second type, the structure has structural rotational symmetry in the circumferential direction. The theory and a corresponding set of DMAP instructions which permits the NASTRAN user to automatically alter the rigid format-8 sequence to solve the intended class of problems are presented. Final results are recovered as with any ordinary rigid format-8 solution, except that the results are only printed for the typical periodic segment of the structure. A simple demonstration problem having a known exact solution is used to illustrate the implementation of the procedure.

Kalinowski, A. J.↗

NASA NTRS1990

Wave interactions in a three-dimensional attachment-line boundary layer

The three-dimensional boundary layer on a swept wing can support different types of hydrodynamic instability. Here attention is focused on the so-called 'spanwise instability' problem which occurs when the attachment-line boundary layer on the leading edge becomes unstable to Tollmien-Schlichting waves. In order to gain insight into the interactions that are important in that problem a simplified basic state is considered. This simplified flow corresponds to the swept attachemnt-line boundary layer on an infinite flat plate. The basic flow here is an exact solution of the Navier-Stokes equations and its stability to two-dimensional waves propagating along the attachment line can be considered exactly at finite Reynolds number. The corresponding problem is studied for oblique waves, and their interaction with two-dimensional waves is investigated. It is shown that there are two types of oblique wave which, if excited, cause the destabilization of the two-dimensional mode and the breakdown of the disturbed flow at a finite distance from the leading edge.

Hall, Philip↗

DOE OSTI2021

Critical percolation on the kagome hypergraph

In this report we study the percolation critical surface of the kagome lattice in which each triangle is allowed an arbitrary connectivity. Using the method of critical polynomials, we find points along this critical surface to high precision. This kagome hypergraph contains many unsolved problems as special cases, including bond percolation on the kagome and (3, 12 2 ) lattices, and site percolation on the hexagonal, or honeycomb, lattice, as well as a single point for which there is an exact solution. We are able to compute enough points along the critical surface to find a very accurate fit, essentially a Taylor series about the exact point, that allows estimations of the critical point of any system that lies on the surface to precision rivaling Monte Carlo and traditional techniques of similar accuracy. We find also that this system sheds light on some of the surprising aspects of the method of critical polynomials, such as why it is so accurate for certain problems, like the kagome and (3, 12 2 ) lattices. The bond percolation critical points of these lattices can be found to 17 and 18 digits, respectively, because they are in close proximity, in a sense that can be made quantitative, to the exact point on the critical surface. We also discuss in detail a parallel implementation of the method which we use here for a few calculations.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1987

Rapid temporal evolution of radiation from non-thermal electrons in solar flares

Solutions of the time dependent Fokker-Planck equation was found for accelerated electrons undergoing Coulomb collisions in a magnetized, fully ionized plasma. An exact solution was found for arbitrary pitch angle and energy distribution in a uniform background plasma. Then, for an inhomogeneous plasma, a solution was found for particles with small pitch angles. These solutions were used to calculate the temporal evolution of bremsstrahlung x-rays from short bursts of nonthermal electron beams, and these spectra were compared with observed high time resolution spectra of short timescale solar hard x-ray bursts. It is shown that the observed softening in time of the spectra rules out a homogeneous background and therefore the possibility of electrons being confined to the corona either because of converging magnetic field or high densities. The inhomogeneous solution was also applied to a model with constant coronal density and exponentially rising chromospheric density. The spectra are shown to be consistent with that produced by a collimated beam of electrons accelerated in the corona with certain given conditions. These conditions could be violated if large pitch angle electrons are present.

Lu, Edward T.↗

NASA NTRS1988

Rapid temporal evolution of radiation from nonthermal electrons in solar flares

Solutions of the time dependent Fokker-Planck equation was found for accelerated electrons undergoing Coulomb collisions in a magnetized, fully ionized plasma. An exact solution was found for arbitrary pitch angle and energy distribution in a uniform background plasma. Then, for an inhomogeneous plasma, a solution was found for particles with small pitch angles. These solutions were used to calculate the temporal evolution of bremsstrahlung X-rays from short bursts of nonthermal electron beams, and these spectra were compared with observed high time resolution spectra of short timescale solar hard X-ray bursts. It is shown that the observed softening in time of the spectra rules out a homogeneous background and therefore the possibility of electrons being confined to the corona either because of converging magnetic field or high densities. The inhomogeneous solution was also applied to a model with constant coronal density and exponentially rising chromospheric density. The spectra are shown to be consistent with that produced by a collimated beam of electrons accelerated in the corona with certain given conditions. These conditions could be violated if large pitch angle electrons are present.

Lu, Edward T.↗

NASA NTRS1975

Nonlinear acoustic behavior at a caustic - An approximate analytical solution

The present paper discusses an approximate analytical solution to the nonlinear behavior of a discontinuous acoustic signal near a caustic. The Seebass transformation (1970) is refined to provide results which satisfy the governing equation to any prescribed accuracy, except across the shock wave produced by reflection of the simple wave at the caustic. The solution is approximate in the sense that the basic equation is satisfied wherever the solution is continuous but can satisfy only one of the two jump conditions at the reflected shock. The results give essential geometric features of the exact solution and provide a quantitative estimate of the strength of the so-called superboom.

Gill, P. M.↗

DOE OSTI2021

Variational Discrete Action Theory

In this work, we propose the variational discrete action theory (VDAT) to study the ground state properties of quantum many-body Hamiltonians. VDAT is a variational theory based on the sequential product density matrix (SPD) ansatz, characterized by an integer $\mathscr{N}$, which monotonically approaches the exact solution with increasing $\mathscr{N}$. To evaluate the SPD, we introduce a discrete action and a corresponding integer time Green’s function. We use VDAT to exactly evaluate the SPD in two canonical models of interacting electrons: the Anderson impurity model and the d = ∞ Hubbard model. For the latter, we evaluate $\mathscr{N}$ = 2 – 4, where $\mathscr{N}$ = 2 recovers the Gutzwiller approximation (GA), and we show that $\mathscr{N}$ = 3, which exactly evaluates the Gutzwiller-Baeriswyl wave function, provides a truly minimal yet precise description of Mott physics with a cost similar to that of the GA. VDAT is a flexible theory for studying quantum Hamiltonians, competing both with state-of-the-art methods and simple, efficient approaches all within a single framework.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Comments on 'Do ultrafine cloud condensation nuclei invigorate deep convection?'

Here we elaborated on the deficiencies associated with the theoretical arguments and model simulations in Grawbowski and Morrison (2020; GM20), and showed that the concept of convective invigoration by aerosols can be supported by both accurate theoretical analysis and explicit physics modeling with prognostic aerosols and supersaturation. Above the freezing level, droplet freezing itself does not change mass loading and the latent heat release resulting from freezing leads to an increase in the buoyancy. The derivation in GM20 is erroneous because of omitting the liquid water sink term. They also ignored the effect of enhanced latent heating from the subsequent growth processes (deposition and riming) after freezing. Regarding aerosol effect on condensational growth, the quasi-steady assumption for supersaturation as adopted in GM20 is invalidated especially when droplet number concentration is low and updraft is strong from both theoretical analysis and our bin microphysics modeling results. The quasi-steady assumption makes condensation independent of droplet number and size and therefore reduces the overall aerosol effects on cloud dynamics. Studying aerosol effects on clouds particularly deep convective clouds requires the exact solution of supersaturation. Any assumption imposed to supersaturation such as saturation adjustment and quasi-steady approximation leads to incorrect aerosol effects on diffusional growth. The exact supersaturation equation clearly shows that condensation depends on droplet number and size, and more droplets in the polluted clouds increase condensation (thus lower supersaturation), leading to enhanced buoyancy and updraft speeds.

Convective invigoration, deep convective clouds, a↗

NASA NTRS1983

Direct determination of surface albedos from satellite imagery

An empirical method to measure the spectral surface albedo of surfaces from Landsat imagery is presented and analyzed. The empiricism in the method is due only to the fact that three parameters of the solution must be determined for each spectral photograph of an image on the basis of independently known albedos at three points. The approach is otherwise based on exact solutions of the radiative transfer equation for upwelling intensity. Application of the method allows the routine construction of spectral albedo maps from satelite imagery, without requiring detailed knowledge of the atmospheric aerosol content, as long as the optical depth is less than 0.75, and of the calibration of the satellite sensor.

Mekler, Y.↗

NASA NTRS1989

Reflections, diffractions, and surface waves for an interior impedance wedge of arbitrary angle

The asymptotic-impedance wedge solution for plane-wave illumination at normal incidence is examined for interior wedge diffraction. An efficient method for calculating the diffraction coefficient for arbitrary wedge angle is presented. The asymptotic solution isolates the incident, singly reflected, multiply reflected, diffracted, surface-wave, and associated-surface-wave transition fields. Multiply reflected fields (of any order) from the exact solution arise as ratios of auxiliary Maliuzhinets functions; however, by using properties of these functions, the representation can be reduced to products of reflection coefficients, much more efficient for calculation. A surface-wave transition field is added to the surface wave to retain continuity of the total field at the surface wave boundaries. This formulation is equally valid for both exterior and interior wedges with uniform but different impedances on each face, for both soft and hard polarizations.

Griesser, Timothy↗

NASA NTRS1990

An adaptive refinement procedure for transient thermal analysis using nodeless variable finite elements

An adaptive mesh refinement procedure that uses nodeless variables and quadratic interpolation functions is presented for analyzing transient thermal problems. A temperature based finite element scheme with Crank-Nicolson time marching is used to obtain the thermal solution. The strategies used for mesh adaption, computing refinement indicators, and time marching are described. Examples in one and two dimensions are presented and comparisons are made with exact solutions. The effectiveness of this procedure for transient thermal analysis is reflected in good solution accuracy, reduction in number of elements used, and computational efficiency.

Ramakrishnan, R.↗

NASA NTRS1992

Hierarchical flux-based thermal-structural finite element analysis method

A hierarchical flux-based finite element method is developed for both a one and two dimensional thermal structural analyses. Derivation of the finite element equations is presented. The resulting finite element matrices associated with the flux based formulation are evaluated in a closed form. The hierarchical finite elements include additional degrees of freedom in the approximation of the element variable distributions by the use of nodeless variables. The nodeless variables offer increased solution accuracy without the need for defining actual nodes and rediscretizing the finite element model. Thermal and structural responses are obtained from a conventional linear finite element method and exact solutions. Results show that the hierarchical flux-based method can provide improved thermal and structural solution accuracy with fewer elements when compared to results for the conventional linear element method.

Polesky, Sandra P.↗

NASA NTRS1987

Conservative finite volume solutions of a linear hyperbolic transport equation in two and three dimensions using multiple grids

The feasibility of the multiple grid technique is investigated by solving linear hyperbolic equations for simple two- and three-dimensional cases. The results are compared with exact solutions and those obtained from the single grid calculations. It is demonstrated that the technique works reasonably well when two grid systems contain grid cells of comparative sizes. The study indicates that use of the multiple grid does not introduce any significant error and that it can be used to attack more complex problems.

Tiwari, Surendra N.↗

NASA NTRS1985

A three-dimensional boundary-layer method for flow over delta wings with leading-edge separation

A three-dimensional, laminar boundary-layer method is applied to the incompressible flow over a slender delta wing at incidence. The predictor-corrector finite-difference scheme of Matsuno is used to difference the governing equations. The method has the advantages that no iterations are required to advance the solution and the cross-flow derivatives are formed independent of the cross-field direction. The difference scheme is demonstrated to yield accurate numerical results when compared to the exact solution of the three-dimensional boundary-layer equations for parabolic flow over a moving flat plate. The method is applied to delta wings of various sweep angles at angles of attack up to 20 deg., with the inviscid solution determined using a higher-order, three-dimensional panel method.

Woodson, S. H.↗

NASA NTRS1973

Analysis of heat transfer for a normally impinging liquid-metal slot jet

A two-dimensional liquid-metal slot jet that is impinging normally against a uniformly heated flat plate is analyzed. The distributions of wall temperature and heat-transfer coefficient are obtained as functions of position along the plate. The liquid-metal assumptions are made that the jet is inviscid and that molecular condition is dominating heat diffusion. The solution is obtained by mapping the jet flow region into a potential plane where it occupies a strip of uniform width. The energy equation is transformed into potential coordinates, and an exact solution obtained in the strip region. Conformal mapping is then used to transform the solution into the physical plane.

Siegel, R.↗

NASA NTRS1989

Comparison between pressure gradient method and MAC method on high Re calculation

A cavity flow driven by shear and buoyancy forces is used as a test problem in the application of a nonstaggered pressure gradient (PG) method in solving the two-dimensional incompressible Navier-Stokes equations. Twelve finite differencing schemes are used to solve the cavity flow problem. The schemes consist of various combinations of grid arrangements, upwinding treatments, and conservativeness of convection terms. An artificial source term is introduced, and the solutions are compared with those obtained by the conventional marker-and-cell (MAC) method. The comparisons favor the PG method. Numerical results obtained by the twelve schemes are compared with exact solutions in order to assess the stability and accuracy of each scheme.

Tan, C.-H.↗

NASA NTRS2000

Adjoint Techniques for Topology Optimization of Structures Under Damage Conditions

The objective of this cooperative agreement was to seek computationally efficient ways to optimize aerospace structures subject to damage tolerance criteria. Optimization was to involve sizing as well as topology optimization. The work was done in collaboration with Steve Scotti, Chauncey Wu and Joanne Walsh at the NASA Langley Research Center. Computation of constraint sensitivity is normally the most time-consuming step of an optimization procedure. The cooperative work first focused on this issue and implemented the adjoint method of sensitivity computation (Haftka and Gurdal, 1992) in an optimization code (runstream) written in Engineering Analysis Language (EAL). The method was implemented both for bar and plate elements including buckling sensitivity for the latter. Lumping of constraints was investigated as a means to reduce the computational cost. Adjoint sensitivity computation was developed and implemented for lumped stress and buckling constraints. Cost of the direct method and the adjoint method was compared for various structures with and without lumping. The results were reported in two papers (Akgun et al., 1998a and 1999). It is desirable to optimize topology of an aerospace structure subject to a large number of damage scenarios so that a damage tolerant structure is obtained. Including damage scenarios in the design procedure is critical in order to avoid large mass penalties at later stages (Haftka et al., 1983). A common method for topology optimization is that of compliance minimization (Bendsoe, 1995) which has not been used for damage tolerant design. In the present work, topology optimization is treated as a conventional problem aiming to minimize the weight subject to stress constraints. Multiple damage configurations (scenarios) are considered. Each configuration has its own structural stiffness matrix and, normally, requires factoring of the matrix and solution of the system of equations. Damage that is expected to be tolerated is local and represents a small change in the stiffness matrix compared to the baseline (undamaged) structure. The exact solution to a slightly modified set of equations can be obtained from the baseline solution economically without actually solving the modified system.. Shennan-Morrison-Woodbury (SMW) formulas are matrix update formulas that allow this (Akgun et al., 1998b). SMW formulas were therefore used here to compute adjoint displacements for sensitivity computation and structural displacements in damaged configurations.

Akgun, Mehmet A.↗