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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 271 records · Page 15
All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

MANN: A program to transfer designs for diffractive optical elements to a MANN photolithographic mask generator

There are two basic areas of interest for diffractive optics. In the first, the property of wavefront division is exploited for achieving optical fanout, analogous to the more familiar electrical fanout of electronic circuitry. The basic problem here is that when using a simple uniform diffraction grating the energy input is divided unevenly among the output beams. The other area of interest is the use of diffractive elements to replace or supplement standard refractive elements such as lenses. Again, local grating variations can be used to control the amount of bending imparted to optical rays, and the efficiency of the diffractive element will depend on how closely the element can be matched to the design requirements. In general, production restrictions limit how closely the element approaches the design, and for the common case of photolithographic production, a series of binary masks is required to achieve high efficiency. The actual design process is much more involved than in the case of elements for optical fanout, as the desired phase of the optical wavefront over some reference plane must be specified and the phase alteration to be introduced at each point by the diffraction element must be known. This generally requires the utilization of a standard optical design program. Two approaches are possible. In the first approach, the diffractive element is treated as a special type of lens and the ordinary optical design equations are used. Optical design programs tend to follow a second approach, namely, using the equations of optical interference derived from holographic theory and then allowing the introduction of phase front corrections in the form of polynomial equations. By using either of these two methods, diffractive elements can be used not only to compensate for distortions such as chromatic or spherical aberration, but also to perform the work of a variety of other optical elements such as null correctors, beam shapers, etc. The main focus of the project described in this report is how the design information from the lens design program is incorporated into the photolithographic process. It is shown that the MANN program, a photolithographic mask generator, fills the need for a link between lens design programs and mask generation controllers.The generated masks can be used to expose a resist-coated substrate which is etched and then must be re-coated, re-exposed, and re-etched for making copies, just as in the electronics industry.

Matthys, Donald R.↗
NASA NTRS1986

Higher order parabolic approximations of the reduced wave equation

Asymptotic solutions of order k to the nth are developed for the reduced wave equation. Here k is a dimensionless wave number and n is the arbitrary order of the approximation. These approximations are an extension of geometric acoustics theory, and provide corrections to that theory in the form of multiplicative functions which satisfy parabolic partial differential equations. These corrections account for the diffraction effects caused by variation of the field normal to the ray path and the interaction of these transverse variations with the variation of the field along the ray. The theory is applied to the example of radiation from a piston, and it is demonstrated that the higher order approximations are more accurate for decreasing values of k.

Mcaninch, G. L.↗
NASA NTRS1959

A Generalized Hydrodynamic-Impact Theory for the Loads and Motions of Deeply Immersed Prismatic Bodies

A theory is derived for determining the loads and motions of a deeply immersed prismatic body. The method makes use of a two-dimensional water-mass variation and an aspect-ratio correction for three-dimensional flow. The equations of motion are generalized by using a mean value of the aspect-ratio correction and by assuming a variation of the two-dimensional water mass for the deeply immersed body. These equations lead to impact coefficients that depend on an approach parameter which, in turn, depends upon the initial trim and flight-path angles. Comparison of experiment with theory is shown at maximum load and maximum penetration for the flat-bottom (0 deg dead-rise angle) model with bean-loading coefficients from 36.5 to 133.7 over a wide range of initial conditions. A dead-rise angle correction is applied and maximum-load data are compared with theory for the case of a model with 300 dead-rise angle and beam-loading coefficients from 208 to 530.

Markey, Melvin F.↗
NASA NTRS1960

Effect on Gaseous Film Cooling of Coolant Injection Through Angled Slots and Normal Holes

A study was made to determine the effect of coolant injection angularity on gaseous film-cooling effectiveness. In the correlation of experimental data an effective injection angle was defined by a vector summation of the coolant and mainstream gas flows. The cosine of this angle was used as a parameter to empirically develop a corrective term to qualify a correlating equation presented in Technical Note D-130 that was limited to tangential injection of the coolant. Data were also obtained for coolant injection through rows of holes normal to the test plate. The slot correlating equation was adapted to fit these data by the definition of an effective slot height. An additional corrective term was then determined to correlate these data.

Papell, S. Stephen↗
NASA NTRS1995

Convergence properties of finite-difference hydrodynamics schemes in the presence of shocks

We investigate the asymptotic convergence of finite-difference schemes for the Euler equations when the limiting solution contains shocks. The Lax-Wendroff theorem guarantees that certain conservative schemes converge to correct, physically valid solutions. We focus on two one-dimensional operator-split schemes with explicit artificial-viscosity terms. One, an internal-energy scheme, does not satisfy the assumptions of Lax-Wendroff; the other, a conservative total-energy scheme, does. With viscous lengths chosen proportional to the grid size, we find that both schemes converge to their zero-grid-size limits at the theoretically expected rate, but only the conversative scheme converges toward correct solutions of the inviscid fluid equations. We show that the difference in their behaviors results directly from the presence of shocks in the limiting solution. Empirically, we find that when the viscous lenghts tend toward zero more slowly than the grid size, however the nonconservative scheme also converges toward correct solutions. We characterize the asymptotic behavior of the total-energy scheme in a particular problem in which a shock forms. As the grid is refined, a Cauchy error approaches the expected rate of change slowly. We show that the changes in the artificial viscosity alter the diffusion of small-amplitude waves. The differences associated with such waves make the dominant contribution to the Cauchy error. We formulate an analytic model to relate the rate of approach to the effect of varying diffusion in waves and find quantitative agreement with our numerical results.

Kimoto, Paul A.↗
DOE OSTI2024

Quantum-gravitational null Raychaudhuri equation

Abstract We consider a congruence of null geodesics in the presence of a quantized spacetime metric. The coupling to a quantum metric induces fluctuations in the congruence; we calculate the change in the area of a pencil of geodesics induced by such fluctuations. For the gravitational field in its vacuum state, we find that quantum gravity contributes a correction to the null Raychaudhuri equation which is of the same sign as the classical terms. We thus derive a quantum-gravitational focusing theorem valid for linearized quantum gravity.

Physics↗
NASA NTRS1972

Orbit improvement from satellite imaging data obtainable from the Mariner/Jupiter/Saturn 1977 missions

Equations of motion are established for a dynamic system in which a spacecraft flies close to and interacts with an outer planet and one or more of its satellites. For the computation of the state and mass partials needed in the orbit corrections, a set of variational equations is derived. These assume a notably compact form through the introduction of a matrix operator D. The above system of differential equations is integrated numerically on a computer.

Aksnes, K.↗
DOE OSTI2022

Single-Logarithmic Corrections to Small-$x$ Helicity Evolution

The small- x x quark helicity evolution equations at double-logarithmic order, with the kernel \sim\alpha_s\ln^2(1/x) ∼ α s ln 2 ( 1 / x ) , have been derived previously. In this work, we derive the single-logarithmic corrections to the equations, to order \alpha_s\ln(1/x) α s ln ( 1 / x ) of the evolution kernel. The new equations include the effects of the running coupling and the unpolarized small- x x evolution, both of which are parametrically significant at single-logarithmic order. The large- N_c N c and large- N_c\ N_f N c N f approximations to the equation are computed. (Here, N_c N c and N_f N f are the numbers of quark colors and flavors, respectively.) Their solutions will provide more precise estimates of the quark helicity distribution at small x x , contributing to the resolution of the proton spin puzzle.

Tawabutr, Yossathorn↗
DOE OSTI2021

A variational multiscale immersed meshfree method for heterogeneous materials

Abstract We introduce an immersed meshfree formulation for modeling heterogeneous materials with flexible non-body-fitted discretizations, approximations, and quadrature rules. The interfacial compatibility condition is imposed by a volumetric constraint, which avoids a tedious contour integral for complex material geometry. The proposed immersed approach is formulated under a variational multiscale based formulation, termed the variational multiscale immersed method (VMIM). Under this framework, the solution approximation on either the foreground or the background can be decoupled into coarse-scale and fine-scale in the variational equations, where the fine-scale approximation represents a correction to the residual of the coarse-scale equations. The resulting fine-scale solution leads to a residual-based stabilization in the VMIM discrete equations. The employment of reproducing kernel (RK) approximation for the coarse- and fine-scale variables allows arbitrary order of continuity in the approximation, which is particularly advantageous for modeling heterogeneous materials. The effectiveness of VMIM is demonstrated with several numerical examples, showing accuracy, stability, and discretization efficiency of the proposed method.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Correction to “Frank–Kasper Phases of Diblock Copolymer Melts Studied with the DPD Model: SCF Results”

In a recent study, (1) we performed the self-consistent field (SCF) calculations of a dissipative particle dynamics (DPD) model of conformationally asymmetric diblock copolymer A-B melts to study the stability of various Frank–Kasper (FK) phases formed by such A-B melts. While all the equations in ref (1) are correct, we did not calculate the dimensionless (mean-field) Helmholtz free energies per chain βf c = βu c,χ + βu c,κ – s c /k B and the dimensionless stresses dβf c /dθ of ordered phases accurately. For example, the dimensionless (nonbonded) internal energy per chain due to the system compressibility βu c,κ = (N/2κ)∑ q βû 0 (|q|)(ϕ̂ A (q) + ϕ̂ B (q) – δ q,0 )(ϕ̂ A (−q) + ϕ̂ B (−q) – δ– q,0 ) should be calculated in the reciprocal space as given (note that βû 0 (|q|) is obtained analytically, and that since ϕ A (r) is real 𝜙̂ A (−𝐪)= $\overline{𝜙̂ A}$(𝐪), i.e., the complex conjugate of ϕ̂ A (q)); this corresponds to numerically evaluating βu c,κ = (N/2κ)∫dr(ϕ A (r) + ϕ B (r) – 1)ψ(r) with ψ(r) ≡ (∫dr′βu 0 (|r–r′|)(ϕ A (r′) + ϕ B (r′) – 1)/V in the real space via the (composite) trapezoidal rule. While the trapezoidal rule is less accurate than the Romberg integration (2) in general, the opposite occurs when the integration domain is periodic; this was pointed out by Matsen in ref (3) and explained in detail in refs (4and5). With our ignorance of this exceptional convergence property of the trapezoidal rule for periodic functions, we calculated all convolutions having the form of ψ(r) in the reciprocal space via the fast Fourier transforms (6) but those having the form of βu c,κ in the real space via Romberg integration; the latter leads to inaccurate results and unfortunately incorrect phase diagrams reported in ref (1), which are corrected here.

He, Juntong↗
NASA NTRS1993

Application and development of the Schwinger multichannel scattering theory and the partial differential equation theory of electron-molecule scattering

One version of the multichannel theory for electron-target scattering based on the Schwinger variational principle, the SMC method, requires the introduction of a projection parameter. The role of the projection parameter a is investigated and it is shown that the principal-value operator in the SMC equation is Hermitian regardless of the value of a as long as it is real and nonzero. In a basis that is properly orthonormalizable, the matrix representation of this operator is also Hermitian. The use of such basis is consistent with the Schwinger variational principle because the Lippmann-Schwinger equation automatically builds in the correct boundary conditions. Otherwise, an auxiliary condition needs to be introduced, and Takatsuka and McKoy's original value of a is one of the three possible ways to achieve Hermiticity. In all cases but one, a can be uncoupled from the Hermiticity condition and becomes a free parameter. An equation for a based on the variational stability of the scattering amplitude is derived; its solution has an interesting property that the scattering amplitude from a converged SMC calculation is independent of the choice of a even though the SMC operator itself is a-dependent. This property provides a sensitive test of the convergence of the calculation. For a static-exchange calculation, the convergence requirement only depends on the completeness of the one-electron basis, but for a general multichannel case, the a-invariance in the scattering amplitude requires both the one-electron basis and the N plus 1-electron basis to be complete. The role of a in the SMC equation and the convergence property are illustrated using two examples: e-CO elastic scattering in the static-exchange approximation, and a two-state treatment of the e-H2 Chi(sup 1)Sigma(sub g)(+) yields b(sup 3)Sigma(sub u)(+) excitation.

Weatherford, Charles A.↗
NASA NTRS1993

Stress stiffening and approximate equations in flexible multibody dynamics

A useful model for open chains of flexible bodies undergoing large rigid body motions, but small elastic deformations, is one in which the equations of motion are linearized in the small elastic deformations and deformation rates. For slow rigid body motions, the correctly linearized, or consistent, set of equations can be compared to prematurely linearized, or inconsistent, equations and to 'oversimplified,' or ruthless, equations through the use of open loop dynamic simulations. It has been shown that the inconsistent model should never be used, while the ruthless model should be used whenever possible. The consistent and inconsistent models differ by stress stiffening terms. These are due to zeroth-order stresses effecting virtual work via nonlinear strain-displacement terms. In this paper we examine in detail the nature of these stress stiffening terms and conclude that they are significant only when the associated zeroth-order stresses approach 'buckling' stresses. Finally it is emphasized that when the stress stiffening terms are negligible the ruthlessly linearized equations should be used.

Padilla, Carlos E.↗
NASA NTRS1986

Efficient numerical techniques for complex fluid flows

The central feature in any flow prediction method is the treatment of the coupling between the momentum and continuity equations. In natural-convection flows, the energy equation also becomes strongly coupled with the momentum equations. Because of the nonlinear nature of the coupling, these equations are solved iteratively. Iterative methods are often prone to slow convergence, divergence, and extreme sensitivity to underrelaxation factors. The aim of the present research is to develop more efficient and reliable solution schemes for the coupled flow equations. Such schemes will significantly reduce the expense of computing complex flows encountered in combustion chambers, gas turbines, heat exchangers, and other practical equipment. In the work completed so far, a technique employing norm reduction in conjunction with the successive-substitution and Newton-Raphson techniques was developed. Also, a block-correction procedure for the flow equations is currently being formulated and tested.

Patankar, Suhas V.↗
NASA NTRS1998

Analysis and Design of Variable Stiffness Composite Cylinders

An investigation of the possible performance improvements of thin circular cylindrical shells through the use of the variable stiffness concept is presented. The variable stiffness concept implies that the stiffness parameters change spatially throughout the structure. This situation is achieved mainly through the use of curvilinear fibers within a fiber-reinforced composite laminate, though the possibility of thickness variations and discrete stiffening elements is also allowed. These three mechanisms are incorporated into the constitutive laws for thin shells through the use of Classical Lamination Theory. The existence of stiffness variation within the structure warrants a formulation of the static equilibrium equations from the most basic principles. The governing equations include sufficient detail to correctly model several types of nonlinearity, including the formation of a nonlinear shell boundary layer as well as the Brazier effect due to nonlinear bending of long cylinders. Stress analysis and initial buckling estimates are formulated for a general variable stiffness cylinder. Results and comparisons for several simplifications of these highly complex governing equations are presented so that the ensuing numerical solutions are considered reliable and efficient enough for in-depth optimization studies. Four distinct cases of loading and stiffness variation are chosen to investigate possible areas of improvement that the variable stiffness concept may offer over traditional constant stiffness and/or stiffened structures. The initial investigation deals with the simplest solution for cylindrical shells in which all quantities are constant around the circumference of the cylinder. This axisymmetric case includes a stiffness variation exclusively in the axial direction, and the only pertinent loading scenarios include constant loads of axial compression, pressure, and torsion. The results for these cases indicate that little improvement over traditional laminates exists through the use of curvilinear fibers, mainly due to the presence of a weak link area within the stiffness variation that limits the ultimate load that the structure can withstand. Rigorous optimization studies reveal that even though slight increases in the critical loads can be produced for designs with an arbitrary variation of the fiber orientation angle, the improvements are not significant when compared to traditional design techniques that utilize ring stiffeners and frames. The second problem that is studied involves arbitrary loading of a cylinder with a stiffness variation that changes only in the circumferential direction. The end effects of the cylinder are ignored, so that the problem takes the form of an analysis of a cross-section for a short cylinder segment. Various load cases including axial compression, pressure, torsion, bending, and transverse shear forces are investigated. It is found that the most significant improvements in load-carrying capability exist for cases which involve loads that also vary around the circumference of the shell, namely bending and shear forces. The stiffness variation of the optimal designs contribute to the increased performance in two ways: lowering the stresses in the critical areas through redistribution of the stresses; and providing a relatively stiff region that alters the buckling behavior of the structure. These results lead to an in-depth optimization study involving weight optimization of a fuselage structure subjected to typical design constraints. Comparisons of the curvilinear fiber format to traditional stiffened structures constructed of isotropic and composite materials are included. It is found that standard variable stiffness designs are quite comparable in terms of weight and load-carrying capability yet offer the added advantage of tailorability of distinct regions of the structure that experience drastically different loading conditions. The last two problems presented in this work involve the nonlinear phenomenon of long tubes under bending. Though this scenario is not as applicable to fuselage structures as the previous problems, the mechanisms that produce the nonlinear effect are ideally suited to be controlled by the variable stiffness concept. This is due to the fact that the dominating influence for long cylinders under bending is the ovalization of the cross-section, which is governed mainly by the stiffness parameters of the cylindrical shell. Possible improvement of the critical buckling moments for these structures is investigated using either a circumferential or axial stiffness variation. For the circumferential case involving infinite length cylinders, it is found that slight improvements can be observed by designing structures that resist the cross-sectional deformation yet do not detract from the buckling resistance at the critical location. The results also indicate that buckling behavior is extremely dependent on cylinder length. This effect is most easily seen in the solution of finite length cylinders under bending that contain an axial stiffness variation. For these structures, the only mechanism that exhibits improved response are those that effectively shorten the length of the cylinder, thus reducing the cross-sectional deformation due to the forced restraint at the ends. It was found that the use of curvilinear fibers was not able to achieve this effect in sufficient degree to resist the deformation, but that ring stiffeners produced the desired response admirably. Thus, it is shown that the variable stiffness concept is most effective at improving the bending response of long cylinders through the use of a circumferential stiffness variation.

Tatting, Brian F.↗
NASA NTRS1987

Studies of pressure-velocity coupling schemes for analysis of incompressible and compressible flows

Two pressure-velocity coupling schemes, both of which solve the fully implicit discretized equations governing the flow of fluids were examined, and the capability of performing large Reynolds number, low Mach number compressible flow calculations were assessed. The semi-implicit iterative SIMPLE algorithm is extended to handle transient compressible flow calculations. This extension takes into account a strong coupling between the pressure and temperature through a correction procedure, based on the equations of state. Results obtained from the extended SIMPLE algorithm are then compared to similar results obtained from the non-iterative PISO algorithm. Both time-dependent and steady state calculations were performed using an axisymmetric 2:1 pipe expansion geometry and laminar flow conditions corresponding to Reynolds number of 1000 and Mach number of 2.0. For calculations simulating a time-dependent compression/expansion process, both schemes exhibit transient features in excellent agreement with each other, and moreover, the PISO method shows a significant computational time reduction of 60 percent over the SIMPLE scheme, regardless of the time step size or grid size employed. The effects of numerical diffusion are shown to be significant in these calculations. For steady state compressible flows, however, the SIMPLE algorithm displays increasing computational efficiency over the PISO method as the time step sizes employed to reach steady state conditions are decreased.

Burgreen, Gregory Wayne↗
DOE OSTI2023

WeakIdent: Weak formulation for identifying differential equation using narrow-fit and trimming

Data-driven identification of differential equations is an interesting but challenging problem, especially when the given data are corrupted by noise. When the governing differential equation is a linear combination of various differential terms, the identification problem can be formulated as solving a linear system, with the feature matrix consisting of linear and nonlinear terms multiplied by a coefficient vector. This product is equal to the time derivative term, and thus generates dynamical behaviors. The goal is to identify the correct terms that form the equation to capture the dynamics of the given data. We propose a general and robust framework to recover differential equations using a weak formulation with two new mechanisms, narrow-fit and trimming, for both ordinary and partial differential equations (ODEs and PDEs). The weak formulation facilitates an efficient and robust way to handle noise, and two new mechanisms, narrow-fit and trimming, improve the coefficient support and value recoveries respectively. For each sparsity level, Subspace Pursuit is utilized to find an initial set of support from the large dictionary. Then, we focus on highly dynamic regions (rows of the feature matrix), and error normalize the feature matrix in the narrow-fit step. The support is further updated via trimming the terms that contribute the least. Finally, the support set of features with the smallest Cross-Validation error is chosen as the result. A comprehensive set of numerical experiments are presented for both systems of ODEs and PDEs with various noise levels. The proposed method gives a robust recovery of the coefficients, and a significant denoising effect which can handle up to 100% noise-to-signal ratio for some equations. We compare the proposed method with several state-of-the-art algorithms for the recovery of differential equations.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1947

Equations for Adiabatic but Rotational Steady Gas Flows without Friction

This paper makes the following assumptions: 1) The flowing gases are assumed to have uniform energy distribution. ("Isoenergetic gas flows," that is valid with the same constants for the the energy equation entire flow.) This is correct, for example, for gas flows issuing from a region of constant pressure, density, temperature, end velocity. This property is not destroyed by compression shocks because of the universal validity of the energy law. 2) The gas behaves adiabatically, not during the compression shock itself but both before and after the shock. However, the adiabatic equation (p/rho(sup kappa) = C) is not valid for the entire gas flow with the same constant C but rather with an appropriate individual constant for each portion of the gas. For steady flows, this means that the constant C of the adiabatic equation is a function of the stream function. Consequently, a gas that has been flowing "isentropically",that is, with the same constant C of the adiabatic equation throughout (for example, in origination from a region of constant density, temperature, and velocity) no longer remains isentropic after a compression shock if the compression shock is not extremely simple (wedge shaped in a two-dimensional flow or cone shaped in a rotationally symmetrical flow). The solution of nonisentropic flows is therefore an urgent necessity.

Schaefer, Manfred↗