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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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NASA NTRS1993

A hybrid Pade-Galerkin technique for differential equations

A three-step hybrid analysis technique, which successively uses the regular perturbation expansion method, the Pade expansion method, and then a Galerkin approximation, is presented and applied to some model boundary value problems. In the first step of the method, the regular perturbation method is used to construct an approximation to the solution in the form of a finite power series in a small parameter epsilon associated with the problem. In the second step of the method, the series approximation obtained in step one is used to construct a Pade approximation in the form of a rational function in the parameter epsilon. In the third step, the various powers of epsilon which appear in the Pade approximation are replaced by new (unknown) parameters (delta(sub j)). These new parameters are determined by requiring that the residual formed by substituting the new approximation into the governing differential equation is orthogonal to each of the perturbation coordinate functions used in step one. The technique is applied to model problems involving ordinary or partial differential equations. In general, the technique appears to provide good approximations to the solution even when the perturbation and Pade approximations fail to do so. The method is discussed and topics for future investigations are indicated.

Geer, James F.↗
NASA NTRS1976

The Relation of Finite Element and Finite Difference Methods

Finite element and finite difference methods are examined in order to bring out their relationship. It is shown that both methods use two types of discrete representations of continuous functions. They differ in that finite difference methods emphasize the discretization of independent variable, while finite element methods emphasize the discretization of dependent variable (referred to as functional approximations). An important point is that finite element methods use global piecewise functional approximations, while finite difference methods normally use local functional approximations. A general conclusion is that finite element methods are best designed to handle complex boundaries, while finite difference methods are superior for complex equations. It is also shown that finite volume difference methods possess many of the advantages attributed to finite element methods.

Vinokur, M.↗
NASA NTRS1985

A hybrid numerical technique for predicting the aerodynamic and acoustic fields of advanced turboprops

A hybrid numerical procedure is presented for the prediction of the aerodynamic and acoustic performance of advanced turboprops. A hybrid scheme is proposed which in principle leads to a consistent simultaneous prediction of both fields. In the inner flow a finite difference method, the Approximate-Factorization Alternating-Direction-Implicit (ADI) scheme, is used to solve the nonlinear Euler equations. In the outer flow the linearized acoustic equations are solved via a Boundary-Integral Equation (BIE) method. The two solutions are iteratively matched across a fictitious interface in the flow so as to maintain continuity. At convergence the resulting aerodynamic load prediction will automatically satisfy the appropriate free-field boundary conditions at the edge of the finite difference grid, while the acoustic predictions will reflect the back-reaction of the radiated field on the magnitude of the loading source terms, as well as refractive effects in the inner flow. The equations and logic needed to match the two solutions are developed and the computer program implementing the procedure is described. Unfortunately, no converged solutions were obtained, due to unexpectedly large running times. The reasons for this are discussed and several means to alleviate the situation are suggested.

Homicz, G. F.↗
NASA NTRS1993

On the prediction of multigrid efficiency through local mode analysis

A single grid local mode analysis is used to predict the smoothing properties of numerical schemes for solving the Navier-Stokes equations with factorization based on Stone's Strongly Implicit Method. Four difference approximations for the convection terms are considered, namely, hybrid, central, second-order upwind, and third-order upwind. Smoothing factors from the analysis are compared with practical convergence factors in a multigrid method for flow over a backward facing step and it is found that the local mode analysis correctly predicts the effects of Reynolds number and higher-order schemes.

Wilson, R. V.↗
NASA NTRS1977

Composite methods for hyperbolic equations

A composite approximation procedure combining the properties of the Lax-Wendroff and leapfrog algorithms is proposed for solving hyperbolic equations. For a one-dimensional equation, a three-step approximation consisting of a two-step Richtmeyer method followed by a leapfrog step is considered. This is a two-level scheme, so all difficulties, including storage requirements, associated with the three-level leapfrog are eliminated. For two-dimensional problems a generalization of the preceding method is used consisting of a rotated Richtmeyer method followed by a modified leapfrog step. It is found that the composite schemes are effective in reducing oscillations and nonlinear instabilities that affect the leapfrog method. The dissipation in the composite schemes is much less than in the Richtmeyer algorithm, and hence can be used for long term integrations.

Turkel, E.↗
NASA NTRS1995

Calculated coupling efficiency between an elliptical-core optical fiber and an optical waveguide over temperature

To determine the feasibility of coupling the output of a single-mode optical fiber into a single-mode rib waveguide in a temperature varying environment, a theoretical calculation of the coupling efficiency between the two was investigated. Due to the complex geometry of the rib guide, there is no analytical solution to the wave equation for the guided modes, thus, approximation and/or numerical techniques must be utilized to determine the field patterns of the guide. In this study, three solution methods were used for both the fiber and guide fields; the effective-index method (EIM), Marcatili's approximation, and a Fourier method. These methods were utilized independently to calculate the electric field profile of each component at two temperatures, 20 C and 300 C, representing a nominal and high temperature. Using the electric field profile calculated from each method, the theoretical coupling efficiency between an elliptical-core optical fiber and a rib waveguide was calculated using the overlap integral and the results were compared. It was determined that a high coupling efficiency can be achieved when the two components are aligned. The coupling efficiency was more sensitive to alignment offsets in the y direction than the x, due to the elliptical modal field profile of both components. Changes in the coupling efficiency over temperature were found to be minimal.

Tuma, Margaret L.↗
NASA NTRS1992

Comparison of the Unified Perturbation Method with the two-scale expansion

The Unified Perturbation Method (UPM) converges faster over a wider domain of surface roughness than other perturbations, such as the small perturbation method, the phase perturbation method, the Kirchhoff approximation, and the momentum transfer expansion. It can be shown that UPM intrinsically possess characteristics similar to the two-scale expansion without requiring a free parameter. This paper considers whether the UPM can be improved by applying the two-scale concept to the method. In order to do so, the unknown source current is expanded in a two-scale manner starting from the extinction theorem. Several two-scale expansions are derived and are compared with the conventional two-scale approximation. It is shown that the UPM performs best without two-scale expansions.

Kim, Yunjin↗
NASA NTRS2014

Interpolation Method Needed for Numerical Uncertainty Analysis of Computational Fluid Dynamics

Using Computational Fluid Dynamics (CFD) to predict a flow field is an approximation to the exact problem and uncertainties exist. There is a method to approximate the errors in CFD via Richardson's Extrapolation. This method is based off of progressive grid refinement. To estimate the errors in an unstructured grid, the analyst must interpolate between at least three grids. This paper describes a study to find an appropriate interpolation scheme that can be used in Richardson's extrapolation or other uncertainty method to approximate errors. Nomenclature

CFD↗
NASA NTRS2014

High-Order Implicit-Explicit Multi-Block Time-stepping Method for Hyperbolic PDEs

This work seeks to explore and improve the current time-stepping schemes used in computational fluid dynamics (CFD) in order to reduce overall computational time. A high-order scheme has been developed using a combination of implicit and explicit (IMEX) time-stepping Runge-Kutta (RK) schemes which increases numerical stability with respect to the time step size, resulting in decreased computational time. The IMEX scheme alone does not yield the desired increase in numerical stability, but when used in conjunction with an overlapping partitioned (multi-block) domain significant increase in stability is observed. To show this, the Overlapping-Partition IMEX (OP IMEX) scheme is applied to both one-dimensional (1D) and two-dimensional (2D) problems, the nonlinear viscous Burger's equation and 2D advection equation, respectively. The method uses two different summation by parts (SBP) derivative approximations, second-order and fourth-order accurate. The Dirichlet boundary conditions are imposed using the Simultaneous Approximation Term (SAT) penalty method. The 6-stage additive Runge-Kutta IMEX time integration schemes are fourth-order accurate in time. An increase in numerical stability 65 times greater than the fully explicit scheme is demonstrated to be achievable with the OP IMEX method applied to 1D Burger's equation. Results from the 2D, purely convective, advection equation show stability increases on the order of 10 times the explicit scheme using the OP IMEX method. Also, the domain partitioning method in this work shows potential for breaking the computational domain into manageable sizes such that implicit solutions for full three-dimensional CFD simulations can be computed using direct solving methods rather than the standard iterative methods currently used.

Nielsen, Tanner B.↗
NASA NTRS1972

Development of a continuous linear model of a d-c to d-c flyback converter.

The analytical design of the feedback circuit for a d-c flyback converter requires the formulation of a model defining the static and dynamic performance of the forward loop. This paper describes the steps which were taken to develop a linear continuous model of a typical flyback circuit. Although the method uses several approximations to simplify the work, the resulting model was found to duplicate the performance of the actual circuit very closely. The model makes it possible to design the feedback circuit using well known linear feedback techniques. The method is an extension of prior work in the modeling of pulse-width controlled circuits.

Wells, B. A.↗
NASA NTRS1979

Solving the current element problem over lossy half-space without Sommerfeld integrals

An approach is presented for efficient computation of the vector potentials arising in the problem of a current element radiating over a lossy half-space. The present approach departs from the conventional ones in that it works primarily with the transform domain representations rather than with the Sommerfeld integrals which are the corresponding spatial domain counterparts. The key step in the present method is to approximate the transforms using a suitable approximation which is valid for a wide range of parameters of practical interest. The approximated transforms can be inverted in a closed form for the horizontal component of the vector potentials and can be expressed in a computationally efficient form for the vertical component. Numerical results illustrating the accuracy of the method are presented and some estimates of comparative computational times are also included.

Mittra, R.↗
NASA NTRS2004

A Most Probable Point-Based Method for Reliability Analysis, Sensitivity Analysis and Design Optimization

A major step in a most probable point (MPP)-based method for reliability analysis is to determine the MPP. This is usually accomplished by using an optimization search algorithm. The minimum distance associated with the MPP provides a measurement of safety probability, which can be obtained by approximate probability integration methods such as FORM or SORM. The reliability sensitivity equations are derived first in this paper, based on the derivatives of the optimal solution. Examples are provided later to demonstrate the use of these derivatives for better reliability analysis and reliability-based design optimization (RBDO).

Hou, Gene J.-W↗
NASA NTRS1996

Relativistic Navigation: A Theoretical Foundation

We present a theoretical foundation for relativistic astronomical measurements in curved space-time. In particular, we discuss a new iterative approach for describing the dynamics of an isolated astronomical N-body system in metric theories of gravity. To do this, we generalize the Fock-Chandrasekhar method of the weak-field and slow-motion approximation (WFSMA) and develop a theory of relativistic reference frames (RF's) for a gravitationally bounded many-extended-body problem. In any proper RF constructed in the immediate vicinity of an arbitrary body, the N-body solutions of the gravitational field equations are formally presented as a sum of the Riemann-flat inertial space-time, the gravitational field generated by the body itself, the unperturbed solutions for each body in the system transformed to the coordinates of this proper RF, and the gravitational interaction term. We develop the basic concept of a general WFSMA theory of the celestial RF's applicable to a wide class of metric theories of gravity and an arbitrary model of matter distribution. We apply the proposed method to general relativity. Celestial bodies are described using a perfect fluid model; as such, they possess any number of internal mass and current multipole moments that explicitly characterize their internal structures. The obtained relativistic corrections to the geodetic equations of motion arise because of a coupling of the bodies' multiple moments to the surrounding gravitational field. The resulting relativistic transformations between the different RF's extend the Poincare group to the motion of deformable self-gravitating bodies. Within the present accuracy of astronomical measurements we discuss the properties of the Fermi-normal-like proper RF that is defined in the immediate vicinity of the extended compact bodies. We further generalize the proposed approximation method and include two Eddington parameters (gamma, Beta). This generalized approach was used to derive the relativistic equations of satellite motion in the vicinity of the extended bodies. Anticipating improvements in radio and laser tracking technologies over the next few decades, we apply this method to spacecraft orbit determination. We emphasize the number of feasible relativistic gravity tests that may be performed within the context of the parameterized WFSMA. Based on the planeto-centric equations of motion of a spacecraft around the planet, we suggested a new null test of the Strong Equivalence Principle (SEP). The experiment to measure the corresponding SEP violation effect could be performed with the future Mercury Orbiter mission. We discuss other relativistic effects, including the perihelion advance and the redshift and geodetic precession of the orbiter's orbital plane about Mercury, as well as the possible future implementation of the proposed formalism in software codes developed for solar-system orbit determination. All the important calculations are completely documented, and the references contain an extensive list of cited literature.

Turyshev, Slava G.↗
NASA NTRS2004

A Most Probable Point-Based Method for Reliability Analysis, Sensitivity Analysis and Design Optimization

A major step in a most probable point (MPP)-based method for reliability analysis is to determine the MPP. This is usually accomplished by using an optimization search algorithm. The optimal solutions associated with the MPP provide measurements related to safety probability. This study focuses on two commonly used approximate probability integration methods; i.e., the Reliability Index Approach (RIA) and the Performance Measurement Approach (PMA). Their reliability sensitivity equations are first derived in this paper, based on the derivatives of their respective optimal solutions. Examples are then provided to demonstrate the use of these derivatives for better reliability analysis and Reliability-Based Design Optimization (RBDO).

Hou, Gene J.-W.↗
NASA NTRS1992

Viscous simulation method for unsteady flows past multicomponent configurations

The present algorithm for the numerical simulation of flows about complex configurations (whose multiple and nonsimilar components have arbitraty geometries) employs a hybridization of the domain decomposition techniques for grid generation as well as to reduce computer-memory requirements. A fully vectorized, finite-volume, upwind-biased, approximately factored multigrid method is used to solve 3D Reynolds-averaged unsteady and compressible Navier-Stokes equations simulating supersonic flows past an ogive-nose-cylinder near or within a cavity. The time-averaged surface pressures obtained compare favorably with the wind tunnel data.

Fouladi, Kamran↗
NASA NTRS1984

Supersonic nonlinear potential analysis

The NCOREL computer code was established to compute supersonic flow fields of wings and bodies. The method encompasses an implicit finite difference transonic relaxation method to solve the full potential equation in a spherical coordinate system. Two basic topic to broaden the applicability and usefulness of the present method which is encompassed within the computer code NCOREL for the treatment of supersonic flow problems were studied. The first topic is that of computing efficiency. Accelerated schemes are in use for transonic flow problems. One such scheme is the approximate factorization (AF) method and an AF scheme to the supersonic flow problem is developed. The second topic is the computation of wake flows. The proper modeling of wake flows is important for multicomponent configurations such as wing-body and multiple lifting surfaces where the wake of one lifting surface has a pronounced effect on a downstream body or other lifting surfaces.

Siclari, M. J.↗
NASA NTRS1986

Upwind relaxation algorithms for Euler/Navier-Stokes equations

A description of and results from a solution algorithm for the compressible Navier-Stokes equations are presented. The main features of the algorithm are second or third order accurate upwind discretization of the convection and pressure derivatives and a relaxation scheme for the unfactored implicit backward Euler time method, implemented in a finite-volume formulation. Upwind methods were successfully used to obtain solutions to the Euler equations for flows with strong shock waves. The particular upwind method being used is based on the flux vector splitting technique developed by Van Leer and both second and third order accurate discretizations were developed. Currently, the most widely used implicit solution technique for the Navier-Stokes equations use approximate factorization (AF) methods to treat multidimensional problems. The time integration scheme being used in the present algorithm corresponds to a line Gauss-Seidel relaxation method. This method produces good convergence rates for steady-state flows, and most of the algorithm was vectorized on the NASA Langley VPS 32 computer. The Navier-Stokes algorithm was tested for several two-dimensional flow problems. Solutions for the problems gave excellent results. The presented effort is directed toward the extension of the scheme to the full three-dimensional Navier-Stokes equations.

Thomas, J. L.↗