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Results for “Closed form solution”

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 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
NASA NTRS1991

A study of the response of nonlinear springs

The various phases to developing a methodology for studying the response of a spring-reinforced arch subjected to a point load are discussed. The arch is simply supported at its ends with both the spring and the point load assumed to be at midspan. The spring is present to off-set the typical snap through behavior normally associated with arches, and to provide a structure that responds with constant resistance over a finite displacement. The various phases discussed consist of the following: (1) development of the closed-form solution for the shallow arch case; (2) development of a finite difference analysis to study (shallow) arches; and (3) development of a finite element analysis for studying more general shallow and nonshallow arches. The two numerical analyses rely on a continuation scheme to move the solution past limit points, and to move onto bifurcated paths, both characteristics being common to the arch problem. An eigenvalue method is used for a continuation scheme. The finite difference analysis is based on a mixed formulation (force and displacement variables) of the governing equations. The governing equations for the mixed formulation are in first order form, making the finite difference implementation convenient. However, the mixed formulation is not well-suited for the eigenvalue continuation scheme. This provided the motivation for the displacement based finite element analysis. Both the finite difference and the finite element analyses are compared with the closed form shallow arch solution. Agreement is excellent, except for the potential problems with the finite difference analysis and the continuation scheme. Agreement between the finite element analysis and another investigator's numerical analysis for deep arches is also good.

Hyer, M. W.↗
NASA NTRS1992

Composite laminates with spatially varying fiber orientations - 'Variable stiffness panel concept'

A solution has been obtained to the plane elasticity problem for a symmetrically laminated composite panel with spatially varying fiber orientations. Since variation of the fiber angles along the length of a composite laminate results in stiffness properties that change as a function of location, the laminates are called variable stiffness panels. An analysis of the stiffness variation and its effect on the elastic response of the panel is presented here. A numerical solution has been obtained using an iterative collocation technique. Corresponding closed-form solutions are given for three different sets of boundary conditions. Two of the cases considered have exact solutions and thus serve to validate the numerical model.

Guerdal, Zafer↗
NASA NTRS1982

Study of laser heated propulsion devices. Part 2: Assessment of laser propulsion modeling and simulation requirements

Theoretical models for the laser heated thruster are reviewed. The requirements for an improved laser heated thruster theoretical model are discussed. The application of a time dependent finite difference Navier-Stokes equation solution to the laser heated thruster problem is described, along with a simple closed form solution which was developed in order to gain insight into the difficulties encountered in the pursuit of the numerical solution.

Thoenes, J.↗
NASA NTRS1995

Stokes problems for moving half-planes

New exact solutions of the Navier-Stokes equations are obtained for the unbounded and bounded oscillatory and impulsive tangential edgewise motion of touching half-infinite plates in their own plane. In contrast to Stokes classical solutions for the harmonic and impulsive motion of an infinite plane wall, where the solutions are separable or have a simple similarity form, the present solutions have a two-dimensional structure in the near region of the contact between the half-infinite plates. Nevertheless, it is possible to obtain relatively simple closed-form solutions for the flow field in each case by defining new variables which greatly simplify the r- and theta- dependence of the solutions in the vicinity of the contact region. These solutions for flow in a half-infinite space are then extended to bounded flows in a channel using an image superposition technique. The impulsive motion has application to the motion near geophysical faults, whereas the oscillatory motion has arisen in the design of a novel oscillating half-plate flow chamber for examining the effect of fluid shear stress on cultured cell monolayers.

NASA Discipline Musculoskeletal↗
NASA NTRS1988

High Rayleigh number convection in rectangular enclosures with differentially heated vertical walls and aspect ratios between zero and unity

High Rayleigh number convection in a rectangular cavity with insulated horizontal surfaces and differentially heated vertical walls was analyzed for an arbitrary aspect ratio smaller than or equal to unity. Unlike previous analytical studies, a systematic method of solution based on linearization technique and analytical iteration procedure was developed to obtain approximate closed-form solutions for a wide range of aspect ratios. The predicted velocity and temperature fields are shown to be in excellent agreement with available experimental and numerical data.

Kassemi, Siavash A.↗
NASA NTRS1988

A Mathematical Formulation of the SCOLE Control Problem. Part 2: Optimal Compensator Design

The study initiated in Part 1 of this report is concluded and optimal feedback control (compensator) design for stability augmentation is considered, following the mathematical formulation developed in Part 1. Co-located (rate) sensors and (force and moment) actuators are assumed, and allowing for both sensor and actuator noise, stabilization is formulated as a stochastic regulator problem. Specializing the general theory developed by the author, a complete, closed form solution (believed to be new with this report) is obtained, taking advantage of the fact that the inherent structural damping is light. In particular, it is possible to solve in closed form the associated infinite-dimensional steady-state Riccati equations. The SCOLE model involves associated partial differential equations in a single space variable, but the compensator design theory developed is far more general since it is given in the abstract wave equation formulation. The results thus hold for any multibody system so long as the basic model is linear.

Balakrishnan, A. V.↗
NASA NTRS1985

Disturbance-accommodating tracking maneuvers of flexible spacecraft

In this paper the problem of maneuvering a flexible spacecraft through a large angle is considered, where the disturbance-accommodating feedback control tracks a desired output state. The desired output state is provided from an open-loop solution for the linear system model. The components of the disturbance vector are assumed to be represented in terms of Fourier series. Closed-form solutions are provided for the Ricati, prefilter, state trajectory, and residual state trajectory equations which define the optimal control. Example maneuvers are presented where control-rate penalties have been included in the performance index for frequency-shaping, in order to smooth both the open- and closed-loop control commands.

Turner, J. D.↗
DOE OSTI2025

CUDO: closed-form universal dwell-time optimization for computer-controlled optical surfacing

Precision optical figuring demands fast and accurate dwell time optimization to reach nanometer- and sub-nanometer-level accuracy in next-generation optical systems. We introduce CUDO (closed-form universal dwell-time optimization), the first, to the best of our knowledge, unified closed-form analytical framework that supports both function-form and matrix-form dwell time models in computer-controlled optical surfacing (CCOS). In contrast to traditional methods, which rely on iterative optimization and hyperparameter tuning, our framework derives direct analytical solutions with no adjustable parameters. This approach unifies the solution principles of existing methods within a single mathematical model, delivering three key advantages: (1) accuracy on par with, or superior to, iterative solvers, (2) substantial reduction in computation time, and (3) numerical robustness. Comparative studies with prior art confirm that closed-form solutions achieve equivalent residual error while removing runtime bottlenecks. By simplifying the implementation and enabling real-time, scalable deployment, CUDO establishes a practical foundation for future deterministic fabrication of large-aperture and high-performance optics.

36 MATERIALS SCIENCE↗
NASA NTRS1990

Parallel flows with Soret effect in tilted cylinders

Henry and Roux (1986, 1987, 1988) have conducted extensive numerical studies on the interaction of Soret separation with convection in cylindrical geometry. Many of their solutions exhibit parallel flow away from end walls. Their parallel flow results can be matched by closed-form solutions. Solutions are nonunique in some parameter regions. Disappearance of one branch of solutions correlates with a sudden transition of Henry and Roux's results from a separated to a well-mixed flow.

Jacqmin, David↗
NASA NTRS1986

On self-preserving, variable-density, turbulent free jets

Published experimental data on incompressible, compresible, free binary, and confined binary turbulent axisymmetric jet flows are compiled and characterized, and the effect of varying turbulent diffusivity across the mixing region of a free jet is investigated analytically, applying the similarity-solution approach of So and Hwang (1986) to the self-preserving region. It is shown that closed-form solutions, represented by Gaussian error functions and having the turbulent Reynolds number and a profile-shape factor as free parameters, can be obtained if the turbulent diffusivities of momentum, mass, or heat are assumed to be different and to vary in both the streamwise and radial directions. An entrainment function uniquely related to the turbulent Reynolds number is derived, and good agreement between theoretical predictions and experimental measurements is demonstrated in graphs.

So, R. M. C.↗
NASA NTRS1987

A theory for turbulent curved wall jets

An attempt is made to formulate the simplest possible model for the flow of a wall jet emanating from a two dimensional source into quiescent surroundings in the presence of a wall of arbitrary shape. The method used self similar profiles for the mean velocity together with a simple eddy viscosity model. The streamwise and radial momentum equations are integrated across the wall jet flow to give an expression for the momentum balance including the effect of the pressure gradient induced by the rate of change of surface curvature. The streamwise momentum equation is also evaluated at the point of maximum velocity to provide a second equation and thereby permit a solution for the two unknown quantities, the jet half width and the maximum velocity. This approach provides approximate closed form solutions for the flow of the wall jet over surfaces of various shapes, and in particular permits a direct comparison with the available experimental results for plane, cylindrical or logarithmic spiral surfaces.

Roberts, Leonard↗
NASA NTRS1993

Trajectory optimization for the National aerospace plane

While continuing the application of the inverse dynamics approach in obtaining the optimal numerical solutions, the research during the past six months has been focused on the formulation and derivation of closed-form solutions for constrained hypersonic flight trajectories. Since it was found in the research of the first year that a dominant portion of the optimal ascent trajectory of the aerospace plane is constrained by dynamic pressure and heating constraints, the application of the analytical solutions significantly enhances the efficiency in trajectory optimization, provides a better insight to understanding of the trajectory and conceivably has great potential in guidance of the vehicle. Work of this period has been reported in four technical papers. Two of the papers were presented in the AIAA Guidance, Navigation, and Control Conference (Hilton Head, SC, August, 1992) and Fourth International Aerospace Planes Conference (Orlando, FL, December, 1992). The other two papers have been accepted for publication by Journal of Guidance, Control, and Dynamics, and will appear in 1993. This report briefly summarizes the work done in the past six months and work currently underway.

Lu, Ping↗
NASA NTRS1998

Attitude Determination Using Two Vector Measurements

Many spacecraft attitude determination methods use exactly two vector measurements. The two vectors are typically the unit vector to the Sun and the Earth's magnetic field vector for coarse "sun-mag" attitude determination or unit vectors to two stars tracked by two star trackers for fine attitude determination. TRIAD, the earliest published algorithm for determining spacecraft attitude from two vector measurements, has been widely used in both ground-based and onboard attitude determination. Later attitude determination methods have been based on Wahba's optimality criterion for n arbitrarily weighted observations. The solution of Wahba's problem is somewhat difficult in the general case, but there is a simple closed-form solution in the two-observation case. This solution reduces to the TRIAD solution for certain choices of measurement weights. This paper presents and compares these algorithms as well as sub-optimal algorithms proposed by Bar-Itzhack, Harman, and Reynolds. Some new results will be presented, but the paper is primarily a review and tutorial.

Markley, F. Landis↗
NASA NTRS1999

Attitude Determination Using Two Vector Measurements

Many spacecraft attitude determination methods use exactly two vector measurements. The two vectors are typically the unit vector to the Sun and the Earth's magnetic field vector for coarse "sun-mag" attitude determination or unit vectors to two stars tracked by two star trackers for fine attitude determination. TRIAD, the earliest published algorithm for determining spacecraft attitude from two vector measurements, has been widely used in both ground-based and onboard attitude determination. Later attitude determination methods have been based on Wahba's optimality criterion for n arbitrarily weighted observations. The solution of Wahba's problem is somewhat difficult in the general case, but there is a simple closed-form solution in the two-observation case. This solution reduces to the TRIAD solution for certain choices of measurement weights. This paper presents and compares these algorithms as well as sub-optimal algorithms proposed by Bar-Itzhack, Harman, and Reynolds. Some new results will be presented, but the paper is primarily a review and tutorial.

Markley, F. Landis↗
NASA NTRS1972

Interpolation using surface splines.

A surface spline is a mathematical tool for interpolating a function of two variables. It is based upon the small deflection equation of an infinite plate. The surface spline depends upon the solution of a system of linear equations, and thus, will ordinarily require the use of a digital computer. The closed form solution involves no functions more complicated than logarithms, and is easily coded. Several modifications which can be incorporated are discussed.

Harder, R. L.↗
NASA NTRS1981

Radiative transfer theory for passive microwave remote sensing of random media

The radiative transfer theory is applied to a random medium for the case of passive remote sensing. A general numerical solution with Gaussian quadrature method is studied. For a medium with laminar structure or a medium with cylindrical structure, closed form solutions are obtained by solving the radiative transfer equations of a two-layer random medium. Numerical results for the general case and the two limiting cases are compared.

Chuang, S. L.↗
DOE OSTI2020

Computationally Efficient Learning of Large Scale Dynamical Systems: A Koopman Theoretic Approach

In recent years there has been a considerable drive towards data-driven analysis, discovery and control of dynamical systems. To this end, operator theoretic methods, namely, Koopman operator methods have gained a lot of interest. In general, the Koopman operator is obtained as a solution to a least-squares problem, and as such, the Koopman operator can be expressed as a closed-form solution that involves the computation of Moore-Penrose inverse of a matrix. For high dimensional systems and also if the size of the obtained data-set is large, the computation of the Moore-Penrose inverse becomes computationally challenging. In this paper, we provide an algorithm for computing the Koopman operator for high dimensional systems in a time-efficient manner. We further demonstrate the efficacy of the proposed approach on two different systems, namely a network of coupled oscillators (with state-space dimension up to 2500) and IEEE 68 bus system (with state-space dimension 204 and up to 24,000 time-points).

Sinha, Subhrajit↗
DOE OSTI2021

A simple model of a strong shock driven by a spherical or cylindrical piston

A simple model of piston-driven spherical and cylindrical shocks is suggested. The model is based on a consistent use of two factors: (a) an almost uniform pressure across the shocked layer and, (b) continuous geometrical stretching of the surface elements of the expanding piston. It turns out that for a uniform pre-shock medium the gas between the piston and the shock behaves essentially as an incompressible fluid. An algebraic equation for the shock vs piston position is obtained. Detailed evaluation of the accuracy of the proposed solution shows that its accuracy is a few percent for the adiabatic index γ = 5/3 (as in ideal plasma). A closed-form solution describing enhancement of a weak ambient magnetic field by the shock is presented. The proposed model of piston-driven shocks goes beyond the classical self-similar solutions in that it: (1) naturally covers an early, non-asymptotic dynamics and its transition to asymptotic regime; (2) allows for smooth radial density variation of an ambient gas of the form of bumps, dimples or ramps between two constant values; and (3) allows for smooth temporal variation of piston velocity of the form of bumps, dimples, or ramps. Furthermore, this simple and versatile model provides some new insights into a classical hydrodynamical problem.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗