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NASA NTRS2012

A Robust Initialization Scheme for a Lateral Trajectory Optimization Problem with Time of Arrival Windows

We present a robust initialization scheme that estimates parameter values for the numerical solution of a two-point boundary value problem. The two-point boundary value problem formulation stems from the optimization of a cost functional subject to the dynamics of a simplified lateral aircraft model and other constraints. Leveraging regular perturbation methods, initial parameter estimates are analytically determined and used to initialize a gradient descent optimization routine which is shown to rapidly converge over a range of initial aircraft positions and heading angles. Additionally, the velocity of the aircraft is optimized to ensure the trajectory of the aircraft terminates within a desired region in both time and space.

lateral trajectory optimization↗

NASA NTRS1972

Calculation of free fall trajectories based on numerical optimization techniques

The development of a means of computing free-fall (nonthrusting) trajectories from one specified point in the solar system to another specified point in the solar system in a given amount of time was studied. The problem is that of solving a two-point boundary value problem for which the initial slope is unknown. Two standard methods of attack exist for solving two-point boundary value problems. The first method is known as the initial value or shooting method. The second method of attack for two-point boundary value problems is to approximate the nonlinear differential equations by an appropriate linearized set. Parts of both boundary value problem solution techniques described above are used. A complete velocity history is guessed such that the corresponding position history satisfies the given boundary conditions at the appropriate times. An iterative procedure is then followed until the last guessed velocity history and the velocity history obtained from integrating the acceleration history agree to some specified tolerance everywhere along the trajectory.

Source record↗

NASA NTRS1974

User's guide to four-body and three-body trajectory optimization programs

A collection of computer programs and subroutines written in FORTRAN to calculate 4-body (sun-earth-moon-space) and 3-body (earth-moon-space) optimal trajectories is presented. The programs incorporate a variable step integration technique and a quadrature formula to correct single step errors. The programs provide capability to solve initial value problem, two point boundary value problem of a transfer from a given initial position to a given final position in fixed time, optimal 2-impulse transfer from an earth parking orbit of given inclination to a given final position and velocity in fixed time and optimal 3-impulse transfer from a given position to a given final position and velocity in fixed time.

Pu, C. L.↗

NASA NTRS1971

On some problems in a theory of thermally and mechanically interacting continuous media

Using a linearized theory of thermally and mechanically interacting mixture of linear elastic solid and viscous fluid, we derive a fundamental relation in an integral form called a reciprocity relation. This reciprocity relation relates the solution of one initial-boundary value problem with a given set of initial and boundary data to the solution of a second initial-boundary value problem corresponding to a different initial and boundary data for a given interacting mixture. From this general integral relation, reciprocity relations are derived for a heat-conducting linear elastic solid, and for a heat-conducting viscous fluid. An initial-boundary value problem is posed and solved for the mixture of linear elastic solid and viscous fluid. With the aid of the Laplace transform and the contour integration, a real integral representation for the displacement of the solid constituent is obtained as one of the principal results of the analysis.

Lee, Y. M.↗

NASA NTRS1975

Computer analysis of multicircuit shells of revolution by the field method

The field method, presented previously for the solution of even-order linear boundary value problems defined on one-dimensional open branch domains, is extended to boundary value problems defined on one-dimensional domains containing circuits. This method converts the boundary value problem into two successive numerically stable initial value problems, which may be solved by standard forward integration techniques. In addition, a new method for the treatment of singular boundary conditions is presented. This method, which amounts to a partial interchange of the roles of force and displacement variables, is problem independent with respect to both accuracy and speed of execution. This method was implemented in a computer program to calculate the static response of ring stiffened orthotropic multicircuit shells of revolution to asymmetric loads. Solutions are presented for sample problems which illustrate the accuracy and efficiency of the method.

Cohen, G. A.↗

NASA NTRS1989

A hybrid-perturbation-Galerkin technique which combines multiple expansions

A two-step hybrid perturbation-Galerkin method for the solution of a variety of differential equations type problems is found to give better results when multiple perturbation expansions are employed. The method assumes that there is parameter in the problem formulation and that a perturbation method can be sued to construct one or more expansions in this perturbation coefficient functions multiplied by computed amplitudes. In step one, regular and/or singular perturbation methods are used to determine the perturbation coefficient functions. The results of step one are in the form of one or more expansions each expressed as a sum of perturbation coefficient functions multiplied by a priori known gauge functions. In step two the classical Bubnov-Galerkin method uses the perturbation coefficient functions computed in step one to determine a set of amplitudes which replace and improve upon the gauge functions. The hybrid method has the potential of overcoming some of the drawbacks of the perturbation and Galerkin methods as applied separately, while combining some of their better features. The proposed method is applied, with two perturbation expansions in each case, to a variety of model ordinary differential equations problems including: a family of linear two-boundary-value problems, a nonlinear two-point boundary-value problem, a quantum mechanical eigenvalue problem and a nonlinear free oscillation problem. The results obtained from the hybrid methods are compared with approximate solutions obtained by other methods, and the applicability of the hybrid method to broader problem areas is discussed.

Geer, James F.↗

NASA NTRS1990

A hybrid perturbation-Galerkin technique that combines multiple expansions

A two-step hybrid perturbation-Galerkin method for the solution of a variety of differential equations type problems is found to give better results when multiple perturbation expansions are employed. The method assumes that there is parameter in the problem formulation and that a perturbation method can be used to construct one or more expansions in this perturbation coefficient functions multiplied by computed amplitudes. In step one, regular and/or singular perturbation methods are used to determine the perturbation coefficient functions. The results of step one are in the form of one or more expansions each expressed as a sum of perturbation coefficient functions multiplied by a priori known gauge functions. In step two the classical Bubnov-Galerkin method uses the perturbation coefficient functions computed in step one to determine a set of amplitudes which replace and improve upon the gauge functions. The hybrid method has the potential of overcoming some of the drawbacks of the perturbation and Galerkin methods as applied separately, while combining some of their better features. The proposed method is applied, with two perturbation expansions in each case, to a variety of model ordinary differential equations problems including: a family of linear two-boundary-value problems, a nonlinear two-point boundary-value problem, a quantum mechanical eigenvalue problem and a nonlinear free oscillation problem. The results obtained from the hybrid methods are compared with approximate solutions obtained by other methods, and the applicability of the hybrid method to broader problem areas is discussed.

Geer, James F.↗

NASA NTRS1988

Solving time-dependent two-dimensional eddy current problems

Results of transient eddy current calculations are reported. For simplicity, a two-dimensional transverse magnetic field which is incident on an infinitely long conductor is considered. The conductor is assumed to be a good but not perfect conductor. The resulting problem is an interface initial boundary value problem with the boundary of the conductor being the interface. A finite difference method is used to march the solution explicitly in time. The method is shown. Treatment of appropriate radiation conditions is given special consideration. Results are validated with approximate analytic solutions. Two stringent test cases of high and low frequency incident waves are considered to validate the results.

Lee, Min Eig↗

NASA NTRS1992

Vision-based stereo ranging as an optimal control problem

The recent interest in the use of machine vision for flight vehicle guidance is motivated by the need to automate the nap-of-the-earth flight regime of helicopters. Vision-based stereo ranging problem is cast as an optimal control problem in this paper. A quadratic performance index consisting of the integral of the error between observed image irradiances and those predicted by a Pade approximation of the correspondence hypothesis is then used to define an optimization problem. The necessary conditions for optimality yield a set of linear two-point boundary-value problems. These two-point boundary-value problems are solved in feedback form using a version of the backward sweep method. Application of the ranging algorithm is illustrated using a laboratory image pair.

Menon, P. K. A.↗

NASA NTRS1975

Theoretical investigations of plasma processes in the ion bombardment thruster

A physical model for a thruster discharge was developed, consisting of a spatially diverging plasma sustained electrically between a small ring cathode and a larger ring anode in a cylindrical chamber with an axial magnetic field. The associated boundary-value problem for the coupled partial differential equations with mixed boundary conditions, which describe the electric potential and the plasma velocity fields, was solved in closed form. By means of quantum-mechanical perturbation theory, a formula for the number S(E) of atoms sputtered on the average by an ion of energy E was derived from first principles. The boundary-value problem describing the diffusion of the sputtered atoms through the surrounding rarefied electron-ion plasma to the system surfaces of ion propulsion systems was formulated and treated analytically. It is shown that outer boundary-value problems of this type lead to a complex integral equation, which requires numerical resolution.

Wilhelm, H. E.↗

NASA NTRS1973

Asymptotic solution of the problem for a thin axisymmetric cavern

The boundary value problem which describes the axisymmetric separation of the flow around a body by a stationary infinite stream is considered. It is understood that the cavitation number varies over the length of the cavern. Using the asymptotic expansions for the potential of a thin body, the orders of magnitude of terms in the equations of the problem are estimated. Neglecting small quantities, a simplified boundary value problem is obtained.

Serebriakov, V. V.↗

NASA NTRS1999

Autolanding Trajectory Design for the X-34

An Autolanding I-load Program (ALIP) is developed to design unpowered autolanding trajectories for the X-34 Mach 8 vehicle. The trajectory is comprised of geometric flight segments that are based on the shuttle approach and landing design (steep glideslope, circular flare, exponential flare to shallow glideslope). Enforcing physical constraints such as loads, vertical descent rate, continuity and smoothness reduces the design problem to a two point boundary value problem with conditions on the initial and final dynamic pressure. Finding a solution required the development of trajectory simulation techniques that constrained the flight profile to a prescribed geometry. The design methodology can be extended beyond the autolanding flight regime by repeating the series of geometric segments and solving multiple two-point boundary values problems (one for each series). The techniques described in this paper facilitate rapid design of reference trajectories.

Barton, Gregg H.↗

NASA NTRS1990

Launch flexibility using NLP guidance and remote wind sensing

This paper examines the use of lidar wind measurements in the implementation of a guidance strategy for a nonlinear programming (NLP) launch guidance algorithm. The NLP algorithm uses B-spline command function representation for flexibility in the design of the guidance steering commands. Using this algorithm, the guidance system solves a two-point boundary value problem at each guidance update. The specification of different boundary value problems at each guidance update provides flexibility that can be used in the design of the guidance strategy. The algorithm can use lidar wind measurements for on pad guidance retargeting and for load limiting guidance steering commands. Examples presented in the paper use simulated wind updates to correct wind induced final orbit errors and to adjust the guidance steering commands to limit the product of the dynamic pressure and angle-of-attack for launch vehicle load alleviation.

Cramer, Evin J.↗

NASA NTRS1993

Accurate computation and continuation of homoclinic and heteroclinic orbits for singular perturbation problems

In earlier papers, Doedel and the authors have developed a numerical method and derived error estimates for the computation of branches of heteroclinic orbits for a system of autonomous ordinary differential equations in R(exp n). The idea of the method is to reduce a boundary value problem on the real line to a boundary value problem on a finite interval by using a local (linear or higher order) approximation of the stable and unstable manifolds. A practical limitation for the computation of homoclinic and heteroclinic orbits has been the difficulty in obtaining starting orbits. Typically these were obtained from a closed form solution or via a homotopy from a known solution. Here we consider extensions of our algorithm which allow us to obtain starting orbits on the continuation branch in a more systematic way as well as make the continuation algorithm more flexible. In applications, we use the continuation software package AUTO in combination with some initial value software. The examples considered include computation of homoclinic orbits in a singular perturbation problem and in a turbulent fluid boundary layer in the wall region problem.

Vaughan, William W.↗

NASA NTRS1982

Aerodynamics of an airfoil with a jet issuing from its surface

A simple, two dimensional, incompressible and inviscid model for the problem posed by a two dimensional wing with a jet issuing from its lower surface is considered and a parametric analysis is carried out to observe how the aerodynamic characteristics depend on the different parameters. The mathematical problem constitutes a boundary value problem where the position of part of the boundary is not known a priori. A nonlinear optimization approach was used to solve the problem, and the analysis reveals interesting characteristics that may help to better understand the physics involved in more complex situations in connection with high lift systems.

Tavella, D. A.↗

NASA NTRS1994

Completed Beltrami-Michell Formulation for Analyzing Radially Symmetrical Bodies

A force method formulation, the completed Beltrami-Michell formulation (CBMF), has been developed for analyzing boundary value problems in elastic continua. The CBMF is obtained by augmenting the classical Beltrami-Michell formulation with novel boundary compatibility conditions. It can analyze general elastic continua with stress, displacement, or mixed boundary conditions. The CBMF alleviates the limitations of the classical formulation, which can solve stress boundary value problems only. In this report, the CBMF is specialized for plates and shells. All equations of the CBMF, including the boundary compatibility conditions, are derived from the variational formulation of the integrated force method (IFM). These equations are defined only in terms of stresses. Their solution for kinematically stable elastic continua provides stress fields without any reference to displacements. In addition, a stress function formulation for plates and shells is developed by augmenting the classical Airy's formulation with boundary compatibility conditions expressed in terms of the stress function. The versatility of the CBMF and the augmented stress function formulation is demonstrated through analytical solutions of several mixed boundary value problems. The example problems include a composite circular plate and a composite circular cylindrical shell under the simultaneous actions of mechanical and thermal loads.

Kaljevic, Igor↗

NASA NTRS1989

A hybrid perturbation-Galerkin method for differential equations containing a parameter

A two-step hybrid perturbation-Galerkin method to solve a variety of differential equations which involve a parameter is presented and discussed. The method consists of: (1) the use of a perturbation method to determine the asymptotic expansion of the solution about one or more values of the parameter; and (2) the use of some of the perturbation coefficient functions as trial functions in the classical Bubnov-Galerkin method. This hybrid method has the potential of overcoming some of the drawbacks of the perturbation method and the Bubnov-Galerkin method when they are applied by themselves, while combining some of the good features of both. The proposed method is illustrated first with a simple linear two-point boundary value problem and is then applied to a nonlinear two-point boundary value problem in lubrication theory. The results obtained from the hybrid method are compared with approximate solutions obtained by purely numerical methods. Some general features of the method, as well as some special tips for its implementation, are discussed. A survey of some current research application areas is presented and its degree of applicability to broader problem areas is discussed.

Geer, James F.↗

NASA NTRS1991

The semi-infinite strip problem in the mechanics of composite materials

Many problems, ranging from interface-fiber fracture to determining the strength of fiber-matrix interfaces in composites, can be greatly assisted by a complete understanding of the boundary value problem of a semi-infinite strip. The specific boundary value problem of a semi-infinite strip with symmetric tractions on the transverse edges and normal loads on the end is investigated in the present study. The nature of the singular behavior in the slope of the end vertical displacement at the corner points is found. An asymptotic analysis of the solution shows that, depending on the nature of the applied stresses in the vicinity of the corner points, the singularity in the slope of the end vertical displacement may either be a power or logarithmic type, both, or not exist at all. Illustrative examples, for which exact or numerical results are known, are given. Numerical procedures are also given. The direct application of the results of the paper are illustrated for several problems in the mechanics of composite materials.

Kaw, Autar K.↗