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All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

Optimal return-to-launchsite abort trajectories for an HL-20 Personnel Launch System vehicle

The Personnel Launch System (PLS) being studied by NASA is a system to complement the Space Shuttle and provide alternative access to space. The PLS consists of a manned spacecraft launched by an expendable launch vehicle (ELV). A candidate for the manned spacecraft is the HL-20 lifting body. In the event of an ELV malfunction during the initial portion of the ascent trajectory, the HL-20 will separate from the rocket and perform an unpowered return-to-launchsite (RTLS) abort. This paper describes an investigation of the RTLS abort scenario using optimal control theory. The objective of the abort trajectory is to maximize final altitude at a point near the runway. The assumption is then made that there exists a control history to steer the vehicle to any final altitude lower than the final optimal altitude. With this selection of cost function, and with this assumption, the feasibility of an RTLS abort at different times along the ascent trajectory can be determined. The method of differential inclusions, which allows the determination of optimal states and eliminates the need for determining the optimal controls, is used to determine the optimal trajectories.

Dutton, Kevin E.↗
NASA NTRS1990

On domains of convergence in optimization problems

Numerical optimization algorithms require the knowledge of an initial set of design variables. Starting from an initial design x(sup 0), improved solutions are obtained by updating the design iteratively in a way prescribed by the particular algorithm used. If the algorithm is successful, convergence is achieved to a local optimal solution. Let A denote the iterative procedure that characterizes a typical optimization algorithm, applied to the problem: Find x belonging to R(sup n) that maximizes f(x) subject to x belonging to Omega contained in R(sup n). We are interested in problems with several local maxima (x(sub j))(sup *), j=1, ..., m, in the feasible design space Omega. In general, convergence of the algorithm A to a specific solution (x(sub j))(sup *) is determined by the choice of initial design x(sup 0). The domain of convergence D(sub j) of A associated with a local maximum (x(sub j))(sup *) is a subset of initial designs x(sup 0) in Omega such that the sequence (x(sup k)), k=0,1,2,... defined by x(sup k+1) = A(x(sup k)), k=0,1,... converges to (x(sub j))(sup *). The set D(sub j) is also called the basin of attraction of (x(sub j))(sup *). Cayley first proposed the problem of finding the basin of attraction for Newton's method in 1897. It has been shown that the basin of attraction for Newton's method exhibits chaotic behavior in problems with polynomial objective. This implies that there may be regions in the feasible design space where arbitrarily close starting points will converge to different local optimal solutions. Furthermore, the boundaries of the domains of convergence may have a very complex, even fractal structure. In this paper we show that even simple structural optimization problems solved using standard gradient based (first order) algorithms exhibit similar features.

Diaz, Alejandro R.↗
NASA NTRS1993

Design optimization studies using COSMIC NASTRAN

The purpose of this study is to create, test and document a procedure to integrate mathematical optimization algorithms with COSMIC NASTRAN. This procedure is very important to structural design engineers who wish to capitalize on optimization methods to ensure that their design is optimized for its intended application. The OPTNAST computer program was created to link NASTRAN and design optimization codes into one package. This implementation was tested using two truss structure models and optimizing their designs for minimum weight, subject to multiple loading conditions and displacement and stress constraints. However, the process is generalized so that an engineer could design other types of elements by adding to or modifying some parts of the code.

Pitrof, Stephen M.↗
NASA NTRS1994

Theory and computation of optimal low- and medium-thrust transfers

This report presents two numerical methods considered for the computation of fuel-optimal, low-thrust orbit transfers in large numbers of burns. The origins of these methods are observations made with the extremal solutions of transfers in small numbers of burns; there seems to exist a trend such that the longer the time allowed to perform an optimal transfer the less fuel that is used. These longer transfers are obviously of interest since they require a motor of low thrust; however, we also find a trend that the longer the time allowed to perform the optimal transfer the more burns are required to satisfy optimality. Unfortunately, this usually increases the difficulty of computation. Both of the methods described use small-numbered burn solutions to determine solutions in large numbers of burns. One method is a homotopy method that corrects for problems that arise when a solution requires a new burn or coast arc for optimality. The other method is to simply patch together long transfers from smaller ones. An orbit correction problem is solved to develop this method. This method may also lead to a good guidance law for transfer orbits with long transfer times.

Chuang, C.-H.↗
NASA NTRS1992

Systematic Propulsion Optimization Tools (SPOT)

This paper describes a computer program written by senior-level Mechanical Engineering students at the University of Alabama in Huntsville which is capable of optimizing user-defined delivery systems for carrying payloads into orbit. The custom propulsion system is designed by the user through the input of configuration, payload, and orbital parameters. The primary advantages of the software, called Systematic Propulsion Optimization Tools (SPOT), are a user-friendly interface and a modular FORTRAN 77 code designed for ease of modification. The optimization of variables in an orbital delivery system is of critical concern in the propulsion environment. The mass of the overall system must be minimized within the maximum stress, force, and pressure constraints. SPOT utilizes the Design Optimization Tools (DOT) program for the optimization techniques. The SPOT program is divided into a main program and five modules: aerodynamic losses, orbital parameters, liquid engines, solid engines, and nozzles. The program is designed to be upgraded easily and expanded to meet specific user needs. A user's manual and a programmer's manual are currently being developed to facilitate implementation and modification.

Bower, Mark↗
NASA NTRS1994

Optimal control theory determination of feasible return-to-launch-site aborts for the HL-20 Personnel Launch System vehicle

The personnel launch system (PLS) being studied by NASA is a system to complement the space shuttle and provide alternative access to space. The PLS consists of a manned spacecraft launched by an expendable launch vehicle (ELV). A candidate for the manned spacecraft is the HL-20 lifting body. In the event of an ELV malfunction during the initial portion of the ascent trajectory, the HL-20 will separate from the rocket and perform an unpowered return to launch site (RTLS) abort. This work details an investigation, using optimal control theory, of the RTLS abort scenario. The objective of the optimization was to maximize final altitude. With final altitude as the cost function, the feasibility of an RTLS abort at different times during the ascent was determined. The method of differential inclusions was used to determine the optimal state trajectories, and the optimal controls were then calculated from the optimal states and state rates.

Dutton, Kevin E.↗
NASA NTRS1994

Finite element approximation of an optimal control problem for the von Karman equations

This paper is concerned with optimal control problems for the von Karman equations with distributed controls. We first show that optimal solutions exist. We then show that Lagrange multipliers may be used to enforce the constraints and derive an optimality system from which optimal states and controls may be deduced. Finally we define finite element approximations of solutions for the optimality system and derive error estimates for the approximations.

Hou, L. Steven↗
NASA NTRS1994

Algorithms for bilevel optimization

General multilevel nonlinear optimization problems arise in design of complex systems and can be used as a means of regularization for multi-criteria optimization problems. Here, for clarity in displaying our ideas, we restrict ourselves to general bi-level optimization problems, and we present two solution approaches. Both approaches use a trust-region globalization strategy, and they can be easily extended to handle the general multilevel problem. We make no convexity assumptions, but we do assume that the problem has a nondegenerate feasible set. We consider necessary optimality conditions for the bi-level problem formulations and discuss results that can be extended to obtain multilevel optimization formulations with constraints at each level.

Alexandrov, Natalia↗
NASA NTRS1995

Multidisciplinary design optimization using multiobjective formulation techniques

This report addresses the development of a multidisciplinary optimization procedure using an efficient semi-analytical sensitivity analysis technique and multilevel decomposition for the design of aerospace vehicles. A semi-analytical sensitivity analysis procedure is developed for calculating computational grid sensitivities and aerodynamic design sensitivities. Accuracy and efficiency of the sensitivity analysis procedure is established through comparison of the results with those obtained using a finite difference technique. The developed sensitivity analysis technique are then used within a multidisciplinary optimization procedure for designing aerospace vehicles. The optimization problem, with the integration of aerodynamics and structures, is decomposed into two levels. Optimization is performed for improved aerodynamic performance at the first level and improved structural performance at the second level. Aerodynamic analysis is performed by solving the three-dimensional parabolized Navier Stokes equations. A nonlinear programming technique and an approximate analysis procedure are used for optimization. The proceduredeveloped is applied to design the wing of a high speed aircraft. Results obtained show significant improvements in the aircraft aerodynamic and structural performance when compared to a reference or baseline configuration. The use of the semi-analytical sensitivity technique provides significant computational savings.

Chattopadhyay, Aditi↗
NASA NTRS1995

Wing-section optimization for supersonic viscous flow

To improve the shape of a supersonic wing, an automated method that also includes higher fidelity to the flow physics is desirable. With this impetus, an aerodynamic optimization methodology incorporating thin-layer Navier-Stokes equations and sensitivity analysis had been previously developed. Prior to embarking upon the wind design task, the present investigation concentrated on testing the feasibility of the methodology, and the identification of adequate problem formulations, by defining two-dimensional, cost-effective test cases. Starting with two distinctly different initial airfoils, two independent shape optimizations resulted in shapes with similar features: slightly cambered, parabolic profiles with sharp leading- and trailing-edges. Secondly, the normal section to the subsonic portion of the leading edge, which had a high normal angle-of-attack, was considered. The optimization resulted in a shape with twist and camber which eliminated the adverse pressure gradient, hence, exploiting the leading-edge thrust. The wing section shapes obtained in all the test cases had the features predicted by previous studies. Therefore, it was concluded that the flowfield analyses and sensitivity coefficients were computed and fed to the present gradient-based optimizer correctly. Also, as a result of the present two-dimensional study, suggestions were made for the problem formulations which should contribute to an effective wing shape optimization.

Item, Cem C.↗
NASA NTRS1995

Shape optimization of single- and two-element airfoils on multiblock grids

A multiblock, discrete sensitivity analysis method is used to couple a direct optimization method and a flow analysis method. The domain is divided into smaller subdomains for which the sensitivities are obtained separately. Then, an effective sensitivity equation is solved to complete the coupling of all the sensitivity information. The flow analysis is based on the thin-layer Navier-Stokes equations solved by an implicit, upwind-biased, finite-volume method. The method of feasible directions is used for the present gradient-based optimization approach. First, a transonic airfoil is optimized to investigate the behavior of the method in highly nonlinear flows as well as the effect of different blocking strategies on the procedure. A supercritical airfoil is produced from an initially symmetric airfoil with multiblocking affecting the path but not the final shape. Secondly, a two-element airfoil is shape optimized in subsonic flow to demonstrate the present method's capability of shaping aerodynamically interfering elements simultaneously. For a very low and a very high Reynolds number cases, the shape of the main airfoil and the flap are optimized to yield improved lift-to-drag ratios.

Lacasse, James M.↗
NASA NTRS1995

Optimal Signal Processing of Frequency-Stepped CW Radar Data

An optimal signal processing algorithm is derived for estimating the time delay and amplitude of each scatterer reflection using a frequency-stepped CW system. The channel is assumed to be composed of abrupt changes in the reflection coefficient profile. The optimization technique is intended to maximize the target range resolution achievable from any set of frequency-stepped CW radar measurements made in such an environment. The algorithm is composed of an iterative two-step procedure. First, the amplitudes of the echoes are optimized by solving an overdetermined least squares set of equations. Then, a nonlinear objective function is scanned in an organized fashion to find its global minimum. The result is a set of echo strengths and time delay estimates. Although this paper addresses the specific problem of resolving the time delay between the two echoes, the derivation is general in the number of echoes. Performance of the optimization approach is illustrated using measured data obtained from an HP-851O network analyzer. It is demonstrated that the optimization approach offers a significant resolution enhancement over the standard processing approach that employs an IFFT. Degradation in the performance of the algorithm due to suboptimal model order selection and the effects of additive white Gaussion noise are addressed.

Ybarra, Gary A.↗
NASA NTRS1996

Aerodynamic shape optimization using control theory

Aerodynamic shape design has long persisted as a difficult scientific challenge due its highly nonlinear flow physics and daunting geometric complexity. However, with the emergence of Computational Fluid Dynamics (CFD) it has become possible to make accurate predictions of flows which are not dominated by viscous effects. It is thus worthwhile to explore the extension of CFD methods for flow analysis to the treatment of aerodynamic shape design. Two new aerodynamic shape design methods are developed which combine existing CFD technology, optimal control theory, and numerical optimization techniques. Flow analysis methods for the potential flow equation and the Euler equations form the basis of the two respective design methods. In each case, optimal control theory is used to derive the adjoint differential equations, the solution of which provides the necessary gradient information to a numerical optimization method much more efficiently then by conventional finite differencing. Each technique uses a quasi-Newton numerical optimization algorithm to drive an aerodynamic objective function toward a minimum. An analytic grid perturbation method is developed to modify body fitted meshes to accommodate shape changes during the design process. Both Hicks-Henne perturbation functions and B-spline control points are explored as suitable design variables. The new methods prove to be computationally efficient and robust, and can be used for practical airfoil design including geometric and aerodynamic constraints. Objective functions are chosen to allow both inverse design to a target pressure distribution and wave drag minimization. Several design cases are presented for each method illustrating its practicality and efficiency. These include non-lifting and lifting airfoils operating at both subsonic and transonic conditions.

Reuther, James↗
NASA NTRS1996

Optimization Methodology for Unconventional Rocket Nozzle Design

Several current rocket engine concepts such as the bell-annular tripropellant engine, and the linear aerospike being proposed for the X-33, require unconventional three-dimensional rocket nozzles which must conform to rectangular or sector-shaped envelopes to meet integration constraints. These types of nozzles exist outside the current experience database, therefore, development of efficient design methods for these propulsion concepts is critical to the success of launch vehicle programs. Several approaches for optimizing rocket nozzles, including streamline tracing techniques, and the coupling of CFD analysis to optimization algorithms are described. The relative strengths and weaknesses of four classes of optimization algorithms are discussed: Gradient based methods, genetic algorithms, simplex methods, and surface response methods. Additionally, a streamline tracing technique, which provides a very computationally efficient means of defining a three-dimensional contour, is discussed. The performance of the various optimization methods on thrust optimization problems for tripropellant and aerospike concepts is assessed and recommendations are made for future development efforts.

Follett, W.↗
NASA NTRS1996

A General-Purpose Optimization Engine for Multi-Disciplinary Design Applications

A general purpose optimization tool for multidisciplinary applications, which in the literature is known as COMETBOARDS, is being developed at NASA Lewis Research Center. The modular organization of COMETBOARDS includes several analyzers and state-of-the-art optimization algorithms along with their cascading strategy. The code structure allows quick integration of new analyzers and optimizers. The COMETBOARDS code reads input information from a number of data files, formulates a design as a set of multidisciplinary nonlinear programming problems, and then solves the resulting problems. COMETBOARDS can be used to solve a large problem which can be defined through multiple disciplines, each of which can be further broken down into several subproblems. Alternatively, a small portion of a large problem can be optimized in an effort to improve an existing system. Some of the other unique features of COMETBOARDS include design variable formulation, constraint formulation, subproblem coupling strategy, global scaling technique, analysis approximation, use of either sequential or parallel computational modes, and so forth. The special features and unique strengths of COMETBOARDS assist convergence and reduce the amount of CPU time used to solve the difficult optimization problems of aerospace industries. COMETBOARDS has been successfully used to solve a number of problems, including structural design of space station components, design of nozzle components of an air-breathing engine, configuration design of subsonic and supersonic aircraft, mixed flow turbofan engines, wave rotor topped engines, and so forth. This paper introduces the COMETBOARDS design tool and its versatility, which is illustrated by citing examples from structures, aircraft design, and air-breathing propulsion engine design.

Patnaik, Surya N.↗
NASA NTRS1996

Optimization of Residual Stresses in MMC's through Process Parameter Control and the use of Heterogeneous Compensating/Compliant Interfacial Layers. OPTCOMP2 User's Guide

A user's guide for the computer program OPTCOMP2 is presented in this report. This program provides a capability to optimize the fabrication or service-induced residual stresses in unidirectional metal matrix composites subjected to combined thermomechanical axisymmetric loading by altering the processing history, as well as through the microstructural design of interfacial fiber coatings. The user specifies the initial architecture of the composite and the load history, with the constituent materials being elastic, plastic, viscoplastic, or as defined by the 'user-defined' constitutive model, in addition to the objective function and constraints, through a user-friendly data input interface. The optimization procedure is based on an efficient solution methodology for the inelastic response of a fiber/interface layer(s)/matrix concentric cylinder model where the interface layers can be either homogeneous or heterogeneous. The response of heterogeneous layers is modeled using Aboudi's three-dimensional method of cells micromechanics model. The commercial optimization package DOT is used for the nonlinear optimization problem. The solution methodology for the arbitrarily layered cylinder is based on the local-global stiffness matrix formulation and Mendelson's iterative technique of successive elastic solutions developed for elastoplastic boundary-value problems. The optimization algorithm employed in DOT is based on the method of feasible directions.

Pindera, Marek-Jerzy↗
NASA NTRS1995

Optimal Signal Processing of Frequency-Stepped CW Radar Data

An optimal signal processing algorithm is derived for estimating the time delay and amplitude of each scatterer reflection using a frequency-stepped CW system. The channel is assumed to be composed of abrupt changes in the reflection coefficient profile. The optimization technique is intended to maximize the target range resolution achievable from any set of frequency-stepped CW radar measurements made in such an environment. The algorithm is composed of an iterative two-step procedure. First, the amplitudes of the echoes are optimized by solving an overdetermined least squares set of equations. Then, a nonlinear objective function is scanned in an organized fashion to find its global minimum. The result is a set of echo strengths and time delay estimates. Although this paper addresses the specific problem of resolving the time delay between the first two echoes, the derivation is general in the number of echoes. Performance of the optimization approach is illustrated using measured data obtained from an HP-X510 network analyzer. It is demonstrated that the optimization approach offers a significant resolution enhancement over the standard processing approach that employs an IFFT. Degradation in the performance of the algorithm due to suboptimal model order selection and the effects of additive white Gaussion noise are addressed.

Ybarra, Gary A.↗
NASA NTRS1995

Comparison of Two Multidisciplinary Optimization Strategies for Launch-Vehicle Design

The investigation focuses on development of a rapid multidisciplinary analysis and optimization capability for launch-vehicle design. Two multidisciplinary optimization strategies in which the analyses are integrated in different manners are implemented and evaluated for solution of a single-stage-to-orbit launch-vehicle design problem. Weights and sizing, propulsion, and trajectory issues are directly addressed in each optimization process. Additionally, the need to maintain a consistent vehicle model across the disciplines is discussed. Both solution strategies were shown to obtain similar solutions from two different starting points. These solutions suggests that a dual-fuel, single-stage-to-orbit vehicle with a dry weight of approximately 1.927 x 10(exp 5)lb, gross liftoff weight of 2.165 x 10(exp 6)lb, and length of 181 ft is attainable. A comparison of the two approaches demonstrates that treatment or disciplinary coupling has a direct effect on optimization convergence and the required computational effort. In comparison with the first solution strategy, which is of the general form typically used within the launch vehicle design community at present, the second optimization approach is shown to he 3-4 times more computationally efficient.

Braun, R. D.↗