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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 19 records

Improving Automated Strategies for Univariate Quantifier Elimination

This report discusses improved support for univariate quantifier elimination in the Prototype Verification System (PVS). Previously, PVS had three strategies for quantifier elimination—hutch, tarski, and sturm. Of these, only hutch is able to decide queries in any input format—sturm only works on queries regarding a single polynomial on an interval and tarski resolves queries in the universal existential fragment. This paper describes an extended version of tarski. The extension is accomplished by formally verifying a disjunctive normal form transformation in PVS and using tarski on each conjunctive clause. Additionally, a preprocessing step is added to the decision procedure underlying tarski. This preprocessing is designed to exploit properties of polynomial structure to quickly resolve queries that have certain formats. The preprocessing produces dramatic speedup when it succeeds in resolving a query, and seems to introduce negligible overhead when it does not resolve a query. Finally, testing reveals some ways to improve the hutch and tarski strategies.

Polynomial Constraints↗

Symbolic discrete event system specification

Extending discrete event modeling formalisms to facilitate greater symbol manipulation capabilities is important to further their use in intelligent control and design of high autonomy systems. An extension to the DEVS formalism that facilitates symbolic expression of event times by extending the time base from the real numbers to the field of linear polynomials over the reals is defined. A simulation algorithm is developed to generate the branching trajectories resulting from the underlying nondeterminism. To efficiently manage symbolic constraints, a consistency checking algorithm for linear polynomial constraints based on feasibility checking algorithms borrowed from linear programming has been developed. The extended formalism offers a convenient means to conduct multiple, simultaneous explorations of model behaviors. Examples of application are given with concentration on fault model analysis.

Zeigler, Bernard P.↗

Stacking optimization of compressor blades of gas turbine engines

A procedure is presented to obtain optimal designs of axial compressor blades with structural design constraints. Coefficients of the polynomials defining the circumferential tilting angles and the axial leaning distances of the airfoil cross sections from the initial design geometry are used as design variables. The compressor blades are modeled by 20-node solid elements. An efficient finite element method is developed for modal analysis and sensitivity analysis with respect to the design variables. Based on this information, a sequential linear programming method is applied to calculate the required change of geometry for the desired structural design constraints.

Cheu, Tsu-Chien↗

Application of response surface techniques to helicopter rotor blade optimization procedure

In multidisciplinary optimization problems, response surface techniques can be used to replace the complex analyses that define the objective function and/or constraints with simple functions, typically polynomials. In this work a response surface is applied to the design optimization of a helicopter rotor blade. In previous work, this problem has been formulated with a multilevel approach. Here, the response surface takes advantage of this decomposition and is used to replace the lower level, a structural optimization of the blade. Problems that were encountered and important considerations in applying the response surface are discussed. Preliminary results are also presented that illustrate the benefits of using the response surface.

Henderson, Joseph Lynn↗

Structural design utilizing updated, approximate sensitivity derivatives

A method to improve the computational efficiency of structural optimization algorithms is investigated. In this method, the calculations of 'exact' sensitivity derivatives of constraint functions are performed only at selected iterations during the optimization process. The sensitivity derivatives utilized within other iterations are approximate derivatives which are calculated using an inexpensive derivative update formula. Optimization results are presented for an analytic optimization problem (i.e., one having simple polynomial expressions for the objective and constraint functions) and for two structural optimization problems. The structural optimization results indicate that up to a factor of three improvement in computation time is possible when using the updated sensitivity derivatives.

Scotti, Stephen J.↗

Robustness Analysis and Optimally Robust Control Design via Sum-of-Squares

A control analysis and design framework is proposed for systems subject to parametric uncertainty. The underlying strategies are based on sum-of-squares (SOS) polynomial analysis and nonlinear optimization to design an optimally robust controller. The approach determines a maximum uncertainty range for which the closed-loop system satisfies a set of stability and performance requirements. These requirements, de ned as inequality constraints on several metrics, are restricted to polynomial functions of the uncertainty. To quantify robustness, SOS analysis is used to prove that the closed-loop system complies with the requirements for a given uncertainty range. The maximum uncertainty range, calculated by assessing a sequence of increasingly larger ranges, serves as a robustness metric for the closed-loop system. To optimize the control design, nonlinear optimization is used to enlarge the maximum uncertainty range by tuning the controller gains. Hence, the resulting controller is optimally robust to parametric uncertainty. This approach balances the robustness margins corresponding to each requirement in order to maximize the aggregate system robustness. The proposed framework is applied to a simple linear short-period aircraft model with uncertain aerodynamic coefficients.

Dorobantu, Andrei↗

Combined Bernstein Polynomial, Optimal Reciprocal Collision Avoidance, Differential Dynamic Programming for Trajectory Replanning and Collision Avoidance for UAM Vehicles

This paper presents an integration of Differential Dynamic Programming (DDP) with the Optimal Reciprocal Collision Avoidance (ORCA) algorithm as the basis for a new algorithm, titled Combined Bernstein Polynomial Optimal Reciprocal Collision Avoidance DDP (COBRA-DDP), for trajectory replanning and collision avoidance for Urban Air Mobility (UAM) vehicles. State-constrained variants of DDP provide the ability to plan trajectories while avoiding obstacles, but these methods require a large increase in computational time per iteration which hinders the overall speed of the algorithm. ORCA utilizes simplified dynamics to recognize potential collisions along a trajectory and provides an optimal velocity for the avoidance of multiple vehicles. These velocity commands, however, may not result in a dynamically feasible trajectory for DDP to plan around. As such, a Bernstein polynomial curve that considers the general dynamic constraints of the vehicle is generated to approximate a trajectory based on the velocity commands. COBRA-DDP optimizes this suggested trajectory via unconstrained DDP to provide a dynamically feasible trajectory that provides collision avoidance. This new trajectory can be applied to the vehicle or used to warm start the state constrained DDP algorithms to decrease computation time. Its benefits and effectiveness of the algorithm are demonstrated on a UAM Vertical Takeoff and Landing (VTOL) vehicle simulation with highly nonlinear dynamics.

Optimal Reciprocal Collision Avoidance↗

Focal Plane Geometry Characterization of the Kepler Mission

The Kepler Mission focal plane contains 42 charge-coupled device (CCD) photodetectors. Each CCD is composed of 2.2 million square pixels, 27 micrometers on a side, arranged in a grid of 2,200 columns by 1,044 rows. The science goals of the Kepler Mission require that the position of each CCD be determined with an accuracy of 0.1 pixels, corresponding to 2.7 micrometers or 0.4 seconds of arc, a level which is not achievable through pre-flight metrology. We describe a technique for determining the CCD positioning using images of the Kepler field of view (FOV) obtained in flight. The technique uses the fitted centroid row and column positions of 400 pre-selected stars on each CCD to obtain empirical polynomials which relate sky coordinates (right ascension and declination) to chip coordinates (row and column). The polynomials are in turn evaluated to produce constraints for a nonlinear model fit which directly determines the model parameters describing the location and orientation of each CCD. The focal plane geometry characterization algorithm is itself embedded in an iterative process which determines the focal plane geometry and the Pixel Response Function for each CCD in a self-consistent manner. In addition to the fully automated calculation, a person-in-the-loop implementation was developed to allow an initial determination of the geometry in the event of large misalignments, achieving a much looser capture tolerance for more modest accuracy and reduced automation.

ccd↗

Trajectory Generation for Noise-Constrained Autonomous Flight Operations

One of the major factors in acceptance of aircraft operating in urban areas is noise. In this work, we build on a framework for trajectory generation in order to account for limits on acousticmetrics at one ormore observer locations. The spatial trajectories are generated using Bézier polynomials and satisfy dynamic, acoustic, and mission constraints. The trajectories also guarantee spatial or temporal separation between vehicles for multi-vehicle operations. A simulation example is provided that demonstrates the reduction in noise levels at a set of measurement locations.

Ackerman, Kasey A.↗

Optimization of Car Body under Constraints of Noise, Vibration, and Harshness (NVH), and Crash

To be competitive on the today's market, cars have to be as light as possible while meeting the Noise, Vibration, and Harshness (NVH) requirements and conforming to Government-man dated crash survival regulations. The latter are difficult to meet because they involve very compute-intensive, nonlinear analysis, e.g., the code RADIOSS capable of simulation of the dynamics, and the geometrical and material nonlinearities of a thin-walled car structure in crash, would require over 12 days of elapsed time for a single design of a 390K elastic degrees of freedom model, if executed on a single processor of the state-of-the-art SGI Origin2000 computer. Of course, in optimization that crash analysis would have to be invoked many times. Needless to say, that has rendered such optimization intractable until now. The car finite element model is shown. The advent of computers that comprise large numbers of concurrently operating processors has created a new environment wherein the above optimization, and other engineering problems heretofore regarded as intractable may be solved. The procedure, shown, is a piecewise approximation based method and involves using a sensitivity based Taylor series approximation model for NVH and a polynomial response surface model for Crash. In that method the NVH constraints are evaluated using a finite element code (MSC/NASTRAN) that yields the constraint values and their derivatives with respect to design variables. The crash constraints are evaluated using the explicit code RADIOSS on the Origin 2000 operating on 256 processors simultaneously to generate data for a polynomial response surface in the design variable domain. The NVH constraints and their derivatives combined with the response surface for the crash constraints form an approximation to the system analysis (surrogate analysis) that enables a cycle of multidisciplinary optimization within move limits. In the inner loop, the NVH sensitivities are recomputed to update the NVH approximation model while keeping the Crash response surface constant. In every outer loop, the Crash response surface approximation is updated, including a gradual increase in the order of the response surface and the response surface extension in the direction of the search. In this optimization task, the NVH discipline has 30 design variables while the crash discipline has 20 design variables. A subset of these design variables (10) are common to both the NVH and crash disciplines. In order to construct a linear response surface for the Crash discipline constraints, a minimum of 21 design points would have to be analyzed using the RADIOSS code. On a single processor in Origin 2000 that amount of computing would require over 9 months! In this work, these runs were carried out concurrently on the Origin 2000 using multiple processors, ranging from 8 to 16, for each crash (RADIOSS) analysis. Another figure shows the wall time required for a single RADIOSS analysis using varying number of processors, as well as provides a comparison of 2 different common data placement procedures within the allotted memories for each analysis. The initial design is an infeasible design with NVH discipline Static Torsion constraint violations of over 10%. The final optimized design is a feasible design with a weight reduction of 15 kg compared to the initial design. This work demonstrates how advanced methodology for optimization combined with the technology of concurrent processing enables applications that until now were out of reach because of very long time-to-solution.

Kodiyalam, Srinivas↗

Re-entry vehicle shape for enhanced performance

A convex shell structure for enhanced aerodynamic performance and/or reduced heat transfer requirements for a space vehicle that re-enters an atmosphere. The structure has a fore-body, an aft-body, a longitudinal axis and a transverse cross sectional shape, projected on a plane containing the longitudinal axis, that includes: first and second linear segments, smoothly joined at a first end of each the first and second linear segments to an end of a third linear segment by respective first and second curvilinear segments; and a fourth linear segment, joined to a second end of each of the first and second segments by curvilinear segments, including first and second ellipses having unequal ellipse parameters. The cross sectional shape is non-symmetric about the longitudinal axis. The fourth linear segment can be replaced by a sum of one or more polynomials, trigonometric functions or other functions satisfying certain constraints.

Brown, James L.↗

Sizing and Topology Design of an Aeroelastic Wingbox Under Uncertainty

The goals of this work are to use a nested optimizer to conduct simultaneous sizing (inner level) and topology (outer level) design of a wingbox, considering uncertainties in the safety factors used to define the aeroelastic constraints. These uncertainties, propagated via sampling-driven polynomial chaos, are explicitly introduced at the inner level of the method, during gradient-based sizing optimization, resulting in a stochastic optimal sizing distribution. Measures of robustness in the total structural mass are then passed to the outer level, where a global optimizer evolves the topology parameters. The results demonstrate design choices needed to improve robustness in the face of uncertain safety factors, and the various physical mechanisms driving this process.

Stanford, Bret K.↗

Orion Exploration Mission Entry Interface Target Line

The Orion Multi-Purpose Crew Vehicle is required to return to the continental United States at any time during the month. In addition, it is required to provide a survivable entry from a wide range of trans-lunar abort trajectories. The Entry Interface (EI) state must be targeted to ensure that all requirements are met for all possible return scenarios, even in the event of no communication with the Mission Control Center to provide an updated EI target. The challenge then is to functionalize an EI state constraint manifold that can be used in the on-board targeting algorithm, as well as the ground-based trajectory optimization programs. This paper presents the techniques used to define the EI constraint manifold and to functionalize it as a set of polynomials in several dimensions.

Rea, Jeremy R.↗

Comparison of polynomial approximations and artificial neural nets for response surfaces in engineering optimization

Engineering optimization problems involve minimizing some function subject to constraints. In areas such as aircraft optimization, the constraint equations may be from numerous disciplines such as transfer of information between these disciplines and the optimization algorithm. They are also suited to problems which may require numerous re-optimizations such as in multi-objective function optimization or to problems where the design space contains numerous local minima, thus requiring repeated optimizations from different initial designs. Their use has been limited, however, by the fact that development of response surfaces randomly selected or preselected points in the design space. Thus, they have been thought to be inefficient compared to algorithms to the optimum solution. A development has taken place in the last several years which may effect the desirability of using response surfaces. It may be possible that artificial neural nets are more efficient in developing response surfaces than polynomial approximations which have been used in the past. This development is the concern of the work.

Carpenter, William C.↗

The constraint method: A new finite element technique

An approch to the finite element method which utilizes families of conforming finite elements based on complete polynomials is presented. Finite element approximations based on this method converge with respect to progressively reduced element sizes as well as with respect to progressively increasing orders of approximation. Numerical results of static and dynamic applications of plates are presented to demonstrate the efficiency of the method. Comparisons are made with plate elements in NASTRAN and the high-precision plate element developed by Cowper and his co-workers. Some considerations are given to implementation of the constraint method into general purpose computer programs such as NASTRAN.

Tsai, C.↗

Displacement Based Multilevel Structural Optimization

In the complex environment of true multidisciplinary design optimization (MDO), efficiency is one of the most desirable attributes of any approach. In the present research, a new and highly efficient methodology for the MDO subset of structural optimization is proposed and detailed, i.e., for the weight minimization of a given structure under size, strength, and displacement constraints. Specifically, finite element based multilevel optimization of structures is performed. In the system level optimization, the design variables are the coefficients of assumed polynomially based global displacement functions, and the load unbalance resulting from the solution of the global stiffness equations is minimized. In the subsystems level optimizations, the weight of each element is minimized under the action of stress constraints, with the cross sectional dimensions as design variables. The approach is expected to prove very efficient since the design task is broken down into a large number of small and efficient subtasks, each with a small number of variables, which are amenable to parallel computing.

Sobieszezanski-Sobieski, J.↗

Smoothing of functions of range and range rate measurements from earth orbiting satellites

It is shown that for satellites in circular Earth orbits with altitudes of 500 kilometers to 1500 kilometers, and for satellites in elliptical orbits with an approximate 4000 kilometer height of perigee, a high degree least squares polynomial (e.g., a 9th or 10th degree in some cases) is required to smooth both range and range rate data for purposes of input to orbit determination programs. In order to circumvent this problem, functions of range and range rate are smoothed with lower degree least squares polynomials (e.g., 3rd and 4th degree) and it is shown that under the above geometric constraints the standard deviation of fit can be reduced to levels commensurate with typical S-band tracking system resolution which is 1 to 2 meters in range and 0.005 meters/second in range rate for a 1 per second data rate. Also shown are the effects of Gaussian random noise, biases, and periodic noise. This analysis includes numerous examples applied to the 44 point data smoothing interval currently used in much of the operational preprocessing at the Goddard Space Flight Center.

Grenchik, T. J.↗

A triangular thin shell finite element: Linear analysis

The formulation of the linear stiffness matrix for a doubly-curved triangular thin shell element, using a modified potential energy principle, is described. The strain energy component of the potential energy is expressed in terms of displacements and displacement gradients by use of consistent Koiter strain-displacement equations. The element inplane and normal displacement fields are approximated by complete cubic polynomials. The interelement displacement admissibility conditions are met in the global representation by imposition of constraint conditions on the interelement boundaries; the constraints represent the modification of the potential energy. Errors due to the nonzero strains under rigid body motion are shown to be of small importance for practical grid refinements through performance of extensive comparison analyses.

Thomas, G. R.↗