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All sourcesNASA NTRSDOE OSTI
NASA NTRS1982

Climate studies with a multi-layer energy balance model. I - Model description and sensitivity to the solar constant. II - The role of feedback mechanisms in the CO2 problem

A nine-layer zonally averaged, steady-state model, based upon thermal energy balance, is developed for use in climate sensitivity studies and includes an accurate treatment of radiative transfer, parameterized meridional and vertical energy transport, and thermodynamic interaction between the surface and the atmosphere. A high degree of nonlinearity is exhibited by the model in a study of sensitivity to changes in the solar constant. The change in the hemispheric mean surface temperature is +3.1 C in response to a 2% increase in the solar constant and -4.3 C in response to a 2% decrease in the solar constant. The sensitivity varies with latitude, and the response of atmospheric temperature varies with height. In addition, the model is used to study the sensitivity of climate to a doubling of the atmospheric CO2 content. It is found that the tropospheric temperature lapse rate decreases at low latitudes but increases at high latitudes in response to a doubled CO2 content. Averaged over the Northern Hemisphere, the change is +2.3 C in the surface temperature and +0.47 C in the earth's brightness temperature. The effects of some feedback mechanisms on the climate sensitivity to a doubled CO2 content show that the sensitivity of surface temperature approximately doubles at all latitudes due to the change in water vapor content.

Peng, L.↗
NASA NTRS1987

Investigation, development and application of optimal output feedback theory. Vol. 4: Measures of eigenvalue/eigenvector sensitivity to system parameters and unmodeled dynamics

Some measures of eigenvalue and eigenvector sensitivity applicable to both continuous and discrete linear systems are developed and investigated. An infinite series representation is developed for the eigenvalues and eigenvectors of a system. The coefficients of the series are coupled, but can be obtained recursively using a nonlinear coupled vector difference equation. A new sensitivity measure is developed by considering the effects of unmodeled dynamics. It is shown that the sensitivity is high when any unmodeled eigenvalue is near a modeled eigenvalue. Using a simple example where the sensor dynamics have been neglected, it is shown that high feedback gains produce high eigenvalue/eigenvector sensitivity. The smallest singular value of the return difference is shown not to reflect eigenvalue sensitivity since it increases with the feedback gains. Using an upper bound obtained from the infinite series, a procedure to evaluate whether the sensitivity to parameter variations is within given acceptable bounds is developed and demonstrated by an example.

Halyo, Nesim↗
NASA NTRS1988

An investigation of new methods for estimating parameter sensitivities

Parameter sensitivity is defined as the estimation of changes in the modeling functions and the design variables due to small changes in the fixed parameters of the formulation. There are currently several methods for estimating parameter sensitivities requiring either difficult to obtain second order information, or do not return reliable estimates for the derivatives. Additionally, all the methods assume that the set of active constraints does not change in a neighborhood of the estimation point. If the active set does in fact change, than any extrapolations based on these derivatives may be in error. The objective here is to investigate more efficient new methods for estimating parameter sensitivities when the active set changes. The new method is based on the recursive quadratic programming (RQP) method and in conjunction a differencing formula to produce estimates of the sensitivities. This is compared to existing methods and is shown to be very competitive in terms of the number of function evaluations required. In terms of accuracy, the method is shown to be equivalent to a modified version of the Kuhn-Tucker method, where the Hessian of the Lagrangian is estimated using the BFS method employed by the RPQ algorithm. Inital testing on a test set with known sensitivities demonstrates that the method can accurately calculate the parameter sensitivity. To handle changes in the active set, a deflection algorithm is proposed for those cases where the new set of active constraints remains linearly independent. For those cases where dependencies occur, a directional derivative is proposed. A few simple examples are included for the algorithm, but extensive testing has not yet been performed.

Beltracchi, Todd J.↗
NASA NTRS1989

An investigation of using an RQP based method to calculate parameter sensitivity derivatives

Estimation of the sensitivity of problem functions with respect to problem variables forms the basis for many of our modern day algorithms for engineering optimization. The most common application of problem sensitivities has been in the calculation of objective function and constraint partial derivatives for determining search directions and optimality conditions. A second form of sensitivity analysis, parameter sensitivity, has also become an important topic in recent years. By parameter sensitivity, researchers refer to the estimation of changes in the modeling functions and current design point due to small changes in the fixed parameters of the formulation. Methods for calculating these derivatives have been proposed by several authors (Armacost and Fiacco 1974, Sobieski et al 1981, Schmit and Chang 1984, and Vanderplaats and Yoshida 1985). Two drawbacks to estimating parameter sensitivities by current methods have been: (1) the need for second order information about the Lagrangian at the current point, and (2) the estimates assume no change in the active set of constraints. The first of these two problems is addressed here and a new algorithm is proposed that does not require explicit calculation of second order information.

Beltracchi, Todd J.↗
NASA NTRS1990

Sensitivity of the transmission zeros of flexible space structures

The new pole/zero cancellation technique developed for the problem of sensitivity and robustness in vibration isolation systems for flexible spacecraft requires an analysis of the sensitivity of transmission zeros. This paper analyzes the sensitivity of the transmission zeros of flexible structures with compatible sensors and actuators, in terms of partial derivatives and condition numbers. It is shown that, in both measures, the sensitivities of the zeros of such a system are closely related to those of its poles. It is also shown that the closed-loop poles produced by applying the pole/zero cancellation to the structure have sensitivities approaching those of the zeros, so that these are given by the sensitivities of the open-loop poles. Examples are presented that illustrate these points.

Williams, Trevor↗
NASA NTRS1991

A discourse on sensitivity analysis for discretely-modeled structures

A descriptive review is presented of the most recent methods for performing sensitivity analysis of the structural behavior of discretely-modeled systems. The methods are generally but not exclusively aimed at finite element modeled structures. Topics included are: selections of finite difference step sizes; special consideration for finite difference sensitivity of iteratively-solved response problems; first and second derivatives of static structural response; sensitivity of stresses; nonlinear static response sensitivity; eigenvalue and eigenvector sensitivities for both distinct and repeated eigenvalues; and sensitivity of transient response for both linear and nonlinear structural response.

Adelman, Howard M.↗
NASA NTRS1990

The physics of the environmental sensitivity of rubidium gas cell atomic frequency standards

Environmental sensitivity is often the most significant limitation to the practical stability of rubidium frequency standards (RFS). For example, temperature sensitivity can cause a rapid frequency change of several parts in 10(exp 10) for a tactical RFS that has an aging of only 1 times 10(exp -11)/month. Other important environmental factors are barometric pressure, vibration, magnetic field, and nuclear radiation. The physical mechanisms that lie behind these environmental sensitivities are considered. These physical mechanisms are related to the performance of actual rubidium frequency standards. For the user of these devices, a better knowledge of the causes for Rb clock instability will aid in their testing and proper application. For the time frequency specialist, a review of these factors may prove useful toward improving RFS design. Some of the RFS environmental sensitivities are due to simple physical mechanisms like the effect of dc magnetic field on the Rb hyperfine resonance frequency. For these, an analysis can be based on physical principles and straightforward design factors. Other environment factors, like temperature sensitivity, are more complex combinations of many effects, both physical and practical, and the analysis often takes the form of an error budget with large unit-to-unit variations. Today's rubidium frequency standards span a wide performance range from small, inexpensive units with pp 10(exp 10) error budgets to larger, higher performance versions offering pp 10(exp 10) stabilities. For both extremes, however, environmental sensitivity can be the most significant performance limitation. Why this is the case and how to make improvements are discussed.

Riley, William J.↗
NASA NTRS1991

Sensitivity analysis of a wing aeroelastic response

A variation of Sobieski's Global Sensitivity Equations (GSE) approach is implemented to obtain the sensitivity of the static aeroelastic response of a three-dimensional wing model. The formulation is quite general and accepts any aerodynamics and structural analysis capability. An interface code is written to convert one analysis's output to the other's input, and visa versa. Local sensitivity derivatives are calculated by either analytic methods or finite difference techniques. A program to combine the local sensitivities, such as the sensitivity of the stiffness matrix or the aerodynamic kernel matrix, into global sensitivity derivatives is developed. The aerodynamic analysis package FAST, using a lifting surface theory, and a structural package, ELAPS, implementing Giles' equivalent plate model are used.

Kapania, Rakesh K.↗
NASA NTRS1992

An analytical approach to grid sensitivity analysis for NACA four-digit wing sections

Sensitivity analysis in computational fluid dynamics with emphasis on grids and surface parameterization is described. An interactive algebraic grid-generation technique is employed to generate C-type grids around NACA four-digit wing sections. An analytical procedure is developed for calculating grid sensitivity with respect to design parameters of a wing section. A comparison of the sensitivity with that obtained using a finite difference approach is made. Grid sensitivity with respect to grid parameters, such as grid-stretching coefficients, are also investigated. Using the resultant grid sensitivity, aerodynamic sensitivity is obtained using the compressible two-dimensional thin-layer Navier-Stokes equations.

Sadrehaghighi, I.↗
NASA NTRS1992

An analytical approach to grid sensitivity analysis

Sensitivity analysis in Computational Fluid Dynamics with emphasis on grids and surface parameterization is described. An interactive algebraic grid-generation technique is employed to generate C-type grids around NACA four-digit wing sections. An analytical procedure is developed for calculating grid sensitivity with respect to design parameters of a wing section. A comparison of the sensitivity with that obtained using a finite-difference approach is made. Grid sensitivity with respect to grid parameters, such as grid-stretching coefficients, are also investigated. Using the resultant grid sensitivity, aerodynamic sensitivity is obtained using the compressible two-dimensional thin-layer Navier-Stokes equations.

Sadrehaghighi, Ideen↗
NASA NTRS1993

Sensitivity of discrete systems

Sensitivity techniques are examined for structures discretized by finite element analysis. The discussion covers finite difference sensitivities, sensitivity of static response, sensitivity of eigenvalues and eigenvectors, sensitivity of linear transient response, and an example of use of transient sensitivity.

Haftka, Raphael T.↗
NASA NTRS1993

An adjoint sensitivity study of blocking in a two-layer isentropic model

This paper presents a new methodology for adjoint sensitivity analysis, previously developed in general terms by Cacuci, into a form directly applicable 10 meteorological problems. This technique is illustrated by examining the sensitivity of a blocking index in a two-layer isentropic model. The index represents a response function for the sensitivity analysis that unlike previous meteorological applications, is an operator and not a functional, and thus extends the scope of adjoint sensitivity to general operator-type responses depending on time and/or space. The sensitivity of the blocking index to perturbations introduced into the model atmosphere, as well as to model parameters, is discussed. The methodology of generalized adjoint sensitivity analysis described in this paper constitutes a prototype for further applications in the atmospheric and/or oceanic sciences.

Zou, X.↗
NASA NTRS1994

Method development for compensating temperature effects in pressure sensitive paint measurements

Pressure sensitive luminescent paints (PSP) have recently emerged as a viable technique for aerodynamic pressure measurements. The technique uses a surface coating which contains probe molecules that luminesce when excited by light of an appropriate wavelength. The photoluminescence of these materials is known to be quenched by the presence of molecular oxygen. Since oxygen is a fixed mole fraction of the air, the coating's luminescence intensity varies inversely with air pressure. Digital imaging of the luminescence varying across a coated surface produces a pressure distribution map over that surface. One difficulty encountered with this technique is the temperature effect on the luminescence intensity. Present PSP formulations have significant sensitivity to temperature. At the moment, the most practical way of correcting for temperature effects is to calibrate the paint in place at the operating temperatures by using a few well-placed pressure taps. This study is looking at development of temperature indicating coatings that can be applied and measured concurrently with PSP, and use the temperature measurement to compute the correct pressure. Two methods for this dual paint formulation are proposed. One method will use a coating that consists of temperature sensitive phosphors in a polymer matrix. This is similar in construction to PSP, except that the probe molecules used are selected primarily for their temperature sensitivity. Both organic phosphors (e.g., europium thenoyltrifluoroacetonate, bioprobes) and inorganic phosphors (e.g., Mg4(F)GeO6:Mn, La2O2S:Eu, Radelin Type phosphors, Sylvania Type phosphors) will be evaluated for their temperature sensing potential. The next method will involve a novel coating composing of five membered heterocyclic conducting polymers which are known to show temperature dependent luminescence (e.g., poly(3-alkylthiopene), poly(3-alkylselenophene), poly(3-alkylfuran)). Both methods will involve applying a bottom layer of temperature sensitive coating followed by a top coating of PSP. An oxygen-impermeable polymer can be used as the temperature sensitive coating matrix, or it can be layered in between the coatings to prevent oxygen quenching of the bottom coating's luminescence. The probe molecules of the coatings will be excited by a broad band of light, with the different emissions detected and measured at their distinct wavelengths. Developmental research of these coatings is still in progress; however, preliminary results look very promising.

Demandante, Carlo Greg N.↗
NASA NTRS1995

Sensitivity analysis for large-deflection and postbuckling responses on distributed-memory computers

A computational strategy is presented for calculating sensitivity coefficients for the nonlinear large-deflection and postbuckling responses of laminated composite structures on distributed-memory parallel computers. The strategy is applicable to any message-passing distributed computational environment. The key elements of the proposed strategy are: (1) a multiple-parameter reduced basis technique; (2) a parallel sparse equation solver based on a nested dissection (or multilevel substructuring) node ordering scheme; and (3) a multilevel parallel procedure for evaluating hierarchical sensitivity coefficients. The hierarchical sensitivity coefficients measure the sensitivity of the composite structure response to variations in three sets of interrelated parameters; namely, laminate, layer and micromechanical (fiber, matrix, and interface/interphase) parameters. The effectiveness of the strategy is assessed by performing hierarchical sensitivity analysis for the large-deflection and postbuckling responses of stiffened composite panels with cutouts on three distributed-memory computers. The panels are subjected to combined mechanical and thermal loads. The numerical studies presented demonstrate the advantages of the reduced basis technique for hierarchical sensitivity analysis on distributed-memory machines.

Watson, Brian C.↗
NASA NTRS1997

Variational Methods in Design Optimization and Sensitivity Analysis for Two-Dimensional Euler Equations

Variational methods (VM) sensitivity analysis employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

Ibrahim, A. H.↗
NASA NTRS1998

Overview of Sensitivity Analysis and Shape Optimization for Complex Aerodynamic Configurations

This paper presents a brief overview of some of the more recent advances in steady aerodynamic shape-design sensitivity analysis and optimization, based on advanced computational fluid dynamics. The focus here is on those methods particularly well- suited to the study of geometrically complex configurations and their potentially complex associated flow physics. When nonlinear state equations are considered in the optimization process, difficulties are found in the application of sensitivity analysis. Some techniques for circumventing such difficulties are currently being explored and are included here. Attention is directed to methods that utilize automatic differentiation to obtain aerodynamic sensitivity derivatives for both complex configurations and complex flow physics. Various examples of shape-design sensitivity analysis for unstructured-grid computational fluid dynamics algorithms are demonstrated for different formulations of the sensitivity equations. Finally, the use of advanced, unstructured-grid computational fluid dynamics in multidisciplinary analyses and multidisciplinary sensitivity analyses within future optimization processes is recommended and encouraged.

Newman, Perry A.↗
NASA NTRS1999

Overview of Sensitivity Analysis and Shape Optimization for Complex Aerodynamic Configurations

This paper presents a brief overview of some of the more recent advances in steady aerodynamic shape-design sensitivity analysis and optimization, based on advanced computational fluid dynamics (CFD). The focus here is on those methods particularly well-suited to the study of geometrically complex configurations and their potentially complex associated flow physics. When nonlinear state equations are considered in the optimization process, difficulties are found in the application of sensitivity analysis. Some techniques for circumventing such difficulties are currently being explored and are included here. Attention is directed to methods that utilize automatic differentiation to obtain aerodynamic sensitivity derivatives for both complex configurations and complex flow physics. Various examples of shape-design sensitivity analysis for unstructured-grid CFD algorithms are demonstrated for different formulations of the sensitivity equations. Finally, the use of advanced, unstructured-grid CFDs in multidisciplinary analyses and multidisciplinary sensitivity analyses within future optimization processes is recommended and encouraged.

Newman, James C., III↗
NASA NTRS2004

Employing Sensitivity Derivatives to Estimate Uncertainty Propagation in CFD

Two methods that exploit the availability of sensitivity derivatives are successfully employed to predict uncertainty propagation through Computational Fluid Dynamics (CFD) code for an inviscid airfoil problem. An approximate statistical second-moment method and a Sensitivity Derivative Enhanced Monte Carlo (SDEMC) method are successfully demonstrated on a two-dimensional problem. First- and second-order sensitivity derivatives of code output with respect to code input are obtained through an efficient incremental iterative approach. Given uncertainties in statistically independent, random, normally distributed flow parameters (input variables); these sensitivity derivatives enable one to formulate first- and second-order Taylor Series approximations for the mean and variance of CFD output quantities. Additionally, incorporation of the first-order sensitivity derivatives into the data reduction phase of a conventional Monte Carlo (MC) simulation allows for improved accuracy in determining the first moment of the CFD output. Both methods are compared to results generated using a conventional MC method. The methods that exploit the availability of sensitivity derivatives are found to be valid when considering small deviations from input mean values.

Putko, Michele M.↗