Results for “Analytic derivatives”

NASA NTRS2018

Determination of Uncertainties for Analytically Derived Material Properties to Be Used in Monte Carlo Based Orion Heatshield Sizing

Ablative materials are often used for spacecraft heatshields to protect underlying structures from the extreme environments associated with atmospheric reentry. NASA's Orion EM-1 capsule has been designed to use a molded Avcoat material system. In order to determine the required heatshield thickness, a Monte Carlo approach to the sizing process was proposed. To perform the Monte Carlo simulation, statistical uncertainties on all material property input parameters were required. Obtaining these values for measured properties is straightforward, however input parameters that are derived analytically have historically used uncertainties based on engineering judgment. A MATLAB program was created to use laboratory generated thermogravimetric analysis (TGA) data to calculate uncertainties on the Arrhenius parameters for molded Avcoat. Uncertainties associated with the normalized ablation rate and pyrolysis gas enthalpy were also generated using a wrapper script and the ACE code. These uncertainties could then be tied directly to measured values of individual elemental constituents. The resulting uncertainty values will allow for a probabilistic sizing approach on molded Avcoat with a higher level of confidence in the input parameters.

material property uncertainties↗

NASA NTRS1991

Planetary accretion rates - Analytical derivation

New analytical formulae that are valid even in cases where Keplerian shear determines the approach velocity in planetary accretion rates are presently shown to accurately reproduce Monte Carlo study results. The formulae yield accretion rates over a great variety of planet formation-conducive conditions. Since the absolute values of impact rates yielded by the formulae were expected to have factor-of-2-to-3 uncertainties, the good agreement with Monte Carlo experiments obtained suggests the reasonableness of many of the estimates adopted, or the mutual cancelling-out of various uncertainties.

Greenberg, Richard↗

NASA NTRSDate not supplied

Impact of Analytic Derivatives on Optimization of N-Impulse Orbit Transfer

Several formulations are possible for the optimization of N-impulse two-body orbit transfers. One formulation that assumes the firstN−1impulses are design variables and implements Lambert’s algorithm in the final leg is here considered. This paper presents a derivation for the analytic expressions of the gradients needed to optimize this formulation. The impact of using these analytic expressions on the optimization computational cost is also presented. A linear system of equations is developed that approximates the Lambert’s problem solution in a way that is suitable for computing the analytic gradients. The derivations of the analytic gradients, as well as numerical case studies for 2-impulse and 3-impulse orbit transfers, are presented. The numerical case studies highlights a significant reduction in the computational cost, measured in terms of the number of function calls.

Ahmed Ellithy↗

NASA NTRS2016

Parallel Aircraft Trajectory Optimization with Analytic Derivatives

Trajectory optimization is an integral component for the design of aerospace vehicles, but emerging aircraft technologies have introduced new demands on trajectory analysis that current tools are not well suited to address. Designing aircraft with technologies such as hybrid electric propulsion and morphing wings requires consideration of the operational behavior as well as the physical design characteristics of the aircraft. The addition of operational variables can dramatically increase the number of design variables which motivates the use of gradient based optimization with analytic derivatives to solve the larger optimization problems. In this work we develop an aircraft trajectory analysis tool using a Legendre-Gauss-Lobatto based collocation scheme, providing analytic derivatives via the OpenMDAO multidisciplinary optimization framework. This collocation method uses an implicit time integration scheme that provides a high degree of sparsity and thus several potential options for parallelization. The performance of the new implementation was investigated via a series of single and multi-trajectory optimizations using a combination of parallel computing and constraint aggregation. The computational performance results show that in order to take full advantage of the sparsity in the problem it is vital to parallelize both the non-linear analysis evaluations and the derivative computations themselves. The constraint aggregation results showed a significant numerical challenge due to difficulty in achieving tight convergence tolerances. Overall, the results demonstrate the value of applying analytic derivatives to trajectory optimization problems and lay the foundation for future application of this collocation based method to the design of aircraft with where operational scheduling of technologies is key to achieving good performance.

aircraft↗

NASA NTRSDate not supplied

End-To-End Uncertainty Quantification with Analytical Derivatives for Design Under Uncertainty

Uncertainty quantification (UQ) is a rapidly growing and evolving discipline, especially within the aerospace community. Performing analysis with UQ can provide decision makers with a wealth of information about a candidate design. However, the value of UQ is fully realized when the information gained during UQ analysis is leveraged in a feedback loop of a design optimization process, often referred to as design under uncertainty. Although design under uncertainty can be a powerful risk mitigation technique, there are a number of roadblocks that prevent its implementation. Two primary factors are computational costs and added complexity of the analysis. High fidelity simulations on the order tens of uncertain variables quickly become computationally infeasible. Also, implementing UQ into an existing multidisciplinary design and optimization (MDO) process often requires extensive knowledge of the UQ methods and careful treatment of the problem formulation. The objective of this work is to address these two primary roadblocks and enable practitioners to efficiently perform design under uncertainty with limited knowledge of the UQ discipline. Methods outlined in this paper demonstrate MDO incorporating UQ into the design process, leveraging an analytic derivative tool chain through the entire optimization. The proposed approach leverages machine learning techniques to generate a differentiable confidence interval output from polynomial chaos models. This technique, coupled with the incorporation of analytical derivatives through the Polynomial Chaos Expansion (PCE) process, eliminates the need to estimate derivatives which are usually obtained from finite difference, complex step, or similar methods. Developing a differentiable confidence interval allows mixed uncertainty problems (both epistemic and aleatory) to be modeled. Without such modeling, these problems cannot accurately predict objective functions containing statistical quantities such as mean and variance. The addition of analytic derivatives to a polynomial chaos-based UQ method decreases the computational costs of performing design under uncertainty by orders of magnitude in comparison with methods such as complex step. The method and codes developed are modular in nature and are a drop-in solution for design under uncertainty within existing MDO problems. A low-fidelity analytical multidisciplinary optimization under uncertainty for a wing design in OpenMDAO is detailed in this paper. This demonstration case will include both objective functions and constraints which are influenced by uncertain parameters.

Ben D Phillips↗

NASA NTRSDate not supplied

End-To-End Uncertainty Quantification with Analytical Derivatives for Design Under Uncertainty

Uncertainty quantification (UQ) is a rapidly growing and evolving discipline, especially within the aerospace community. Performing analysis with UQ can provide decision makers with a wealth of information about a candidate design. However, the value of UQ is fully realized when the information gained during UQ analysis is leveraged in a feedback loop of a design optimization process, often referred to as design under uncertainty. Although design under uncertainty can be a powerful risk mitigation technique, there are a number of roadblocks that prevent its implementation. Two primary factors are computational costs and added complexity of the analysis. High fidelity simulations on the order tens of uncertain variables quickly become computationally infeasible. Also, implementing UQ into an existing multidisciplinary design and optimization (MDO) process often requires extensive knowledge of the UQ methods and careful treatment of the problem formulation. The objective of this work is to address these two primary roadblocks and enable practitioners to efficiently perform design under uncertainty with limited knowledge of the UQ discipline. Methods outlined in this paper demonstrate MDO incorporating UQ into the design process, leveraging an analytic derivative tool chain through the entire optimization. The proposed approach leverages machine learning techniques to generate a differentiable confidence interval output from polynomial chaos models. This technique, coupled with the incorporation of analytical derivatives through the Polynomial Chaos Expansion (PCE) process, eliminates the need to estimate derivatives which are usually obtained from finite difference, complex step, or similar methods. Developing a differentiable confidence interval allows mixed uncertainty problems (both epistemic and aleatory) to be modeled. Without such modeling, these problems cannot accurately predict objective functions containing statistical quantities such as mean and variance. The addition of analytic derivatives to a polynomial chaos-based UQ method decreases the computational costs of performing design under uncertainty by orders of magnitude in comparison with methods such as complex step. The method and codes developed are modular in nature and are a drop-in solution for design under uncertainty within existing MDO problems. A low-fidelity analytical multidisciplinary optimization under uncertainty for a wing design in OpenMDAO is detailed in this paper. This demonstration case will include both objective functions and constraints which are influenced by uncertain parameters.

Ben Phillips↗

NASA NTRS2016

Thermodynamics of Gas Turbine Cycles with Analytic Derivatives in OpenMDAO

A new equilibrium thermodynamics analysis tool was built based on the CEA method using the OpenMDAO framework. The new tool provides forward and adjoint analytic derivatives for use with gradient based optimization algorithms. The new tool was validated against the original CEA code to ensure an accurate analysis and the analytic derivatives were validated against finite-difference approximations. Performance comparisons between analytic and finite difference methods showed a significant speed advantage for the analytic methods. To further test the new analysis tool, a sample optimization was performed to find the optimal air-fuel equivalence ratio, , maximizing combustion temperature for a range of different pressures. Collectively, the results demonstrate the viability of the new tool to serve as the thermodynamic backbone for future work on a full propulsion modeling tool.

Propulsion↗

NASA NTRS2016

Optimization of Turbine Engine Cycle Analysis with Analytic Derivatives

A new engine cycle analysis tool, called Pycycle, was built using the OpenMDAO framework. Pycycle provides analytic derivatives allowing for an efficient use of gradient-based optimization methods on engine cycle models, without requiring the use of finite difference derivative approximation methods. To demonstrate this, a gradient-based design optimization was performed on a turbofan engine model. Results demonstrate very favorable performance compared to an optimization of an identical model using finite-difference approximated derivatives.

OpenMDAO↗

NASA NTRS2016

Optimization of Turbine Engine Cycle Analysis with Analytic Derivatives

A new engine cycle analysis tool, called Pycycle, was recently built using the OpenMDAO framework. This tool uses equilibrium chemistry based thermodynamics, and provides analytic derivatives. This allows for stable and efficient use of gradient-based optimization and sensitivity analysis methods on engine cycle models, without requiring the use of finite difference derivative approximation methods. To demonstrate this, a gradient-based design optimization was performed on a multi-point turbofan engine model. Results demonstrate very favorable performance compared to an optimization of an identical model using finite-difference approximated derivatives.

Turbine engine↗

NASA NTRS2019

Load Flow Analysis with Analytic Derivatives for Electric Aircraft Design Optimization

Many of the aircraft concepts of the future are exploring the use of hybrid-, turbo- or all-electric propulsion systems to improve performance and decrease environmental impacts. These aircraft concepts range from small rotorcraft for urban air mobility to conventional commercial transports to large blended wing body designs. Developing the conceptual design for these vehicles presents a challenge, however, as traditional aircraft design tools often were not developed to handle these unique propulsion system architectures. Previous studies on these vehicles have therefore relied on relatively simple models of the electrical transmission and distribution system. This paper presents the development of a hybrid AC-DC load flow (or power flow) analysis capability to enhance the conceptual design of these concept vehicles. Specifically, the desire was to create a load flow analysis capability within the OpenMDAO framework that is also being used to develop a set of compatible tools for rapid optimization of conceptual designs. This load flow analysis capability is unique in its flexible object-oriented structure and implementation of analytic derivatives to facilitate the use of solvers and gradient based optimization in the design process. The developed hybrid load flow analysis capability is first verified against a published 13-bus example then used to model the electrical distribution system for a turbo-electric tiltwing aircraft.

Load flow↗

NASA NTRS2017

A Requirements-Driven Optimization Method for Acoustic Liners Using Analytic Derivatives

More than ever, there is flexibility and freedom in acoustic liner design. Subject to practical considerations, liner design variables may be manipulated to achieve a target attenuation spectrum. But characteristics of the ideal attenuation spectrum can be difficult to know. Many multidisciplinary system effects govern how engine noise sources contribute to community noise. Given a hardwall fan noise source to be suppressed, and using an analytical certification noise model to compute a community noise measure of merit, the optimal attenuation spectrum can be derived using multidisciplinary systems analysis methods. In a previous paper on this subject, a method deriving the ideal target attenuation spectrum that minimizes noise perceived by observers on the ground was described. A simple code-wrapping approach was used to evaluate a community noise objective function for an external optimizer. Gradients were evaluated using a finite difference formula. The subject of this paper is an application of analytic derivatives that supply precise gradients to an optimization process. Analytic derivatives improve the efficiency and accuracy of gradient-based optimization methods and allow consideration of more design variables. In addition, the benefit of variable impedance liners is explored using a multi-objective optimization.

Aircraft Noise↗

NASA NTRS1953

Analytical Derivation and Experimental Evaluation of Short-Bearing Approximation for Full Journal Bearing

An approximate analytical solution including the effect of end leakage from the oil film of short plain bearings is presented because of the importance of endwise flow in sleeve bearings of the short lengths commonly used. The analytical approximation is supported by experimental data, resulting in charts which facilitate analysis of short plain bearings. The analytical approximation includes the endwise flow and that part of the circumferential flow which is related to surface velocity and film thickness but neglects the effect of film pressure on the circumferential flow. In practical use, this approximation applies best to bearings having a length-diameter ratio up to 1, and the effects of elastic deflection, inlet oil pressure, and changes of clearance with temperature minimize the relative importance of the neglected term. The analytical approximation was found to be an extension of a little-known pressure-distribution function originally proposed by Michell and Cardullo.

Dubois, George B↗

NASA NTRS1973

A theoretical analysis of the free vibrations of ring- and/or stringer-stiffened elliptical cylinders with arbitrary end conditions. Volume 1: Analytical derivation and applications

The derivation and application of a Rayleigh-Ritz modal vibration analysis are presented for ring and/or stringer stiffened noncircular cylindrical shells with arbitrary end conditions. Comparisons with previous results from experimental and analytical studies showed this method of analysis to be accurate for a variety of end conditions. Results indicate a greater effect of rings on natural frequencies than of stringers.

Boyd, D. E.↗

NASA NTRS1985

Experimental and analytical derivation of arc-heater scaling laws for simulating high-enthalpy environments for Aeroassisted Orbital Transfer Vehicle application

The computer code ARCFLO II was used as a guide to increase the performance of the Interaction Heating Facility at Ames Research Center. A closed-form scaling law relation was derived that provides an understanding of the factors that affect enthalpy in the constricted-arc heater. From a study of this scaling law, it is concluded that at constant pressure, enthalpy is proportional to current density raised to the 0.60 power for current densities from 80 to 150 A/sq cm. At constant current density, enthalpy is inversely proportional to pressure to the nth power, where n varies from 0.14 to 0.43, depending on the current density. Radiative heat losses are responsible for the falloff in performance at combinations of high current density and high pressure. An analytical, closed form scaling law based on a constant-temperature arc-core model agrees qualitatively with the scaling law deduced from ARCFLO II.

Winovich, W.↗

NASA NTRS1969

Analytical derivation of the time domain expression for a non-periodic frequency modulated square wave.

Time domain expression derived for frequency varying nonperiodic square wave in FDM and FM systems, using orthogonal functions to obtain frequency spectrum

Smith, C. E.↗

NASA NTRS1977

Analytic derivation and evaluation of a state-trajectory control law for dc-to-dc converters

Mathematical representations of a state-plane switching boundary employed in a state-trajectory control law for dc-to-dc converters are derived. Several levels of approximation to the switching boundary equations are presented, together with an evaluation of the effects of nonideal operating characteristics of converter power stage components on the shape and location of the boundary and the behavior of a system controlled by it. Digital computer simulations of dc-to-dc converters operating in conjunction with each of these levels of control are presented and evaluated with respect to changes in transient and steady-state performance.

Burns, W. W., III↗

NASA NTRS1981

A comparison of observed and analytically derived remote sensing penetration depths for turbid water

The depth to which sunlight will penetrate in turbid waters was investigated. The tests were conducted in water with a single scattering albedo range, and over a range of solar elevation angles. Two different techniques were used to determine the depth of light penetration. It showed little change in the depth of sunlight penetration with changing solar elevation angle. A comparison of the penetration depths indicates that the best agreement between the two methods was achieved when the quasisingle scattering relationship was not corrected for solar angle. It is concluded that sunlight penetration is dependent on inherent water properties only.

Morris, W. D.↗

NASA NTRS1988

Accuracy problems associated with semi-analytical derivatives of static response

The semianalytical method is widely used for calculating derivatives of static response with respect to design variables for structures modeled by finite elements. This paper shows that the method can have serious accuracy problems for shape design variables in structures modeled by beam elements. The problem was first discovered in the analysis of a complex car model which is described. Next, it is shown that the same accuracy problem occurs for the simplest beam structures. Finally, the problem is shown to be associated with a high ratio of rigid body rotation to elastic deformation, and a test is developed to diagnose the severity of the problem in a given structure. Several examples are presented to show the validity of this test.

Barthelemy, Bruno↗