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NASA NTRS2005

Advanced Turbulence Modeling Concepts

The ZCET program developed at NASA Glenn Research Center is to study hydrogen/air injection concepts for aircraft gas turbine engines that meet conventional gas turbine performance levels and provide low levels of harmful NOx emissions. A CFD study for ZCET program has been successfully carried out. It uses the most recently enhanced National combustion code (NCC) to perform CFD simulations for two configurations of hydrogen fuel injectors (GRC- and Sandia-injector). The results can be used to assist experimental studies to provide quick mixing, low emission and high performance fuel injector designs. The work started with the configuration of the single-hole injector. The computational models were taken from the experimental designs. For example, the GRC single-hole injector consists of one air tube (0.78 inches long and 0.265 inches in diameter) and two hydrogen tubes (0.3 inches long and 0.0226 inches in diameter opposed at 180 degree). The hydrogen tubes are located 0.3 inches upstream from the exit of the air element (the inlet location for the combustor). To do the simulation, the single-hole injector is connected to a combustor model (8.16 inches long and 0.5 inches in diameter). The inlet conditions for air and hydrogen elements are defined according to actual experimental designs. Two crossing jets of hydrogen/air are simulated in detail in the injector. The cold flow, reacting flow, flame temperature, combustor pressure and possible flashback phenomena are studied. Two grid resolutions of the numerical model have been adopted. The first computational grid contains 0.52 million elements, the second one contains over 1.3 million elements. The CFD results have shown only about 5% difference between the two grid resolutions. Therefore, the CFD result obtained from the model of 1.3-million grid resolution can be considered as a grid independent numerical solution. Turbulence models built in NCC are consolidated and well tested. They can handle both coarse and fine grids near the wall. They can model the effect of anisotropy of turbulent stresses and the effect of swirling. The chemical reactions of Magnusson model and ILDM method were both used in this study.

Shih, Tsan-Hsing↗
NASA NTRS2016

Photosynthetic Efficiency of Northern Forest Ecosystems Using a MODIS-Derived Photochemical Reflectance Index (PRI)

This study evaluates a direct remote sensing approach from space for the determination of ecosystem photosynthetic light use efficiency (LUE), through measurement of vegetation reflectance changes expressed with the Photochemical Reflectance Index (PRI). The PRI is a normalized difference index based on spectral changes at a physiologically active wavelength (approximately 531 nanometers) as compared to a reference waveband, and is only available from a very few satellites. These include the two Moderate-Resolution Imaging Spectroradiometers (MODIS) on the Aqua and Terra satellites each of which have a narrow (10-nanometer) ocean band centered at 531 nanometers. We examined several PRI variations computed with candidate reference bands, since MODIS lacks the traditional 570-nanometer reference band. The PRI computed using MODIS land band 1 (620-670 nanometers) gave the best performance for daily LUE estimation. Through rigorous statistical analyses over a large image collection (n equals 420), the success of relating in situ daily tower-derived LUE to MODIS observations for northern forests was strongly influenced by satellite viewing geometry. LUE was calculated from CO2 fluxes (moles per moles of carbon absorbed quanta) measured at instrumented Canadian Carbon Program flux towers in four Canadian forests: a mature fir site in British Columbia, mature aspen and black spruce sites in Saskatchewan, and a mixed deciduous/coniferous forest site in Ontario. All aspects of the viewing geometry had significant effects on the MODIS-PRI, including the view zenith angle (VZA), the view azimuth angle, and the displacement of the view azimuth relative to the solar principal plane, in addition to illumination related variables.Nevertheless, we show that forward scatter sector views (VZA, 16 degrees-45 degrees) provided the strongest relationships to daily LUE, especially those collected in the early afternoon by Aqua (r squared = 0.83, RMSE (root mean square error) equals 0.003 moles per moles of carbon absorbed quanta). Nadir (VZA, 0 degrees plus or minus 15 degrees) and backscatter views (VZA, −16 degrees to −45 degrees) had lower performance in estimating LUE (nadir: r squared approximately equal to 0.62-0.67; backscatter: r squared approximately equal to 0.54-0.59) and similar estimation error (RMSE equals 0.004-0.005).When directional effects were not considered, only a moderately successful MODIS-PRI vs. LUE relationship (r squared equals 0.34, RMSE equals 0.007) was obtained in the full dataset (all views & sites, both satellites), but site-specific relationships were able to discriminate between coniferous and deciduous forests. Overall, MODIS-PRI values from Terra (late morning) were higher than those from Aqua (early afternoon), before/after the onset of diurnal stress responses expressed spectrally. Therefore, we identified ninety-two Terra-Aqua "same day" pairs, for which the sum of Terra morning and Aqua afternoon MODIS-PRI values (PRI (sub sum) using all available directional observations was linearly correlated with daily tower LUE (r squared equals 0.622, RMSE equals 0.013) and independent of site differences or meteorological information. Our study highlights the value of off-nadir directional reflectance observations, and the value of pairing morning and afternoon satellite observations to monitor stress responses that inhibit carbon uptake in Canadian forest ecosystems. In addition, we show that MODIS-PRI values, when derived from either: (i) forward views only, or (ii) Terra/Aqua same day (any view) combined observations, provided more accurate estimates of tower-measured daily LUE than those derived from either nadir or backscatter views or those calculated by the widely used semi-operational MODIS GPP model (MOD17) which is based on a theoretical maximum LUE and environmental data. Consequently, we demonstrate the importance of diurnal as well as off-nadir satellite observations for detecting vegetation physiological processes.

Middleton, E. M.↗
NASA NTRSDate not supplied

Clearance Sensitivity Mitigation in Small Core Compressors

As future aircraft become lighter and more aerodynamically efficient, thrust requirements will decrease, reducing the core size of the engine. Furthermore, in the pursuit of improved fuel burn, engine overall pressure ratio and bypass ratio will increase, further driving down engine core size. These drivers together mean that the core size, or corrected mass flow rate at the compressor exit, for future single-aisle aircraft applications will shrink below 3.0 lb/s. Traditionally, this small core compressor size is in the domain of axi-centrifugal designs, machines that are typically less efficient and limited to pressure ratios of ~25 due to stress and thermomechanical fatigue in the centrifugal impeller. In this light, NASA, Pratt & Whitney and the Raytheon Technologies Research Center embarked upon a program to develop technologies to enable an all-axial high-pressure compressor with a core size below 3.0 lb/s and an overall pressure ratio greater than 50. One of the challenges with an all-axial high-pressure compressor at this core size is the small span at the rear of the compressor. As core size is scaled down, the rotor tip clearances do not scale with span, leading to significant efficiency penalties. This paper documents a numerical and experimental effort to mitigate this penalty through design optimization and technology insertion, enabling an estimated 5 to 10% fuel burn reduction relative to 2020 best-in-class. The experimental program consisted of two builds of a high-speed rig: a baseline build and a second build to insert technology to manage large rotor tip gap/span ratios. The results demonstrated a reduction in the sensitivity of the compressor to large clearance ratios, which would be likely at the end-of-life of a small core application. The test campaign demonstrated that, with the insertion of technology, small core compressors can maintain a similar efficiency to current best-in-class large core size compressors. In addition to advancing the state-of-the-art of technology, the program also advanced the modeling standards for multistage compressors with large clearance-to-span ratios. The validation of this approach is described in the paper.

Axial Compressors↗
NASA NTRS1990

A new continuum model for suspensions of gyrotactic micro-organisms

A new continuum model is formulated for dilute suspensions of swimming micro-organisms with asymmetric mass distributions. Account is taken of randomness in a cell's swimming direction, p, by postulating that the probability density function for p satisfies a Fokker-Planck equation analogous to that obtained for colloid suspensions in the presence of rotational Brownian motion. The deterministic torques on a cell, viscous and gravitational, are balanced by diffusion, represented by an isotropic rotary diffusivity Dr, which is unknown a priori, but presumably reflects stochastic influences on the cell's internal workings. When the Fokker-Planck equation is solved, macroscopic quantities such as the average cell velocity Vc, the particle diffusivity tensor D and the effective stress tensor sigma can be computed; Vc and D are required in the cell conservation equation, and sigma in the momentum equation. The Fokker-Planck equation contains two dimensionless parameters, lambda and epsilon; lambda is the ratio of the rotary diffusion time Dr-1 to the torque relaxation time B (balancing gravitational and viscous torques), while epsilon is a scale for the local vorticity or strain rate made dimensionless with B. In this paper we solve the Fokker-Planck equation exactly for epsilon = 0 (lambda arbitrary) and also obtain the first-order solution for small epsilon. Using experimental data on Vc and D obtained with the swimming alga, Chlamydomonas nivalis, in the absence of bulk flow, the epsilon = 0 results can be used to estimate the value of lambda for that species (lambda approximately 2.2; Dr approximately 0.13 s-1). The continuum model for small epsilon is then used to reanalyse the instability of a uniform suspension, previously investigated by Pedley, Hill & Kessler (1988). The only qualitatively different result is that there no longer seem to be circumstances in which disturbances with a non-zero vertical wavenumber are more unstable than purely horizontal disturbances. On the way, it is demonstrated that the only significant contribution to sigma, other than the basic Newtonian stress, is that derived from the stresslets associated with the cells' intrinsic swimming motions.

NASA Discipline Number 28-20↗
NASA NTRSDate not supplied

Clearance Sensitivity Mitigation in Small Core Compressors

As future aircraft become lighter and more aerodynamically efficient, thrust requirements will decrease, reducing the core size of the engine. Furthermore, in the pursuit of improved fuel burn, engine overall pressure ratio and bypass ratio will increase, further driving down engine core size. These drivers together mean that the core size, or corrected mass flow rate at the compressor exit, for future single-aisle aircraft applications will shrink below 3.0 lbm/s. Traditionally, this small core compressor size is in the domain of axi-centrifugal designs, machines that are typically less efficient and limited to pressure ratios of ~25 due to stress and thermomechanical fatigue in the centrifugal impeller. In this light, NASA, Pratt & Whitney and the Raytheon Technologies Research Center embarked upon a program to develop technologies to enable an all-axial high-pressure compressor with a core size below 3.0 lbm/s and an overall pressure ratio greater than 50. One of the challenges with an all-axial high-pressure compressor at this core size is the small span at the rear of the compressor. As core size is scaled down, the rotor tip clearances do not scale with span, leading to significant efficiency penalties. This paper documents a numerical and experimental effort to mitigate this penalty through design optimization and technology insertion, enabling an estimated 5 to 10% fuel burn reduction relative to 2020 best-in-class. The experimental program consisted of two builds of a high-speed rig: a baseline build and a second build to insert technology to manage large rotor tip gap/span ratios. The results demonstrated a reduction in the sensitivity of the compressor to large clearance ratios, which would be likely at the end-of-life of a small core application. The test campaign demonstrated that, with the insertion of technology, small core compressors can maintain a similar efficiency to current best-in-class large core size compressors. In addition to advancing the state-of-the-art of technology, the program also advanced the modeling standards for multistage compressors with large clearance-to-span ratios. The validation of this approach is described in the paper.

Axial Compressors↗
NASA NTRS2017

NeMO-Net The Neural Multi-Modal Observation Training Network for Global Coral Reef Assessment

In the past decade, coral reefs worldwide have experienced unprecedented stresses due to climate change, ocean acidification, and anthropomorphic pressures, instigating massive bleaching and die-off of these fragile and diverse ecosystems. Furthermore, remote sensing of these shallow marine habitats is hindered by ocean wave distortion, refraction and optical attenuation, leading invariably to data products that are often of low resolution and signal-to-noise (SNR) ratio. However, recent advances in UAV and Fluid Lensing technology have allowed us to capture multispectral 3D imagery of these systems at sub-cm scales from above the water surface, giving us an unprecedented view of their growth and decay. Exploiting the fine-scaled features of these datasets, machine learning methods such as MAP, PCA, and SVM can not only accurately classify the living cover and morphology of these reef systems (below 8 error), but are also able to map the spectral space between airborne and satellite imagery, augmenting and improving the classification accuracy of previously low-resolution datasets.We are currently implementing NeMO-Net, the first open-source deep convolutional neural network (CNN) and interactive active learning and training software to accurately assess the present and past dynamics of coral reef ecosystems through determination of percent living cover and morphology. NeMO-Net will be built upon the QGIS platform to ingest UAV, airborne and satellite datasets from various sources and sensor capabilities, and through data-fusion determine the coral reef ecosystem makeup globally at unprecedented spatial and temporal scales. To achieve this, we will exploit virtual data augmentation, the use of semi-supervised learning, and active learning through a tablet platform allowing for users to manually train uncertain or difficult to classify datasets. The project will make use of Pythons extensive libraries for machine learning, as well as extending integration to GPU and High-End Computing Capability (HECC) on the Pleiades supercomputing cluster, located at NASA Ames. The project is being supported by NASAs Earth Science Technology Office (ESTO) Advanced Information Systems Technology (AIST-16) Program.

Remote Sensin↗
NASA NTRS2017

NeMO-Net: The Neural Multi-Modal Observation and Training Network for Global Coral Reef Assessment

In the past decade, coral reefs worldwide have experienced unprecedented stresses due to climate change, ocean acidification, and anthropomorphic pressures, instigating massive bleaching and die-off of these fragile and diverse ecosystems. Furthermore, remote sensing of these shallow marine habitats is hindered by ocean wave distortion, refraction and optical attenuation, leading invariably to data products that are often of low resolution and signal-to-noise (SNR) ratio. However, recent advances in UAV and Fluid Lensing technology have allowed us to capture multispectral 3D imagery of these systems at sub-cm scales from above the water surface, giving us an unprecedented view of their growth and decay. Exploiting the fine-scaled features of these datasets, machine learning methods such as MAP, PCA, and SVM can not only accurately classify the living cover and morphology of these reef systems (below 8 percent error), but are also able to map the spectral space between airborne and satellite imagery, augmenting and improving the classification accuracy of previously low-resolution datasets. We are currently implementing NeMO-Net, the first open-source deep convolutional neural network (CNN) and interactive active learning and training software to accurately assess the present and past dynamics of coral reef ecosystems through determination of percent living cover and morphology. NeMO-Net will be built upon the QGIS platform to ingest UAV, airborne and satellite datasets from various sources and sensor capabilities, and through data-fusion determine the coral reef ecosystem makeup globally at unprecedented spatial and temporal scales. To achieve this, we will exploit virtual data augmentation, the use of semi-supervised learning, and active learning through a tablet platform allowing for users to manually train uncertain or difficult to classify datasets. The project will make use of Pythons extensive libraries for machine learning, as well as extending integration to GPU and High-End Computing Capability (HECC) on the Pleiades supercomputing cluster, located at NASA Ames. The project is being supported by NASAs Earth Science Technology Office (ESTO) Advanced Information Systems Technology (AIST-16) Program.

NeMO-Net↗
NASA NTRS2014

Turbulence Modeling Verification and Validation

Computational fluid dynamics (CFD) software that solves the Reynolds-averaged Navier-Stokes (RANS) equations has been in routine use for more than a quarter of a century. It is currently employed not only for basic research in fluid dynamics, but also for the analysis and design processes in many industries worldwide, including aerospace, automotive, power generation, chemical manufacturing, polymer processing, and petroleum exploration. A key feature of RANS CFD is the turbulence model. Because the RANS equations are unclosed, a model is necessary to describe the effects of the turbulence on the mean flow, through the Reynolds stress terms. The turbulence model is one of the largest sources of uncertainty in RANS CFD, and most models are known to be flawed in one way or another. Alternative methods such as direct numerical simulations (DNS) and large eddy simulations (LES) rely less on modeling and hence include more physics than RANS. In DNS all turbulent scales are resolved, and in LES the large scales are resolved and the effects of the smallest turbulence scales are modeled. However, both DNS and LES are too expensive for most routine industrial usage on today's computers. Hybrid RANS-LES, which blends RANS near walls with LES away from walls, helps to moderate the cost while still retaining some of the scale-resolving capability of LES, but for some applications it can still be too expensive. Even considering its associated uncertainties, RANS turbulence modeling has proved to be very useful for a wide variety of applications. For example, in the aerospace field, many RANS models are considered to be reliable for computing attached flows. However, existing turbulence models are known to be inaccurate for many flows involving separation. Research has been ongoing for decades in an attempt to improve turbulence models for separated and other nonequilibrium flows. When developing or improving turbulence models, both verification and validation are important steps in the process. Verification insures that the CFD code is solving the equations as intended (no errors in the implementation). This is typically done either through the method of manufactured solutions (MMS) or through careful step-by-step comparisons with other verified codes. After the verification step is concluded, validation is performed to document the ability of the turbulence model to represent different types of flow physics. Validation can involve a large number of test case comparisons with experiments, theory, or DNS. Organized workshops have proved to be valuable resources for the turbulence modeling community in its pursuit of turbulence modeling verification and validation. Workshop contributors using different CFD codes run the same cases, often according to strict guidelines, and compare results. Through these comparisons, it is often possible to (1) identify codes that have likely implementation errors, and (2) gain insight into the capabilities and shortcomings of different turbulence models to predict the flow physics associated with particular types of flows. These are valuable lessons because they help bring consistency to CFD codes by encouraging the correction of faulty programming and facilitating the adoption of better models. They also sometimes point to specific areas needed for improvement in the models. In this paper, several recent workshops are summarized primarily from the point of view of turbulence modeling verification and validation. Furthermore, the NASA Langley Turbulence Modeling Resource website is described. The purpose of this site is to provide a central location where RANS turbulence models are documented, and test cases, grids, and data are provided. The goal of this paper is to provide an abbreviated survey of turbulence modeling verification and validation efforts, summarize some of the outcomes, and give some ideas for future endeavors in this area.

Rumsey, Christopher L.↗
NASA NTRS2017

Low-Thrust Trajectory Optimization with Simplified SQP Algorithm

The problem of low-thrust trajectory optimization in highly perturbed dynamics is a stressing case for many optimization tools. Highly nonlinear dynamics and continuous thrust are each, separately, non-trivial problems in the field of optimal control, and when combined, the problem is even more difficult. This paper de-scribes a fast, robust method to design a trajectory in the CRTBP (circular restricted three body problem), beginning with no or very little knowledge of the system. The approach is inspired by the SQP (sequential quadratic programming) algorithm, in which a general nonlinear programming problem is solved via a sequence of quadratic problems. A few key simplifications make the algorithm presented fast and robust to initial guess: a quadratic cost function, neglecting the line search step when the solution is known to be far away, judicious use of end-point constraints, and mesh refinement on multiple shooting with fixed-step integration.In comparison to the traditional approach of plugging the problem into a “black-box” NLP solver, the methods shown converge even when given no knowledge of the solution at all. It was found that the only piece of information that the user needs to provide is a rough guess for the time of flight, as the transfer time guess will dictate which set of local solutions the algorithm could converge on. This robustness to initial guess is a compelling feature, as three-body orbit transfers are challenging to design with intuition alone. Of course, if a high-quality initial guess is available, the methods shown are still valid.We have shown that endpoints can be efficiently constrained to lie on 3-body repeating orbits, and that time of flight can be optimized as well. When optimizing the endpoints, we must make a trade between converging quickly on sub-optimal endpoints or converging more slowly on end-points that are arbitrarily close to optimal. It is easy for the mission design engineer to adjust this trade based on the problem at hand.The biggest limitation to the algorithm at this point is that multi-revolution transfers (greater than 2 revolutions) do not work nearly as well. This restriction comes in because the relationship between node 1 and node N becomes increasingly nonlinear as the angular distance grows. Trans-fers with more than about 1.5 complete revolutions generally require the line search to improve convergence. Future work includes: Comparison of this algorithm with other established tools; improvements to how multiple-revolution transfers are handled; parallelization of the Jacobian computation; in-creased efficiency for the line search; and optimization of many more trajectories between a variety of 3-body orbits.

Parrish, Nathan L.↗
NASA NTRS2012

Characterization of Deficiencies in the Frequency Domain Forced Response Analysis Technique for Supersonic Turbine Bladed Disks

Turbine blades in rocket and jet engine turbomachinery experience enormous harmonic loading conditions. These loads result from the integer number of upstream and downstream stator vanes as well as the other turbine stages. Assessing the blade structural integrity is a complex task requiring an initial characterization of whether resonance is possible and then performing a forced response analysis if that condition is met. The standard technique for forced response analysis in rocket engine turbines is to decompose a computational fluid dynamics (CFD).generated flow field into its harmonic components, and to then perform a frequency response analysis at the problematic natural frequencies using cyclically symmetric structural dynamic models. Recent CFD analysis and water-flow testing at NASA/MSFC, though, indicates that this technique may miss substantial harmonic and non ]harmonic excitation sources that become present in complex flows. This complex content can only be captured by a CFD flow field encompassing at least an entire revolution. A substantial development effort to create a series of software programs to enable application of the 360 degree forcing function in a frequency response analysis on cyclic symmetric models has been completed (to be described in a future paper), but the question still remains whether the frequency response analysis itself is capable of capturing the excitation content sufficiently. Two studies comparing frequency response analysis with transient response analysis, therefore, of bladed-disks undergoing this complex flow environment have been performed. The first is of a bladed disk with each blade modeled by simple beam elements and the disk modeled with plates (using the finite element code MSC/NASTRAN). The focus of this model is to be representative of response of realistic bladed disks, and so the dimensions are roughly equivalent to the new J2X rocket engine 1st stage fuel pump turbine. The simplicity of the model allows the CFD load to be able to be readily applied, along with analytical and experimental variations in both the temporal and spatial fourier components of the excitation. In addition, this model is a first step in identifying response differences between transient and frequency forced response analysis techniques. The second phase assesses this difference for a much more realistic solid model of a bladed-disk in order to evaluate the effect of the spatial variation in loading on blade dominated modes. Neither research on the accuracy of the frequency response method when used in this context or a comprehensive study of the effect of test-observed variation on blade forced response have been found in the literature, so this research is a new contribution to practical structural dynamic analysis of gas turbines. The primary excitation of the upstream nozzles interacts with the blades on fuel pump of the J2X causes the 5th Nodal diameter modes to be excited, as explained by Tyler and Sofrin1, so a modal analysis was first performed on the beam/plate model and the 5ND bladed-disk mode at 40167 hz was identified and chosen to be the one excited at resonance (see figure 1). The first forced response analysis with this model focuses on identifying differences between frequency and transient response analyses. A hypothesis going into the analysis was that perhaps the frequency response was enforcing a temporal periodicity that did not really exist, and so therefore it would overestimate the response. As high dynamic response was a considerable source of stress in the J2X, examining this concept could potentially be beneficial for the program.

Brown, Andrew M.↗
NASA NTRS1999

Engineering Overview of a Multidisciplinary HSCT Design Framework Using Medium-Fidelity Analysis Codes

An objective of the HPCC Program at NASA Langley has been to promote the use of advanced computing techniques to more rapidly solve the problem of multidisciplinary optimization of a supersonic transport configuration. As a result, a software system has been designed and is being implemented to integrate a set of existing discipline analysis codes, some of them CPU-intensive, into a distributed computational framework for the design of a High Speed Civil Transport (HSCT) configuration. The proposed paper will describe the engineering aspects of integrating these analysis codes and additional interface codes into an automated design system. The objective of the design problem is to optimize the aircraft weight for given mission conditions, range, and payload requirements, subject to aerodynamic, structural, and performance constraints. The design variables include both thicknesses of structural elements and geometric parameters that define the external aircraft shape. An optimization model has been adopted that uses the multidisciplinary analysis results and the derivatives of the solution with respect to the design variables to formulate a linearized model that provides input to the CONMIN optimization code, which outputs new values for the design variables. The analysis process begins by deriving the updated geometries and grids from the baseline geometries and grids using the new values for the design variables. This free-form deformation approach provides internal FEM (finite element method) grids that are consistent with aerodynamic surface grids. The next step involves using the derived FEM and section properties in a weights process to calculate detailed weights and the center of gravity location for specified flight conditions. The weights process computes the as-built weight, weight distribution, and weight sensitivities for given aircraft configurations at various mass cases. Currently, two mass cases are considered: cruise and gross take-off weight (GTOW). Weights information is obtained from correlations of data from three sources: 1) as-built initial structural and non-structural weights from an existing database, 2) theoretical FEM structural weights and sensitivities from Genesis, and 3) empirical as-built weight increments, non-structural weights, and weight sensitivities from FLOPS. For the aeroelastic analysis, a variable-fidelity aerodynamic analysis has been adopted. This approach uses infrequent CPU-intensive non-linear CFD to calculate a non-linear correction relative to a linear aero calculation for the same aerodynamic surface at an angle of attack that results in the same configuration lift. For efficiency, this nonlinear correction is applied after each subsequent linear aero solution during the iterations between the aerodynamic and structural analyses. Convergence is achieved when the vehicle shape being used for the aerodynamic calculations is consistent with the structural deformations caused by the aerodynamic loads. To make the structural analyses more efficient, a linearized structural deformation model has been adopted, in which a single stiffness matrix can be used to solve for the deformations under all the load conditions. Using the converged aerodynamic loads, a final set of structural analyses are performed to determine the stress distributions and the buckling conditions for constraint calculation. Performance constraints are obtained by running FLOPS using drag polars that are computed using results from non-linear corrections to the linear aero code plus several codes to provide drag increments due to skin friction, wave drag, and other miscellaneous drag contributions. The status of the integration effort will be presented in the proposed paper, and results will be provided that illustrate the degree of accuracy in the linearizations that have been employed.

Weston, R. P.↗