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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 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
NASA NTRS1979

A high order accurate numerical solution algorithm for turbulent boundary layer flow

A fourth-order accurate numerical solution algorithm is derived using finite element interpolation theory for the non-linear parabolic equations governing turbulent boundary layer flow including a two-equation turbulence closure model. The results of carefully controlled numerical experiments firmly quantize for the first time performance differences between finite element and finite difference solution methodology for this type of equation. The developed algorithm takes advantage of the apparent semi-analytical formulational procedure, in establishment of a single, retarded-evaluation Jacobian matrix iterative solution algorithm. Numerical results document performance of solution economy features in terms of computer requirements and solution accuracy. The developed algorithm should find wide application in aerodynamics analysis.

Soliman, M. O.↗
NASA NTRS2019

Regularization and Error Characterization of GRACE Mascons

We present a new global time-variable gravity mascon solution derived from Gravity Recovery and Climate Experiment (GRACE) Level 1B data. The new product from the NASA Goddard Space Flight Center (GSFC) results from a novel approach that combines an iterative solution strategy with geographical binning of inter-satellite range-acceleration residuals in the construction of time-dependent regularization matrices applied in the inversion of mascon parameters. This estimation strategy is intentionally conservative as it seeks to maximize the role of the GRACE measurements on the final solution while minimizing the influence of the regularization design process. We fully reprocess the Level 1B data in the presence of the final mascon solution to generate true post-fit inter-satellite residuals, which are utilized to confirm solution convergence and to validate the mascon noise uncertainties. We also present the mathematical case that regularized mascon solutions are biased, and that this bias, or leakage, must be combined with the estimated noise variance to accurately assess total mascon uncertainties. The estimated leakage errors are determined from the monthly resolution operators. We present a simple approach to compute the total uncertainty for both individual mascon and regional analysis of the GSFC mascon product, and validate the results in comparison with independent mascon solutions and calibrated Stokes uncertainties. Lastly, we present the new solution and uncertainties with global analyses of the mass trends and annual amplitudes, and compute updated trends for the global ocean, and the respective contributions of the Greenland Ice Sheet, Antarctic Ice Sheet, Gulf of Alaska, and terrestrial water storage. This analysis highlights the successful closure of the global mean sea level budget, that is, the sum of global ocean mass from the GSFC mascons and the steric component from Argo floats agrees well with the total determined from sea surface altimetry.

B. D. Loomis↗
NASA NTRS2019

Regularization and Error Characterization of GRACE Mascons

We present a new global time-variable gravity mascon solution derived from Gravity Recovery and Climate Experiment (GRACE) Level 1B data. The new product from the NASA Goddard Space Flight Center (GSFC) results from a novel approach that combines an iterative solution strategy with geographical binning of inter-satellite range-acceleration residuals in the construction of time-dependent regularization matrices applied in the inversion of mascon parameters. This estimation strategy is intentionally conservative as it seeks to maximize the role of the GRACE measurements on the final solution while minimizing the influence of the regularization design process. We fully reprocess the Level 1B data in the presence of the final mascon solution to generate true post-fit inter-satellite residuals, which are utilized to confirm solution convergence and to validate the mascon noise uncertainties. We also present the mathematical case that regularized mascon solutions are biased, and that this bias, or leakage, must be combined with the estimated noise variance to accurately assess total mascon uncertainties. The estimated leakage errors are determined from the monthly resolution operators. We present a simple approach to compute the total uncertainty for both individual mascon and regional analysis of the GSFC mascon product, and validate the results in comparison with independent mascon solutions and calibrated Stokes uncertainties. Lastly, we present the new solution and uncertainties with global analyses of the mass trends and annual amplitudes, and compute updated trends for the global ocean, and the respective contributions of the Greenland Ice Sheet, Antarctic Ice Sheet, Gulf of Alaska, and terrestrial water storage. This analysis highlights the successful closure of the global mean sea level budget, that is, the sum of global ocean mass from the GSFC mascons and the steric component from Argo floats agrees well with the total determined from sea surface altimetry.

Loomis, B. D.↗
NASA NTRS1973

First-order corrections to approximate solutions to two-point boundary-condition problems

A method, applicable to real time guidance, is developed for accurate solution to exo-atmospheric space flight optimization problems. In principle the method is applicable to many other two-point boundary-condition (TPBC) problems. The first step of the method is the iterative solution (using a shooting method) of a TPBC problem with differential equations simplified so that they may be solved analytically by means of a single closed-form solution over each stage of the flight. The second step is the addition of a closed-form correction to the solution to the TPBC problem obtained in the first step. The correction accounts (to first-order accuracy) for the errors due to the aforementioned simplifications. Numerical results are given for several orbital injection problems.

Andrus, J. F.↗
NASA NTRS1982

Relaxation solution of the full Euler equations

A numerical procedure for the relaxation solution of the full steady Euler equations is described. By embedding the Euler system in a second order surrogate system, central differencing may be used in subsonic regions while retaining matrix forms well suited to iterative solution procedures and convergence acceleration techniques. Hence, this method allows the development of stable, fully conservative differencing schemes for the solution of quite general inviscid flow problems. Results are presented for both subcritical and shocked supercritical internal flows. Comparisons are made with a standard time dependent solution algorithm.

Johnson, G. M.↗
NASA NTRS1982

Relaxation solution of the full Euler equations

A numerical procedure for the relaxation solution of the full steady Euler equations is described. By embedding the Euler system in a second order surrogate system, central differencing may be used in subsonic regions while retaining matrix forms well suited to iterative solution procedures and convergence acceleration techniques. Hence, this method allows the development of stable, fully conservative differencing schemes for the solution of quite general inviscid flow problems. Results are presented for both subcritical and shocked supercritical internal flows. Comparisons are made with a standard time dependent solution algorithm. Previously announced in STAR as N82-24859

Johnson, G. M.↗
NASA NTRS1993

A comparative study of an ABC and an artificial absorber for truncating finite element meshes

The type of mesh termination used in the context of finite element formulations plays a major role on the efficiency and accuracy of the field solution. The performance of an absorbing boundary condition (ABC) and an artificial absorber (a new concept) for terminating the finite element mesh was evaluated. This analysis is done in connection with the problem of scattering by a finite slot array in a thick ground plane. The two approximate mesh truncation schemes are compared with the exact finite element-boundary integral (FEM-BI) method in terms of accuracy and efficiency. It is demonstrated that both approximate truncation schemes yield reasonably accurate results even when the mesh is extended only 0.3 wavelengths away from the array aperture. However, the artificial absorber termination method leads to a substantially more efficient solution. Moreover, it is shown that the FEM-BI method remains quite competitive with the FEM-artificial absorber method when the FFT is used for computing the matrix-vector products in the iterative solution algorithm. These conclusions are indeed surprising and of major importance in electromagnetic simulations based on the finite element method.

Oezdemir, T.↗
NASA NTRS1976

The calculation of steady non-linear transonic flow over finite wings with linear theory aerodynamics

The feasibility of calculating steady mean flow solutions for nonlinear transonic flow over finite wings with a linear theory aerodynamic computer program is studied. The methodology is based on independent solutions for upper and lower surface pressures that are coupled through the external flow fields. Two approaches for coupling the solutions are investigated which include the diaphragm and the edge singularity method. The final method is a combination of both where a line source along the wing leading edge is used to account for blunt nose airfoil effects; and the upper and lower surface flow fields are coupled through a diaphragm in the plane of the wing. An iterative solution is used to arrive at the nonuniform flow solution for both nonlifting and lifting cases. Final results for a swept tapered wing in subcritical flow show that the method converges in three iterations and gives excellent agreement with experiment at alpha = 0 deg and 2 deg. Recommendations are made for development of a procedure for routine application.

Cunningham, A. M., Jr.↗
NASA NTRS1972

Analysis of the flow field generated near an aircraft engine operating in reverse thrust

A computer solution is developed to the exhaust gas reingestion problem for aircraft operating in the reverse thrust mode on a crosswind-free runway. The computer program determines the location of the inlet flow pattern, whether the exhaust efflux lies within the inlet flow pattern or not, and if so, the approximate time before the reversed flow reaches the engine inlet. The program is written so that the user is free to select discrete runway speeds or to study the entire aircraft deceleration process for both the far field and cross-ingestion problems. While developed with STOL applications in mind, the solution is equally applicable to conventional designs. The inlet and reversed jet flow fields involved in the problem are assumed to be noninteracting. The nacelle model used in determining the inlet flow field is generated using an iterative solution to the Neuman problem from potential flow theory while the reversed jet flow field is adapted using an empirical correlation from the literature. Sample results obtained using the program are included.

Ledwith, W. A., Jr.↗
NASA NTRS1982

Aerothermal modeling

The objectives, approach, and status of a program to develop the computational fluid dynamics tools needed to improve combustor design and analysis are outlined. The calculation procedure selected consists of a finite difference solution of the time averaged, steady state, primitive variable, elliptic form of the Reynolds equations. Standard TEACH type numerics are used to solve the resulting equations. These include hybrid differencing, SIMPLE algorithm for the pressure field, line by line iterative solution using the ADI method and the tridiagonal matrix algorithm (TDMA). Convergence is facilitated by using under relaxation. The physical processes are modeled by a two equation eddy viscosity model for turbulence; combustion is represented by a simple, irreversible, one step chemical reaction whose rate is influenced only by the time scale of the turbulence. The model evaluation procedure is also described.

Sturgess, G. J.↗
NASA NTRS1979

A split finite element algorithm for the compressible Navier-Stokes equations

An accurate and efficient numerical solution algorithm is established for solution of the high Reynolds number limit of the Navier-Stokes equations governing the multidimensional flow of a compressible essentially inviscid fluid. Finite element interpolation theory is used within a dissipative formulation established using Galerkin criteria within the Method of Weighted Residuals. An implicit iterative solution algorithm is developed, employing tensor product bases within a fractional steps integration procedure, that significantly enhances solution economy concurrent with sharply reduced computer hardware demands. The algorithm is evaluated for resolution of steep field gradients and coarse grid accuracy using both linear and quadratic tensor product interpolation bases. Numerical solutions for linear and nonlinear, one, two and three dimensional examples confirm and extend the linearized theoretical analyses, and results are compared to competitive finite difference derived algorithms.

Baker, A. J.↗
NASA NTRS1996

Development of a Computer Architecture to Support the Optical Plume Anomaly Detection (OPAD) System

The NASA OPAD spectrometer system relies heavily on extensive software which repetitively extracts spectral information from the engine plume and reports the amounts of metals which are present in the plume. The development of this software is at a sufficiently advanced stage where it can be used in actual engine tests to provide valuable data on engine operation and health. This activity will continue and, in addition, the OPAD system is planned to be used in flight aboard space vehicles. The two implementations, test-stand and in-flight, may have some differing requirements. For example, the data stored during a test-stand experiment are much more extensive than in the in-flight case. In both cases though, the majority of the requirements are similar. New data from the spectrograph is generated at a rate of once every 0.5 sec or faster. All processing must be completed within this period of time to maintain real-time performance. Every 0.5 sec, the OPAD system must report the amounts of specific metals within the engine plume, given the spectral data. At present, the software in the OPAD system performs this function by solving the inverse problem. It uses powerful physics-based computational models (the SPECTRA code), which receive amounts of metals as inputs to produce the spectral data that would have been observed, had the same metal amounts been present in the engine plume. During the experiment, for every spectrum that is observed, an initial approximation is performed using neural networks to establish an initial metal composition which approximates as accurately as possible the real one. Then, using optimization techniques, the SPECTRA code is repetitively used to produce a fit to the data, by adjusting the metal input amounts until the produced spectrum matches the observed one to within a given level of tolerance. This iterative solution to the original problem of determining the metal composition in the plume requires a relatively long period of time to execute the software in a modern single-processor workstation, and therefore real-time operation is currently not possible. A different number of iterations may be required to perform spectral data fitting per spectral sample. Yet, the OPAD system must be designed to maintain real-time performance in all cases. Although faster single-processor workstations are available for execution of the fitting and SPECTRA software, this option is unattractive due to the excessive cost associated with very fast workstations and also due to the fact that such hardware is not easily expandable to accommodate future versions of the software which may require more processing power. Initial research has already demonstrated that the OPAD software can take advantage of a parallel computer architecture to achieve the necessary speedup. Current work has improved the software by converting it into a form which is easily parallelizable. Timing experiments have been performed to establish the computational complexity and execution speed of major components of the software. This work provides the foundation of future work which will create a fully parallel version of the software executing in a shared-memory multiprocessor system.

Katsinis, Constantine↗
NASA NTRS1999

Bidirectional Reflectance of Flat, Optically Thick Particulate Layers: An Efficient Radiative Transfer Solution and Applications to Snow and Soil Surfaces

We describe a simple and highly efficient and accurate radiative transfer technique for computing bidirectional reflectance of a macroscopically flat scattering layer composed of nonabsorbing or weakly absorbing, arbitrarily shaped, randomly oriented and randomly distributed particles. The layer is assumed to be homogeneous and optically semi-infinite, and the bidirectional reflection function (BRF) is found by a simple iterative solution of the Ambartsumian's nonlinear integral equation. As an exact Solution of the radiative transfer equation, the reflection function thus obtained fully obeys the fundamental physical laws of energy conservation and reciprocity. Since this technique bypasses the computation of the internal radiation field, it is by far the fastest numerical approach available and can be used as an ideal input for Monte Carlo procedures calculating BRFs of scattering layers with macroscopically rough surfaces. Although the effects of packing density and coherent backscattering are currently neglected, they can also be incorporated. The FORTRAN implementation of the technique is available on the World Wide Web at http://ww,,v.giss.nasa.gov/-crmim/brf.html and can be applied to a wide range of remote sensing, engineering, and biophysical problems. We also examine the potential effect of ice crystal shape on the bidirectional reflectance of flat snow surfaces and the applicability of the Henyey-Greenstein phase function and the 6-Eddington approximation in calculations for soil surfaces.

Mishchenko, Michael I.↗
NASA NTRS1987

Program VSAERO theory document: A computer program for calculating nonlinear aerodynamic characteristics of arbitrary configurations

The VSAERO low order panel method formulation is described for the calculation of subsonic aerodynamic characteristics of general configurations. The method is based on piecewise constant doublet and source singularities. Two forms of the internal Dirichlet boundary condition are discussed and the source distribution is determined by the external Neumann boundary condition. A number of basic test cases are examined. Calculations are compared with higher order solutions for a number of cases. It is demonstrated that for comparable density of control points where the boundary conditions are satisfied, the low order method gives comparable accuracy to the higher order solutions. It is also shown that problems associated with some earlier low order panel methods, e.g., leakage in internal flows and junctions and also poor trailing edge solutions, do not appear for the present method. Further, the application of the Kutta conditions is extremely simple; no extra equation or trailing edge velocity point is required. The method has very low computing costs and this has made it practical for application to nonlinear problems requiring iterative solutions for wake shape and surface boundary layer effects.

Maskew, Brian↗
NASA NTRS1996

Multigrid Methods for Fully Implicit Oil Reservoir Simulation

In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for two-phase flow problems with strong heterogeneities and anisotropies is studied. Here we consider both possibilities. Moreover we present a novel way for constructing the coarse grid correction operator in linear multigrid algorithms. This approach has the advantage in that it preserves the sparsity pattern of the fine grid matrix and it can be extended to systems of equations in a straightforward manner. We compare the linear and nonlinear multigrid algorithms by means of a numerical experiment.

Molenaar, J.↗
NASA NTRS2022

NDARC - NASA Design and Analysis of Rotorcraft: Input and Data Structures - Vol 2

The NDARC code performs design and analysis tasks. The design task involves sizing the rotorcraft to satisfy specified design conditions and missions. The analysis tasks can include off-design mission performance analysis, flight performance calculation for point operating conditions, and generation of subsystem or component performance maps. Figure 1-1 illustrates the tasks. The principal tasks (sizing, mission analysis, flight performance analysis) are shown in the figure as boxes with heavy borders. Heavy arrows show control of subordinate tasks. The aircraft description (figure 1-1) consists of all the information, input and derived, that defines the aircraft. The aircraft consists of a set of components, including fuselage, rotors, wings, tails, and propulsion. This information can be the result of the sizing task; can come entirely from input, for a fixed model; or can come from the sizing task in a previous case or previous job. The aircraft description information is available to all tasks and all solutions (indicated by light arrows). The sizing task determines the dimensions, power, and weight of a rotorcraft that can perform a specified set of design conditions and missions. The aircraft size is characterized by parameters such as design gross weight, weight empty, rotor radius, and engine power available. The relations between dimensions, power, and weight generally require an iterative solution. From the design flight conditions and missions, the task can determine the total engine power or the rotor radius (or both power and radius can be fixed), as well as the design gross weight, maximum takeoff weight, drive system torque limit, and fuel tank capacity. For each propulsion group, the engine power or the rotor radius can be sized. Missions are defined for the sizing task, and for the mission performance analysis. A mission consists of a number of mission segments, for which time, distance, and fuel burn are evaluated. For the sizing task, certain missions are designated to be used for design gross weight calculations; for transmission sizing; and for fuel tank sizing. The mission parameters include mission takeoff gross weight and useful load. For specified takeoff fuel weight with adjustable segments, the mission time or distance is adjusted so the fuel required for the mission (burned plus reserve) equals the takeoff fuel weight. The mission iteration is on fuel weight or energy. Flight conditions are specified for the sizing task, and for the flight performance analysis. For the sizing task, certain flight conditions are designated to be used for design gross weight calculations; for transmission sizing; for maximum takeoff weight calculations; and for antitorque or auxiliary thrust rotor sizing. The flight condition parameters include gross weight and useful load. For flight conditions and mission takeoff, the gross weight can be maximized, such that the power required equals the power available. A flight state is defined for each mission segment and each flight condition. The aircraft performance can be analyzed for the specified state, or a maximum effort performance can be identified. The maximum effort is specified in terms of a quantity such as best endurance or best range, and a variable such as speed, rate of climb, or altitude. The aircraft must be trimmed, by solving for the controls and motion that produce equilibrium in the specified flight state. Different trim solution definitions are required for various flight states. Evaluating the rotor hub forces may require solution of the blade flap equations of motion.

NDARC↗
NASA NTRS2022

NDARC - NASA Design and Analysis of Rotorcraft: Input - Vol 3

The NDARC code performs design and analysis tasks. The design task involves sizing the rotorcraft to satisfy specified design conditions and missions. The analysis tasks can include off-design mission performance analysis, flight performance calculation for point operating conditions, and generation of subsystem or component performance maps. Figure 1-1 illustrates the tasks. The principal tasks (sizing, mission analysis, flight performance analysis) are shown in the figure as boxes with heavy borders. Heavy arrows show control of subordinate tasks. The aircraft description (figure 1-1) consists of all the information, input and derived, that defines the aircraft. The aircraft consists of a set of components, including fuselage, rotors, wings, tails, and propulsion. This information can be the result of the sizing task; can come entirely from input, for a fixed model; or can come from the sizing task in a previous case or previous job. The aircraft description information is available to all tasks and all solutions (indicated by light arrows). The sizing task determines the dimensions, power, and weight of a rotorcraft that can perform a specified set of design conditions and missions. The aircraft size is characterized by parameters such as design gross weight, weight empty, rotor radius, and engine power available. The relations between dimensions, power, and weight generally require an iterative solution. From the design flight conditions and missions, the task can determine the total engine power or the rotor radius (or both power and radius can be fixed), as well as the design gross weight, maximum takeoff weight, drive system torque limit, and fuel tank capacity. For each propulsion group, the engine power or the rotor radius can be sized. Missions are defined for the sizing task, and for the mission performance analysis. A mission consists of a number of mission segments, for which time, distance, and fuel burn are evaluated. For the sizing task, certain missions are designated to be used for design gross weight calculations; for transmission sizing; and for fuel tank sizing. The mission parameters include mission takeoff gross weight and useful load. For specified takeoff fuel weight with adjustable segments, the mission time or distance is adjusted so the fuel required for the mission (burned plus reserve) equals the takeoff fuel weight. The mission iteration is on fuel weight or energy. Flight conditions are specified for the sizing task, and for the flight performance analysis. For the sizing task, certain flight conditions are designated to be used for design gross weight calculations; for transmission sizing; for maximum takeoff weight calculations; and for antitorque or auxiliary thrust rotor sizing. The flight condition parameters include gross weight and useful load. For flight conditions and mission takeoff, the gross weight can be maximized, such that the power required equals the power available. A flight state is defined for each mission segment and each flight condition. The aircraft performance can be analyzed for the specified state, or a maximum effort performance can be identified. The maximum effort is specified in terms of a quantity such as best endurance or best range, and a variable such as speed, rate of climb, or altitude. The aircraft must be trimmed, by solving for the controls and motion that produce equilibrium in the specified flight state. Different trim solution definitions are required for various flight states. Evaluating the rotor hub forces may require solution of the blade flap equations of motion.

NDARC↗
NASA NTRS2023

NDARC - NASA Design and Analysis of Rotorcraft: Input - Appendix 3

The NDARC code performs design and analysis tasks. The design task involves sizing the rotorcraft to satisfy specified design conditions and missions. The analysis tasks can include off-design mission performance analysis, flight performance calculation for point operating conditions, and generation of subsystem or component performance maps. Figure 1-1 illustrates the tasks. The principal tasks (sizing, mission analysis, flight performance analysis) are shown in the figure as boxes with heavy borders. Heavy arrows show control of subordinate tasks. The aircraft description (figure 1-1) consists of all the information, input and derived, that defines the aircraft. The aircraft consists of a set of components, including fuselage, rotors, wings, tails, and propulsion. This information can be the result of the sizing task; can come entirely from input, for a fixed model; or can come from the sizing task in a previous case or previous job. The aircraft description information is available to all tasks and all solutions (indicated by light arrows). The sizing task determines the dimensions, power, and weight of a rotorcraft that can perform a specified set of design conditions and missions. The aircraft size is characterized by parameters such as design gross weight, weight empty, rotor radius, and engine power available. The relations between dimensions, power, and weight generally require an iterative solution. From the design flight conditions and missions, the task can determine the total engine power or the rotor radius (or both power and radius can be fixed), as well as the design gross weight, maximum takeoff weight, drive system torque limit, and fuel tank capacity. For each propulsion group, the engine power or the rotor radius can be sized. Missions are defined for the sizing task, and for the mission performance analysis. A mission consists of a number of mission segments, for which time, distance, and fuel burn are evaluated. For the sizing task, certain missions are designated to be used for design gross weight calculations; for transmission sizing; and for fuel tank sizing. The mission parameters include mission takeoff gross weight and useful load. For specified takeoff fuel weight with adjustable segments, the mission time or distance is adjusted so the fuel required for the mission (burned plus reserve) equals the takeoff fuel weight. The mission iteration is on fuel weight or energy. Flight conditions are specified for the sizing task, and for the flight performance analysis. For the sizing task, certain flight conditions are designated to be used for design gross weight calculations; for transmission sizing; for maximum takeoff weight calculations; and for antitorque or auxiliary thrust rotor sizing. The flight condition parameters include gross weight and useful load. For flight conditions and mission takeoff, the gross weight can be maximized, such that the power required equals the power available. A flight state is defined for each mission segment and each flight condition. The aircraft performance can be analyzed for the specified state, or a maximum effort performance can be identified. The maximum effort is specified in terms of a quantity such as best endurance or best range, and a variable such as speed, rate of climb, or altitude. The aircraft must be trimmed, by solving for the controls and motion that produce equilibrium in the specified flight state. Different trim solution definitions are required for various flight states. Evaluating the rotor hub forces may require solution of the blade flap equations of motion.

NDARC↗