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Numerical integration based on definition of Riemann integral, describing application to nonlinear integral equations of isotropic scattering of radiation in plane parallel atmosphere
Composite materials have become ubiquitous in the aerospace industry due to their exceptionally light weight and high strength characteristics, as well as their unique ability to be engineered and tailored to meet specific loading conditions and performance requirements. These advanced materials offer superior strength-to-weight ratios compared to traditional metallic materials, making them particularly valuable in weight-critical aerospace applications where every pound saved translates to improved efficiency and performance. In currently operating fleets of commercial and military aircraft, composite materials have been successfully applied to critical structural components, including primary load-bearing elements such as the fuselage sections and flooring structures, which must withstand significant in-flight loads and provide passenger safety. Additionally, these materials have been specifically tailored and optimized for aerodynamic components such as wings and tail assemblies, where their ability to be molded into complex shapes while maintaining structural integrity is particularly advantageous. The application of composite materials extends beyond primary structural elements into the realm of internal cabin components, most notably in innovative seat designs where weight reduction and structural integration are paramount concerns. Modern composite seat structures can be designed to integrate multiple functions, including structural support, comfort features, and safety systems, all while maintaining the lightweight characteristics essential for aircraft performance.
Integrated Modeling has been a key component of verifying optical requirements for the Nancy Grace Roman Space Telescope (RST) that are either impossible or impractical to verify exclusively through ground testing. Two major areas for integrated Modeling are Jitter and Thermal Distortion that require the exchanges of model performance predictions across disciplines. In both cases, distortions are impressed on optical models to evaluate the impact on boresight alignment and wave front error. In the case of Jitter, the disturbances are driven by reactions to motions most often from actuators; however, in the case of thermal distortion, the motions are driven by thermal expansion or contraction as a result of changing temperatures. This then requires a link further upstream to the thermal model, which is used to predict the thermal performance and temperature gradients and stability. The process for mapping temperatures from a thermal model to a corresponding structural model has been performed numerous times through the RST project lifecycle, with improvements in the accuracy, verification, and effort sought throughout. This paper describes some of the recent improvements to the process, including: capture of the visualization parameters, automatic generation of the mapped images for both the thermal and structural model groupings, and reduction in the effort to assemble the full set of mapped temperatures. These upgrades have greatly reduced the manual effort associated with thermal mapping and allowed for faster turn-around of Integrated Modeling predictions.
Integrated circuit with multiple collector current source achieves the equivalent of a large number of resistors in a small area. Functional equivalents of a transistor reduce the size requirement for low power integrated circuits, providing an efficient alternative to the conventional diffused resistor process in integrated circuit fabrication.
The Integrated Life Support System was conceived to study the problems of integrating regenerative equipment designed to operate in a negligible gravitational field. It is the first to fully integrate the three major contributors to atmospheric contamination: man, machine, and materials.
Values of coefficients for defining the effectiveness of integral stiffeners in resisting shear deformations of the plate of which they are an integral part are presented for a variety of proportions of rectangular stiffeners with circular fillets. Formulas are given in which these coefficients may be employed to calculate the elastic constants associated with the twisting and shearing of integrally stiffened plates. The size of fillet radius is shown to contribute appreciably to the degree of penetration of the stresses from the skin into the stiffener.
The Navier-Stokes equations of motion and the equation of continuity are transformed so as to apply to an orthogonal curvilinear coordinate system rotating with a uniform angular velocity about an arbitrary axis in space. A usual simplification of these equations as consistent with the accepted boundary-layer theory and an integration of these equations through the boundary layer result in boundary-layer momentum-integral equations for three-dimensional flows that are applicable to either rotating or nonrotating fluid boundaries. These equations are simplified and an approximate solution in closed integral form is obtained for a generalized boundary-layer momentum-loss thickness and flow deflection at the wall in the turbulent case. A numerical evaluation of this solution carried out for data obtained in a curving nonrotating duct shows a fair quantitative agreement with the measures values. The form in which the equations are presented is readily adaptable to cases of steady, three-dimensional, incompressible boundary-layer flow like that over curved ducts or yawed wings; and it also may be used to describe the boundary-layer flow over various rotating surfaces, thus applying to turbomachinery, propellers, and helicopter blades.
Digital engineering (DE) and integrated computational materials engineering (ICME) are widely recognized as critical enablers of faster, more affordable, and more reliable aerospace systems. However, many organizations have struggled to realize the promised return on investment (ROI) from digital initiatives. A primary reason is the absence of a shared, decision-focused framework that distinguishes simple digitization of existing workflows from true digital transformation that fundamentally changes how engineering decisions are made. This paper introduces an ICME capability maturity framework that fills this gap. The framework defines six cumulative ICME capability maturity levels (CMLs), explicitly tied to decision authority, engineering integration, optimization, and uncertainty management across material, process, structure, and performance scales. It is designed to complement established readiness metrics such as technology readiness levels (TRLs), manufacturing readiness levels (MRLs), and integration readiness levels (IRLs), by addressing a missing dimension: the conditions required for model-informed decision authority across scales. A unifying figure and capability table illustrate the six-level ICME Capability Maturity Framework, showing how organizations progress from digitization—with limited or negative ROI—to true digital transformation, where ICME-enabled workflows deliver measurable improvements in decision quality, cycle time, risk reduction, and reuse. The framework is intended for both technical practitioners and executive leadership, providing a common language to assess current state, guide roadmaps, align software ecosystem investments, and set realistic expectations for digital transformation outcomes. A regulatory-relevant statement clarifying the relationship between ICME capability and existing certification frameworks is provided.
The Water Recovery System (WRS) is a critical component of life support aboard the International Space Station (ISS) and will play an essential role in future missions beyond Low Earth Orbit (LEO). Its primary functional units – the Urine Processor Assembly (UPA), Brine Processor Assembly (BPA), and Water Processor Assembly (WPA) – must be evaluated for extended operation, dormancy resilience, material obsolescence, and reliability under exploration-driven constraints. Ground testing is vital for developing these technologies and generating statistically relevant reliability assessments, which requires extended runtime under integrated, Flight-like conditions. Currently, no high-fidelity, fully integrated WPA ground test bed exists to support these objectives. To address this gap, NASA is developing a WPA test bed at Marshall Space Flight Center (MSFC) that combines downgraded ISS flight hardware with functionally flight-like components in a cost-effective configuration while maintaining priority hardware investigations. This paper describes the current status of hardware assembly and integration, outlines key challenges such as simulating microgravity effects and mitigating obsolescence, and presents future test objectives including software development, reliability assessments, dormancy studies, and exploration-oriented upgrades.
The Carbothermal Reduction Demonstration (CaRD) project was an effort to develop a prototype system to demonstrate the extraction of oxygen from simulated lunar regolith using concentrated solar energy and a carbothermal reaction. The prototype consisted of a deployable solar concentrator capable of tracking the sun, carbothermal reactor, fluid system, gas analysis, and a solar concentrator control system consisting of avionics and software. These subsystems were developed by multiple NASA centers and a private industry partner, Sierra Space. The various teams worked together to define requirements and interfaces to successfully assemble the complex system and demonstrate an integrated solar carbothermal process. The solar concentrator developed at Glenn Research Center (GRC) was designed to be stowed for a launch environment then deployed on the lunar surface. It utilized a crossed dragone configuration of composite mirrors to direct horizontal sunlight onto a target 90° from the incoming sunlight. The key performance parameters for the solar concentrator were efficiency and power density. The carbothermal reactor was developed by Sierra Space through a separate project called the Carbothermal Oxygen Production Reactor (COPR) where it successfully demonstrated a fully automated process in a thermal vacuum environment. The fluid system needed for the carbothermal reaction was also developed by Sierra Space and successfully demonstrated in the same thermal vacuum test. The gas analysis system was developed at Kennedy Space Center (KSC) and was required to determine the amount of oxygen extracted during each test. The gas analysis system was based on the Mass Spectrometer Observing Lunar Operations (MSOLO) instrument. Avionics and software for the CaRD prototype were also developed at KSC and based on experience with MSOLO avionics and software. The control system was designed to stow, deploy, track the sun, and perform beam alignment of the concentrated light. The prototype subsystems were integrated and tested at Johnson Space Center’s (JSC) Energy Systems Test Area. A heliostat was used to direct sunlight toward the prototype in a way that is representative of the sunlight conditions at the south pole of the Moon. When concentrated sunlight was focused on simulated lunar regolith within the reactor, the gas analysis team confirmed the presence of carbon monoxide gas, which confirmed that a solar carbothermal reaction took place. The key performance parameter for the integrated prototype was grams of oxygen extracted per kilowatt hour of energy arriving at the concentrator primary mirror. The prototype design successfully demonstrated end-to-end capability and further steps to achieve a flight capable system have been defined. With lunar data, engineers would be able to design a scaled-up system capable of extracting oxygen from regolith at useful quantities for crew life support and rocket propellant. On the long term, this method of In-Situ Resource Utilization could be used to drastically reduce the cost and risk of a sustained human presence on the Moon by reducing the amount of oxygen that would have to be delivered.
The Water Recovery System (WRS) is a critical component of life support aboard the International Space Station (ISS) and will play an essential role in future missions beyond Low Earth Orbit (LEO). Its primary functional units – the Urine Processor Assembly (UPA), Brine Processor Assembly (BPA), and Water Processor Assembly (WPA) – must be evaluated for extended operation, dormancy resilience, material obsolescence, and reliability under exploration-driven constraints. Ground testing is vital for developing these technologies and generating statistically relevant reliability assessments, which requires extended runtime under integrated, Flight-like conditions. Currently, no high-fidelity, fully integrated WPA ground test bed exists to support these objectives. To address this gap, NASA is developing a WPA test bed at Marshall Space Flight Center (MSFC) that combines downgraded ISS flight hardware with functionally flight-like components in a cost-effective configuration while maintaining priority hardware investigations. This paper describes the current status of hardware assembly and integration, outlines key challenges such as simulating microgravity effects and mitigating obsolescence, and presents future test objectives including software development, reliability assessments, dormancy studies, and exploration-oriented upgrades.
Dorodnitsyn's integral method is used to obtain an approximate solution to the supersonic nonequilibrium flow over pointed bodies with attached shock waves. The partial differential equations governing the flow are converted to an approximate set of ordinary equations, which are solved by numerical integration starting at the body tip. Detailed analytical and numerical results for the first approximation are presented, considering the vibrational relaxation of a diatomic gas over a wedge or cone. It is shown that the first approximation yields: (1) The exact flow-variable gradients at the wedge tip (2) Expressions for the flow-variable gradients at the cone tip which are in agreement with extrapolations of characteristics calculations (3) A good approximate algebraic solution for frozen or equilibrium conical flow (4) An approximate expression for the nonequilibrium-flow stream function which affords a means of obtaining variations across the shock layer of the temperature and vibrational energy. Numerical results for both the wedge and cone compare favorably with identical cases computed by the method of characteristics.
Gauss quadrature formula for approximate integration by computer - numerical integration
Elliptic integral tables for evaluation of integrals with algebraic or trigonometric integrands
Numerical integration calculation of collision integrals for exponential attractive potential of gases
Atomic integral obtained with magnetic dipole- dipole operator in form of finite sum of integrals over hypergeometric functions
Numerical integration method for analyzing geomagnetic field for 1955 using poisson integral
Metal panels with integral ribs are formed by the cold-press extrusion method without material loss. Integral ribs in aluminum-alloy panels are formed by this process.