Search NASA⌕ Search

SEARCH · Search NASA

Results for “Test guidelines”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 325 records · Page 18

Controls for agility research in the NASA High-Alpha Technology Program

The research process being used to develop control law design methodologies and guidelines in the NASA High-Alpha Technology Program are discussed. This step-by-step process consists of four basic elements: (1) control law architecture definition and linear synthesis, (2) nonlinear batch simulation, (3) piloted simulation evaluation, and (4) flight test validation. This paper discusses the research tools being used in this effort and provides a status report on design methodologies and guidelines being developed for each of these elements.

Foster, John V.↗

Field Test: Results of Tandem Walk Performance Following Long-Duration Spaceflight

BACKGROUND: Coordinated locomotion has proven to be challenging for many astronauts following long duration spaceflight. As NASA's vision for spaceflight points toward interplanetary travel, we must prepare for unassisted landings, where crewmembers may need to perform mission critical tasks within minutes of landing. Thus, it is vital to develop a knowledge base from which operational guidelines can be written that define when astronauts can be expected to safely perform certain tasks. Data obtained during the Field Test experiment (FT) will add important insight to this knowledge base. Specifically, we aim to develop a recovery timeline of functional sensorimotor performance during the first 24 hours and several days after landing. METHODS: FT is an ongoing study of 30 long-duration ISS crewmembers. Thus far, 9 have completed the full FT (5 U.S. Orbital Segment [USOS] astronauts and 4 Russian cosmonauts) and 4 more consented and launching within the next year. This is in addition to the eighteen crewmembers that participated in the pilot FT (11 USOS and 7 Russian crewmembers). The FT is conducted three times preflight and three times during the first 24 hours after landing. All crewmembers were tested in Kazakhstan in either the medical tent at the Soyuz landing site (~one hour post-landing), or at the airport (~four hours post-landing). The USOS crewmembers were also tested at the refueling stop (~12 hours post-landing) and at the NASA Johnson Space Center (~24 hours post-landing) and a final session 7 days post-landing. Crewmembers are instrumented with 9 inertial measurement unit sensors that measure acceleration and angular displacement (APDM's Emerald Sensors) and foot pressure-sensing insoles that measure force, acceleration, and center of pressure (Moticon GmbH, Munich, Germany) along with heart rate and blood pressure recording instrumentation. The FT consists of 12 tasks, but here we will focus on the most challenging task, the Tandem Walk, which was also performed as part of pilot FT. To perform the Tandem Walk, subjects begin with their feet together, their arms crossed at their chest and eyes closed. When ready, they brought one foot forward and touched the heel of their foot to their toe, repeating with the other foot, and continuing for about 10 steps. Three trials were collected with the eyes closed and a fourth trial was collected with eyes open. There are four metrics which are used to determine the performance level of the Tandem Walk. The first is percent correct steps. For a step to be counted as correct, the foot could not touch the ground while bringing it forward (no side stepping), eyes must stay closed during the eyes closed trials, the heel and toe should be touching, or almost touching (no large gaps) and there shouldn't be more than a three second pause between steps. Three judges score each step and the median of the three scores is kept. The second metric is the average step speed, or the number of steps/time to complete them. Thirdly, the root mean squared (RMS) error in the resultant trunk acceleration is used to determine the amount of upper body instability observed during the task. Finally, the RMS error of the mediolateral center of pressure as measured by the Moticon insoles is used to determine the mediolateral instability at the foot level. These four parameters are combined into a new overall Tandem Walk Parameter. RESULTS: Preliminary results show that crewmembers perform the Tandem Walk significantly worse the first 24 hours after landing as compared to their baseline performance. We find that each of the four performance metrics is significantly worse immediately after landing. We will present the results of tandem walk performance during the FT thus far. We will also combine these with the 18 crewmembers that participated in the pilot FT, concentrating on the level of performance and recovery rate. CONCLUSION: The Tandem Walk data collected as part of the FT experiment will provide invaluable information on the performance capabilities of astronauts during the first 24 hours after returning from long-duration spaceflight that can be used in planning future Mars, or other deep-space missions with unassisted landings. FT will determine the average sensorimotor recovery timeline and inform return-to-duty guidelines for unassisted landings.

Rosenberg, M. J. F.↗

Design, manufacture, development, test, and evaluation of boron/aluminum structural components for space shuttle. Volume 4: Repairability

The repairability of boron/aluminum structural components was investigated. It was demonstrated that metal matrix composite material, damaged in service, can be repaired by techniques that are not very different from those currently in use for conventional materials. A list of repair guidelines was prepared to aid in determining the proper repair techniques for a given structure. The guidelines include specifying types of repair material and their applicability, corrosion prevention procedures, design criteria, and inspection criteria. Boron/aluminum structural components were repaired and tested to compare as-fabricated and repaired performance. All but one set of specimens, when repaired, exceeded the strength of the original specimens.

Miller, M. F.↗

Pressure control and analysis report: Hydrogen Thermal Test Article (HTTA)

Tasks accomplished during the HTTA Program study period included: (1) performance of a literature review to provide system guidelines; (2) development of analytical procedures needed to predict system performance; (3) design and analysis of the HTTA pressurization system considering (a) future utilization of results in the design of a spacecraft maneuvering system propellant package, (b) ease of control and operation, (c) system safety, and (d) hardware cost; and (4) making conclusions and recommendations for systems design.

Source record↗

CisLunar Habitat Internal Architecture Design Criteria

BACKGROUND: In preparation for human exploration to Mars, there is a need to define the development and test program that will validate deep space operations and systems. In that context, a Proving Grounds CisLunar habitat spacecraft is being defined as the next step towards this goal. This spacecraft will operate differently from the ISS or other spacecraft in human history. The performance envelope of this spacecraft (mass, volume, power, specifications, etc.) is being defined by the Future Capabilities Study Team. This team has recognized the need for a human-centered approach for the internal architecture of this spacecraft and has commissioned a CisLunar Phase-1 Habitat Internal Architecture Study Team to develop a NASA reference configuration, providing the Agency with a "smart buyer" approach for future acquisition. THE CISLUNAR HABITAT INTERNAL ARCHITECTURE STUDY: Overall, the CisLunar Habitat Internal Architecture study will address the most significant questions and risks in the current CisLunar architecture, habitation, and operations concept development. This effort is achieved through definition of design criteria, evaluation criteria and process, design of the CisLunar Habitat Phase-1 internal architecture, and the development and fabrication of internal architecture concepts combined with rigorous and methodical Human-in-the-Loop (HITL) evaluations and testing of the conceptual innovations in a controlled test environment. The vision of the CisLunar Habitat Internal Architecture Study is to design, build, and test a CisLunar Phase-1 Habitat Internal Architecture that will be used for habitation (e.g. habitability and human factors) evaluations. The evaluations will mature CisLunar habitat evaluation tools, guidelines, and standards, and will interface with other projects such as the Advanced Exploration Systems (AES) Program integrated Power, Avionics, Software (iPAS), and Logistics for integrated human-in-the-loop testing. The mission of the CisLunar Habitat Internal Architecture Study is to become a forcing function to establish a common understanding of CisLunar Phase-1 Habitation Internal Architecture design criteria, processes, and tools. The scope of the CisLunar Habitat Internal Architecture study is to design, develop, demonstrate, and evaluate a Phase-1 CisLunar Habitat common module internal architecture based on design criteria agreed to by NASA, the International Partners, and Commercial Exploration teams. This task is to define the CisLunar Phase-1 Internal Architecture Government Reference Design, assist NASA in becoming a "smart buyer" for Phase-1 Habitat Concepts, and ultimately to derive standards and requirements from the Internal Architecture Design Process. The first step was to define a Habitat Internal Architecture Design Criteria and create a structured philosophy to be used by design teams as a filter by which critical aspects of consideration would be identified for the purpose of organizing and utilizing interior spaces. With design criteria in place, the team will develop a series of iterative internal architecture concept designs which will be assessed by means of an evaluation criteria and process. These assessments will successively drive and refine the design, leading to the combination and down-selection of design concepts. A single refined reference design configuration will be developed into in a medium-to-high fidelity mockup. A multi-day human-in-the-loop mission test will fully evaluate the reference design and validate its configuration. Lessons learned from the design and evaluation will enable the team to identify appropriate standards for Phase-1 CisLunar Habitat Internal Architecture and will enable NASA to develop derived requirements in support of maturing CisLunar Habitation capabilities. This paper will describe the criteria definition process, workshop event, and resulting CisLunar Phase-1 Habitat Internal Architecture Design Criteria.

Jones, R.↗

Low Intensity Low Temperature (LILT) measurements and coefficients on new photovoltaic structures

Past NASA missions to Mars, Jupiter, and the outer planets were powered by radioisotope thermal generators (RTG's). Although these devices proved to be reliable, their high cost and highly toxic radioactive heat source has made them far less desirable for future planetary missions. This has resulted in a renewed search for alternate energy sources, some of them being photovoltaic (PV) and thermophotovoltaic (TPV). Both of these alternate energy sources convert light/thermal energy directly into electricity. In order to create a viable PV and TPV data base for planetary mission planners and cell designers, we have compiled low temperature low intensity (LILT) I-V data on single junction and multi-junction high efficiency solar cells. The cells tested here represent the latest photovoltaic technology. Using this LILT data to calculate dI(sub SC)/dT, dV(sub OC)/dT, dFF/dT, and also as a function of intensity, an accurate prediction of cell performance under the AMO spectrum can be determined. When combined with QUantum efficiency at Low Temperature (QULT) data, one can further enhance the data by adding spectral variations to the measurements. This paper presents an overview of LILT measurements and is only intended to be used as a guideline for material selection and performance predictions. As single junction and multi-junction cell technologies emerge, new test data must be collected. Cell materials included are Si, GaAs/Ge, GainP/GaAs/Ge, InP, InGaAs/InP, InP/InGaAs/InP, and GainP. Temperatures range as low as -175 C and intensities range from 1 sun to .02 suns.

Schelman, David A.↗

Inflatable Softgoods Testing Capability Development for Habitation

The Habitation Systems Development Office (HP40) at NASA Marshall Space Flight Center supports systems engineering, integration, and project management for next generation space habitats. The office is responsible for formulation activities for Lunar Surface Habitat and Mars Transit Habitat. HP40 also manages commercial partnerships for exploration habitat development. The Lunar Surface Habitat is intended to initially support a minimum of two crew on the lunar surface for an approximately 30-day mission, with the ability to accommodate four crew for short durations. The Mars Transit Habitat will transport four crew to and from Mars orbit on a potentially 1,200 day mission. Both concepts are envisioned to leverage inflatable softgoods as the primary structural material for the habitation portion of the architecture. Inflatable softgoods are a complex, multi-material system consisting of multiple layers which perform specific functions. A bladder layer (polymer) contains the habitat’s atmosphere. The bladder is indexed to a woven restraint layer, usually Kevlar or Vectran, which serves as the structural layer. The remaining outer layers are for micro-meteoroid orbital debris (MMOD) shielding, protection from atomic Oxygen (if the material system is to be deployed in low earth orbit), and thermal insulation. The primary advantage of inflatable softgoods is that they can offer a larger volume per unit mass relative to a rigid metal structure. Inflatable softgoods have a strong development heritage within NASA, beginning with work in the 1960s in partnership with Goodyear Aerospace and continuing with Johnson Space Center’s TransHab project in the 1990s . Some development efforts subsequently transitioned into the private sector to mature the material technology for flight applications, including human habitation. In August 2022, NASA published its guidelines for certification of crewed inflatable softgoods structures (JSC-67721). This document outlines a series of tests which will be needed to certify inflatable softgoods material systems for use in crewed mission applications. It includes burst testing at the subscale and full-scale to determine the ultimate burst pressure of the softgoods architecture. Creep (long term deformation of the material under stress) is a key failure mode which must be characterized. Creep testing at the component and module levels is also recommended to generate time to failure curves and verify that the inflatable system has a predicted operational life which exceeds mission parameters. ET (Test Engineering) at MSFC and HP40 have worked together to establish softgoods testing capabilities for the center. This includes the ability to conduct module level burst tests and creep tests. The poster will highlight these capabilities and provide examples of testing which has been performed to date by ET. NASA MSFC anticipates that these test capabilities will continue to offer critical support for companies pursuing inflatable softgoods for exploration applications (lunar surface habitation, Mars transit) as well as commercial low earth orbit platforms.

habitation↗

Assessment of Electrical, Electronic, and Electromechanical (EEE) Parts Copper Wire Bonds for Space Programs

The NASA Electronic Parts and Packaging Program co-manager, requested the NASA Engineering and Safety Center (NESC) to compile a body of publicly available knowledge on copper (Cu) wire bonds and perform reliability testing and analysis on these bonds, including environment tests, sample destructive physical analysis, and bond pull/shear tests during environment test intervals. The goal was to understand the risks of using Cu wire bonds for space applications and develop guidelines on Cu bond pull/shear limits for NASA applications. This document contains the results of the NESC assessment.

Electrical, Electronic, and Electromechanical↗

Low Intensity Low Temperature (LILT) Measurements and Coefficients on New Photovoltaic Structures

Past NASA missions to Mars, Jupiter and the outer planets were powered by radioisotope thermal generators (RTGs). Although these devices proved to be reliable, their high cost and highly toxic radioactive heat source has made them far less desirable for future planetary missions. This has resulted in a renewed search for alternate energy sources, some of them being photovoltaics (PV) and thermophotovoltaics (TPV). Both of these alternate energy sources convert light/thermal energy directly into electricity. In order to create a viable PV data base for planetary mission planners and cell designers, we have compiled low intensity low temperature (LILT) I-V data on single junction and multi-junction high efficiency solar cells. The cells tested here represent the latest photovoltaic technology. Using this LILT data to calculate Short Circuit Current (I(sub sc)), Open Circuit Voltage (V(sub os)), and Fill Factor (FF) as a function of temperature and intensity, an accurate prediction of cell performance under the AM0 spectrum can be determined. When combined with QUantum efficiency at Low Temperature (QULT) data, one can further enhance the data by adding spectral variations to the measurements. This paper presents an overview of LILT measurements and is only intended to be used as a guideline for material selection and performance predictions. As single junction and multi-junction cell technologies emerge, new test data must be collected. Cell materials included are Si, GaAs/Ge, GaInP/GaAs/GaAs, InP, InGaAs/InP, InP/InGaAs/InP, and GaInP. Temperatures range down to as low as -180 C and intensities range from 1 sun down to 0.02 suns. The coefficients presented in this paper represent experimental results and are intended to provide the user with approximate numbers.

Scheiman, David A.↗

Visual Impairment/lntracranial Pressure Risk Clinical Care Data Tools

Prior to 2010, several ISS crewmembers returned from spaceflight with changes to their vision, ranging from a mild hyperopic shift to frank disc edema. As a result, NASA expanded clinical vision testing to include more comprehensive medical imaging, including Optical Coherence Tomography and 3 Tesla Brain and Orbit MRIs. The Space and Clinical Operations (SCO) Division developed a clinical practice guideline that classified individuals based on their symptoms and diagnoses to facilitate clinical care. For the purposes of clinical surveillance, this classification was applied retrospectively to all crewmembers who had sufficient testing for classification. This classification is also a tool that has been leveraged for researchers to identify potential risk factors. In March 2014, driven in part by a more comprehensive understanding of the imaging data and increased imaging capability on orbit, the SCO Division revised their clinical care guidance to outline in‐flight care and increase post‐flight follow up. The new clinical guidance does not include a classification scheme

Van Baalen, Mary↗

SMS/GOES cell and battery data analysis report

The nickel-cadmium battery design developed for the Synchronous Meteorological Satellite (SMS) and Geostationary Operational Environmental Satellite (GOES) provided background and guidelines for future development, manufacture, and application of spacecraft batteries. SMS/GOES battery design, development, qualification testing, acceptance testing, and life testing/mission performance characteristics were evaluated for correlation with battery cell manufacturing process variables.

Armantrout, J. D.↗

Study to develop improved fire resistant aircraft passenger seat materials

The Phase 3 study of the NASA 'Improved Fire Resistant Aircraft Seat Materials' involved fire tests of improved materials in multilayered combinations representative of cushion configurations. Tests were conducted to determine their thermal, smoke, and fire resistance characteristics. Additionally, a 'Design Guideline' for Fire Resistant Passenger Seats was written outlining general seat design considerations. Finally, a three-abreast 'Tourist Class' passenger seat assembly fabricated from the most advanced fire-resistant materials was delivered.

Duskin, F. E.↗

Recent progress in NASA Langley Research Center textile reinforced composites program

Research was conducted to explore the benefits of textile reinforced composites for transport aircraft primary structures. The objective is to develop and demonstrate the potential of affordable textile reinforced composite materials to meet design properties and damage tolerance requirements of advanced aircraft structural concepts. Some program elements include development of textile preforms, processing science, mechanics of materials, experimental characterization of materials, and development and evaluation of textile reinforced composite structural elements and subcomponents. Textile 3-D weaving, 3-D braiding, and knitting and/or stitching are being compared with conventional laminated tape processes for improved damage tolerance. Through-the-thickness reinforcements offer significant damage tolerance improvements. However, these gains must be weighted against potential loss in in-plane properties such as strength and stiffness. Analytical trade studies are underway to establish design guidelines for the application of textile material forms to meet specific loading requirements. Fabrication and testing of large structural parts are required to establish the potential of textile reinforced composite materials.

Dexter, H. Benson↗

Second Annual Transformative Vertical Flight Concepts Workshop: Enabling New Flight Concepts Through Novel Propulsion and Energy Architectures

On August 3rd and 4th, 2015, a workshop was held at the NASA Ames Research Center, located at the Moffett Federal Airfield in California to explore the aviation communities interest in Transformative Vertical Flight (TVF) Concepts. The Workshop was sponsored by the AHS International (AHS), the American Institute of Aeronautics and Astronautics (AIAA), the National Aeronautics and Space Administration (NASA), and hosted by the NASA Aeronautics Research Institute (NARI). This second annual workshop built on the success and enthusiasm generated by the first TVF Workshop held in Washington, DC in August of 2014. The previous Workshop identified the existence of a multi-disciplinary community interested in this topic and established a consensus among the participants that opportunities to establish further collaborations in this area are warranted. The desire to conduct a series of annual workshops augmented by online virtual technical seminars to strengthen the TVF community and continue planning for advocacy and collaboration was a direct outcome of the first Workshop. The second Workshop organizers focused on four desired action-oriented outcomes. The first was to establish and document common stakeholder needs and areas of potential collaborations. This includes advocacy strategies to encourage the future success of unconventional vertiport capable flight concept solutions that are enabled by emerging technologies. The second was to assemble a community that can collaborate on new conceptual design and analysis tools to permit novel configuration paths with far greater multi-disciplinary coupling (i.e., aero-propulsive-control) to be investigated. The third was to establish a community to develop and deploy regulatory guidelines. This community would have the potential to initiate formation of an American Society for Testing and Materials (ASTM) F44 Committee Subgroup for the development of consensus-based certification standards for General Aviation scale vertiport capable flight systems. These standards need to accommodate novel fixed wing concepts that do not fit within the existing Federal Aviation Administration (FAA) rotorcraft certification framework (Code of Federal Regulations, Title 14, Chapter I, Subchapter C, Part 27). The fourth desired outcome was to launch an information campaign to ensure key U.S. Government agencies understand the potential benefits and industry interest in establishing new vertiport capable flight markets. This record of the Workshop proceedings documents Workshop activities and products including summaries of the video recorded technical presentations, overviews of three breakout sessions (Missions Operational Concepts, Prioritized Technical Challenges, Regulatory Roadmap), and a preliminary draft roadmap framework for TVF.

V/STOL↗

Lightning protection technology for small general aviation composite material aircraft

An on going NASA (Small Business Innovative Research) SBIR Phase II design and development program will produce the first lightning protected, fiberglass, General Aviation aircraft that is available as a kit. The results obtained so far in development testing of typical components of the aircraft kit, such as the wing and fuselage panels indicate that the lightning protection design methodology and materials chosen are capable of protecting such small composite airframes from lightning puncture and structural damage associated with severe threat lightning strikes. The primary objective of the program has been to develop a lightening protection design for full scale test airframe and verify its adequacy with full scale laboratory testing, thus enabling production and sale of owner-built, lightning-protected, Stoddard-Hamilton Aircraft, Inc. Glasair II airplanes. A second objective has been to provide lightning protection design guidelines for the General Aviation industry, and to enable these airplanes to meet lightening protection requirements for certification of small airplanes. This paper describes the protection design approaches and development testing results obtained thus far in the program, together with design methodology which can achieve the design goals listed above. The presentation of this paper will also include results of some of the full scale verification tests, which will have been completed by the time of this conference.

Plumer, J. A.↗

Guidelines for Reliable DC/DC Converters for Space Use

NESC saw the need to study the persistent failure of DC/DC Converters during ground testing and in flight, motivated investigation of causes and mitigation options. Research indicated misapplication and device quality to be root causes. The study took 20 months. Team included multiple NASA Centers : JPL, JSC, MSFC, GSFC Project Background

Plante, Jeannette↗

Guidelines for Verification Strategies to Minimize RISK Based on Mission Environment, -Application and -Lifetime (MEAL)

There is a trend of compromising verification testing to address the cost and schedule constraints, which poses a high-risk posture for programs/projects. Current and emerging aerospace scientific and/or human exploration programs continue to pose new technological challenges. These technological challenges combined with finite budgets and truncated schedules are forcing designers, scientists, engineers, and managers to push technologies to their physical limits. In addition, budget and schedule pressures challenge how those technologies/missions are verified. A clear understanding of the different verification processes is needed to ensure the proper verification of the technology within the mission (i.e., capabilities, advantages, and limitations). The goal of verification is to prove through test, analysis, inspection, and/or demonstration that a product provides its required function while meeting the performance requirements. It is important that verification yield understanding of representative performance under worst-case conditions so that margins to failure can be evaluated for proposed applications. The capabilities, advantages, and limitations of the testing and inspection performed at each level are different, and the risk incurred by omitting a verification step depends on the level of integration as well as Mission, Environment, Application and Lifetime (MEAL). This paper focuses on verification processes. The goal of the verification process is to ensure the given avionics technology could be safely implemented on the given MEAL consistent with the program/project risk posture.

Gonzalez, Oscar↗

Test and Analysis of the 8-foot Diameter Cylindrical Sandwich Composite Test Article CTA8.2B: As part of the NASA Engineering and Safety Center Shell Buckling Knockdown Factor Project

The buckling response of thin-walled cylindrical structures have long been shown to be sensitive to various imperfections. That is, imperfections related to load introduction, boundary conditions, material property variation, and geometric deviations can all contribute to experimentally obtained buckling loads being lower than the buckling loads predicted for perfect representations of the same thin-walled cylindrical structure. To account for the influence of geometry-based imperfections of a thin-walled cylindrical structure on buckling response, a design factor (also known as a buckling knockdown factor) is used during the design phase. Guidelines for buckling knockdown factors are readily available in NASA SP-8007, Buckling of Thin Walled Circular Cylinders, which was updated in 2020. Prior to this recent update, NASA SP-8007 had not been updated since 1969, and since that time, computational analysis, experimental testing, and manufacturing methods have significantly improved. The NASA Engineering and Safety Center (NESC) Shell Buckling Knockdown Factor Project (SBKF, NESC Assessment 07-010-E) had the goal of developing buckling design recommendations for select classes of metallic and composite launch-vehicle structures [3]. Specifically, the test and analysis results from the fourth SBKF composite test article SBKF-P3-CYL-CTA8.2B, which is referred to as CTA8.2B, are described in this paper. This test was the fourth in a series of four tests on sandwich composite cylinders that will be used to experimentally validate analysis methods, which in turn can be used to develop new analysis-based shell buckling design guidelines for typical sandwich composite launch vehicle cylindrical structures. Publications describing previous test validation with analysis are available in Refs. CTA8.2B was designed to occupy one corner of the design space with approximately equal bending stiffnesses in the axial and transverse directions and have a larger ratio of radius to effective shell thickness. Each test article had a unique design and was designed to occupy a specific region of the design space. CTA8.2B was an 8-foot diameter honeycomb-core sandwich cylinder that was fabricated and tested at the NASA Marshall Space Flight Center (MSFC). The primary objectives of this test were to interrogate the structural capability of a composite test article, and to verify the test article design and analysis approach for cylinders subjected to axial compression and combined axial and bending loads. First, descriptions of the test article design and test are given in Section 2.0, and modeling and analysis methods used in support of the test article design and testing activities are described briefly in Section 3.0. Then, select test results are presented and compared to predicted results in Section 4.0, and concluding remarks are presented in Section 5.0. Finally, drawings of CTA8.2B and references used to support the test and analysis are provided in the Appendix.

Nonlinear analysis↗