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A Machine Learning Approach to Design Shape Memory Alloys for NASA Applications

Shape memory alloys (SMAs) are a unique class of materials with several remarkable properties including shape recovery, superelasticity, etc. Nickel-titanium (NiTi) based alloys are the most widely studied of this class, with compositions including ternary, quaternary, or higher additions being considered. Especially important for many NASA applications is the ability to tune the martensitic phase transition temperature of NiTi alloys by varying the alloy composition and processing conditions. In addition, low hysteresis and an acceptable recoverable transformation strain are required. Over the past several years, a significant database of SMA properties has been assembled by NASA researchers. Such a database is ideal for data science-based approaches. We present results from our machine learning approach for designing new SMAs with target properties within our range of interest. Our developed models are capable of accurately predicting the transition temperature, hysteresis, and transformation strain of SMAs across a wide range of compositions. This approach has the potential to significantly accelerate the discovery and design of new SMA materials.

Shreyas Jaikumar Honrao↗

NASA NTRS1989

Real-time closed-loop simulation and upset evaluation of control systems in harsh electromagnetic environments

Digital control systems for applications such as aircraft avionics and multibody systems must maintain adequate control integrity in adverse as well as nominal operating conditions. For example, control systems for advanced aircraft, and especially those with relaxed static stability, will be critical to flight and will, therefore, have very high reliability specifications which must be met regardless of operating conditions. In addition, multibody systems such as robotic manipulators performing critical functions must have control systems capable of robust performance in any operating environment in order to complete the assigned task reliably. Severe operating conditions for electronic control systems can result from electromagnetic disturbances caused by lightning, high energy radio frequency (HERF) transmitters, and nuclear electromagnetic pulses (NEMP). For this reason, techniques must be developed to evaluate the integrity of the control system in adverse operating environments. The most difficult and illusive perturbations to computer-based control systems that can be caused by an electromagnetic environment (EME) are functional error modes that involve no component damage. These error modes are collectively known as upset, can occur simultaneously in all of the channels of a redundant control system, and are software dependent. Upset studies performed to date have not addressed the assessment of fault tolerant systems and do not involve the evaluation of a control system operating in a closed-loop with the plant. A methodology for performing a real-time simulation of the closed-loop dynamics of a fault tolerant control system with a simulated plant operating in an electromagnetically harsh environment is presented. In particular, considerations for performing upset tests on the controller are discussed. Some of these considerations are the generation and coupling of analog signals representative of electromagnetic disturbances to a control system under test, analog data acquisition, and digital data acquisition from fault tolerant systems. In addition, a case study of an upset test methodology for a fault tolerant electromagnetic aircraft engine control system is presented.

Belcastro, Celeste M.↗

DOE OSTI2023

Ammonia generation in Ns pulse and Ns pulse/RF discharges over a catalytic surface

Plasma-catalytic ammonia synthesis in a ns pulse discharge and a 'hybrid' ns pulse/RF discharge in plane-to-plane geometry is studied by Fourier Transform infrared absorption spectroscopy. The data are taken in a preheated H 2 –N 2 mixture, with and without Ni/γ-Al 2 O 3 or Co/γ-Al 2 O 3 catalyst placed in the discharge section. The measurement results are taken using two different approaches. The first is a 'single-stage' process, where a ns pulse discharge in the H 2 –N 2 mixture is sustained continuously. In this case, the ammonia yield increases slowly, over a period of tens of minutes. The second is a 'two-stage' process, where the catalyst is first activated by the ns pulse discharge sustained in pure nitrogen, and then the activated catalyst is exposed to the H 2 –N 2 flow, with or without the discharge. In this case, a strong overshoot of the NH 3 number density at the reactor exit is detected, by over a factor of two compared to the single-stage process. After the initial overshoot, the ammonia yield gradually decreases to the 'single stage' value (with the discharge on), or to near zero (with the discharge off). The results demonstrate that the ammonia yield in the plasma-catalytic reactor is controlled by the N atom accumulation on the catalyst surface, which reacts with H atoms thermally dissociated on the catalyst or generated in the plasma. The results also show that the plasma-catalytic ammonia yield is significantly higher compared to that in the ns pulse discharge without the catalyst. The accumulation of H atoms on the catalyst, with their subsequent reactions with N atoms generated in the plasma, is of relatively minor importance at the present conditions. An additional series of measurements was made with a sub-breakdown RF waveform overlapped with the ns pulse discharge train, to enhance the vibrational excitation of nitrogen. The ammonia yield measured with the RF waveform added is approximately 20% higher compared to that at the baseline ns pulse discharge conditions, both with and without the catalyst. This effect is weaker compared to that of the catalyst activation by N atoms. Additional data are necessary to isolate the possible effect of the vibrationally excited N 2 molecules on the ammonia synthesis in the plasma catalytic reactions.

surface reactions↗

DOE OSTI2021

Influence of Pelletization and Moisture Content of Oil Palm Empty Fruit Bunches (EFBs) on Dynamic Gasification Performance

Wet-ground empty fruit bunches (EFBs), dry-ground EFB, and pelletized EFB were gasified in a laboratory-scale updraft fixed-bed reactor at differing temperatures and equivalent ratios (ERs) between 0.19 and 0.78 to evaluate the performance and quality of the syngas under autothermal operating conditions. In addition, the theoretical estimation of the key performance indicator of the ashes on the performance of the gasification process was obtained. The results show that for ERs > 0.4 the autothermal operating condition was achieved for pretreatments with pelletized and dry-ground EFBs, with reactor operating temperatures T > 700 °C. With wet EFB, this condition is achieved at any ER value. The average composition of the syngas from pelletized EFB presented the best average concentrations of the combustible species CO, H2, and CH4 when the ER values were between 0.25 and 0.40. The highest hot and cold thermal efficiency values were obtained for the gasification of EFB pellets with ER values between 0.25 and 0.40. For ground EFB, increases in the ER value led to improvements in hot thermal efficiency. Finally, during the EFB pellet gasification tests, no agglomeration and sintering problems were evidenced in comparison with wet-ground and dry-ground EFBs that showed slag formation, especially at ER < 0.46.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

DOE OSTI2026

Mechanical behaviour of additively manufactured metals

Additive manufacturing is reshaping the production of engineering components in diverse industries, such as the automotive, aerospace, defense, and biomedical sectors, by offering unprecedented design flexibility. The non-equilibrium processing conditions of additive manufacturing generate materials with unique microstructures and tailored mechanical properties that are often unattainable through conventional routes. This review focuses on recent advances in additively manufactured metals that demonstrate distinctive mechanical behaviors, including strength-ductility synergy, microstresses and gradient plasticity, fracture and fatigue resistance, and high-temperature creep performance. Here, we examine the mechanisms and micromechanical effects arising from the heterogeneous microstructures fabricated by additive manufacturing, to guide the design of a wide range of high-performance structural materials. Furthermore, we discuss critical research needs and emerging opportunities in process control, alloy design, advanced characterization, high-fidelity computational modeling, and machine learning aimed at achieving exceptional mechanical properties in additively manufactured metals.

Additive Manufacturing↗

NASA NTRS2015

Creep Behavior and Durability of Cracked CMC

To understand failure mechanisms and durability of cracked Ceramic matrix composites (CMCs), Melt Infiltration (MI) SiCSiC composites with Sylramic-iBN fibers and full Chemical vapour infiltration SiCSiC composites with Sylramic-ion bombarded BN (iBN) and Hi-Nicalon -S fibers were pre-cracked between 150 to 200 megapascal and then creep and Sustained Peak Low Cycle Fatigue (SPLCF) tested at 13150 C at stress levels from 35 to 103 megapascal for up to 200 hours under furnace and burner rig conditions. In addition creep testing was also conducted on pre-cracked full Chemical vapour infiltration SiCSiC composites at 14500 C between 35 and 103 megapascal for up to 200 hours under furnace conditions. If the specimens survived the 200 hour durability tests, then they were tensile tested at room temperature to determine their residual tensile properties. The failed specimens were examined by Scanning electron microscope (SEM) to determine the failure modes and mechanisms. The influence of crack healing matrix, fiber types, crack density, testing modes and interface oxidation on durability of cracked Ceramic matrix composites (CMCs) will be discussed.

Creep↗

DOE OSTI2020

Method used in Tokunaga, T. et al. 2018 for simulating time- and depth- dependent temperature variations in soil and sediment in Rifle, Colorado, 2013-2016.

This data package contains temperature, moisture content, matric potential, and carbon dioxide (CO2) concentration data from the Rifle, Colorado field site and from a 2 meter tall laboratory soil/sediment column designed to simulate field conditions. In addition, redox potential data from the laboratory column are included. The purpose of the data and paper is to demonstrate how a laboratory column was constructed to reproduce the dynamics of nonisothermal conditions in the field, such that CO2 production above a variable-depth water table can be studied in the laboratory. The data package contains 3 data files corresponding to the figures in the related paper (Tokunaga, T. et al. 2018) and 1 README file with the paper information.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS2005

Diode Laser Sensors for Arc-Jet Characterization

The development and application of tunable diode laser (TDL) absorption sensors to monitor the health and operating conditions in the large-scale 60 MW arc-heated- plasma wind-tunnel at NASA Ames Research Center is reported. The interactive heating facility (THF) produces re-entry flow conditions by expanding the gas heated in a constricted plasma arc-heater to flow at high velocity over a model located in a test cabin. This facility provides the conditions needed to test thermal protective systems for spacecraft re-entering the earth s atmosphere. TDL sensors are developed to monitor gas flows in both the high-temperature constricted flow and the supersonic expansion flow into test cabin. These sensors utilize wavelength-tuned diode lasers to measure absorption transitions of atomic oxygen near 777.2 nm, atomic nitrogen near 856.8 nm, and atomic copper near 793.3 nm. The oxygen and nitrogen sensors measure the population density in exited electronic states of these atoms. The measurements combined with the assumption of local thermal and chemical equilibrium yield gas temperature (typically near 7,000K). The nitrogen and oxygen population temperatures are redundant, and their close agreement provides an important test of the local thermal equilibrium assumption. These temperature sensors provide time-resolved monitors of the operating conditions of the arc-heater and can be used to verify and control the test conditions. An additional TDL sensor was developed to monitor the copper concentration in the arc-heater flow yielding values as high as 13 ppm. Measurements of copper in the flow can identify flow conditions with unacceptably rapid electrode erosion, and hence this sensor provides valuable information needed to schedule maintenance to avoid costly arc-heater failure. TDL sensors were also developed for measurements in the test cabin, where absorption measurements of the populations of argon and molecular nitrogen in excited metastable electronic states established that the number density of these excited species is much lower than estimated using frozen-chemistry approximations. This key finding suggests that in the post-expansion region there is not a significant energy sequestration in electronically excited species. Finally, TDL measurements of atomic potassium seeded into the test cabin flow were used to directly measure the static temperature of the test gas. The results of this study illustrate the high potential of time-resolved TDL measurements for routine and economical sensing of arc-heater health (gas temperature and electrode erosion) as well as the time-resolved test-cabin-flow conditions in front of the model.

Hanson, Ronald K.↗

NASA NTRS1996

Ames Research Center Shear Tests of SLA-561V Heat Shield Material for Mars-Pathfinder

This report describes the results of arc-jet testing at Ames Research Center on behalf of Jet Propulsion Laboratory (JPL) for the development of the Mars-Pathfinder heat shield. The current test series evaluated the performance of the ablating SLA-561V heat shield material under shear conditions. In addition, the effectiveness of several methods of repairing damage to the heat shield were evaluated. A total of 26 tests were performed in March 1994 in the 2 in. X 9 in. arc-heated turbulent Duct Facility, including runs to calibrate the facility to obtain the desired shear stress conditions. A total of eleven models were tested. Three different conditions of shear and heating were used. The non-ablating surface shear stresses and the corresponding, approximate, non-ablating surface heating rates were as follows: Condition 1, 170 N/m(exp 2) and 22 W/cm(exp 2); Condition 2, 240 N/m(exp 2) and 40 W/cm(exp 2); Condition 3, 390 N/m(exp 2) and 51 W/cm(exp 2). The peak shear stress encountered in flight is represented approximately by Condition 1; however, the heating rate was much less than the peak flight value. The peak heating rate that was available in the facility (at Condition 3) was about 30 percent less than the maximum value encountered during flight. Seven standard ablation models were tested, of which three models were instrumented with thermocouples to obtain in-depth temperature profiles and temperature contours. An additional four models contained a variety of repair plugs, gaps, and seams. These models were used to evaluated different repair materials and techniques, and the effect of gaps and construction seams. Mass loss and surface recession measurements were made on all models. The models were visually inspected and photographed before and after each test. The SLA-561 V performed well; even at test Condition 3, the char remained intact. Most of the resins used for repairs and gap fillers performed poorly. However, repair plugs made of SLA-561V performed well. Approximately 70 percent of the thermocouples yielded good data.

Tauber, Michael↗

DOE OSTI2025

Degrees of rate control and AutoDiff-driven direct sensitivity analysis in heterogeneous catalysis

Despite the wide application and benefits of the degree of rate control (DRC) analysis, several details remain argued, particularly about the conservation of DRCs at transient (TR) and steady-state (SS) conditions, especially for complex reaction networks. This work argues that previous proofs about the conservation properties of DRCs have been incomplete, and we provide new mathematical proofs at TR and SS conditions. In addition, we use both analytical (automatic differentiation) and numerical (finite difference) approaches to compute DRCs for the case study of ethane hydrogenolysis (EH) over Pt(111). This work confirms that at both TR and SS conditions, the sum of all DRCs, i.e., sum of the degrees of kinetic (DKRC) and thermodynamic rate control (DTRC), is conserved at zero. At SS conditions, the sum of DKRC is conserved at 1 while the sum of DTRC is conserved at −1. In corroboration of previous works, we show that the DTRC for any adsorbate at SS is equal to the product of the species coverage and a constant. In contrast, at TR conditions, the individual sums of both DTRC and DKRC are not conserved and can be any real number, with potential implications for the novel field of dynamic catalysis. Finally, we show that the conventional finite difference (FD) approach, only useful at SS, is prone to inaccuracy and very sensitive to the value of the differential change applied. The optimal differential value also varies significantly with system and rate definition. Consequently, we describe and illustrate in this work the application of the automatic differentiation (AD) approach for the more accurate determination of DRCs at both TR and SS conditions.

Automatic differentiation↗

DOE OSTI2023

What Makes Passengers Uncomfortable In Vehicles Today? An Exploratory Study of Current Factors that May Influence Acceptance of Future Autonomous Vehicles

Autonomous vehicles have the potential to transform lives by providing transportation to a wider range of users. However, with this new method of transportation, user acceptance and comfort are critical for widespread adoption. This exploratory study aims to investigate what makes passengers uncomfortable in existing vehicles to inform the design of future autonomous vehicles. In order to predict what may impact user acceptance for a diverse rider population for future autonomous vehicles, it is important to understand what makes a broad range of passengers uncomfortable today. In this study, interviews were conducted for a total of 75 participants from three diverse groups, including 20 automotive engineering graduate students who are building an autonomous concept vehicle, 21 non-technical adults, and 34 senior citizens. The results revealed both topics which made different groups of passengers uncomfortable as well as how these varied between the groups. The leading contributors to the highest discomfort for all groups were being a passenger in situations with a distracted driver, being in a vehicle that is following too closely, being near a vehicle that is following too closely, and being in foggy conditions. In addition, the results showed that passenger discomfort can be attributed to a broad range of factors ranging from behaviors of the driver/vehicle that one is traveling with/in, the behaviors of other surrounding vehicles, the environmental conditions and the vehicle’s interior, all of which may differ between different groups of passengers. This research provides important findings and insights into factors that may influence users’ acceptance and use of future autonomous vehicles.

Mims, Lauren K.↗

DOE OSTI2021

Measuring the residual stress and stress-corrosion cracking susceptibility of additively manufactured 316L by ASTM G36-94

Residual stress is a contributor to stress-corrosion cracking (SCC) and a common byproduct of additive manufacturing (AM). In this work, the relationship between residual stress and SCC susceptibility in laser powder bed fusion AM 316L stainless steel was studied through immersion in saturated boiling magnesium chloride per ASTM G36-94. The residual stress was varied by changing the sample height for the as-built condition and additionally by heat treatments at 600, 800, and 1200 °C to control, and in some cases reduce, residual stress. In general, all samples in the as-built condition showed susceptibility to SCC with the thinner, lower residual stress samples showing shallower cracks and crack propagation occurring perpendicular to melt tracks due to local residual stress fields. The heat-treated samples showed a reduction in residual stress for the 800 and 1200 °C samples. Both were free of cracks after >300 hours of immersion in MgCl 2 , while the 600 °C sample showed similar cracking to their as-built counterpart. Geometrically necessary dislocation (GND) density analysis indicates that the dislocation density may play a major role in the SCC susceptibility.

36 MATERIALS SCIENCE↗

NASA NTRS1994

Development of a Model Following Control Law for Inflight Simulation and Flight Controls Research

The U.S. Army and NASA are currently developing the Rotorcraft Aircrew Systems Concepts Airborne Laboratory (RASCAL) at the Ames Research Center. RASCAL, shown in Figure 1, is a UH-60, which is being modified in a phased development program to have a research fly-by-wire flight control system, and an advanced navigation research platform. An important part of the flight controls and handling qualities research on RASCAL will be an FCS design for the aircraft to achieve high bandwidth control responses and disturbance rejection characteristics. Initially, body states will be used as feedbacks, but research into the use of rotor states will also be considered in later stages to maximize agility and maneuverability. In addition to supporting flight controls research, this FCS design will serve as the inflight simulation control law to support basic handling qualities, guidance, and displays research. Research in high bandwidth controls laws is motivated by the desire to improve the handling qualities in aggressive maneuvering and in severely degraded weather conditions. Naturally, these advantages will also improve the quality of the model following, thereby improving the inflight simulation capabilities of the research vehicle. High bandwidth in the control laws provides tighter tracking allowing for higher response bandwidths which can meet handling qualities requirements for aggressive maneuvering. System sensitivity is also reduced preventing variations in the response from the vehicle due to changing flight conditions. In addition, improved gust rejection will result from this reduced sensitivity. The gust rejection coupled with a highly stable system will make more precise maneuvering and pointing possible in severely degraded weather conditions. The difficulty in achieving higher bandwidths from the control laws in the feedback and in the responses arises from the complexity of the models that are needed to produce a satisfactory design. In this case, high quality models that include rotor dynamics in a physically meaningful context must be available. A non-physical accounting of the rotor, such as lumping the effect as a time delay, is not likely to produce the desired results. High order simulation models based on first principals are satisfactory for the initial design phase in order to work out the control law design concept and get an initial set of gains. These models, however, have known deficiencies, which must be resolved in the final control law design. The error in the pitch-roll cross coupling is one notable deficiency that even sophisticated rotorcraft models including complex wake aerodynamics have yet to capture successfully. This error must be accounted for to achieve the desired decoupling. The approach to design the proposed inflight simulation control law is based on using a combination of simulation and identified models. The linear and nonlinear higher order models were used to develop an explicit model following control structure. This structure was developed to accommodate the design of control laws compliant to many of the quantitative requirements in ADS-33C. Furthermore, it also allows for control law research using rotor-state feedback and other design methodologies such as Quantitative Feedback and H-Infinity. Final gain selection will be based on higher order identified models which include rotor degrees of freedom.

Takahashi, Mark↗

NASA NTRS2005

Aeroacoustic Analysis of Fan Noise Reduction With Increased Bypass Nozzle Area

An advanced model turbofan was tested in the NASA Glenn 9-by 15-Foot Low Speed Wind Tunnel (9x15 LSWT) to explore far field acoustic effects of increased bypass nozzle area. This fan stage test was part of the NASA Glenn Fan Broadband Source Diagnostic Test, second entry (SDT2) which acquired aeroacoustic results over a range of test conditions. The baseline nozzle was sized to produce maximum stage performance at cruise condition. However, the wind tunnel testing is conducted near sea level condition. Therefore, in order to simulate and obtain performance at other operating conditions, two additional nozzles were designed and tested one with +5 percent increase in weight flow (+5.4 percent increase in nozzle area compared with the baseline nozzle), sized to simulate the performance at the stage design point (takeoff) condition, and the other with a +7.5 percent increase in weight flow (+10.9 percent increase in nozzle area) sized for maximum weight flow with a fixed nozzle at sea level condition. Measured acoustic benefits with increased nozzle area were very encouraging, showing overall sound power level (OAPWL) reductions of 2 or more dB while the stage thrust actually increased by 2 to 3 percent except for the most open nozzle at takeoff rotor speed where stage performance decreased. Effective perceived noise levels for a 1500 ft engine flyover and 3.35 scale factor showed a similar noise reduction of 2 or more EPNdB. Noise reductions, principally in the level of broadband noise, were observed everywhere in the far field. Laser Doppler Velocimetry measurements taken downstream of the rotor showed that the total turbulent velocity decreased with increasing nozzle flow, which may explain the reduced rotor broadband noise levels.

Woodward, Richard P.↗

DOE OSTI2024

Process–Property–Performance Mapping of Additively Manufactured 316H Stainless Steel Components

The Advanced Materials and Manufacturing Technologies Program is focused on accelerating the development of advanced materials and components fabricated via additive manufacturing, and is using laser powder bed fusion (LPBF) of 316H stainless steel as an initial case study. In the previous fiscal year, miniature high-throughput specimens were printed on multiple LPBF systems to provide initial processing windows to minimize porosity and limit epitaxial grain growth during prints. This fiscal year, scaled builds were completed on three different LPBF systems at ORNL: a GE Concept Laser M2, a Renishaw AM400, and an EOS M290. Builds on the Concept Laser were conducted on multiple powder lots and processing parameter ranges to provide microstructure effects on time-independent and time-dependent mechanical properties. Builds on the Renishaw were produced using Oak Ridge National Laboratory (ORNL)-optimized printing parameters and Argonne National Laboratory (ANL)-optimized printing parameters to compare outcomes of parallel process optimization efforts at different national laboratories on the same LPBF system. Similarly, the build completed on the EOS M290 replicated the processing parameters of builds completed at Los Alamos National Laboratory (LANL). Optical microscopy and electron backscatter diffraction characterization was completed on all builds. In addition to the general round robin characterization, this work-package generated time-independent data, including tensile and fracture toughness test data on scaled Concept Laser builds as a function of processing parameters and post-build heat treatment. This analysis is complimentary to work in parallel work packages aiming to establish heat treatment and processing effects on time-dependent properties. It was found that although the stress-relief heat treatment provides the highest strength at lower-temperatures, tensile strength begins to converge at higher temperatures regardless of heat treatment condition. In addition, the more rigorous solution annealing and hot-isostatic pressing post-build heat treatments result in higher fracture toughness than the stress-relieved condition. The root-causes of the lower fracture toughness of the stress-relieved LPBF 316H material was informed via a stress-relief optimization study on a scaled concept laser print, where it was found that although dislocation recovery was largely complete after only a couple hours at 650°C, the extended hold of the current 24h heat treatment employed on scaled builds likely caused increased carbide volume fractions along the LPBF 316H grain boundaries, thereby deteriorating crack propagation resistance. This trend was seen to become more deleterious with additional increases of stress-relief temperature to 750°C or 850°C. These results have helped inform a new optimal stress-relief annealing condition for LPBF 316H for future campaign testing (650°C for 2h).

36 MATERIALS SCIENCE↗

DOE OSTI2024

Process–Property–Performance Mapping of Additively Manufactured 316H Stainless Steel Components

36 MATERIALS SCIENCE↗

DOE OSTI2020

High temperature steam oxidation dynamics of U 3 Si 2 with alloying additions: Al, Cr, and Y

Uranium silicides are considered for advanced technology reactor fuels due to their enhanced thermal conductivity and high uranium density (U 3 Si and U 3 Si 2 ) compared to traditional UO 2 . Susceptibility to oxidation and wash out, in the event of a cladding breech, could limit the potential for deployment of silicides as accident tolerant fuels. Mitigating the water reaction for U 3 Si 2 could enable its use as an accident tolerant, high uranium density fuel or as a composite fuel constituent. Reported here is the impact of alloying additions of Al, Cr and Y on the high temperature, steam oxidation response of U 3 Si 2 . In addition to the thermogravimetric response, as melted microstructures, phase compositions and post oxidation analysis are also presented. Here, the investigation shows steam oxidation dynamics are altered, from non-alloyed U 3 Si 2 , under thermally ramped conditions. However, additional alloy development of these fuel forms is necessary for further consideration as candidate accident tolerant fuels in water-cooled reactor designs.

36 MATERIALS SCIENCE↗

NASA NTRS1994

Theoretical study of the incompressible Navier-Stokes equations by the least-squares method

Usually the theoretical analysis of the Navier-Stokes equations is conducted via the Galerkin method which leads to difficult saddle-point problems. This paper demonstrates that the least-squares method is a useful alternative tool for the theoretical study of partial differential equations since it leads to minimization problems which can often be treated by an elementary technique. The principal part of the Navier-Stokes equations in the first-order velocity-pressure-vorticity formulation consists of two div-curl systems, so the three-dimensional div-curl system is thoroughly studied at first. By introducing a dummy variable and by using the least-squares method, this paper shows that the div-curl system is properly determined and elliptic, and has a unique solution. The same technique then is employed to prove that the Stokes equations are properly determined and elliptic, and that four boundary conditions on a fixed boundary are required for three-dimensional problems. This paper also shows that under four combinations of non-standard boundary conditions the solution of the Stokes equations is unique. This paper emphasizes the application of the least-squares method and the div-curl method to derive a high-order version of differential equations and additional boundary conditions. In this paper, an elementary method (integration by parts) is used to prove Friedrichs' inequalities related to the div and curl operators which play an essential role in the analysis.

Jiang, Bo-Nan↗