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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
DOE OSTI2022

Advanced Thin Film Core Technology: CIGS Final Technical Report (FTR)

Cu(In,Ga)Se 2 (CIGS) thin-film photovoltaics are a high-efficiency and reliable technology. This project completed research in two important areas and was designed to work collaboratively with industrial partners. Task 1: Alkali Science focused on alkali post-deposition treatments (PDT). PDTs have been instrumental in the dramatic voltage improvements that have moved CIGS device efficiencies from 20% to 23.35% [1]. Based on a survey of CIGS companies at the beginning of the project, the single biggest breakthrough for the CIGS community would be a mechanistic understanding of the role of alkalis in the material system. Significant accomplishments of Task 1: Alkali Science: 1) KF post-deposition treatments were shown to improve lifetime, open-circuit voltage (VOC), and efficiency of industrial partner samples, even when done as a later step, separate from the original CIGS deposition. 2) KF boosted efficiency when incorporated at the end of the third stage of NREL CIGS growth. 3) XPS characterization of CIGS surfaces with and without PDTs led to a proposed mechanism whereby K drives structural transformation at 350 degrees C that is locked in at room temperature even after K is rinsed away. 4) Published recipes for KF and RbF PDTs. Literature to date did not provide enough detail to quickly reproduce experimental results. 5) Identified most important parameters (RbF cell temperature and lamp setpoint temperature) and set boundaries for successful RbF PDTs. The purpose of Task 2: Cell-level Reliability was to overcome the largest challenges to investor confidence and long product lifetime in CIGS-based photovoltaic products: metastability, shading-induced hot spots, and potential-induced degradation (PID). Key findings were made in each of these areas by studying CIGS reliability at the cell level: 1) Published NREL's cell-level reliability testing procedures along with challenges that were encountered while developing them. These were also distributed to the community through an MRS conference presentation. 2) Decreased metastability by adding a CdS hole-injection layer between the CIGS and Zn(O,S) in the device stack. It also improved device performance. Materials other than CdS can be used for the same purpose. 3) Reduced front-glass PID by replacing soda-lime glass with low-Na borosilicate glass. 4) Found that PID depends on leakage current and light/electrical bias. This will help labs avoid test-specific degradation. 5) Discovered that CIGS can suffer from two different types of PID with different mechanisms. Front is slower and leads to shunting ZnO. Back is faster and degrades the p-n junction. 6) Holding cells at open circuit slows PID compared to short circuit. This affects testing protocols for glass/glass modules.

14 SOLAR ENERGY↗
DOE OSTI2020

Long Term Degradation of LSM Based SOFC Cathodes: Use of a Proven Accelerated Test Regimen

The overall goals of the program were: • To understand the microstructural basis of long-term performance loss in solid oxide fuel cells (SOFCs) with cathodes based on lanthanum strontium manganite (LSM, (La 1–x Sr x ) 1–yM nO 3±z ); • To develop strategies for optimizing LSM–based cathodes that achieve improved long-term performance, while exhibiting microstructural and chemical stability, for commercial fuel cell systems. The research project carried out accelerated testing of SOFCs with LSM-based cathodes of selected compositions. After intervals of accelerated testing (and control tests using non-accelerated conditions), the cells underwent extensive microstructural characterization to determine the physical and chemical changes in the cells. By analysis of a) the results of the testing, b) the observed microstructural changes, and c) the differences in these correlations between different cathode compositions and test durations, the research: • Identified a correlation between microstructural change and performance loss: specifically, between decrease of three-phase boundary (TPB) density and increase of area specific re-sistance (ASR) during operation; • Identified a cathode composition parameter — Mn excess, defined as 1/(1–y) in (La 1–x Sr x ) 1–y MnO 3±z ) — and how it affects TPB density, ASR and change of cell performance over time.

08 HYDROGEN↗
DOE OSTI2024

Durable and High-Performance SOECs Based on Proton Conductors for Hydrogen Production

Proton-conducting solid oxide electrolysis cells (P-SOECs) are a promising technology for cost-effective and efficient production of green hydrogen. Breakthroughs in materials development, optimization of cell structure, and achievement of high performance and durability are essential to significantly increase the commercial competitiveness of these technologies. The main objective of this project is to gain scientific knowledge for the rational design, fabrication, and demonstration of a robust, highly efficient, and low-cost SOEC technology based on a proton-conducting electrolyte membrane for hydrogen production. We focused on better understanding the degradation mechanisms of proton-conducting electrolytes, air electrodes, and catalyst materials under electrolysis mode to develop an effective strategy for rationalizing new materials that are vital for enhancing cell performance and durability. The scope includes enhancing the performance and durability of the electrolyte and electrode materials under realistic operating conditions, developing highly active and robust catalysts to minimize electrode losses while improving tolerance to contaminant poisoning, revealing the mechanism of enhanced activity and stability of the catalyst, and understanding the underlying degradation mechanisms. In addition, various characterization techniques were employed to gain a fundamental understanding of the materials’ behavior and their impact on cell performance, providing vital information to guide materials discovery and cell design. After defect chemistry engineering, the optimized donor and acceptor co-doped electrolytes BaMo/W 0.03 Ce 0.71 Yb 0.26 O 3-δ (BM/W03) showed substantially improved chemical stability against high concentrations of CO 2 and H 2 O compared to the state-of-the-art electrolyte (BaZr 0.1 Ce 0.7 Y 0.1 Yb 0.1 O 3-δ , BZCYYb1711) while maintaining comparable ionic conductivity and ionic transference number. To bypass the inherent trade-off between conductivity and chemical stability, we fabricated a bi-layer electrolyte composed of BZCYYb1711 coated with a highly-stable thin layer of BaHf 0.83 Yb 0.17 O 3-δ (BHYb). This bi-layer electrolyte displayed excellent chemical stability against high concentration CO 2 ; there was no detectable formation of BaCO 3 after exposure to 97% CO 2 (with 3% H 2 O) at 500 °C for 1000 hours and the rate of degradation in resistance was about 0.4% per 1,000 hours (kh). In contrast, the same BZCYYb1711 electrolyte without a BHYb coating degraded significantly under the same testing conditions; the degradation rate was increased to 5.1%/kh. In addition, a triple conducting air electrode Ba 0.9 Pr 0.1 Hf 0.1 Y0.1Co 0.8 O 3-δ (BPHYC) was developed by heavily doping transition metal ions into a proton-conducting material. This air electrode material, composed of 3 distinct phases, exhibits superior electrocatalytic activity due to the synergistic effect from the three component phases. Moreover, an active and durable catalyst, La 2 Ni 0.5 Fe 0.5 O 4+δ (LNF), was developed, showing excellent catalytic activity and contaminant tolerance, with a degradation rate of only 0.49%/kh when exposed to high concentrations of steam and Cr. Finally, single cells were constructed from the best electrolytes, electrodes, and catalyst coatings developed in this project. These cells demonstrated superior high current density at a given cell voltage, high roundtrip efficiency, and remarkable durability (up to 1000 hours of operation).

08 HYDROGEN↗
DOE OSTI2026

Relief Zones Enhance the Durability of Ultrathin Membranes in Electrochemical Conversion Devices

Premature failures in electrochemical conversion systems often result when membrane electrode assemblies (MEAs) use ultrathin (≤15 μm-thick) polymer electrolyte membranes, susceptible to mechanical degradation from stress concentrations arising from device-level integration. Herein, relief zones were developed to mitigate mechanical degradation by alleviating excess and nonuniform compression across active areas. Relief zones, created through ablation of carbonaceous diffusion media, enable seamless adaptation across MEA dimensions without need for hardware modifications. Demonstrated using fuel cells as a case study, accelerated stress tests revealed a 6-fold lifetime improvement (∼1500 h) compared to conventional edge-protected MEAs, decoupling device-level engineering effects from material limitations.

accelerated stress test↗
DOE OSTI2025

Field Aging of Photovoltaic Module Packaging Materials: DuraMAT Field Module Library

To understand and develop models for silicon photovoltaic module degradation, accelerated testing is often used, however, outdoor field testing is necessary for validation. Outdoor field testing publications are often limited by the lack of a pristine, control module to compare the fielded module to. In this work, commercially available modules were purchased from seven different manufacturers for outdoor fielding then destructive characterization to investigate packaging material degradation on

14 SOLAR ENERGY↗
DOE OSTI2022

Electrochemical characterization of evolving ionomer/electrocatalyst interactions throughout accelerated stress tests

Here, the degradation of polymer electrolyte membrane fuel cells (PEMFCs) catalyst layers used for heavy-duty vehicles was examined using a catalyst-specific accelerated stress test (AST). High surface area carbon supported dispersed Pt (Pt/HSC), annealed Pt (a-Pt/HSC) and PtCo (PtCo/HSC) alloy catalysts were examined over the course of 90,000 cycles by measuring changes in mass activity, O 2 transport resistance, electrochemical active surface area (ECSA), catalyst accessibility and ionomer-electrocatalyst interactions. Compared to a-Pt/HSC and Pt/HSC catalyst, the PtCo/HSC showed better initial mass activity, a larger initial mass transport loss, and faster degradation after the first 30k AST cycles, as a large portion of Co leached out during potential cycling. Pt/HSC showed higher initial performance relative to a-Pt/HSC but had faster degradation. STEM characterizations show that the ECSA losses are largely related to Pt dissolution resulting in either catalyst particle growth via the Ostwald ripening mechanism or redeposition in the membrane. Catalyst accessibility measurements showed decreased RH sensitivity for all three samples, while CO impedance measurements revealed a decrease in both Pt-water and carbon-water interactions. This implies that, Pt is either preferentially redepositing on the exterior of the carbon support, or that the ionomer is undergoing morphological changes enabling the enhanced intrusion of mesopores.

25 ENERGY STORAGE↗
NASA NTRS1987

Energetic oxygen atom material degradation studies

As part of a study designed to test potential Shuttle surface materials for the extents of degradation and mass loss expected to be suffered in space from the velocity impacts of ambient oxygen atoms, a novel technique was developed for generation of a high flux of energetic oxygen atoms. The generation technique involves laser-induced breakdown of molecular oxygen followed by a rapid expansion of energetic oxygen atoms. The high-velocity streams developed in an evacuated hypersonic nozzle have average O-atom velocities of about 5 to 13 km/s, with an estimated total production of 10 to the 18th atoms per pulse over pulse durations of several microseconds. Results on preliminary material degradation tests conducted with this test facility have been reported by Caledonia et al. (1987). Diagrams of the experimental setup are included.

Caledonia, George E.↗
NASA NTRS1993

Determination of atomic oxygen fluence using spectrophotometric analysis of infrared transparent witness coupons for long duration exposure tests

Atomic oxygen degradation is one of several major threats to the durability of spaceborne systems in low Earth orbit. Ground-based simulations are conducted to learn how to minimize the adverse effects of atomic oxygen exposure. Assessing the fluence of atomic oxygen in test chambers such as a plasma asher over long periods of time is necessary for accurate determination of atomic oxygen exposure. Currently, an atomic oxygen susceptible organic material such as Kapton is placed next to samples as a witness coupon and its mass loss is monitored and used to determine the effective atomic oxygen fluence. However, degradation of the Kapton witness coupons occurs so rapidly in plasma ashers that for any long term test many witness coupons must be used sequentially in order to keep track of the fluence. This necessitates opening vacuum to substitute fresh coupons. A passive dosimetry technique was sought to monitor atomic oxygen exposure over longer periods without the need to open the plasma asher to the atmosphere. This paper investigates the use of spectrophotometric analysis of durable IR transparent witness coupons to measure atomic oxygen exposure for longer duration testing. The method considered would be conductive to making in situ measurements of atomic oxygen fluence.

Podojil, Gregg M.↗
NASA NTRS2020

Cathode & Electromagnet Qualification Status and Power Processing Unit Development Update for the Ascendant Sub-kW Transcelestial Electric Propulsion System

A review of the component-level flight qualification efforts and power processing unit development status of the Ascendant Sub-kW Transcelestial Electric Propulsion System (ASTRAEUS) program is presented. Component-level qualification efforts were undertaken for the system’s ultra-compact heaterless LaB6 hollow cathode and electromagnets, both of which employ designs bespoke to ASTRAEUS, as they represent the highest failure risks for the thruster. Through parallel long-duration wear and ignition tests, the ASTRAEUS cathode demonstrated invariant discharge performance over more than 5000 h of operation at its maximum operating current of 4 A and demonstrated more than 25,000 ignition cycles. The ASTRAEUS electromagnets completed their environmental qualification through a demonstration of more than 1200 deep thermal cycles with no indication of coil degradation (the test articles previously completed qualification-level vibration and shock testing). ASTRAEUS’s prototype power processing unit has demonstrated more than 92% total power conversion efficiency and class-leading power density & specific power density of 4.5 W/cm3 & 1670 W/kg, respectively. The various power converters found in the ASTRAEUS power processing unit are reviewed with a focus on the methods by which such high performance was achieved.

Conversano, Ryan W.↗
NASA NTRS1980

Monitoring land degradation in southern Tunisia: A test of LANDSAT imagery and digital data

The possible use of LANDSAT imagery and digital data for monitoring desertification indicators in Tunisia was studied. Field data were sampled in Tunisia for estimation of mapping accuracy in maps generated through interpretation of LANDSAT false color composites and processing of LANDSAT computer compatible tapes respectively. Temporal change studies were carried out through geometric registration of computer classified windows from 1972 to classified data from 1979. Indications on land degradation were noted in some areas. No important differences, concerning results, between the interpretation approach and the computer processing approach were found.

Hellden, U.↗
NASA NTRS1981

A test and instrumentation system for the investigation of degradation of electrical insulating materials

A comprehensive test apparatus has been developed for the evaluation of the effects of thermal cycling, high temperatures and high electric fields on the quality and stability of electrical insulating materials used in spacecraft systems. The system includes a test chamber in which the specimen can be subjected to thermal and electrical stresses encountered in the space environment (temperatures from -50 to 200 C and voltages from 0 to 10 kV rms, respectively) simultaneously. The system instrumentation includes provisions for the measurement of the change in capacitance, the thermal dissipation factor of the material, and complex dielectric constants with changes in temperature, corona intensity and voltages at frequencies from 22 kHz to 70 MHz, as well as the volume and surface resistivities of the materials.

Doreswamy, C. V.↗
DOE OSTI2024

Investigating the Crosslinking, Degradation, and Adhesion Behavior of Photovoltaic Encapsulants Under Thermal Accelerated Aging

Degradation of photovoltaic (PV) module encapsulant characteristics that lead to mechanical embrittlement and delamination remains a cause of failure in solar installations. A multiscale reliability model connecting the encapsulant mechanical and fracture properties to the degraded molecular structure and interfacial bonding to adjacent solar cell and glass substrates was previously published. The model, developed primarily for poly(ethylene-co-vinyl acetate) acetate (EVA) encapsulants, remains to be experimentally validated. Determining the degradation and crosslinking kinetics of alternative encapsulants, such as polyolefin elastomer (POE) and EVA/POE/EVA composites (EPE), can generalize the model. In this work, we subject fully cured EVA, POE, and EPE encapsulants to accelerated thermal aging to determine how high temperatures impact reaction kinetics. An increase in gel content (crosslinking) and decrease in crystallinity of the encapsulants under hot-aerobic (90 degrees C, 22% RH) and hot-anaerobic (90 degrees C, sealed in N 2 air) aging were observed, even in the absence of UV and crosslinking initiators. Fourier transform infrared spectroscopy (FTIR)-attenuated total reflectance analysis showed insignificant encapsulant degradation, demonstrating the critical role of UV and moisture in accelerating degradation. Adhesion testing performed on coupon-level specimens (cell/encapsulant/glass laminates) showed decreases in adhesion energy, Gc , from 5000 h of hot-dry (90 degrees C, ~1% RH) and hot-humid (90 degrees C, 60% RH) aging. POE coupons demonstrated the best stability, followed by EPE then EVA. For EVA and POE, hot-humid aged coupons experienced a larger decrease in Gc due to enhanced hydrolytic degradation. Hot-dry aging condition demonstrated that thermal degradation of the interface could be significant even if the encapsulant experiences negligible degradation in the absence of UV and elevated humidity.

accelerated aging↗
NASA NTRS1998

Manufacture of porous biodegradable polymer conduits by an extrusion process for guided tissue regeneration

We have fabricated porous, biodegradable tubular conduits for guided tissue regeneration using a combined solvent casting and extrusion technique. The biodegradable polymers used in this study were poly(DL-lactic-co-glycolic acid) (PLGA) and poly(L-lactic acid) (PLLA). A polymer/salt composite was first prepared by a solvent casting process. After drying, the composite was extruded to form a tubular construct. The salt particles in the construct were then leached out leaving a conduit with an open-pore structure. PLGA was studied as a model polymer to analyze the effects of salt weight fraction, salt particle size, and processing temperature on porosity and pore size of the extruded conduits. The porosity and pore size were found to increase with increasing salt weight fraction. Increasing the salt particle size increased the pore diameter but did not affect the porosity. High extrusion temperatures decreased the pore diameter without altering the porosity. Greater decrease in molecular weight was observed for conduits manufactured at higher temperatures. The mechanical properties of both PLGA and PLLA conduits were tested after degradation in vitro for up to 8 weeks. The modulus and failure strength of PLLA conduits were approximately 10 times higher than those of PLGA conduits. Failure strain was similar for both conduits. After degradation for 8 weeks, the molecular weights of the PLGA and PLLA conduits decreased to 38% and 43% of the initial values, respectively. However, both conduits maintained their shape and did not collapse. The PLGA also remained amorphous throughout the time course, while the crystallinity of PLLA increased from 5.2% to 11.5%. The potential of seeding the conduits with cells for transplantation or with biodegradable polymer microparticles for drug delivery was also tested with dyed microspheres. These porous tubular structures hold great promise for the regeneration of tissues which require tubular scaffolds such as peripheral nerve, long bone, intestine, or blood vessel.

NASA Discipline Cell Biology↗
NASA NTRS1987

Degradation mechanisms of sulfur and nitrogen containing compounds during thermal stability testing of model fuels

The degradation behavior of n-dodecane (singly or in combination with S- and N-containing dopants) was studied using a modified Jet Fuel Thermal Oxidation Tester facility between 200 and 400 C. The products were analyzed by gas chromatography and mass spectrometry. The soluble products consisted mainly of n-alkanes and 1-alkenes, aldehydes, tetrahydrofuran derivatives, dodecanol and dodecanone isomers, C21-C24 alkane isomers, and dodecylhydroperoxide (ROOH) decomposition products. The major products were always the same, with and without dopants, but their distributions varied considerably. The 3,4-dimercaptotoluene and dibutylsulfide dopants added individually to n-dodecane interferred with the hydrocarbon oxidation at the alkylperoxy radical and the alkylhydroperoxide link, respectively, while the 2,5-dimethylpyrrole dopant inhibited ROOH formation. Pyridine, pyrrole, and dibenzothiophene added individually showed few significant effects.

Reddy, K. T.↗
DOE OSTI2021

Enhancing Oxygen Reduction Activity and Cr Tolerance of Solid Oxide Fuel Cell Cathodes by a Multiphase Catalyst Coating

Intermediate temperature solid oxide fuel cells (IT-SOFCs) are cost-effective and efficient energy conversion systems. Here, the sluggish oxygen reduction reaction (ORR) and the degradation of cathodes are critical challenges to the commercialization of IT-SOFCs. Here, a highly efficient multiphase (MP) catalyst coating, consisting of Ba 1–x Co 0.7 Fe 0.2 Nb 0.1 O 3–δ (BCFN) and BaCO 3 , to enhance the ORR activity and durability of the state-of-the-art lanthanum strontium cobalt ferrite (La 0.6 Sr 0.4 Co 0.2 Fe 0.8 O 3–δ , LSCF) cathode is reported. The conformal MP catalyst-coated LSCF cathode shows a polarization resistance (R p ) of 0.048 Ω cm 2 at 650 °C, about one order of magnitude smaller than that of the bare LSCF. In an accelerated Cr-poisoning test, the degradation rate of the catalyst-coated LSCF electrode is 10 –3 Ω cm 2 h –1 (0.59% h –1 ) over 200 h, only one fifth of the degradation rate of the bare LSCF electrode at 750 °C. In addition, anode-supported single cells with the MP catalyst-coated LSCF cathode show a dramatically enhanced peak power density (1.4 W cm –2 vs 0.67 W cm –2 at 750 °C) and increased durability against Cr and H 2 O. Both experimental results and density functional theory-based calculations indicate that the BCFN phase improves the ORR activity while the BaCO 3 phase enhances the stability of the LSCF cathode.

solid oxide fuel cells↗
NASA NTRS1987

Environmental degradation of 316 stainless steel in high temperature low cycle fatigue

Procedures based on modification of the conventional Strainrange Partitioning method are proposed to characterize the time-dependent degradation of engineering alloys in high-temperature, low-cycle fatigue. Creep-fatigue experiments were conducted in air using different waveforms of loading on 316 stainless steel at 816 C (1500 F) to determine the effect of exposure time on cyclic life. Reductions in the partitioned cyclic lives were observed with an increase in the time of exposure (or with the corresponding decrease in the steady-state creep rate) for all the waveforms involving creep strain. Excellent correlations of the experimental data were obtained by modifying the Conventional Strainrange Partitioning life relationships involving creep strain using a power-law term of either: (1) time of exposure, or (2) steady-state creep rate of the creep-fatigue test. Environmental degradation due to oxidation, material degradation due to the precipitation of carbides along the grain boundaries and detrimental deformation modes associated with the prolonged periods of creep were observed to be the main mechanisms responsible for life reductions at long exposure times.

Kalluri, Sreeramesh↗
DOE OSTI2021

DOE-GENIORS Dynamic Irradiation Testing

As part of the homogeneous actinides recycling strategy, the EURO-GANEX process is one of the most promising options to achieve the goal of minor actinides recovery. Improvements made to EURO-GANEX system have resulted in the emergence of the so-called New EURO-GANEX process, where the composition of the solvent has been modified by replacing TODGA and DMDOHEMA with cis-mTDDGA in the organic phase and SO3-Ph-BTP with PyTri-Diol in the aqueous phase in order to resolve important issues. The objective of this work is twofold: evaluate the gamma radiolytic resistance of the new EURO-GANEX process by dynamic irradiation conditions simulating the three main steps of the process, and validate the design of CIEMAT Náyade, CEA Marcel and INL irradiation loop devices since each of them mimics different aspects of the real process. Náyade and the INL loops could irradiate the organic and aqueous phases together, whereas in the CEA loop, the irradiated solvent is recycled continuously inside a platform with several stages of mixer-settlers containing aqueous flows simulating the three main steps of the process. The extraction performances and changes in the composition of the solvent have been analyzed during the irradiation experiment by different techniques: gamma spectrometry and ICP-MS/OES for cations or radioactive tracer extraction, and HPLC-MS to identify and quantify the degradation compounds. Despite some differences between the three-irradiation facilities, this inter-institutional study shows that these three comparative tools provide similar trends in the radiolytic stability of a liquid-liquid extraction system. Favourable extraction results for the different steps are obtained according to the static irradiation studies found in literature. However, the degradation of cis-mTDDGA is appreciable leading to degradation compounds, some of which form precipitates and produce important changes in viscosity, important aspects that must be addressed prior to the successful industrial application of the new EURO-GANEX process.

38 - RADIATION CHEMISTRY, RADIOCHEMISTRY, AND NUCL↗
DOE OSTI2024

Highly accelerated life testing (HALT): A review from a statistical perspective

Despite its use in one form or another for at least four decades, HALT and related techniques [e.g., highly accelerated-stress screening (HASS) and stress audits (HASA)] are not well understood within the statistical community and remain controversial. This largely reflects a conflict in motivation between engineers, testing under harsh conditions to discover and eliminate failure modes, and statisticians, taking a more cautious approach to develop quantitative estimates of parameters such as mean time between failures (MTBF). Here, this review article will clarify HALT concepts and methods and explain where it fits within the universe of methods that involve the application of accelerating factors to compress the time required to evaluate or enhance product reliability. A major distinction is between methods such as HALT, a high-stress test-analyze-fix-test iterative process directed at improving reliability by discovering and fixing weak points in a design, and quantitative accelerated life testing (QALT), whose goal is the estimation of product life for a fixed design. We discuss methods such as physics of failure that offer some hope of bridging the gap between the qualitative nature of HALT, and purely quantitative statistical methods. We present a variety of engineering applications of HALT including metal fatigue, piping and pressure vessels, structural damage, radiation damage, and rotating machinery. We also discuss potential synergies between HALT and QALT, such as rapid identification, through HALT, of failure modes requiring quantitative analysis. For further study, extensive references to the applicable literature are provided as well as an appendix that describes related methods.

97 MATHEMATICS AND COMPUTING↗