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Reliability of Digital Communications in Nuclear Facilities and Operations

A comprehensive evaluation of methods to compare the reliability of wired and wireless digital communication networks for use in nuclear power plants. The study underscores the critical role of communication reliability in ensuring operational safety in nuclear power plants. Key objectives include developing a methodology to assess communication technologies’ reliability, particularly for comparing wireless and wired networks, for nuclear facility applications. The framework focuses on technology-agnostic evaluations and emphasizes the importance of reliability metrics spanning safety, security, and monitoring functions. The framework emphasizes that the key performance indicators are application dependent and provides a hierarchy of different network types that will have different reliability requirements. The report considers both existing plants and advanced reactors, including small modular reactors.

22 GENERAL STUDIES OF NUCLEAR REACTORS

Reliability, biological variability, and accuracy of multi-frequency bioelectrical impedance analysis for measuring body composition components

Introduction Bioelectrical impedance analysis (BIA) systems are gaining popularity for use in research and fitness assessments as the technology improves and becomes more affordable and easier to use. Multifrequency BIA (MF-BIA) may improve accuracy and precision using octopolar contacts for segmental analyses. Purpose Evaluate reliability, biological variability, and accuracy of component measures (total body water, mass, and composition) of commercially available MF-BIA system (InBody 770, Cerritos, California, USA). Methods Fourteen healthy military-age adults were assessed by MF-BIA in duplicate on five laboratory visits across 3 weeks (10 measures each). Participants were evaluated at the same time of day after refraining from strenuous exercise (> 48 h), alcohol consumption (> 24 h), and caffeine, nicotine, and food (> 10 h). Systematic error (test–retest reliability) and biological variability (day-to-day reliability) were summarized by intraclass correlation coefficient (ICC) values determined for body mass (fat, fat-free, total) and body water (extracellular, intracellular, total). Body composition measurements derived from BIA on the second visit were also tested for accuracy compared to dual-energy x-ray absorptiometry (DXA). Results Test–retest reliability was very high for all measurements of whole-body water and mass (ICC ≥ 0.999) and high for regional body water and mass (ICC 0.973–1.000). Biological variability was observable with very minor differences between tests (same day) for total and regional body water (0.0–0.2 L) and total and regional body mass measurements (0.0–0.2 kg); while between day differences were slightly higher (0.0–0.5 L and 0.1–0.7 kg). Compared to DXA, the MF-BIA whole-body measures showed an offset in %BF (Bias −4.0 ± 2.8%; Standard error of the estimate (SEE), 2.6%), an overprediction for total body fat-free mass (Bias 2.8 ± 2.1 kg; SEE 2.2 kg) and an underprediction of total body fat mass (Bias −2.9 ± 2.0 kg; SEE 1.9 kg). Conclusion Under controlled conditions with fit and healthy men and women, this MF-BIA system has high methodological reliability and demonstrates stable day-to-day measurements of major body composition components. Previously reported ~3% body fat offset compared to criterion methods was again confirmed. Precision of the InBody 770 shows consistency and supports further testing of this specific device as a new military standards method and suitability across a wider range of %BF.

Nutrition & Dietetics

Time Distribution Analysis for Task Primitives to Support Dynamic Human Reliability Analysis

To support data collection for dynamic human reliability analysis (HRA), this study investigates time distributions for task primitives defined in the Goals, Operators, Methods, and Selection rules (GOMS)–Human Reliability Analysis (HRA) method and Human Reliability data EXtraction (HuREX). GOMS-HRA was developed to provide cognition-based time and human error probability (HEP) information for dynamic HRA calculations within the Human Unimodel for Nuclear Technology to Enhance Reliability (HUNTER) framework, while HuREX is a comprehensive HRA data collection method developed by the Korea Atomic Energy Research Institute (KAERI). In this paper, we examine time distributions by using experimental data collected from the Simplified Human Error Experimental Program (SHEEP) study, which proposes an HRA data collection framework to complement full-scope simulator research and gather input data for dynamic HRA by using simplified simulators such as the Rancor Microworld simulator. This paper investigates whether the time required for GOMS-HRA and HuREX task primitives fits 13 statistical distributions. Additionally, we compare and discuss the time distributions obtained from both student operators and professional operators. The result was that this study identified several time distributions for five GOMS-HRA and four HuREX task primitives. In the future, the results of this study are expected to provide objective reference data on the elapsed time for task primitives and aid in realistically simulating scenarios within dynamic HRA.

Dynamic Human Reliability Analysis

Evaluating grid stress and reliability in future electricity grids across a range of demand, generation mix, and weather trends

The reliability of power grids in the future will depend on how system planners account for the integration of new technologies, extreme weather events, and uncertainties in demand growth from increased electrification and data centers. This study introduces an open-source, multisectoral, multiscale modeling framework that projects grid stress and reliability trends between 2020 and 2055 in the Western Interconnection of the United States. The framework integrates global to national energy-water-land dynamics with power plant siting and hourly grid operations modeling. We analyze future wholesale electricity price shocks and unserved energy events across eight scenarios spanning a range of population growth and economic change, generation mixes, and weather conditions. Our results show future grids with high percentage of non-renewable generation and strong economic growth are characterized by higher reliability and lower wholesale electricity prices than lower growth scenarios because of larger reliance on dispatchable generators and lower fossil fuel extraction costs. Scenarios with high percentage of renewable resources have lower median but more volatile wholesale electricity prices as well as more frequent and severe unserved energy events compared to scenarios relying more on dispatchable generators. These events occur because higher proportion of solar and wind energy causes net demand curves to deepen during midday (duck curves get progressively severe), exacerbating the challenge of meeting demand during summer evening peaks. This study suggests that robust and co-optimized transmission and energy storage planning could help maintain low wholesale electricity prices and high reliability levels in future electricity grids across uncertainties in generation mixes.

Electric grid reliability

Assessing the Reliability Benefits of Energy Storage as a Transmission Asset

Utilizing energy storage solutions to reduce the need for traditional transmission investments has been recognized by system planners and supported by federal policies in recent years. This work demonstrates the need for detailed reliability assessment for quantitative comparison of the reliability benefits of energy storage and traditional transmission investments. First, a mixed-integer linear programming expansion planning model considering candidate transmission lines and storage technologies is solved to find the least-cost investment decisions. Next, operations under the resulting system configuration are simulated in a probabilistic reliability assessment which accounts for weather-dependent forced outages. The outcome of this work, when applied to TPPs, is to further equalize the consideration of energy storage compared to traditional transmission assets by capturing the value of storage for system reliability.

co-optimization

Inferring Reliability Model Parameters from Expert Opinion

Here, we propose a method for constructing bathtub models of reliability from opinion. The method is intended for reliability studies early in the design and prototyping of a new system, before data concerning reliability has become available. A stylized bathtub curve is presented for soliciting best engineering judgement from technical experts. By pooling these stylized curves, we produce data that can be used to infer parameters for a piece-wise Weibull model of reliability. A numerical example demonstrates the practicality of the method while also highlighting potential pitfalls when working with subjective data.

42 ENGINEERING

ACCELERATED DEPLOYMENT OF NOVEL MATERIALS BASED ON RELIABILITY INTEGRITY MANAGEMENT USING CUMULATIVE DAMAGE MODELING

There is currently no widely agreed, detailed general method for licensing a novel plant incorporating novel materials (or materials being deployed in novel environments); in many such situations, there are no directly applicable engineering code cases for decision-makers (including regulators) to rely on. This paper discusses a framework for solving this problem that is based on the Reliability and Integrity Management (RIM) approach delineated in ASME BPVC Section XI Division 2. NRC Regulatory Guide 1.246, Rev. 0, endorses, with conditions, the subject portion of the 2019 ASME Code. The proposed framework is meant to support development of a licensing case by addressing certain remaining technical challenges. The framework discussed here is compatible with the Licensing Modernization Project, but applying it in a specific case will call for advances in the state of practice, if not the state of the art. The RIM approach calls for applicants to (a) allocate reliability targets to plant structures, systems, and components (SSCs), (b) show that they are able to relate the currently observed physical condition of each SSC in the program to its failure probability well enough to determine whether the target reliability allocations are being satisfied, allowing for uncertainty related to the novelty of the materials/designs/operating environments, and (c) be able to demonstrate that the proposed program of surveillances will reliably detect unacceptable degradation of an SSC before SSC failure occurs. A modeling approach potentially applicable to item (b), based on cumulative damage modeling rather than failure rates, is briefly illustrated.

22 - GENERAL STUDIES OF NUCLEAR REACTORS

What Technical Choices Matter to Characterize Heat Wave and Cold Snap Events in Support of Bulk Power Grid Reliability Studies?

Extreme weather events, such as Heat Waves (HW) and Cold Snaps (CS), pose significant risks to the power grid. The United States (U.S.) Federal Energy Regulatory Commission Order No. 896 mandates regional coordination standards that account for extreme thermal events. However, the lack of a universal definition for extreme thermal events may lead to inconsistent compliance efforts among neighboring entities, undermining the reliability of the transmission system. This study directly addresses this challenge by systematically evaluating how varying technical choices in defining HW and CS fundamentally impact the characterization and ranking of extreme events for power grid reliability studies. We used 12 event definitions and multiple temperature spatial aggregation approaches to construct historical (1980–2024) regional extreme thermal event libraries across North American Electric Reliability Corporation (NERC) subregions in the conterminous U.S. We examined the sensitivity of event characteristics (e.g., duration, frequency, intensity, and spatial coverage) to different definitions. While some definitions produced similar libraries and top event rankings, definitions based on moving-window-averaged temperatures yielded markedly different characteristics. Spatial aggregation methods had minimal impact on heat wave or cold snap intensity, frequency and duration but significantly influenced spatial coverage. The top events identified across different aggregation methods were consistent, but their ranking order varied. These findings offer critical insights for characterizing and selecting extreme thermal events and for supporting local and cross-regional coordination as required by reliability standards.

Wan, Heng [Pacific Northwest National Laboratory (

Pool Boiling Reliability Tests and Degradation Mechanisms of Microporous Copper Inverse Opal (CuIOs) Structures

The rising power density in electronic systems requires thermal management solutions that are both high-performing and reliable. Porous materials such as Copper Inverse Opal (CuIOs) have unique structural features, including high permeability and high thermal conductivity, to enhance pool boiling performance. However, there is little understanding of the degradation mechanism of such porous materials under pool boiling conditions. In this study, samples of 10 ..mu..m thick CuIOs with 4.8 ..mu..m diameter, covering silicon substrate of area 11 mm x 11 mm, with various heated areas ranging from 2.5 mm x 2.5 mm to 10 mm x 10 mm, were tested in 100 degrees C deionized water at a constant heat flux of 110 Wcm -2 for 3-to-7 days. The combined effect of erosion and corrosion caused structural degradation of the CuIOs. The directly heated area had the most severe degradation while the edge of the heater and the unheated area showed progressively less degradation, maintaining some CuIOs structure even after the 7-day reliability test. Among all the tested samples with various heater sizes, the 2.5 mm x 2.5 mm heater sample - in which the heater size was designed to be comparable to the water bubble characteristic length - had the largest critical heat flux (CHF) up to 300 Wcm -2 with a superheat ~ 13 degrees C. Additionally, CuIOs with a smaller heated area performed better in terms of reliability. This study offers preliminary insights into CuIOs degradation mechanisms, contributing to the development of more robust thermal management solutions. We expect that electroless plating of CuIOs with gold (Au), nickel (Ni), and atomic layer deposition (ALD) aluminum oxide (Al 2 O 3 ) in combination with appropriate application-specific coolants will further improve the reliability and lifetime of the CuIOs.

boiling-induced degradation

Day-to-day reliability of basal heart rate and short-term and ultra short-term heart rate variability assessment by the Equivital eq02+ LifeMonitor in US Army soldiers

Introduction The present study determined the (1) day-to-day reliability of basal heart rate (HR) and HR variability (HRV) measured by the Equivital eq02+ LifeMonitor and (2) agreement of ultra short-term HRV compared with short-term HRV. Methods Twenty-three active-duty US Army Soldiers (5 females, 18 males) completed two experimental visits separated by >48 hours with restrictions consistent with basal monitoring (eg, exercise, dietary), with measurements after supine rest at minutes 20–21 (ultra short-term) and minutes 20–25 (short-term). HRV was assessed as the SD of R–R intervals (SDNN) and the square root of the mean squared differences between consecutive R–R intervals (RMSSD). Results The day-to-day reliability (intraclass correlation coefficient (ICC)) using linear-mixed model approach was good for HR (0.849, 95% CI: 0.689 to 0.933) and RMSSD (ICC: 0.823, 95% CI: 0.623 to 0.920). SDNN had moderate day-to-day reliability with greater variation (ICC: 0.689, 95% CI: 0.428 to 0.858). The reliability of RMSSD was slightly improved when considering the effect of respiration (ICC: 0.821, 95% CI: 0.672 to 0.944). There was no bias for HR measured for 1 min versus 5 min (p=0.511). For 1 min measurements versus 5 min, there was a very modest mean bias of −4 ms for SDNN and −1 ms for RMSSD (p≤0.023). Conclusion When preceded by a 20 min stabilisation period using restrictions consistent with basal monitoring and measuring respiration, military personnel can rely on the eq02+ for basal HR and RMSSD monitoring but should be more cautious using SDNN. These data also support using ultra short-term measurements when following these procedures.

General & Internal Medicine

Transmission Operator Workflows for Real-Time Reliability Studies: A Review of Control Room Practices and Naturalistic Decision Making

This report provides an overview of real-time reliability study tools and their use by power system operators in the control room environment. After introducing some of the nuances of the control room environment and the differences in perspectives between power system engineers and operators, the roles and responsibilities of key entities involved in RTCA workflows are introduced. These are specifically the transmission system operator (TOP) and reliability coordinator (RC), which are required to run tools such as real-time contingency analysis (RTCA) as part of a real-time reliability assessment every 30 minutes, as dictated by a series of standards issued by the North American Electric Reliability Corporation (NERC). The process by which power systems operators operate the grid is discussed in terms of naturalistic decision making (NDM) and the recognition-primed decision-making (RPD) model. This cognitive model describe how experts working in high-risk, high-stress environments make safety-critical decisions under uncertainty and time pressure. For power system operators, the mental simulations involved in the traditional RPD model are supplemented by physics-based simulations using numerical tools, such as RTCA, to improve situational awareness and effectiveness of control actions. Next, a generic workflow is introduced to describe operator decision making for running RTCA tools and responding to system violations on a pre-contingent basis. The types of analysis performed and control actions chosen by power system operators are described in detail. The overall high-level workflow is then expanded in subsequent sections, with special attention given to high-voltage violations, low-voltage violations, and thermal overloads. Each type of violation is described in detail, with explanations of common causes, impacts on equipment and customers, and mitigation strategies. An additional workflow diagram is provided for each type of violation.

24 POWER TRANSMISSION AND DISTRIBUTION

Reliable Determination of Pulses and Pulse-Shape Instability in Ultrashort Laser Pulse Trains Using Polarization-Gating and Transient-Grating Frequency-Resolved Optical Gating Using the RANA Approach

Devices that measure the presence of instability in the pulse shapes in trains of ultrashort laser pulses do not exist, so this task necessarily falls to pulse-measurement devices, like Frequency-Resolved Optical Gating (FROG) and its variations, which have proven to be a highly reliable class of techniques for measuring stable trains of ultrashort laser pulses. Fortunately, multi-shot versions of FROG have also been shown to sensitively distinguish trains of stable from those of unstable pulse shapes by displaying readily visible systematic discrepancies between the measured and retrieved traces in the presence of unstable pulse trains. However, the effects of pulse-shape instability and algorithm stagnation can be indistinguishable, so a never-stagnating algorithm—even when instability is present—is required and is generally important. In previous work, we demonstrated that our recently introduced Retrieved-Amplitude N-grid Algorithmic (RANA) approach produces highly reliable (100%) pulse-retrieval in the second-harmonic-generation (SHG) version of FROG for thousands of sample trains of pulses with stable pulse shapes. Further, it does so even for trains of unstable pulse shapes and thus both reliably distinguishes between the two cases and provides a rough measure of the degree of instability as well as a reasonable estimate of most typical pulse parameters. Here, we perform the analogous study for the polarization-gating (PG) and transient-grating (TG) versions of FROG, which are often used for higher-energy pulse trains. We conclude that PG and TG FROG, coupled with the RANA approach, also provide reliable indicators of pulse-shape instability. In addition, for PG and TG FROG, the RANA approach provides an even better estimate of a typical pulse in an unstable pulse train than SHG FROG does, even in cases of significant pulse-shape instability.

47 OTHER INSTRUMENTATION

Pool Boiling Reliability Tests and Degradation Mechanisms of Microporous Copper Inverse Opal (CuIOs) Structures: Preprint

The rising power density in electronic systems requires thermal management solutions that are both high-performing and reliable. Porous materials such as Copper Inverse Opals (CuIOs) have unique structural features, including high permeability and high thermal conductivity, to enhance pool boiling performance. However, there is little understanding of the degradation mechanism of such porous materials under pool boiling conditions. In this study, samples of 10-micrometer-thick CuIOs with 4.8-micrometer diameter, covering silicon substrate of area 11-mm by 11-mm, with various heated areas ranging from 2.5-mm by 2.5-mm to 10-mm by 10-mm, were tested in 100 degrees C deionized water at a constant heat flux of 110 watts per square cm for 3-7 days. The combined effect of erosion and corrosion caused structural degradation of the CuIOs. The directly heated area had the most severe degradation while the edge of the heater and the unheated area showed progressively less degradation, maintaining some CuIOs structure even after the 7-day reliability test. Among all the tested samples with various heater sizes, the 2.5-mm by 2.5-mm heater sample - in which the heater size was designed to be comparable to the water bubble characteristic length - had the largest critical heat flux (CHF) up to 300 watts per square cm with a superheat of approximately 13 degrees C. Additionally, CuIOs with a smaller heated area performed better in terms of reliability. This study offers preliminary insights into CuIOs degradation mechanisms, contributing to the development of more robust thermal management solutions. We expect that electroless plating of CuIOs with gold (Au), nickel (Ni), and atomic layer deposition of aluminum oxide in combination with appropriate application-specific coolants will further improve the reliability and lifetime of the CuIOs.

boiling-induced degradation

Risk-informed Graded Approach for Reliability and Performance Assessment of Sensor and Instrumentation Systems within Advanced Condition Monitoring Technologies

Advanced condition monitoring (ACM) technologies, such as digital twins, are innovative strategies designed to provide real-time health insights, including the remaining useful life of components. The primary goal of ACM is to predict and alert operators to potential functional failures before they occur. ACM systems achieve this by integrating predictive models with various sensor instrumentation, analog-to-digital converters, data warehouses, and data pre-processors. These sensor and instrumentation systems (SIS) are essential for forming a comprehensive understanding of component conditions and ensuring the predictive success of ACM programs. Introducing new technologies like ACM involves varying degrees of risk that can impact plant reliability. Therefore, risk mitigation should be commensurate with the performance and reliability of the developed technology, following a risk-informed graded approach (RIGA). Establishing a RIGA process requires a clear understanding of the hazards and reliability of all subsystems, including their interdependencies and potential impacts on the overall system. Given the critical role of SIS in ACM, this work reviews hazard identification and reliability quantification methods for SIS. It also considers these methods' implications when developing a RIGA process for ACM.

22 - GENERAL STUDIES OF NUCLEAR REACTORS

Risk-informed Graded Approach for Reliability and Performance Assessment of Machine Learning and Artificial Intelligence for Advanced Condition Monitoring Techniques

With the shift away from time-based maintenance and toward condition-based maintenance, and to reduce overall maintenance costs, there has been an upsurge in the usage and development of advanced condition monitoring (ACM) techniques for real-time monitoring of nuclear power plant (NPP) components. ACM is particularly useful in the development of digital twins, which are designed to predict the failure or degradation of plant components. Successful implementation of ACM requires an assessment to inform the development of a risk-informed approach to evaluate the use of ACM to meet Nuclear Regulatory Committee (NRC) regulations for in-service testing (IST) programs. This includes the monitoring and diagnostics of reactor components and systems in current, new, and advanced reactors. A key component in ACM is the usage of machine learning (ML) and artificial intelligence (AI) algorithms that can employ real-time data from instrumentation and sensors to detect and predict reactor component degradations. Such predictive capabilities enable early detection of component degradation so as to help plant personnel plan and execute necessary maintenance. For successful implementation of ML/AI in ACM such that regulatory requirements are met, a risk-informed graded approach is needed to assess the reliability and performance of ML/AI for ACM. The American Society for Mechanical Engineers (ASME) developed their Operations and Maintenance (O&M) Code to provide guidance on safe, reliable O&M of NPPs. The IST section of the O&M Code specifically establishes requirements for IST and examination to gauge operational readiness of components in water-cooled NPPs. This paper presents a state-of-the-art review of how reliability and risk assessment can be integrated with ACM to assess component performance by non-nuclear industries. This is followed by different methodologies and approaches for conducting performance and reliability assessments so as to meet IST requirements for NPP components.

97 - MATHEMATICS AND COMPUTING

Risk-informed Graded Approach for Reliability and Performance Assessment for Advanced Condition Monitoring Techniques

With the shift away from time-based maintenance and toward condition-based maintenance, and to reduce overall maintenance costs, there has been an upsurge in the usage and development of advanced condition monitoring (ACM) techniques for real-time monitoring of nuclear power plant (NPP) components. ACM is particularly useful in the development of digital twins, which are designed to predict the failure or degradation of plant components. Successful implementation of ACM requires an assessment to inform the development of a risk-informed approach to evaluate the use of ACM to meet Nuclear Regulatory Committee (NRC) regulations for in-service testing (IST) programs. This includes the monitoring and diagnostics of reactor components and systems in current, new, and advanced reactors. A key component in ACM is the usage of machine learning (ML) and artificial intelligence (AI) algorithms that can employ real-time data from instrumentation and sensors to detect and predict reactor component degradations. Such predictive capabilities enable early detection of component degradation so as to help plant personnel plan and execute necessary maintenance. For successful implementation of ML/AI in ACM such that regulatory requirements are met, a risk-informed graded approach is needed to assess the reliability and performance of ML/AI for ACM. The American Society for Mechanical Engineers (ASME) developed their Operations and Maintenance (O&M) Code to provide guidance on safe, reliable O&M of NPPs. The IST section of the O&M Code specifically establishes requirements for IST and examination to gauge operational readiness of components in water-cooled NPPs. This paper presents a state-of-the-art review of how reliability and risk assessment can be integrated with ACM to assess component performance by non-nuclear industries. This is followed by different methodologies and approaches for conducting performance and reliability assessments so as to meet IST requirements for NPP components.

99 - GENERAL AND MISCELLANEOUS

Portable and Cost-Effective Device for Reliable Detection of Counterfeit and Non-compliant Refrigerants in Diverse Applications

Counterfeit refrigerants pose significant challenges to safety, system reliability, and operational effectiveness due to their harmful contaminants or incompatible chemical compositions. Utilizing these noncompliant products can lead to reduced efficiency, equipment failures, and expensive repairs. Additionally, heightened demand for alternative refrigerants during the industry's transition has created supply gaps, enabling counterfeit products to proliferate. Accurate detection and analysis tools are therefore essential to verify refrigerant authenticity and ensure system integrity in diverse applications. This paper presents the development of a portable device designed for reliable identification and detailed analysis of refrigerant composition. By integrating precision gas sampling, controlled pressure regulation, and automated sensor technology, the device not only detects deviations from standard refrigerant properties but also provides a comprehensive composition breakdown. Pre-calibrated sensors measure the refrigerant gas to identify specific concentrations and contaminants, with an intuitive LED-based indicator system ensuring quick interpretation of results. The user-friendly interface enables operators to select refrigerant types for targeted testing, further enhancing accuracy and usability for field technicians. Comprehensive testing was conducted on mildly flammable A2L refrigerants, showcasing the device’s robustness and adaptability in analyzing composition and detecting discrepancies. The device demonstrated consistent accuracy across a range of refrigerant samples, affirming its reliability in diverse operational environments. Its design minimizes contamination risks during sampling and provides detailed composition results within 90 seconds, ensuring efficient and precise analysis. With a projected price point under $150, the proposed solution delivers affordability alongside its lightweight portability and straightforward operation. Unlike complex and costly alternatives, such as gas chromatography systems, this device provides an accessible option for technicians, customs personnel, and industry operators in need of quick and effective refrigerant verification. Compatible with both current formulations and emerging refrigerant technologies, the device addresses critical counterfeit detection needs across a range of applications. By delivering accurate composition analysis and counterfeit identification, this innovation enhances system performance, safety, and operational reliability in crucial industries.

Cheekatamarla, Praveen [ORNL] (ORCID:0000000248827

A note on the reliability of goal-oriented error estimates for Galerkin finite element methods with nonlinear functionals

Here, we consider estimating the discretization error in a nonlinear functional J (u) in the setting of an abstract variational problem: find u ϵ $\mathscr{V}$ such that B (u, φ) = L (φ) ∀φ ϵ $\mathscr{V}$, as approximated by a Galerkin finite element method. Here, $\mathscr{V}$ is a Hilbert space, B (. , .) is a bilinear form, and L (∙) is a linear functional. We consider well-known error estimates η of the form J (u) - J (u h ) ≈ η = L (z) - B (u h , z), where u h denotes a finite element approximation to u, and z denotes the solution to an auxiliary adjoint variational problem. We show that there exist nonlinear functionals for which error estimates of this form are not reliable, even in the presence of an exact adjoint solution z. An estimate η is said to be reliable if there exists a constant C ϵ $\mathbb{R}$ >0 independent of u h such that |J (u) - J (u h )| ≤ C|η|. We present several example pairs of bilinear forms and nonlinear functionals where reliability of η is not achieved.

A posteriori