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38 pages, 2692 KB  
Article
Observability- and Identifiability-Guided Sensor-Set Design for Digital-Twin-Assisted Consolidated Bioprocessing
by Mark Korang Yeboah, Nana Yaw Asiedu and Ahmad Addo
Sensors 2026, 26(12), 3948; https://doi.org/10.3390/s26123948 (registering DOI) - 21 Jun 2026
Abstract
Consolidated bioprocessing (CBP) is difficult to monitor because enzyme production, lignocellulose degradation, sugar release, and fermentation occur simultaneously under sparse measurement, feedstock variability, and plant–model mismatch conditions. This study proposes a computational sensor-set design framework for digital-twin-assisted CBP monitoring. A five-state virtual plant, [...] Read more.
Consolidated bioprocessing (CBP) is difficult to monitor because enzyme production, lignocellulose degradation, sugar release, and fermentation occur simultaneously under sparse measurement, feedstock variability, and plant–model mismatch conditions. This study proposes a computational sensor-set design framework for digital-twin-assisted CBP monitoring. A five-state virtual plant, consisting of active biomass, cellulolytic enzyme activity, residual insoluble substrate, soluble sugar, and ethanol, was used to evaluate all 16 ethanol-mandatory measurement packages formed from ethanol, sugar, biomass, enzyme, and residual-substrate proxy channels. Candidate sensor sets were assessed using finite-difference output sensitivities, Fisher-information-based state-observability and parameter-identifiability analyses, eigenvalue and parameter-correlation diagnostics, and paired Monte Carlo unscented Kalman filter soft-sensing reconstruction. Within the tested five-state virtual-plant benchmark and with the specified excitation schedule, noise assumptions, burden indices, and scoring objective, ethanol-only sensing provided the weakest support for state-aware CBP digital-twin reconstruction. At a 6h sampling interval, the state-observability log-pseudodeterminant increased from 4.18 with ethanol-only sensing to 8.56 after adding soluble sugar and to 16.42 with full-proxy monitoring. The ethanol–sugar–biomass–substrate package also gave strong reduced state-observability performance, with log-pseudodeterminants of 15.12, 13.76, and 12.51 at 6, 12, and 24h, respectively. Biomass and enzyme proxies contributed strongly to parameter learning, and the ethanol–sugar–biomass–enzyme package gave the strongest active parameter-identifiability performance, with log-pseudodeterminants of 10.82, 9.06, and 6.67 at 6, 12, and 24h, respectively. In the paired soft-sensing analysis, full-proxy monitoring reduced the mean latent-state RMSE from 1.1899 to 0.3756, followed by ethanol–biomass–enzyme–substrate with 0.3843 and ethanol–sugar–biomass–substrate with 0.4121. The primary aggregate ranking identified ethanol–sugar–biomass–substrate as the best overall package, with a sensor-value score of 0.8432 and a burden index of 7.0, followed by full-proxy monitoring with a score of 0.8173 and a burden index of 10.0. Robustness tests showed that ethanol–sugar–biomass–substrate remained top-ranked under uniform noise scaling, full UKF missingness, delay and bias stress test conditions, most scoring-weight scenarios, and all tested sensor-specific burden workflows. Full-proxy monitoring remained a close competitor under independent sensor-specific noise variation conditions and became top-ranked for some alternative operating trajectories. The proposed framework provides a simulation-based method for prioritizing informative measurement packages before implementing CBP digital twins in laboratory and pilot-plant settings. Full article
(This article belongs to the Special Issue Soft Sensors and Sensing Techniques (2nd Edition))
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26 pages, 30333 KB  
Article
Interpretable Attribution of Sentinel-1/2 and Environmental Covariates for Compositionally Closed Soil Mapping and Uncertainty Quantification
by Wenhao Wang, Chao Dong, Bin Zhao, Yanling Li, Zhuoran Wang and Chunyan Chang
Remote Sens. 2026, 18(12), 2051; https://doi.org/10.3390/rs18122051 (registering DOI) - 21 Jun 2026
Abstract
Soil particle size fractions (PSFs)—sand, silt, and clay—are fundamental determinants of soil hydrological behavior, nutrient retention, and erodibility, yet their spatial prediction remains challenging due to the compositional nature of the data, unquantified prediction uncertainty, and limited interpretability of machine learning models. This [...] Read more.
Soil particle size fractions (PSFs)—sand, silt, and clay—are fundamental determinants of soil hydrological behavior, nutrient retention, and erodibility, yet their spatial prediction remains challenging due to the compositional nature of the data, unquantified prediction uncertainty, and limited interpretability of machine learning models. This study develops an integrated compositional mapping framework incorporating multi-source Sentinel-1/2 and topographic covariates, coupling the isometric log-ratio (ILR) transformation with Quantile Regression Forests (QRFs), a Monte Carlo simulation (MCS)-based latent-to-physical space uncertainty propagation strategy, and a Wrapper-SHAP attribution method to jointly address these challenges. The framework was evaluated across regional croplands in the central Shandong mountain-hilly region of China, using an elevation-stratified spatial cross-validation. Validations achieved R2 values of 0.72, 0.61, and 0.59 for sand, silt, and clay, respectively, and a global Aitchison distance of 0.34. Critically, the MCS error propagation strategy effectively compensated for the probability distribution shift introduced by non-linear ILR back-transformation. This ensured that all predicted compositions strictly satisfied compositional closure and the [0, 100%] constraint, while aligning the prediction interval coverage probability (PICP) of each fraction closely with the 90% nominal level. Wrapper-SHAP overcame direct attribution limitations in compositional models, revealing the predictive associations of these multi-source covariates: high remote sensing-derived Bare Soil Index (BSI) and Moisture Stress Index (MSI) values primarily exhibited strong predictive associations with sand enrichment, whereas their lower values, combined with elevated Normalized Difference Moisture Index (NDMI), Enhanced Vegetation Index (EVI), and anthropogenic indicators, favored silt and clay accumulation. The proposed framework provides a transferable methodological reference for remote sensing-integrated compositional soil mapping with reliable uncertainty estimates and interpretable driver identification at regional scales. Full article
14 pages, 2041 KB  
Article
Research on Detection Performance of NaI(Tl) Detector Based on Monte Carlo Method
by Qingbo Du, Yapeng Yang, Xiaoyu Zhao, Qi Lv, Yuyao Tang, Jiapeng He, Yier Liu and Guoqiang Li
Sensors 2026, 26(12), 3913; https://doi.org/10.3390/s26123913 (registering DOI) - 19 Jun 2026
Abstract
The NaI(TI) detector is highly favored in gamma radiation detection and widely applied in fields such as environmental radiation monitoring, nuclear medicine, and laboratory gamma-ray spectroscopy. Its detection performance determines the results of quantitative gamma-ray detection, making it a crucial indicator in detector [...] Read more.
The NaI(TI) detector is highly favored in gamma radiation detection and widely applied in fields such as environmental radiation monitoring, nuclear medicine, and laboratory gamma-ray spectroscopy. Its detection performance determines the results of quantitative gamma-ray detection, making it a crucial indicator in detector design and development. This study employs the Monte Carlo method and utilizes TopMC 1.0 software to establish a NaI(TI) detector model. First, the effects of crystal size, ray energy, cladding thickness, and distance on the detector’s detection efficiency were investigated. Subsequently, the energy resolution and peak-to-total ratio of the detector were simulated and calculated, with comparisons made to experimental values. The results indicate that all three detection efficiencies of the NaI(TI) detector are positively correlated with crystal size and exhibit an initial increase followed by a decrease with rising gamma-ray energy. Both the absolute detection efficiency and full-energy peak detection efficiency first increase and then decrease with increasing cladding thickness, while showing a negative correlation with detection distance. The intrinsic detection efficiency is almost unaffected by cladding thickness and initially rises before declining with increasing detection distance. The simulated values of energy resolution closely match experimental values, improving with higher gamma-ray energy. The deviation between simulated and experimental values for different source peak-to-total ratios remains within 6.25%, verifying the model’s reliability and the accuracy of simulation data. These findings provide valuable references and guidance for optimizing detection performance, conducting source-free efficiency calibration, and structural design of NaI(TI) detectors. Full article
(This article belongs to the Special Issue Nuclear Radiation Detectors and Sensors)
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31 pages, 2702 KB  
Article
Uncertainty Propagation in Curvature-Based Surface Form Metrology: A Monte Carlo and Differential Geometry Approach
by Dmytro Malakhov, Tatiana Kelemenová and Michal Kelemen
Metrology 2026, 6(2), 43; https://doi.org/10.3390/metrology6020043 (registering DOI) - 19 Jun 2026
Abstract
Curvature-based descriptors are increasingly used in surface metrology for the characterization of complex geometries. However, their sensitivity to measurement uncertainty remains insufficiently understood, particularly in comparison with conventional deviation-based metrics. This study investigates the propagation of coordinate measurement noise into curvature estimation using [...] Read more.
Curvature-based descriptors are increasingly used in surface metrology for the characterization of complex geometries. However, their sensitivity to measurement uncertainty remains insufficiently understood, particularly in comparison with conventional deviation-based metrics. This study investigates the propagation of coordinate measurement noise into curvature estimation using a numerical framework combining differential geometry, local quadratic surface fitting, and Monte Carlo simulation. A set of nominal surfaces, including spherical, cylindrical, and free-form geometries, was analyzed under controlled stochastic perturbations. The results show that curvature uncertainty increases nonlinearly with coordinate noise and is significantly more sensitive to measurement errors than point-wise deviations. Even small perturbations in measured coordinates lead to amplified variability in curvature due to its dependence on second-order derivatives. The analysis further reveals the presence of systematic bias in curvature estimation and demonstrates that the resulting distributions deviate from normality, despite Gaussian input noise. This finding highlights the limitations of classical uncertainty evaluation approaches based on linear propagation and normality assumptions. In addition, the study shows that increasing sampling density does not necessarily improve estimation reliability, while the size of the local fitting window plays a key role in stabilizing curvature estimation, acting as an implicit regularization parameter. The comparison with conventional form deviation metrics confirms that curvature-based analysis provides complementary information about local geometric stability, which is not captured by global measures. The proposed simulation-based approach offers a practical framework for evaluating uncertainty in nonlinear geometric measurements and supports the integration of curvature-based descriptors into advanced metrological applications. The proposed framework can support uncertainty-aware evaluation of free-form surfaces in practical measurement tasks, including coordinate measurement of turbine blades and aerodynamic components in the aerospace industry, as well as optical scanning and verification of patient-specific biomedical implants, where accurate curvature characterization is essential for quality assessment. Full article
16 pages, 951 KB  
Article
Faecal Pathogen Survival and Risks of Use of Ecological Sanitation By-Products in Burera District, Rwanda: A Quantitative Microbial Risks Assessment
by Celestin Banamwana, David Musoke, Theoneste Ntakirutimana, Esther Buregyeya, John Ssempebwa, Gakenia Wamuyu Maina, Charles Drago Kato, Lordrick Alinaitwe, Patrick Albert Ipola and Nazarius Mbona Tumwesigye
Int. J. Environ. Res. Public Health 2026, 23(6), 816; https://doi.org/10.3390/ijerph23060816 (registering DOI) - 19 Jun 2026
Abstract
Reuse of human excreta and derivatives is becoming a common practice in areas with agricultural predominance. While in situ treated faeces through ecological sanitation (Ecosan), known as “faecal by-products” are being used to sustain soil nutrients and improve on-site sanitation, the concern remains [...] Read more.
Reuse of human excreta and derivatives is becoming a common practice in areas with agricultural predominance. While in situ treated faeces through ecological sanitation (Ecosan), known as “faecal by-products” are being used to sustain soil nutrients and improve on-site sanitation, the concern remains about the health risks related to the survival of pathogens in these by-products in the community of farmers. This study assessed the survival of faecal pathogens and estimated microbial risks associated with the use of Ecosan faecal by-products in agriculture. The quantitative microbial risks assessment (QMRA) framework was used to estimate the risks posed by each faecal pathogen in solid and semi-solid faecal by-products under the probabilistic model of Monte Carlo simulation. Ascaris lumbricoides (6.5 eggs/gr), Taenia species (0.3 egg/gr), Schistosoma species (9.3 cercariae/gr), Entamoeba species (4.4 cysts/gr), and Escherichia coli (451 Cfu/gr) were detected in semi-solid faecal products. Exposure scenarios were observed throughout four critical points: vault faecal by-products removal/unloading, transport, collection, and application of faecal by-products in the gardens. Due to the presence of eggs and cysts, an estimated annual risk of infections was found in semi-solid faecal by-products with Schistosoma species (88%) and Ascaris lumbricoides (90%). Both concentrations were above World Health organisation (WHO) standards of associated infective risks of 0–10% of helminths in faecal sludge applied in the gardens. The users of faecal by-products, particularly farmers are exposed not only to high concentrations of helminth eggs but also to protozoa and bacteria with infective risks of Entamoeba species (99%) and E. coli species (62%). A stepwise implementation of faecal pathogens die-off during treatment of faecal by-products in compliance with the WHO’s 2018 guidelines can prevent the use of unsanitary faecal by-products. According to these findings, the proper control of intestinal protozoa and soil-transmitted helminths (STHs) should be enforced through personal protective measures in Burera district, Rwanda. Full article
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37 pages, 21335 KB  
Article
A New Reparameterized Weibull-Type Distribution for Asymmetric Lifetime Data: Inference, Simulation, and Applications
by Ahmed Elshahhat, Heba S. Mohammed, Osama E. Abo-Kasem and Asmaa Abdel-Hakim
Symmetry 2026, 18(6), 1057; https://doi.org/10.3390/sym18061057 - 19 Jun 2026
Abstract
This article presents a comprehensive inferential and applied investigation of the newly reparameterized Z-Weibull (ZW) distribution, a flexible Weibull-type lifetime model capable of accommodating both bounded and unbounded support regimes as well as a wide variety of hazard rate shapes. Unified frequentist and [...] Read more.
This article presents a comprehensive inferential and applied investigation of the newly reparameterized Z-Weibull (ZW) distribution, a flexible Weibull-type lifetime model capable of accommodating both bounded and unbounded support regimes as well as a wide variety of hazard rate shapes. Unified frequentist and Bayesian inference procedures are developed for complete and censored samples using maximum likelihood, maximum product spacing, and Markov chain Monte Carlo methods. Theoretical properties of the estimators and their associated interval estimates are established, while extensive Monte Carlo simulations assess their finite-sample performance under diverse parameter configurations and censoring schemes. The results indicate that Bayesian spacing-based procedures generally provide more accurate estimation, lower bias, and improved interval performance than competing classical methods. Applications to biomedical survival and climatological datasets, together with comparisons against several Weibull-type and exponential-based competitors, demonstrate the superior flexibility and goodness-of-fit of the ZW model. These findings highlight the practical value of the reparameterized ZW distribution as a unified and effective tool for modeling complex lifetime and reliability data arising in survival, environmental, and engineering studies. Full article
20 pages, 386 KB  
Article
Classical Estimation Methods and Optimality of Sampling Plans Under Progressive Type-I Censoring Scheme with Application to Reliability Data
by Ahmed R. El-Saeed
Axioms 2026, 15(6), 459; https://doi.org/10.3390/axioms15060459 (registering DOI) - 18 Jun 2026
Viewed by 14
Abstract
In this paper, the maximum product spacing method of estimation has been investigated under progressive Type-I censoring scheme. This estimation method has not previously been considered in the life-testing literature, particularly under this censoring scheme. The optimality of the sampling plans under progressive [...] Read more.
In this paper, the maximum product spacing method of estimation has been investigated under progressive Type-I censoring scheme. This estimation method has not previously been considered in the life-testing literature, particularly under this censoring scheme. The optimality of the sampling plans under progressive Type-I censoring was studied using different criteria and proposed censoring plans. The applicability of the distribution was examined using the Chen distribution, which is capable of modeling various reliability behaviors. A Monte Carlo simulation was conducted to assess the efficiency of the maximum product spacing method and the optimality of the sampling plans. Finally, an engineering application was analyzed considering progressive Type-I censoring. Full article
(This article belongs to the Special Issue Recent Developments in Statistical Research)
23 pages, 5270 KB  
Article
Constraint-Adjusted Nonparametric Inference for Residual-Life Functionals Under Stochastic Precedence
by Abdulmajeed A. R. Alharbi
Mathematics 2026, 14(12), 2196; https://doi.org/10.3390/math14122196 - 18 Jun 2026
Viewed by 48
Abstract
Nonparametric inference for residual-life functionals is a fundamental problem in mathematical statistics, reliability theory, and survival analysis, particularly in studies with limited sample sizes where empirical plug-in estimators may exhibit substantial sampling variability. In comparative lifetime analysis, additional qualitative information is often available [...] Read more.
Nonparametric inference for residual-life functionals is a fundamental problem in mathematical statistics, reliability theory, and survival analysis, particularly in studies with limited sample sizes where empirical plug-in estimators may exhibit substantial sampling variability. In comparative lifetime analysis, additional qualitative information is often available regarding the relative behavior of two populations; however, such information is frequently too weak to justify classical stochastic dominance assumptions. Stochastic precedence provides a natural and interpretable framework for representing this partial ordering through a pairwise probabilistic constraint. This paper develops a constraint-adjusted nonparametric inference framework for estimating the mean residual life (MRL) and quantile residual life (QRL) functions under stochastic precedence information. The proposed approach replaces the ordinary empirical distribution function in standard residual-life plug-in estimators with a constraint-adjusted empirical distribution function that enforces the stochastic precedence relation at the sample level. The adjustment is governed by a data-driven scaling factor and is asymptotically negligible, thereby preserving the large-sample behavior of the ordinary empirical estimators while incorporating meaningful structural information in finite samples. Strong consistency of the proposed MRL and QRL estimators was established under mild regularity conditions. A Monte Carlo study based on Weibull and gamma lifetime models demonstrates that in the simulation settings considered, the proposed estimators provide improved finite-sample stability and generally achieve smaller mean squared errors than their ordinary empirical counterparts, especially for small and moderate sample sizes. The methodology is further illustrated using survival data from patients with squamous cell carcinoma of the oropharynx, highlighting its practical relevance in biomedical survival analysis. The proposed method offers a flexible, interpretable, and computationally simple framework for nonparametric inference with structured lifetime data under weak stochastic ordering information. Full article
(This article belongs to the Special Issue Mathematical Statistics and Nonparametric Inference)
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22 pages, 662 KB  
Article
State-Dependent Asymmetry in Soft-Pity Gacha Waiting-Time Models: Exact Recurrences, Tail Risk, and Featured-Target Extensions
by Saisai Hou, Yunzhi Zhu and Sen Zhang
Symmetry 2026, 18(6), 1051; https://doi.org/10.3390/sym18061051 - 18 Jun 2026
Viewed by 46
Abstract
Randomized reward mechanisms are often described as repeated trials with a fixed success probability. This constant-hazard reference case is symmetric in the limited finite-state sense that, conditional on non-absorption, the next-draw success probability is invariant with respect to the current draw count. Pity [...] Read more.
Randomized reward mechanisms are often described as repeated trials with a fixed success probability. This constant-hazard reference case is symmetric in the limited finite-state sense that, conditional on non-absorption, the next-draw success probability is invariant with respect to the current draw count. Pity and guarantee rules break this draw-count homogeneity by making the hazard depend on the current state. This paper studies that state-dependent asymmetry for a finite soft-pity waiting-time model. The waiting time for one rare item is represented as an absorption time of a Markov chain whose transient state is the pity counter. We write the corresponding absorbing transition matrix explicitly and then derive the equivalent first-step recurrences for the expectation, variance, and full probability mass function. A simple stochastic-ordering proposition shows how increasing the statewise success probabilities decreases the waiting-time distribution in the usual tail order. Repeated convolution then yields the distribution for multiple independent stages. The numerical section reports quantiles, tail probabilities, VaR/CVaR-type summaries, expected excess values, sensitivity analyses, normal-approximation diagnostics, and distributional asymmetry indicators. A featured-target variant with a binary guarantee state is also included. Throughout, the reported quantities are consequences of the stated transition rule; Monte Carlo simulation is used only as a numerical check. Full article
(This article belongs to the Section Mathematics)
29 pages, 5546 KB  
Review
The Charging-Up Phenomenon in Gas Electron Multiplier Detector
by Sayak Chatterjee, Supriya Das and Saikat Biswas
Particles 2026, 9(2), 65; https://doi.org/10.3390/particles9020065 - 17 Jun 2026
Viewed by 308
Abstract
Gas Electron Multiplier (GEM) detectors have become an indispensable component of modern tracking systems. The heart of a GEM detector is a thin polyimide foil (∼50 µm) clad with copper (∼5 µm) on both sides and containing an array of regularly spaced holes [...] Read more.
Gas Electron Multiplier (GEM) detectors have become an indispensable component of modern tracking systems. The heart of a GEM detector is a thin polyimide foil (∼50 µm) clad with copper (∼5 µm) on both sides and containing an array of regularly spaced holes (typically diameter of ∼70 µm and pitch of ∼140 µm) fabricated using photolithographic techniques. The presence of the dielectric substrate (polyimide) within the amplification region introduces a time dependent response when the detector is exposed to external irradiation, a phenomenon commonly referred to as the charging-up effect. This effect arises from the accumulation of charge on the insulating polyimide surfaces, leading to a gradual modification of the local electric field configuration inside the GEM holes and, consequently, a variation in the detector gain over time. The charging-up behaviour has been systematically investigated for triple GEM chamber prototypes using an Fe-55 radioactive source (5.9 keV X-rays) with an activity of ∼20 mCi. The characteristic charging-up time constant has been extracted, and its dependence on detector gain and irradiation rate has been examined. In addition, the uniformity of detector performance in terms of count rate, gain, and energy resolution has been studied both before and after the charging-up process. In this review article, the experimental setup, data acquisition methodology, and analysis procedures developed and carried out by our group are summarised. The key findings reported by other groups, relevant Monte Carlo simulation efforts, and future outlook for the charging-up investigation on GEM based detectors are also discussed in this article. The investigations and their outcomes reviewed here provide valuable insight into the charging-up dynamics of GEM detectors and their dependence on operational parameters. Full article
(This article belongs to the Section Experimental Physics and Instrumentation)
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21 pages, 1608 KB  
Article
Distributed Jamming Method for ASLC Systems Based on Random Phase Perturbation
by Liang Qi and Jianjiang Zhou
Sensors 2026, 26(12), 3857; https://doi.org/10.3390/s26123857 - 17 Jun 2026
Viewed by 217
Abstract
Adaptive Sidelobe Cancellation (ASLC) is a core technology for modern radar systems to suppress active sidelobe jamming. From the perspective of disrupting the ASLC system’s ability to stably track the jamming direction, this paper proposes a distributed jamming method based on random phase [...] Read more.
Adaptive Sidelobe Cancellation (ASLC) is a core technology for modern radar systems to suppress active sidelobe jamming. From the perspective of disrupting the ASLC system’s ability to stably track the jamming direction, this paper proposes a distributed jamming method based on random phase perturbation. The method employs two spatially separated jamming sources that simultaneously transmit coherent signals. By actively applying controllable random jumps to the relative phase between the two sources, the equivalent wavefront direction of the synthesized signal at the radar receiver changes rapidly, forming a non-stationary jamming that destroys the null-tracking capability of ASLC. An analytical model of the ASLC cancellation ratio (CR) under random phase perturbation is established, with a focus on analyzing the effects of time synchronization accuracy and phase synchronization accuracy on jamming performance. Monte Carlo simulation results show that the proposed method can reduce the average ASLC CR from 26.80 dB to 20.29 dB (a decrease of 6.51 dB). Under identical conditions, this performance is comparable to asynchronous blinking jamming while requiring no precise timing matching, and outperforms multi-source saturation jamming in resource efficiency (two vs. four jammers). This study provides promising simulation-level evidence for the effectiveness of the proposed jamming method. The quantitative results and sensitivity analyses offer a simulation-level theoretical reference for parameter design of distributed cooperative jamming. Further validation in semi-physical simulations or field trials is necessary before claiming engineering readiness. Full article
(This article belongs to the Section Radar Sensors)
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26 pages, 2381 KB  
Article
Probabilistic Sensitivity Analysis of a Nonlinear Electrochemical Model as a Virtual Replica for Lithium-Ion Battery Design Under Uncertainty
by Jurgita Dabulytė-Bagdonavičienė, Gintarė Vaidelienė, Edvinas Juozapaitis and Robertas Alzbutas
Mathematics 2026, 14(12), 2162; https://doi.org/10.3390/math14122162 - 17 Jun 2026
Viewed by 167
Abstract
This paper presents a probabilistic sensitivity analysis of a nonlinear electrochemical model for lithium-ion batteries. The model is treated as a reduced virtual replica for uncertainty-aware analysis rather than as a full digital twin. A reduced electrochemical formulation is combined with constrained inverse [...] Read more.
This paper presents a probabilistic sensitivity analysis of a nonlinear electrochemical model for lithium-ion batteries. The model is treated as a reduced virtual replica for uncertainty-aware analysis rather than as a full digital twin. A reduced electrochemical formulation is combined with constrained inverse parameter identification using experimental current–voltage data to relate observable battery behavior to effective model parameters. Predictive variability is assessed through Monte Carlo uncertainty propagation and global sensitivity analysis under both charging and discharging conditions. The results indicate that the particle radius of the positive active material and the effective electrodes area are the dominant contributors to terminal-voltage uncertainty, whereas the electrode thickness parameter and negative electrode active material particle radius have a moderate influence within the studied ranges. Rank-based and variance-based sensitivity measures are more informative than linear indices for this reduced nonlinear system. From a mathematical perspective, the work integrates reduced-order modeling, inverse problem formulation, numerical simulation, and uncertainty quantification in one computational framework for battery analysis. The results support uncertainty-aware parameter prioritization, calibration of reduced electrochemical models, and provide a basis for future work on battery design, control, and digital-twin-oriented extensions under uncertainty. Full article
(This article belongs to the Special Issue Advanced Mathematical Models in Engineering Design Optimization)
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49 pages, 1621 KB  
Article
A New Gompertz Distribution for Modeling Tensile Strength of Carbon Fibers and Single Carbon Fibers Data
by Ayşe Metin Karakaş, Fatma Bulut and Sinan Çalık
Mathematics 2026, 14(12), 2159; https://doi.org/10.3390/math14122159 - 16 Jun 2026
Viewed by 95
Abstract
The Gompertz distribution is a well-known lifetime model in survival and reliability analysis, but its hazard rate is restricted to monotone increasing behavior, which limits its applicability to more complex data structures. In this study, we investigate the New Extended Gompertz (NEG) distribution, [...] Read more.
The Gompertz distribution is a well-known lifetime model in survival and reliability analysis, but its hazard rate is restricted to monotone increasing behavior, which limits its applicability to more complex data structures. In this study, we investigate the New Extended Gompertz (NEG) distribution, which is obtained by applying the existing NE-X generator framework to the classical Gompertz baseline distribution. Thus, the NEG model is a special case within an already established generator family rather than an entirely new family of distributions. The main contribution of this paper is not the introduction of a new generator, but rather a comprehensive and systematic investigation of this particular Gompertz-based extension, including its statistical properties, estimation procedures, and practical applications. The proposed model introduces an additional shape parameter that provides increased flexibility in modeling skewness, tail behavior, and hazard-rate structures, allowing for increasing, decreasing, bathtub-shaped, and unimodal hazard patterns under different parameter configurations. Several mathematical properties of the NEG distribution are derived, including explicit expressions for the density, distribution, survival, and hazard-rate functions, as well as moments, entropy measures, and series representations. Parameter estimation is performed using both maximum likelihood and Bayesian approaches, with numerical optimization and Metropolis–Hastings MCMC procedures employed due to the absence of closed-form estimators. The finite-sample behavior of the estimators is investigated through extensive Monte Carlo simulation studies under three different parameter settings. The practical usefulness of the NEG distribution is illustrated using two real datasets on carbon-fiber tensile strength. Comparative results with several competing Gompertz-type models indicate that the NEG distribution provides competitive performance. However, all comparisons should be interpreted within the context of the considered datasets and parameter settings, rather than as claims of universal superiority. The findings suggest that the NEG distribution offers a flexible and practical extension of the Gompertz model for lifetime data analysis. Full article
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23 pages, 17945 KB  
Article
Analysis of the Delayed Instability Mechanism of Heterogeneous Fractured Rock Slopes Under Rainfall Infiltration
by Yu Zhao, Jun Shen, Yunhou Sun, Xiaolong Wang and Feng Li
Appl. Sci. 2026, 16(12), 6102; https://doi.org/10.3390/app16126102 - 16 Jun 2026
Viewed by 168
Abstract
Rainfall-induced delayed instability of fractured rock slopes is strongly affected by fracture preferential flow, hydro-mechanical coupling, and spatial matrix heterogeneity. However, the coupled influence of stress-dependent fracture aperture evolution and heterogeneous matrix properties on delayed slope deformation remains insufficiently quantified. In this study, [...] Read more.
Rainfall-induced delayed instability of fractured rock slopes is strongly affected by fracture preferential flow, hydro-mechanical coupling, and spatial matrix heterogeneity. However, the coupled influence of stress-dependent fracture aperture evolution and heterogeneous matrix properties on delayed slope deformation remains insufficiently quantified. In this study, a two-dimensional discrete fracture network (DFN)–equivalent continuum coupled model was established using spectral random field theory and a representative Monte Carlo-generated fracture geometry. The spectral exponent β = 1.0–2.5 was adopted to characterize different degrees of matrix heterogeneity, and rainfall infiltration–stress coupling simulations were conducted under an extreme rainfall scenario followed by drainage. The results indicate that the wetting front advances irregularly in the heterogeneous matrix, while fracture preferential flow accelerates rainwater infiltration and promotes local pore-pressure accumulation near the phreatic surface. After rainfall cessation, water stored in fractures continues to recharge the deep matrix, leading to delayed pore-pressure increase and post-rainfall deformation. The simulated fracture aperture shows an initial closure followed by gradual dilation, which is controlled by the competition between saturation-induced stress redistribution and pore-pressure-driven effective stress reduction. Under a common strength reduction factor of FOS = 1.4, stronger matrix heterogeneity results in more pronounced plastic strain concentration and larger displacement amplitude along the potential slip zone. These findings suggest that fracture aperture evolution and matrix heterogeneity jointly influence delayed deformation and potential failure-zone development in rainfall-affected fractured rock slopes. The conclusions should be interpreted within the scope of a two-dimensional DFN–equivalent continuum numerical framework with prescribed rainfall conditions and representative fracture/random-field realizations. Full article
(This article belongs to the Section Civil Engineering)
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27 pages, 21143 KB  
Article
A Hybrid Machine Learning Method for Dynamic Monitoring of CO2 Sequestration Using Pulsed Neutron Logging
by Tianyang Jiao, Xiaying Li, Juntao Liu, Liyuan Sheng, Yixin Zhang, Bin Lei, Jiarong Guo, Fangyang Yao, Fujun Long, Di Wu, Haoyu Zhang, Xin Tong and Zhiyi Liu
Energies 2026, 19(12), 2848; https://doi.org/10.3390/en19122848 - 16 Jun 2026
Viewed by 177
Abstract
This study proposes a hybrid machine learning model based on full-spectrum pulsed neutron logging data to address the monitoring challenges of Carbon Capture, Utilization, and Storage (CCUS) under complex geological conditions. Traditional interpretation models for sequestered CO2 saturation (e.g., macroscopic capture cross-section [...] Read more.
This study proposes a hybrid machine learning model based on full-spectrum pulsed neutron logging data to address the monitoring challenges of Carbon Capture, Utilization, and Storage (CCUS) under complex geological conditions. Traditional interpretation models for sequestered CO2 saturation (e.g., macroscopic capture cross-section model, characteristic peak count model, and ratio model) heavily rely on prior parameters such as porosity, formation water salinity, and lithology. Acquiring these parameters in real time during practical engineering is often costly and difficult. To reduce the rigid dependence of accurate CO2 saturation monitoring on complex prior parameters like porosity and salinity under heterogeneous geological settings, this research focuses on the Pearl River Mouth Basin, a core carbon sequestration target area in the Guangdong-Hong Kong-Macao Greater Bay Area, based on the evaluation results of offshore carbon sequestration macro-regions in China. Taking the primary reservoirs of the Enping and Wenchang Formations as typical geological prototypes, a high-fidelity, full-spectrum neutron–gamma response database was constructed using Monte Carlo simulations. Two machine learning strategies are proposed: a direct regression model (NMF+SVR) and a joint model (NMF+SVC/KMeans+SVR). Based on Monte Carlo simulated data, experimental results demonstrate that, compared with traditional petrophysical baseline models and simple machine learning models, the proposed joint learning method effectively reduces the dependence of CO2 saturation monitoring on lithology and porosity. Furthermore, it is proven that even with a single-detector tool configuration, the method exhibits high prediction accuracy under complex lithological conditions. Notably, the two-step joint model achieves a Root Mean Square Error (RMSE) as low as 4.200%, significantly outperforming traditional physics-based models and single machine learning models such as MLP and RF. This study provides a physically interpretable and accurate technical reference for applying machine learning to pulsed neutron-logging-based CO2 geological sequestration monitoring. Full article
(This article belongs to the Special Issue Advances in the Development of Geoenergy: 3rd Edition)
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