Combining Monte Carlo Simulations Research Articles

Combined DFT and Monte Carlo simulation studies of potential corrosion inhibition properties of coumarin derivatives.

Corrosion, the degradation of materials due to chemical reactions with their environment presents significant challenges both economically and environmentally. It affects various industries, including construction, transportation, and manufacturing, leading to equipment failures, safety hazards, and increased maintenance costs. Coumarin derivatives have shown promise due to their inherent chemical properties and potential for biodegradability. In this study, a series of the coumarin derivatives were examined in silico to reveal their potential corrosion inhibition properties toward the Fe(110) and Cu(111) surfaces. The compounds investigated include coumarin (2H-chromen-2-one, 1), furanocoumarin (7H-furo[3,2-g]chromen-7-one, 2), dihydrofurano coumarin (2,3-dihydro-7H-furo[3,2-g]chromen-7-one, 3), pyrano coumarin-linear type (8,8-dimethyl-2H,8H-pyrano[3,2-g]chromen-2-one, 4), pyrano coumarin-angular type (8,8-dimethyl-2H,8H-pyrano[2,3-f]chromen-2-one, 5), bicoumarin (3,3'-methylenebis(2H-chromen-2-one), 6), and phenyl coumarin (4-phenyl-2H-chromen-2-one, 7). The findings suggest that the bicoumarin derivative 6 exhibits the lowest adsorption energy with the Fe(110) surface, while the same energy absolute value is about two times lower for the Cu(111) surface. This is due to the formation of a planar configuration of a molecule of 6 on the metal surfaces with the participation of both coumarin fragments upon interacting with the Fe(110) surface, while one coumarin fragment interacts with the Cu(111) surface. Density functional theory (DFT) calculations were employed to study the electronic properties of the coumarin derivatives. The specific computational method used was B3LYP, a hybrid functional that combines with the 6-311 + + G(d,p) basis set. Each coumarin derivative was first subjected to a geometry optimization to find the most stable molecular structure. Electronic properties, dipole moments, and molecular electrostatic potential surfaces were calculated. The Monte Carlo simulations were used to model the adsorption behavior of the coumarin derivatives on metal surfaces, namely, Fe(110) and Cu(111). These simulations allowed to visualize interaction of the studied molecules with the metal surfaces, which is crucial for their function as corrosion inhibitors. The present study provides a comprehensive understanding of the corrosion inhibition potential of the applied coumarin derivatives. The insights gained from these methods can inform the development of effective, sustainable corrosion inhibitors that are both environmentally friendly and highly efficient.

Research on Probabilistic Load Modeling and Simulation Model of Electric Vehicle Charging Station

Abstract The stochastic nature of electric vehicle charging station loads poses certain challenges in establishing relevant load models. This paper introduces a probabilistic load modeling approach that considers factors such as the time electric vehicles begin charging, daily travel distance, charging power, and charging duration. By comprehensively studying the influencing factors of charging station load modeling and combining Monte Carlo simulation with probabilistic statistical analysis, a comprehensive probability load model for typical charging stations is developed. Building upon this, a charging load simulation model is constructed using Simulink. This model can generate daily load curves consistent with the probability load model, making a positive contribution to the study of the impact of electric vehicles on the power grid.

Percolation thresholds of disks with random nonoverlapping patches on four regular two-dimensional lattices.

In percolation of patchy disks on lattices, each site is occupied by a disk, and neighboring disks are regarded as connected when their patches contact. Clusters of connected disks become larger as the patchy coverage of each disk χ increases. At the percolation threshold χ_{c}, an incipient cluster begins to span the whole lattice. For systems of disks with n symmetric patches on Archimedean lattices, a recent work [Wang et al., Phys. Rev. E 105, 034118 (2022)2470-004510.1103/PhysRevE.105.034118] found symmetric properties of χ_{c}(n), which are due to the coupling of the patches' symmetry and the lattice geometry. How does χ_{c} behave with increasing n if the patches are randomly distributed on the disks? We consider two typical random distributions of the patches, i.e., the equilibrium distribution and a distribution from random sequential adsorption. Combining Monte Carlo simulations and the critical polynomial method, we numerically determine χ_{c} for 106 models of different n on the square, honeycomb, triangular, and kagome lattices. The rules governing χ_{c}(n) are investigated in detail. They are quite different from those for disks with symmetric patches and could be useful for understanding similar systems.

Numerical Optimization of Variable Blank Holder Force Trajectories in Stamping Process for Multi-Defect Reduction.

An intelligent optimization technology was proposed to mitigate prevalent multi-defects, particularly failure, wrinkling, and springback in sheet metal forming. This method combined deep neural networks (DNNs), genetic algorithms (GAs), and Monte Carlo simulation (MCS), collectively as DNN-GA-MCS. Our primary aim was to determine intricate process parameters while elucidating the intricate relationship between processing methodologies and material properties. To achieve this goal, variable blank holder force (VBHF) trajectories were implemented into five sub-stroke steps, facilitating adjustments to the blank holder force via numerical simulations with an oil pan model. The Forming Limit Diagram (FLD) predicted by machine learning algorithms based on the Generalized Incremental Stress State Dependent Damage (GISSMO) model provided a robust framework for evaluating sheet failure dynamics during the stamping process. Numerical results confirmed significant improvements in formed quality: compared with the average value of training sets, the improvements of 18.89%, 13.59%, and 14.26% are achieved in failure, wrinkling, and springback; in the purposed two-segmented mode VBHF case application, the average value of three defects is improved by 12.62%, and the total summation of VBHF is reduced by 14.07%. Statistical methodologies grounded in material flow analysis were applied, accompanied by the proposal of distinctive optimization strategies for the die structure aimed at enhancing material flow efficiency. In conclusion, our advanced methodology exhibits considerable potential to improve sheet metal forming processes, highlighting its significant effect on defect reduction.

A state-selected continuous wave laser excitation method for determining CO2's rotational state distribution in a supersonic molecular beam.

State-resolved experiments can provide fundamental insight into the mechanisms behind chemical reactions. Here, we describe our methods for characterizing state-resolved experiments probing the outcome of the collision between CO2 molecules and surfaces. We create a molecular beam from a supersonic expansion that passes through an ultra-high vacuum system. The CO2 is vibrationally excited by a continuous wave infrared (IR) laser using rapid adiabatic passage. We attenuate the fractional excitation using a CO2 absorption cell in the IR beam path. We combine Monte Carlo simulations and molecular beam energy measurements to find the initial rotational state distribution of the molecular beam. We find that our pure CO2 beam from a 300K source has a rotational temperature of ∼26K.

Reliability and sensitivity assessment of laminated composite plates with high-dimensional uncertainty variables using active learning-based ensemble metamodels

Abstract Laminated composite materials play a crucial role in various engineering applications due to their exceptional mechanical properties. This article explores the analysis of reliability and the local sensitivity of failure probability associated with laminated composite materials. It delves into the impact of uncertainties on the performance of these materials and presents an active learning-based reliability methodology. This methodology combines Monte Carlo simulation with a metamodel derived from the combination of three metamodels: artificial neural networks, support vector regression, and Kriging. The article illustrates this methodology through two practical applications on laminated composite plates and compares its performance with other reliability methods. This approach offers valuable insights to enhance the analysis of reliability, strengthen the design process, and facilitate decision-making while fully considering uncertainties related to material properties.

Introducing an Artificial Neural Network for Virtually Increasing the Sample Size of Bioequivalence Studies

Sample size is a key factor in bioequivalence and clinical trials. An appropriately large sample is necessary to gain valuable insights into a designated population. However, large sample sizes lead to increased human exposure, costs, and a longer time for completion. In a previous study, we introduced the idea of using variational autoencoders (VAEs), a type of artificial neural network, to synthetically create in clinical studies. In this work, we further elaborate on this idea and expand it in the field of bioequivalence (BE) studies. A computational methodology was developed, combining Monte Carlo simulations of 2 × 2 crossover BE trials with deep learning algorithms, specifically VAEs. Various scenarios, including variability levels, the actual sample size, the VAE-generated sample size, and the difference in performance between the two pharmaceutical products under comparison, were explored. All simulations showed that incorporating AI generative algorithms for creating virtual populations in BE trials has many advantages, as less actual human data can be used to achieve similar, and even better, results. Overall, this work shows how the application of generative AI algorithms, like VAEs, in clinical/bioequivalence studies can be a modern tool to significantly reduce human exposure, costs, and trial completion time.

Simulation Study of Polyethylene Terephthalate Hydrolase Adsorption on Self-Assembled Monolayers.

Polyethylene terephthalate (PET) hydrolase, discovered in Ideonella sakaiensis (IsPETase), is a promising agent for the biodegradation of PET under mild reaction conditions, yet the thermal stability is poor. The efficient immobilization and orientation of IsPETase on different solid substrates are essential for its application. In this work, the combined parallel tempering Monte Carlo simulation with the all-atom molecular dynamics simulation approach was adopted to reveal the adsorption mechanism, orientation, and conformational changes of IsPETase adsorbed on charged self-assembled monolayers (SAMs), including COOH-SAM and NH2-SAM with different surface charge densities (SCDs). The results show that the protein adsorption orientation was determined not only by attraction interactions but also by repulsion interactions. IsPETase is adsorbed on the COOH-SAM surface with an "end-on" orientation, which favors the exposure of the catalyzed triplet to the solution. In addition, the entrance to the catalytic active center is larger on the COOH-SAM surface with a low SCD. This work reveals the controlled orientation and conformational information on IsPETase on charged surfaces at the atomistic level. This study would certainly promote our understanding of the mechanism of IsPETase adsorption and provide theoretical support for the design of substrates for IsPETase immobilization.

Quantifying the Impact of Sea Level on Coastal Cities using Bayesian optimized Monte Carlo Simulation model

The immediate treatment of the increasing of sea levels represent on coastal cities around the globe is the subject of this research. Using an all-encompassing and probabilistic methodology, the study quantifies the complex effects of rising sea levels using a Bayesian-optimized Monte Carlo simulation framework. The current approaches for assessing the effects of rising sea levels frequently do not adequately capture the intrinsic complexity and unpredictability of changing structures. This work presents a novel method combining Monte Carlo simulation and Bayesian optimization to overcome these drawbacks. The present research is innovative since it incorporates Bayesian optimization methods into a Monte Carlo simulation framework. By constantly modifying distributions of probabilities according to observable results, this combination improves simulation performance. This constantly changing optimization methodology closes a significant gap within current methods by ensuring a more accurate portrayal of shifting conditions. In addition, the method used in this work emphasizes cooperation among mathematicians, coastal scientists, and climate researchers, thereby promoting a comprehensive knowledge of the complex issues raised by rising sea levels. This set of variables includes the current level of the sea, land height, frequency of storm surges, population density, and infrastructure resilience. The model’s effectiveness is rigorously assessed, utilizing procedures for verification and calibration that use past information. Calculations are guided by various rising sea-level situations based on scientific forecasts and uncertainty ranges. Sensitivity analysis pinpoints important factors, guiding further efforts to gather data and improve the model. The structure for the performance assessment ensures that the model is dependable and applicable to those making decisions who want to prioritize flexible measures and create resilient communities along the coast.

Estimating external exposure from different source geometries in soil contaminated by gamma-ray emitting radionuclides: A computational model combining Monte Carlo simulation and mathematical transformation

A computational model is developed to estimate the external exposure to an arbitrary scenario describing the spatial distribution of the source in a contaminated soil by gamma-ray emitting radionuclides released after nuclear activities. This model welds together a Monte Carlo simulation approach and a mathematical transformation based on Tait–Bryan angles formalism. For Monte Carlo simulation, a source depth dependent optimized geometry describing a semi-infinite soil-cover-air medium of propagation is mathematically derived based on the physical attenuation of the gamma ray in matter. For each primary energy of interest, the proposed mathematical transformation is applied to the output of the Monte Carlo simulation to define the overlapping region between the contaminated soil volume and the proposed optimized volume. This approach made it possible to estimate the human external exposure associated with the scenario under investigation. Several scenarios were studied. The flux energy distribution of the detected radiation and the corresponding dose quantities are calculated. A benchmarking test with a direct Monte Carlo simulation and a comparison with published results in the literature show good agreement validating therefore the presented model.

Dynamic currency hedging with non-Gaussianity and ambiguity

This paper introduces a non-Gaussian dynamic currency hedging strategy for globally diversified investors with ambiguity. It provides theoretical and empirical evidence that, under the stylized fact of non-Gaussianity of financial returns and for a given optimal portfolio, the investor-specific ambiguity can be estimated from historical asset returns without the need for additional exogenous information. Acknowledging non-Gaussianity, we compute an optimal ambiguity-adjusted mean-variance (dynamic) currency allocation. Next, we propose an extended filtered historical simulation that combines Monte Carlo simulation based on volatility clustering patterns with the semi-parametric non-normal return distribution from historical data. This simulation allows us to incorporate investor's ambiguity into a dynamic currency hedging strategy algorithm that can numerically optimize an arbitrary risk measure, such as the expected shortfall. The out-of-sample backtest demonstrates that, for globally diversified investors, the derived non-Gaussian dynamic currency hedging strategy is stable, robust, and highly risk reductive. It outperforms the benchmarks of constant hedging as well as static/dynamic hedging approaches with Gaussianity in terms of lower maximum drawdown and higher Sharpe and Sortino ratios, net of transaction costs.

Physics informed ensemble learning used for interval prediction of fracture toughness of pipeline steels in hydrogen environments

Hydrogen embrittlement leads to a loss of plasticity and a decrease in fracture toughness in pipeline steels. Fracture toughness is a critical factor in the design and safety assessment of hydrogen-carrying pipelines. However, conducting large-scale fracture toughness testing on hydrogen-charged materials is time-consuming and expensive. Analytical approaches based on micromechanics often involve numerous undetermined (or assumed) parameters. Therefore, this study presents an approach that combines physical information and ensemble learning for predicting the fracture toughness range under hydrogen environments. Initially, detailed experimental data are gathered from a series of fracture tests conducted on pipeline steels under hydrogen environments, as published in the literature. Subsequently, Given the limited availability of experimental data, a feature space was constructed, taking into account the physical information as well as the distributions of yield strength and tensile strength. Furthermore, a predictive model based on Population-Based Training (PBT) optimization and Category Boosting (CatBoost) methodology was developed. Comparative experimentation substantiated its superior predictive accuracy in comparison to Random Forest, Adaptive Boosting (AdaBoost), Gradient Boosting Decision Tree (GBDT), Light Gradient Boosting Machine (LightGBM), and Extreme Gradient Boosting (XGBoost). The MAE, MSE, R2, VAF, a20_index and MAPE, was 0.02139, 0.00096, 0.98741, 98.98%, 0.83 and 0.1552. Building upon these foundations, this study proposes an interval prediction method by combining Monte Carlo simulation with point prediction results to obtain fracture toughness interval prediction results with different confidence intervals (80%, 85%, 90% and 95%).

Data-driven mid-air collision risk modelling using extreme-value theory

Mid-air collision risk estimation is crucial for maintaining aviation safety and improving the efficiency of air traffic procedures. This paper introduces a novel, data-driven methodology for estimating the probability of mid-air collisions between aircraft by combining Monte Carlo simulation and the Peaks Over Threshold approach from Extreme Value Theory. This innovative approach has substantial advantages over traditional methods. Firstly, it reduces the number of assumptions about the traffic flow compared to traditional analytical methods. In fact, data-driven techniques require fewer assumptions, as they inherently capture the structures of the traffic flow within the underlying data. Secondly, it converges faster than methods based on crude Monte Carlo simulation. Notably, by employing Extreme Value Theory, this approach enables efficient evaluation of low-probabilities, which are commonly found in collision risk modelling. The effectiveness of the proposed methodology is demonstrated through estimating the probability of a mid-air collision in a real-world practical example. The case study investigates the risk of collisions between departures and go-arounds in the terminal airspace at Zurich Airport, highlighting the potential for improved safety and efficiency in air traffic management.

Grain refinement induced by grain boundary segregation in FeNiCrCoCu high-entropy alloys using molecular dynamics simulation of nanoindentation

Atomistic simulation of nanoindentation is performed to investigate the effect of Cu segregation at grain boundaries (GBs) on the mechanical properties of FeNiCrCoCu high-entropy alloys. Initially, a combined Monte Carlo and Molecular dynamics simulation technique is adopted to generate the models with Cu segregation, and further the cases of random Cu distribution are considered for the purpose of comparision. The dislocation density, shear strain and GB behavior in the process of nanoindentation are analyzed in detail. The results reveal that the Cu segregation at GBs leads to superiority in hardness, strength, and stiffness. Dislocation analysis shows that the sessile dislocation density is higher at the initial state for the segregation sample, while dislocation nucleation and movements are suppressed so that dislocation-mediated plasticity is weakened. Furthermore, microrotation analysis during the nanoindentation process indicates that segregation of Cu pins the GBs, thereby inhibiting GB rotation and consequently weakening GB-mediated plastic deformation. In this case, grain refinement is observed, and in good agreement with experiment results. Overall, this study provides insights into the influence of GB segregation of Cu on the mechanical properties and deformation response of FeNiCrCoCu HEAs, and highlights the importance of GB composition for tailoring high-strength materials.

Modeling and performance analysis of underwater wireless optical absorption, scattering, and turbulence channels employing Monte Carlo-multiple phase screens.

The absorption, scattering, and turbulence effects have a significant impact on the performance of underwater wireless optical communication (UWOC). Therefore, it is crucial to consider seawater's optical parameters comprehensively when designing UWOC systems. So far, most studies on the UWOC channel have separately modeled the absorption and scattering, and turbulence of seawater, and furthermore, the continuous phase perturbations caused by turbulence are neglected to simplify the model when modeling turbulence channels. Hence, this paper simultaneously considers the absorption, scattering, and turbulence effects of seawater and proposes a UWOC channel modeling method that combines Monte Carlo simulation with multiple phase screen approaches. Subsequently, the impacts of different systems and channel conditions on system performance are explored, and simulation results indicate that as the turbidities and turbulence intensities of the seawater increase, the probability density function of received light signal intensity becomes more dispersed. The turbulence introduces an increase in path loss of approximately 5dB compared to its absence. Furthermore, the channel impulse response (CIR) is obtained, where the turbulence effects cause a 50% decrease in the CIR peak and the noticeable temporal spread.

Reliability analysis of wind power gearbox based on fault tree and monte carlo simulation

The gearbox is an important speed-up component in wind turbine, and the environment in which wind turbine operates is generally harsh. For maintenance and replacement of the gearbox and its difficult operation, it is self-evident that it is important to ensure its reliable operation. Through collecting and analyzing the structural and functional principle of the gearbox and data of common failure modes, the fault tree of the gearbox is established, and failure rate data and corresponding distribution are obtained by manual and data inquiry for each bottom event. Combining Monte Carlo simulation, the reliability curves of each bottom event mode of the gearbox, the gearbox, and important components are solved, and the assumptions of improving the important bottom events are made. By comparing the curves with the previous ones, it is proved that the improvement of important components can effectively improve the system’s reliability.

Robust optimisation of the streamlined shape of a high-speed train in crosswind conditions

Traditional deterministic aerodynamic optimisation cannot consider environmental uncertainty, which may lead to sensitivity issues. The present study proposes a robust design framework for the aerodynamic optimisation of high-speed trains, which accounts for the uncertain wind and its impact on crosswind stability. In this framework, a variance analysis method based on the Non-Intrusive Polynomial Chaos is proposed to determine the deformation area, and a parametric model is subsequently established. The Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is used as the optimiser to minimise the mean and variance of the aerodynamic response. The mean and variance can be quickly predicted by an uncertainty analysis approach combining Monte Carlo simulation and Kriging model. The framework is then applied to the optimisation of a high-speed train under crosswind. The results of the robust optimisation are compared with those of the baseline geometry and deterministic optimisation. The mean and variance of the rolling moment under crosswind are reduced by 2.26% and 3.37% respectively after optimisation, indicating that the performance and robustness are both improved. The proposed framework is effective for the engineering design of high-speed trains and can also provide a reference for the robust design of other aerodynamic shapes.

Photon-Counting Computed Tomography and Scintillator-Based Detectors: A Simulation Analysis With Scintillating and Reflecting Materials Currently on the Market

During the last decade, Photon-Counting Computed Tomography (PCCT) has been quickly developing and it is expected to revolutionize the X-ray imaging field. Rapid development has also been ongoing in the scintillator and photosensor fields: fast inorganic scintillators have become available on the market and the silicon photomultiplier (SiPM) technology has made substantial progress. Therefore, SiPM-based scintillator detectors may soon be successfully applied in challenging applications like PCCT. In this work, a simulation framework is developed to evaluate detectors composed of scintillators coupled to SiPMs. The idea is to combine Monte Carlo simulations and a library to generate SiPM signals. Focus is given to six fast inorganic scintillators currently on the market (Ce:LYSO, Ce:LuAP, Pr:LuAG, Ce:LuAG, Ce:GGAG (ceramic), Ce:GAGG (single crystal)) and different TiO2-epoxy mixtures. The main properties of scintillating and reflecting materials to be defined in the simulation database are characterised experimentally. Using a virtual model created within this framework, an analysis of the performance in PCCT of a SiPM-based scintillator detector composed of the considered materials is performed. Simulated pulses are processed to report on the count-rate capability. Among the studied crystals, Ce:LYSO would enable sustaining the highest rate of interacting X-ray (2.5 Mcps/pixel with 30% of pile-up). Ce:GAGG could also handle a rate ~ 1 Mcps/pixel with identical pile-up conditions. Other materials show a slow decay time in their scintillation kinetics, implicating a < 1 Mcps/pixel count rate. A qualitative evaluation of the energy binning efficiency is also accomplished, by defining this parameter as a quality metric for multiclass classification. Scintillators with high light yield and good energy resolution (Ce:LYSO, Ce:GAGG and Ce:GGAG) present the best energy binning performance, as expected. The dependence of this aspect is explored as a function of the pulse processing method used, the crystals size and the pile-up probability. Resulting trends respect predictions, even though for a more quantitative analysis a more in-depth study is required.

Entropy-driven phases at high coverage adsorption of straight rigid rods on three-dimensional cubic lattices.

Combining Monte Carlo simulations and thermodynamic integration method, we study the configurational entropy per site of straight rigid rods of length k (k-mers) adsorbed on three-dimensional (3D) simple cubic lattices. The process is monitored by following the dependence of the lattice coverage θ on the chemical potential μ (adsorption isotherm). Then, we perform the integration of μ(θ) over θ to calculate the configurational entropy per site of the adsorbed phase s(k,θ) as a function of the coverage. Based on the behavior of the function s(k,θ), different phase diagrams are obtained according to the k values: k≤4, disordered phase; k=5,6, disordered and layered-disordered phases; and k≥7, disordered, nematic and layered-disordered phases. In the limit of θ→1 (full coverage), the configurational entropy per site is determined for values of k ranging between 2 and 8. For k≥6, MC data coincide (within the statistical uncertainty) with recent analytical predictions [D. Dhar and R. Rajesh, Phys. Rev. E 103, 042130 (2021)2470-004510.1103/PhysRevE.103.042130] for very large rods. This finding represents the first numerical validation of the expression obtained by Dhar and Rajesh for d-dimensional lattices with d>2. In addition, for k≥5, the values of s(k,θ→1) for simple cubic lattices are coincident with those values reported in [P. M. Pasinetti et al., Phys. Rev. E 104, 054136 (2021)2470-004510.1103/PhysRevE.104.054136] for two-dimensional (2D) square lattices. This is consistent with the picture that at high densities and k≥5, the layered-disordered phase is formed on the lattice. Under these conditions, the system breaks to 2D layers, and the adsorbed phase becomes essentially 2D. The 2D behavior of the fully covered lattice reinforces the conjecture that the large-k behavior of entropy per site is superuniversal, and holds on d-dimensional hypercubical lattices for all d≥2.

Seismic reliability analysis of soil slope based on Newmark sliding block model with representative slip surfaces and response surfaces

AbstractThis paper presents a framework combining Monte Carlo Simulation (MCS) and the Newmark sliding block model with representative slip surfaces (RSS) (model II) and multiple response surfaces method (MRSM) to conduct seismic reliability analysis and risk assessment of soil slopes. An empirical threshold is introduced to define the limit state function to identify the failure samples in MCS and the sliding area and Newmark sliding displacement are multiplied to quantify the failure consequence. The proposed methodology is illustrated through a soil slope with multiple layers. The calculation results demonstrate that traditional Newmark sliding block model (model I) tend to underestimate the variations of yield acceleration. Both the failure probability and landslide risk exhibit decreasing trends with the increase of threshold. Significant discrepancy in failure probability and landslide risk between two models is found even for a small threshold. Therefore, the proposed methodology is highly recommended in seismic reliability analysis and risk assessment. The contributions of RSSs to the failure probability and landslide risk are insensitive to the variation of displacement thresholds.