Latin Hypercube Sampling Method Research Articles

Robust seismic fragility curves for concrete gravity dams incorporating aleatory, epistemic, and fuzzy uncertainties

Concrete gravity dams are critical infrastructure assets, and their failure in seismically active regions poses significant risks. The primary problem addressed in this research is the need for a reliable assessment of the seismic fragility of these dams to ensure safety in such areas. The research aims to develop a comprehensive methodology for seismic fragility analysis that accounts for various uncertainties affecting dam performance, including uncertainties in material properties and seismic characteristics and the fuzziness of damage state thresholds through the Latin Hypercube Sampling method. The methodology involves a three-stage approach: first, a finite element model is created using Abaqus software, incorporating uncertainties in material properties and earthquake characteristics using the Latin Hypercube Sampling method (LHS). Second, incremental dynamic analysis (IDA) defines damage states based on concrete cracking severity and overall stability, utilizing four performance categories of limit states (minor, moderate, extensive, and heavy). Finally, a mathematical model is introduced to characterize the ambiguity in damage state thresholds, recognizing the gradual nature of damage evolution. The primary findings demonstrate that fuzziness substantially influences the probability of exceedance across all damage states, with varying effects on different damage indices because of their differing frequency distributions. This research enhances the reliability and accuracy of seismic fragility assessments, contributing to more robust design guidelines and effective risk mitigation strategies for concrete gravity dams.

Uncertainty analysis of digital twin model of mine ventilation system

The digital twin model of mine ventilation system (DTMVS) plays an important role in intelligent safety management. However, the uncertainty of the ventilation resistance coefficient, which is the core parameter of the model, makes it challenging to accurately construct a DTMVS. In this study, Latin Hypercube Sampling (LHS) and ventilation resistance coefficient estimation models (VRCEMs) are used to analyze the uncertainty. First, the LHS method was used to explore the effect of uncertainty in the simulated airflow by continuously increasing the level of uncertainty in the ventilation resistance coefficients. Subsequently, the ventilation resistance coefficients were estimated using the VRCEMs, and the uncertainty of the ventilation resistance coefficient and the simulated airflow was analyzed. The results showed that the ventilation resistance coefficients with a 5% coefficient of variation can cause the DTMVS to lose 34% of the real airflow data points. The degree of uncertainty in the ventilation resistance coefficients estimated by the VRCEM-GA (VRCEM using genetic algorithm) and VRCEM-DE (VRCEM using differential evolution algorithm) methods was enhanced by 27.4% and 4.4%, respectively, compared with VRCEM-ES (VRCEM using evolutionary strategy algorithm). The VRCEM-ES model had the least influence on the uncertainty of the simulated airflow of DTMVS. The simulated airflow of the DTMVS constructed based on VRCEMs fluctuated normally within the confidence interval. VRCEMs had a higher sensitivity to the ventilation resistance coefficients of branches with low coefficients of variation.

Reliability-based multi-objective optimization design of composite patch repair structure using artificial neural networks

Due to the parameter differences and the existence of random factors, the performance of composite repair structure shows a large dispersion. In order to obtain stronger, lighter and more reliable repair structure, multi-objective optimization design of carbon fiber reinforced polymers (CFRP) laminate repair structure is investigated by combining the reliability theory, artificial neural networks and the genetic algorithm. First, a 3D simulation model of the composite laminate patch repair structures is established using the 3D Hashin criterion and the cohesive region model. Then, the Latin Hypercube sampling (LHS) method is used to realize the random sampling, and the strength proxy model of the repair structure is established by using Back-propagation artificial neural network, further a multi-objective optimization model with tensile strength, reliability and weight as objective functions is built considering the design parameters and the random parameters. Finally, NSGAⅡalgorithm is used to solve the multi-objective optimization problem, and a set of solutions on the Pareto front surface are obtained, also the optimal design parameters of the composite repair structure meets the requirements is obtained.

The Parameterized Oceanic Front-Guided PIX2PIX Model: A Limited Data-Driven Approach to Oceanic Front Sound Speed Reconstruction

In response to the demand for high-precision acoustic support under the condition of limited data, this study utilized high-resolution reanalysis data and in situ observation data to extract the Kuroshio Extension Front (KEF) section through front-line identification methods. By combining the parameterized oceanic front model and the statistical features of big data, the parameterized oceanic front was reconstructed. A proxy dataset was generated using the Latin hypercube sampling method, and the sound speed reconstruction model based on the PIX2PIX model was trained and validated using single sound speed profiles at different positions of the oceanic front, combined with the parameterized oceanic front model. The experimental results show that the proposed sound speed reconstruction model can significantly improve the reconstruction accuracy by introducing the parameterized front model as an additional input, especially in the shallow-water area. The mean absolute error (MAE) of the full-depth sound speed reconstruction for this model is 0.63~0.95 m·s−1, and the structural similarity index (SSIM) is 0.76~0.78. The MAE of the sound speed section within a 1000 m depth is reduced by 6.50~37.62%, reaching 1.95~3.31 m·s−1. In addition, the acoustic support capabilities and generalization of the model were verified through ray tracing models and in situ data. This study contributes to advancing high-precision acoustic support in data-limited oceanic environments, laying a solid groundwork for future innovations in marine acoustics.

An Analysis of and Improvements in the Gear Conditions of the Automated Mechanical Transmission of a Battery Electric Vehicle Considering Energy Consumption and Power Performance

The design of the gear quantity and transmission parameters of a vehicle has large effects on its economical and power performance. This paper mainly researches the gear conditions (including the gear quantity and each gear’s transmission parameters) of two-gear and three-gear AMT (Automated Mechanical Transmission). This research uses Cruise software to build a multi-gear simulation model of a BEV (Battery Electric Vehicle) and adopts the LHS (Latin hypercube sampling) method to design an experiment plan and conduct a simulation experiment. This paper proposes a systematic method for influencing factor analyses and the optimization of transmission parameters, combining fuzzy theory, multiple regression, and particle swarm optimization. The research results show that the gear quantity allowing for optimal overall performance is three. The highest score obtained in the results of the simulation experiment for three-gear AMT is 11.15% higher than that of the two-gear AMT. The optimal design plan for the two-gear AMT is a small ig1 with a big k1, in which case the highest score of the regression model increases by 2.67% compared with that before modeling. The optimal design plan for the three-gear AMT is a big k1 with a big k2, in which case the highest score of the regression model increases by 12.78% compared with that before modeling. Then, this research uses PSO (particle swarm optimization) to further optimize the regression models and compares the difference between the highest scores in the results of the simulation experiment. The difference between the highest scores of the three-gear and two-gear AMT further increases to 21.95% after optimization. As shown in the results, the key factor influencing the performance of two-gear and three-gear AMT is gear quantity.

System Reliability Analysis of Concrete Arch Dams Considering Foundation Rock Wedges Movement: A Discussion on the Limit Equilibrium Method

In this paper, a discussion on the applicability and limitations of the limit equilibrium method is presented. In this manner, the reliability of a concrete arch dam-foundation system under static loading is evaluated by considering a set of potentially moveable rock wedges in the foundation. The safety of the system is assessed utilizing a quasi-analytical method, which employs the limit equilibrium method and numerical analysis to calculate the sliding safety factors and the dam trust forces, respectively. The reliability is evaluated using the Latin Hypercube Sampling method. Random variables in the system are the friction angle, cohesion, and the Grout Curtain Efficiency Coefficient. In the end, the influence of two parametric variables of discontinuities, elastic slip and rock mass deformability modulus, on the rock wedges’ sliding safety factor is evaluated by comparing the results of the quasi-analytical method with the purely numerical method. The results show that in the case of complicated geotechnical conditions, the limit equilibrium method may not reflect real-world failure scenarios.

Interpretable machine learning models for predicting probabilistic axial buckling strength of steel circular hollow section members considering discreteness of geometries and material

Steel circular hollow section (CHS) members are widely utilized as axial force-resisting structural members in civil engineering structures. The buckling strength under axial loads is one of the critical parameters to determine the performance of the steel CHS members, which is significantly affected by the discreteness introduced by geometries, material, and initial imperfections. However, the reduction factor employed in the modern design codes (i.e. Chinese codes and EC3) only accounts for the reduction caused by all kinds of discreteness and does not reflect the impacts of every single discreteness and imperfection. To fill the gap, this paper proposed an interpretable machine-learning method to provide the probabilistic axial buckling strength of steel CHS members prediction result in a distribution form with the consideration of detailed discreteness. The model to predict the nominal axial buckling strength of steel CHS members was first developed utilizing ten machine learning algorithms after sufficient numerical simulations, where the numerical model was verified using test results. The artificial neural network (ANN) was selected for developing the prediction model due to its highly reliable performance in testing. The developed ANN models were further interpreted utilizing Shapley Additive exPlanations (SHAP) to determine the interrelationship of different parameters. Then, the probabilistic axial bucking strength prediction model was established based on the developed ANN models, where the Latin hypercube sampling method was applied to address the discreteness of geometries, material, and initial imperfections. The generated probabilistic axial bucking strength prediction model’s effectiveness was verified by the evidence that the machine learning prediction results can highly match the numerical results' probability density function and the result from codes while significantly reducing the computation time. Finally, the design parameters’ impact on the axial buckling strength’s discreteness was evaluated using the global sensitivity analysis (GSA) method. The result shows that the discreteness of design parameters substantially influences the distribution of the axial buckling strength of the steel CHS members and the proposed prediction model can provide an accurate probabilistic distribution prediction.

Design and optimization of the exhaust system for an aviation piston engine

Abstract To further enhance the performance of an aviation piston engine, an engine performance simulation analysis model was constructed by using GT-Power software to design improvements to its exhaust system. The impacts of different exhaust system schemes on the exhaust process, the in-cylinder combustion, and the heat release process at a cruising speed of 5500 RPM were analyzed. Based on the results, under cruising speed conditions, a new exhaust system was designed for the engine by referring to the structural form of a resonant exhaust system. The D-optimal Latin hypercube sampling method was used to obtain sample data for the experimental design of the structural parameters of the designed exhaust system. The response surface approximation models characterizing power and fuel consumption rate were established by using the least squares method. The significance of the impact of each structural parameter on engine performance was analyzed through a multifactor analysis of variance. The key structural parameters of the designed exhaust system were then subjected to multi-objective optimization. The results indicate that at 5500 RPM, the designed exhaust system can increase engine power by 2.88%; after optimization, engine power can be further increased by 3.15%, and the fuel consumption rate is reduced by 4.07%.

Research on turbulence model parameters for the prediction of waterjet propulsion pump hydrodynamic performance

Abstract The hydrodynamic performance of waterjet propulsion pumps has an important impact on improving the speed of waterjet propulsion ships. High-precision prediction of the hydrodynamic performance of water-jet propulsion pumps based on CFD is of great significance to improving the pump design quality. Based on the Reynolds-averaged Naiver-Stokes equation, this paper conducts a grid-independence verification on a water jet propulsion pump based on hydrodynamic test data, uses the Latin hypercube sampling method to generate SST k-ω turbulence model parameter space calculation samples, performs steady-state calculation on all samples, analyzes the sensitivity of changes in the parameters of the turbulence model to the prediction of the external characteristics of the waterjet propulsion pump, and finally obtains the influence of changes in the parameters of the turbulence model on the prediction of the external characteristics of the pump. Among them, the sensitivities of the three parameters β *, β1 and a1 are higher. The results show that the prediction accuracy of the external characteristics of the water jet propulsion pump can be effectively improved by correcting these three highly sensitive parameters, and the correction values of the three key parameters are obtained. The prediction error of the external characteristics of the water jet propulsion pump is reduced from 4.3% to less than 2.1%. A feedforward neural network is used to establish a mapping model between the three turbulence model parameters and the head coefficient and power coefficient of the waterjet propulsion pump in order to achieve rapid prediction. The genetic algorithm is applied to optimize the weights, thresholds and other coefficients in the neural network, thereby improving the prediction accuracy. Based on the optimized neural network, the genetic algorithm is used to optimize the three turbulence model parameters, and the optimized coefficients are obtained. The optimized turbulence model parameters are put into the hydrodynamic model for RANS simulation to verify the degree of optimization of the model coefficients by the genetic algorithm.

Study of the effects of shallow-buried charge parameters on the lower limb injuries of occupant in special vehicle body

In the process of military missions, special vehicles often encounter shallow buried charge threats that will greatly affect the safety of the occupants. To study the integrated effects of shallow buried charge detonation point, height-to-diameter ratio, burial inclination, weight, and other key parameters on the lower limb injuries of occupant, this study was based on the special body of a shallow buried explosive test and numerical calculation methods. The reliability of the numerical model and the load effect mechanism of the shallow burial charge were studied. Based on this, the lower limb injuries response law of occupant was analyzed with respect to each parameter. The results showed that the bottom charge load mainly came from two-stage ejection loading, in which the first stage was a strong load with short-duration local impact loading and the second stage was a weak load with long-duration extended range loading. By comparing the lower limb injury response of occupant under different charge parameters, we found that the weight of charge had the greatest effect on the extent of occupant injury, followed by the height-to-diameter ratio and burial inclination, while the detonation position had a relatively small effect on the injury magnitude. Finally, a comprehensive prediction model of the lower limb injuries of occupant was obtained by using the Latin hypercube sampling method and the least-squares method, and it was verified that the model had a good matching accuracy.

Groundwater Pollution Source and Aquifer Parameter Estimation Based on a Stacked Autoencoder Substitute

A concurrent heuristic search iterative process (CHSIP) is used for estimating groundwater pollution sources and aquifer parameters in this work. Frequent calls to carry out a numerical simulation of groundwater pollution have generated a huge calculated load during the CHSIP. Therefore, a valid means to mitigate this is building a substitute to emulate the numerical simulation at a low calculated load. However, there is a complicated nonlinear correlativity between the import and export of the numerical simulation on account of the large quantity of variables. This leads to a poor approach accuracy of the substitute compared to the simulation when using shallow learning methods. Therefore, we first built a stacked autoencoder substitute, using the deep learning method, to boost the approach accuracy of the substitute compared to the numerical simulation. In total, 400 training samples and 100 testing samples for the substitute were collected by employing the Latin hypercube sampling method and running the numerical simulator. The CHSIP was then employed for estimating the groundwater pollution sources and aquifer parameters, and the estimated outcome was obtained when the CHSIP was terminated. The data analysis, including interval estimation and point estimation, was implemented on the MATLAB platform. A relevant hypothetical case is set to verify our approaches, which shows that the CHSIP is helpful for estimating the groundwater pollution source and aquifer parameters and that the stacked autoencoder method can effectively boost the approach precision of the substitute for the simulator.

Multi-objective robust optimization design framework for low-pollution emission burners

The optimal design of low-pollution emission burners plays an important role in controlling pollutant emissions of industrial equipment, and is crucial for the sustainable development of the national economy and environmental protection. However, many uncertain factors challenge the optimal design of low-pollution emission burners. The Latin hypercube sampling (LHS) method was used to obtain sampling data representing the distribution of the uncertain variable. The training dataset was obtained using the turbulent combustion coupling model. A high-precision sparse polynomial chaos expansion (PCE) model was constructed by the degree-adaptive scheme and least angle regression (LAR) algorithm. Furthermore, the Legendre polynomial is introduced to establish a continuous robust optimization model. The model is carried out by the non-dominated sorting genetic algorithm II (NSGA-II). The results show that the excess air coefficient of 1.227 is optimal. Compared with the excess air coefficient of 1.20 under the discrete robust optimization, the optimal coefficient can further reduce pollutant emissions and bring strong robustness to the ethylene cracking furnace. It has also been proven that the continuous robust optimization scheme improves the optimization granularity. Compared with discrete robust optimization, this method reduces the number of samples by 66.7 %.

Hybrid uncertainty propagation for mechanical dynamics problems via polynomial chaos expansion and Legendre interval inclusion function

This paper investigates a non-intrusive hybrid uncertainty propagation framework in mechanical dynamic systems, utilizing polynomial chaos expansion (PCE) and Legendre inclusion function. Uncertainties with substantial knowledge and information are conceptualized as stochastic parameters and described using PCE, while Legendre polynomials are employed to represent uncertain, bounded parameters, specifically interval uncertainties. During the computation, the PCE model for each time step is developed through Galerkin projection and sparse grid numerical integration. Consequently, the statistical moments of the dynamic responses relative to the stochastic parameters are derived from the orthogonality of the PCE and transformed into functions reflecting the interval parameters. The interval bounds for these statistical moments are further determined using the Legendre inclusion function, which is obtained through optimal Latin hypercube sampling and collocation methods. Three dynamics examples, respectively, modeled by differential–algebraic equations, finite element method, and commercial software proves its applicability and superiority. Detailed assessment demonstrates that this method offers high computational efficiency and accuracy. As a non-intrusive approach, it poses no particular limitations on numerical methods for solving differential equations, making it universally applicable.

Fragility evaluation of bridge shallow foundation piers under floods by coupling simulation in structural and hydraulic fields

The erosion of soil around bridge shallow foundation piers can substantially increase the fragility of bridges during flood events. The available research for assessing the fragility of shallow foundation piers under floods is predominantly qualitative, while ignoring the influence of coupling effects arising from complex hydrodynamic-soil-structure interactions. To fill these gaps, a numerical solver for coupled simulations accounting for hydrodynamic-soil-structure interaction is developed, facilitating boundary conversions for different physical fields and updating them over time. The numerical solver is validated in terms of subgrade reactions, pier displacements and scour extent based on the results of open-channel tests of a reference shallow foundation pier. The validation results show that the developed coupled solver has satisfactory accuracy in predicting the morphology of scour-hole and failure process of shallow foundation piers. Then all potential failure modes of shallow foundation piers under floods are analyzed within the framework of deterministic analysis. Finally, a probabilistic fragility analysis using the coupled solver accounting for uncertainties in hydraulic, structural, and geological parameters is performed through the Latin Hypercube Sampling method. The study concludes that the deterministic analysis without considering the uncertainties in model parameters leads to underestimating the risk due to overturning failure and vertical failure.

Inverse design of TPMS piezoelectric metamaterial based on deep learning

Piezoelectric metamaterials are a class of composite structural materials with a piezoelectric effect that achieve efficient coupling between electrical and mechanical energy through their unique microscopic design. Typically, the tailoring of cellular architectures paves the way for unlocking the potential to achieve unprecedentedly designed materials with specific properties. We have developed three different inverse design frameworks for metamaterials, using deep learning (DL) techniques and underpinned by Bayesian optimisation. These frameworks reveal precise one-to-many correlations between material properties and geometric parameters within the triple periodic minimal surface (TPMS) piezoelectric metamaterial. For the first time, we have introduced the Progressive Latin Hypercube Sampling (PLHS) method to the inverse design process of metamaterials, highlighting the effectiveness of uniform sampling to improve the efficiency of deep learning models. The conditional variational autoencoder (cVAE), the conditional generative adversarial network (cGAN) and the multilayer perceptron (MLP) are studied for comparative analysis. All three models incorporate an inverse design network and a feature predictor to construct a continuous latent representation of a large number of hybrid TPMS (H-TPMS). After extensive training, the models can generate various piezoelectric porous structures that accurately meet the required material property parameters. Our results demonstrate the high accuracy of cVAE in inverse design, with 98.92% agreement for piezoelectric values and 96.44% agreement for mechanical properties. The three inverse design frameworks show different generalisation capabilities, with cVAE performing best. This study thoroughly analyses the strengths and limitations of each model, providing valuable insights for researchers seeking inverse design recommendations.

A New Latin Hypercube Sampling with Maximum Diversity Factor for Reliability-Based Design Optimization of HLM

This research paper presents a new Latin hypercube sampling method, aimed at enhancing its performance in quantifying uncertainty and reducing computation time. The new Latin hypercube sampling (LHS) method serves as a tool in reliability-based design optimization (RBDO). The quantification technique is termed LHSMDF (LHS with maximum diversity factor). The quantification techniques, such as Latin hypercube sampling (LHS), optimum Latin hypercube sampling (OLHS), and Latin hypercube sampling with maximum diversity factor (LHSMDF), are tested against mechanical components, including a circular shaft housing, a connecting rod, and a cantilever beam, to evaluate its comparative performance. Subsequently, the new method is employed as the basis of RBDO in the synthesis of a six-bar high-lift mechanism (HLM) example to enhance the reliability of the resulting mechanism compared to Monte Carlo simulation (MCS). The design problem of this mechanism is classified as a motion generation problem, incorporating angle and position of the flap as an objective function. The six-bar linkage is first adapted to be a high-lift mechanism (HLM), which is a symmetrical device of the aircraft. Furthermore, a deterministic design, without consideration of uncertainty, may lead to unacceptable performance during the manufacturing step due to link length tolerances. The techniques are combined with an efficient metaheuristic known as teaching–learning-based optimization with a diversity archive (ATLBO-DA) to identify a reliable HLM. Performance testing of the new LHSMDF reveals that it outperforms the original LHS and OLHS. The HLM problem test results demonstrate that achieving optimum HLM with high reliability necessitates precision without sacrificing accuracy in the manufacturing process. Moreover, it is suggested that the six-bar HLM could emerge as a viable option for developing a new high-lift device in aircraft mechanisms for the future.

Reliability assessment for pipelines corroded by longitudinally aligned defects

Internal corrosion poses a significant threat to offshore pipeline services. Toward offshore pipeline integrity management, this paper aims to use Finite Element Method (FEM) to assess the reliability of corroded pipelines. The FEM analysis was conducted using ABAQUS software, facilitated through Python scripts, to optimize the modeling process on a large scale. Additionally, the input database was generated through a random sampling method using the Latin Hypercube Sampling (LHS) method. The reliability of 32” oil and gas offshore internally corroded pipelines has been evaluated by fragility curve assessment along with Incremental Pressure Analysis (IPA) which is a new method inspired by Incremental Dynamic Analysis (IDA). Multiple corrosion defects pose a higher risk to offshore pipelines during service, with potentially more severe consequences than a single defect alone. Therefore, the interaction of corrosion defects has been considered in this study. Following the reliability assessment the most appropriate Probability Density Function (PDF) for the Maximum Von Mises Stress (MVMS) and failure probability at various Internal Pressure (IP) and longitudinal spacing (Sl) levels has been evaluated.

Hydrodynamic shape optimization of an autonomous and remotely-operated vehicle via a multi-surrogate model

The design requirements for the hydrodynamic performance of underwater vehicles vary significantly depending on the application. Optimization without an initial model results in challenges such as large design domains, nonlinear complexity, and high data requirements. To optimize the hydrodynamic performance during the design process, in this paper, a multi-surrogate model was employed to progressively design the shape of an autonomous and remotely-operated vehicle. Based on the data characteristics of different stages, this approach strikes a balance among data quantity, prediction accuracy, and multi-objective requirements. An artificial neural network prediction surrogate model was constructed based on the principle of the minimum prediction factor using the optimal Latin hypercube sampling method. During the optimization of the design domain, the optimum objective was to minimize the dimensionless force (G(x)). During the multi-objective optimization stage, a regression kriging surrogate model was constructed based on a support vector product. The optimization objectives were to maximize volume and minimize dimensionless forces. This enabled the overall design process to attain optimal Pareto solutions within the design domain while simultaneously ensuring high prediction accuracy and minimum data requirements. The results obtained are consistent with the simulation comparison, thus verifying the reliability of the entire optimization process.

Optimization Design and Internal Flow Characteristics Analysis Based on Latin Hypercube Sampling Method

Optimization Design and Internal Flow Characteristics Analysis Based on Latin Hypercube Sampling Method

A New Fuzzy Bayesian Inference Approach for Risk Assessments

Bayesian network (BN) inference is an important statistical tool with additive symmetry. However, BN inference cannot deal with the uncertain, fuzzy, random, and conflicting information from experts’ knowledge in the process of conducting a risk assessment. To tackle this issue, a new fuzzy BN inference approach for risk assessments was proposed based on cloud model (CM), interval type-2 fuzzy set (IT2 FS), interval type-2 fuzzy logic system (IT2 FLS), modified Dempster–Shafer (D-S) evidence theory (ET), and Latin hypercube sampling (LHS) methods along the following lines. Firstly, CM was integrated into IT2 FS, and CM-based IT2 FS (CM-IT2 FS) was defined in the IT2 FLS. Secondly, modified D-S ET was utilized to determine the CM-IT2 FS-based a priori probabilities, and the CM-IT2 FS-based BN model was established. Thirdly, the CM-IT2 FS-based a priori probabilities were reduced to the CM-IT1 FS-based ones using a type reducer in the IT2 FLS, LHS was applied to propose a new fuzzy BN inference algorithm, and then, the new algorithm was used in a typical case to perform the fuzzy BN positive inference for risk prediction and the fuzzy BN reverse inference for risk sensitivity analysis. Finally, the BN inference results were analyzed using the proposed algorithm and the two common BN inference algorithms, and the effectiveness of the proposed approach was validated. It can be concluded that the proposed approach was both accurate and promising.