Results Of Numerical Modeling Research Articles

Self-consistent dynamical models with a finite extent – III. Truncated power-law spheres

ABSTRACT Fully analytical dynamical models usually have an infinite extent, while real star clusters, galaxies, and dark matter haloes have a finite extent. The standard method for generating dynamical models with a finite extent consists of taking a model with an infinite extent and applying a truncation in binding energy. This method, however, cannot be used to generate models with a preset analytical mass density profile. We investigate the self-consistency and dynamical properties of a family of power-law spheres with a general tangential Cuddeford (TC) orbital structure. By varying the density power-law slope γ and the central anisotropy β0, these models cover a wide parameter space in density and anisotropy profiles. We explicitly calculate the phase–space distribution function for various parameter combinations, and interpret our results in terms of the energy distribution of bound orbits. We find that truncated power-law spheres can be supported by a TC orbital structure if, and only if, γ ≥ 2β0, which means that the central density slope–anisotropy inequality is both a sufficient and a necessary condition for this family. We provide closed expressions for structural and dynamical properties such as the radial and tangential velocity dispersion profiles, which can be compared against more complex numerical modelling results. This work significantly adds to the available suite of self-consistent dynamical models with a finite extent and an analytical description.

Algorithm for automation of the method of testing statistical hypotheses according to the Pearson criterion by the means of MATLAB

The purpose of the work is to develop and improve the methods of computer modeling and digitalization of procedures for testing statistical hypotheses according to consent criteria for use in high-tech software and measuring complexes of automated control systems. In most cases that are important for practice, the analysis of the statistical properties of distributions requires multiple checks performed using numerical methods, which require the tools of powerful software and computing environments. Examples of such cases are the procedures for finding information subsets of features in regression and statistical analysis, as well as identifying parameter deviations in automated control systems. It is noted that the tasks of processing experimental data do not always fit into the framework of the theory of normal processes and have exclusively such hypotheses. To test such hypotheses, modern mathematical statistics has developed criteria based on specially developed probabilistic rules that distinguish between zero and alternative hypotheses. Moreover, if for the testable (zero) hypothesis the distribution of statistics is known exactly or approximately, then for the alternative it has a greater degree of uncertainty and it (the alternative) itself is not a complete negation of the null hypothesis. As the zero and alternative hypotheses move away from each other, the statistics of the consent criterion begins to take a larger absolute value than with the null hypothesis, which allows us to build a critical area on the basis of estimating the set of this statistics values. However, the use of analytical methods for statistical analysis to determine the discrepancy between empirical and hypothetical statistics in practice is associated with the use, as a rule, of complex computational calculation algorithms and the implementation of high requirements for the assessment accuracy, which they cannot provide. To solve this problem, it is proposed to use numerical methods of computer modeling and integration of probability density of distribution according to the Simpson algorithm using the tools of built-in functions of the Statistics Toolbox of the MATLAB environment. The proposed method for automation of statistical analysis of zero and alternative hypotheses is implemented on the example of applying the Pearson criterion to verify compliance with the normal distribution of two random data samples, one of which meets the normal distribution law. The obtained graphic and numerical results of computer modeling confirm the correspondence of one of the testable distributions to a given normal distribution according to the null hypothesis.

Influence of Blade Exit Angle on the Performance and Internal Flow Pattern of a High-Speed Electric Submersible Pump

The impeller vane exit placement angle has a critical role in the flow characteristics of the fluid inside the lobe, thus having a profound effect on the overall pump performance. The purpose of this study is to investigate the effect of the impeller exit angle on the operating characteristics of a high-speed well submersible pump, and the numerical calculation results of the original model are in good agreement with the experimental results. In this paper, five different impeller vane exit angles, namely 10°, 15°, 20°, 25° and 30°, are selected for numerical analysis based on the original model, and the flow conditions of 0.6 Q, 1.0 Q and 1.4 Q are analyzed for each angle. The results show that the impeller vane exit placement angle not only affects the static pressure distribution, velocity distribution and streamline distribution within the impeller and guide vane, but also has a significant effect on the head curve, power curve and efficiency curve of the well submersible pump. As the flow slip inside the impeller of high-speed well submersible pumps intensifies, the large impeller outlet angle will cause the power of the impeller to increase linearly with the flow rate, thus reducing the pump efficiency. In the low-flow and high-flow conditions, a small outlet angle of 10° will make the efficiency of high-speed submersible pumps higher than in other conditions, and these findings can provide some reference for the optimal design of high-speed submersible pumps.

Insight into influence of thermal radiation and Cattaneo–Christov model on ternary hybrid nanofluid (TiO2–Al2O3–MoS2/kerosene oil) and hybrid nanofluid (TiO2–Al2O3/kerosene oil) flow and heat transfer over a stretching sheet

The present research reports the impact of the Cattaneo–Christov (C-C) model and thermal radiation in concert with the suction/injection on boundary conditions on TiO2–Al2O3–MoS2/Kerosene oil ternary hybrid nanofluid (THNF) over a stretching sheet. The authors have also done a comparative analysis of the heat and mass transfer rate of ternary hybrid nanofluid and TiO2–Al2O3/Kerosene oil hybrid nanofluid flow. The code validation is done with a published study as a special case to validate the numerical code used to derive the numerical results of the present flow model. The flow equations are altered into ODEs with the help of similarity transformation, which are solved in MATLAB by the “bvp4c technique.” The graphs are drawn w.r.t. various parameters: heat source/sink, radiation, chemical reaction, suction/injection, thermal relaxation, and magnetic parameter. The influence of the aforementioned parameters on the velocity profile, thermal profile, concentration profile, heat, and mass transfer profile is displayed by graphs and tabular values. The heat and mass transfer rate enhance by elevating the value of the heat source/sink and radiation parameter. On boosting the magnitude of injection at the surface, the rate of mass transfer elevates. The results of this study can be used in various applications including drainage issues, chemical engineering, solar panels, solar absorption and filtration, ground stream hydrology, solar cells, and others, where flow over a sheet illustrates its significance and applicability.

Modelling of subsurface geological structure related to anomalous magnetotelluric phase data exceeding 90° in Central India

SUMMARY We note anomalous phase (>90°) in magnetotelluric (MT) measurements at several stations along the Narmada river course normal to the Tan Shear Zone (TSZ) in central India. We made efforts to derive a possible cause of the anomalous phase by comparing the results of numerical modelling based on the tectonics of the TSZ with the measured impedance tensor data. A zone of multiple reactivations of the TSZ leads to the formation of a damage zone and conjugate Riedel shear faults parallel and normal to the TSZ, respectively. The conjugate Riedel shear faults act as a pathway for the Narmada River course. The multiple tectonic activities associated with the TSZ induce surficial heterogeneity that results in distortion in the MT data. The Mohr circle and phase tensor analysis establish the effect of distortion in the data. To understand the cause of distortion as well as anomalous phase in the present data, we carried out synthetic 3-D modelling. We modelled the damage zone and conjugate Riedel shear faults by a cross shape. From the results, we noted that the near-surface 3-D heterogeneity across the TSZ critically distorted the YX component of the MT impedance tensor due to the strong current channelling. When the current flows across the conjugate Riedel shear zone (Narmada river course) charges are accumulated along the boundaries of near-surface heterogeneity. The charges associated with the current channelling brought out the reversion of the electric field direction that is reflected in the form of an anomalous phase in the YX component of the impedance tensor in measured data.

Remote Sensing and Geovisualization of Rock Slopes and Landslides

Over the past two decades, advances in remote sensing methods and technology have enabled larger and more sophisticated datasets to be collected. Due to these advances, the need to effectively and efficiently communicate and visualize data is becoming increasingly important. We demonstrate that the use of mixed- (MR) and virtual reality (VR) systems has provided very promising results, allowing the visualization of complex datasets with unprecedented levels of detail and user experience. However, as of today, such visualization techniques have been largely used for communication purposes, and limited applications have been developed to allow for data processing and collection, particularly within the engineering–geology field. In this paper, we demonstrate the potential use of MR and VR not only for the visualization of multi-sensor remote sensing data but also for the collection and analysis of geological data. In this paper, we present a conceptual workflow showing the approach used for the processing of remote sensing datasets and the subsequent visualization using MR and VR headsets. We demonstrate the use of computer applications built in-house to visualize datasets and numerical modelling results, and to perform rock core logging (XRCoreShack) and rock mass characterization (EasyMineXR). While important limitations still exist in terms of hardware capabilities, portability, and accessibility, the expected technological advances and cost reduction will ensure this technology forms a standard mapping and data analysis tool for future engineers and geoscientists.

Impact of Geotechnical Barrier Modelled in the Vicinity of a Building Structures Located in Mining Area

The paper presents a new geotechnical solution indicating a possibility of effective building structures protection. The presented solutions enable minimization of negative effects of underground mining operations. Results of numerical modelling have been presented for an example of design of preventive ditches reducing the influence of mining operations on the ground surface. To minimize the mining damage or to reduce its reach it is reasonable to look for technical solutions, which would enable effective protection of building structures. So far authors concentrated primarily on the development of building structure protection methods to minimize the damage caused by the underground mining. The application of geotechnical methods, which could protect building structures against the mining damage, was not considered so far in scientific papers. It should be noticed that relatively few publications are directly related to those issues and there are no practical examples of effective geotechnical protection. This paper presents a geotechnical solution indicating a possibility of effective protection of building structures. The presented solutions enable minimization of negative effects of underground mining operations. Results of numerical modelling have been presented for an example of design of preventive ditches reducing the influence of mining operations on the ground surface. The calculations were carried out in the Abaqus software, based on the finite element method.

Influence of the Anhydrite Interbeds on a Stability of the Storage Caverns in the Mechelinki Salt Deposit (Northern Poland)

This paper presents a complex study of anhydrite interbeds influence on the cavern stability in the Mechelinki salt deposit. The impact of interbeds on the cavern shape and the stress concentrations were also considered. The stability analysis was based on the 3D numerical modelling. Numerical simulations were performed with use of the Finite Difference Method (FDM) and the FLAC3D v. 6.00 software. The numerical model in a cuboidal shape and the following dimensions: length 1400, width 1400, height 1400 m, comprised the part of the Mechelinki salt deposit. Three (K-6, K-8, K-9) caverns were projected inside this model. The mesh of the numerical model contained about 15 million tetrahedral elements. The occurrence of anhydrite interbeds within the rock salt beds had contributed to the reduction in a diameter and irregular shape of the analysed caverns. The results of the 3D numerical modelling had indicated that the contact area between the rock salt beds and the anhydrite interbeds is likely to the occurrence of displacements. Irregularities in a shape of the analysed caverns are prone to the stress concentration. However, the stability of the analysed caverns are not expected to be affected in the assumed operation conditions and time period (9.5 years).

Analytical model of vertical load acting on jacked pipe considering soil arching effect in cohesionless soil

For the project of pipe jacking in cohesionless soil, it is key to determine the vertical load on jacked pipe so as to predict the jacking force accurately. In this paper, a new parabolic soil arching model was proposed to calculate the vertical load on jacked pipe. This proposed analytical model was composed of parabolic soil arching zone, parabola-typed collapse zone and friction arch zone. Combined with existing literature, the key parameters (i.e., height of parabolic soil arching, horizontal pressure coefficient and width and height of friction arch) were determined. In addition, considering that the trajectory of major stress is parabola, the formula of horizontal pressure coefficient was deduced in the friction arch. The parabolic soil arching zone is assumed as a three-hinged arch with reasonable arch axis, and the formula of load transfer was derived considering the transition effect of parabolic soil arching. The results of experiment, theoretical models and numerical model were adopted to verify the proposed analytical model. Finally, the influence of the key parameters on the vertical load on jacked pipe were also discussed in detail. This work provides a meaningful reference for evaluating the vertical load on jacked pipe for design of pipe jacking.

Traveling salesman problem solution using magnonic combinatorial device

Traveling salesman problem (TSP) is a decision-making problem that is essential for a number of practical applications. Today, this problem is solved on digital computers exploiting Boolean-type architecture by checking one by one a number of possible routes. In this work, we describe a special type of hardware for the TSP solution. It is a magnonic combinatorial device comprising magnetic and electric parts connected in the active ring circuit. There is a number of possible propagation routes in the magnetic mesh made of phase shifters, frequency filters, and attenuators. The phase shifters mimic cities in TSP while the distance between the cities is encoded in the signal attenuation. The set of frequency filters makes the waves on different frequencies propagate through the different routes. The principle of operation is based on the classical wave superposition. There is a number of waves coming in all possible routes in parallel accumulating different phase shifts and amplitude damping. However, only the wave(s) that accumulates the certain phase shift will be amplified by the electric part. The amplification comes first to the waves that possess the minimum propagation losses. It makes this type of device suitable for TSP solution, where waves are similar to the salesmen traveling in all possible routes at a time. We present the results of numerical modeling illustrating the TSP solutions for four and six cities. Also, we present experimental data for the TSP solution with four cities. The prototype device is built of commercially available components including magnetic phase shifters/filters, coaxial cables, splitters, attenuators, and a broadband amplifier. There are three examples of finding the shortest route between the cities for three different sets of city-to-city distances. The proposed approach is scalable to TSP with a larger number of cities. Physical limits and challenges are also discussed.

Defect Detection in CFRP Concrete Reinforcement Using the Microwave Infrared Thermography (MIRT) Method—A Numerical Modeling and Experimental Approach

This research paper presents the application of the microwave infrared thermography (MIRT) technique for the purpose of detecting and characterizing defects in the carbon-fiber-reinforced polymer (CFRP) composite reinforcement of concrete specimens. Initially, a numerical model was constructed, which consisted of a broadband pyramidal horn antenna and the specimen. The present study investigated the application of a 360 W power system that operated at a frequency of 2.4 GHz, specifically focusing on two different operational modes: continuous and modulated. The specimen being examined consisted of a solid concrete slab that was coated with an adhesive layer, which was then overlaid with a layer of CFRP. Within the adhesive layer, at the interface between the concrete and CFRP, there was a defect in the form of an air gap. The study examined three distinct scenarios: a sample without any defects, a sample with a defect positioned at the center, and a sample with a defect positioned outside the center. The subsequent stage of the investigation incorporated experimental verification of the numerical modeling results. The experiment involved the utilization of two concrete specimens reinforced using CFRP, one without any defects and the other with a defect. Numerical modeling was used in this study to analyze the phenomenon of microwave heating in complex structures. The objective was to evaluate the selected antenna geometry and determine the optimal experimental configuration. Subsequently, these findings were experimentally validated. The observations conducted during the heating phase were particularly noteworthy, as they differed from previous studies that only performed observation of the sample after the heating phase. The results show that MIRT has the potential to be utilized as a method for identifying defects in concrete structures that are reinforced with CFRP.

Using Non-Standard Finite Difference Scheme to Study Classical and Fractional Order SEIVR Model

In this study, we considered a model for novel COVID-19 consisting on five classes, namely S, susceptible; E, exposed; I, infected; V, vaccinated; and R, recovered. We derived the expression for the basic reproductive rate R0 and studied disease-free and endemic equilibrium as well as local and global stability. In addition, we extended the nonstandard finite difference scheme to simulate our model using some real data. Moreover, keeping in mind the importance of fractional order derivatives, we also attempted to extend our numerical results for the fractional order model. In this regard, we considered the proposed model under the concept of a fractional order derivative using the Caputo concept. We extended the nonstandard finite difference scheme for fractional order and simulated our results. Moreover, we also compared the numerical scheme with the traditional RK4 both in CPU time as well as graphically. Our results have close resemblance to those of the RK4 method. Also, in the case of the infected class, we compared our simulated results with the real data.

Effects of strain rate and strain softening on the installation of helical pile in soft clay

Helical piles are alternative foundation for offshore wind turbine as a green foundation. When a helical pile is screwed into soft clay, the soils exhibit strain-rate dependency as well as strain softening effect when sheared and remolded. In this study, large deformation finite-element (LDFE) model is established to investigate the penetration behavior of helical pile, where the soil flow mechanism and penetration resistance are studied considering strain rate effect and strain softening effect in soil. The numerical model is validated by comparing with available centrifuge results. Parametric studies are then conducted covering a wide range of strain rate and strain softening parameters including the strain-rate parameter, the sensitivity and ductility of the soil, and the original shear strength. It is found that the penetration behavior of helical pile is significantly influenced by the strain rate and strain softening of soil. The difference of the torque profiles between the ideal Tresca soil and the rate-dependent, strain-softening soil is 0–30%. Based on the numerical modelling results, the theoretical model to predict torque requirements for the installation of helical piles is extended by accounting for the strain rate effect and strain softening effect.

A multi-scale control framework for urban traffic control with connected and automated vehicles

Urban traffic control is multi-scale in both temporal and spatial domains, which is of great importance to improve the quality and efficiency of our transportation system. In this paper, we propose a general multi-scale modeling and control framework for urban traffic control with connected and automated vehicles (CAVs). We apply the proposed multi-scale framework to explicitly model and solve the two-scale signal-vehicle coupled control (SVCC) problem under the environment of full penetration of CAVs. To solve the SVCC problem, we propose a model predictive control (MPC) scheme and develop a stability analysis method based on the concept of consistency of key states between the two scales. We show numerical results of the SVCC model and comparisons with some benchmark methods. We also discuss how to extend the SVCC model to the mixed traffic flow and tri-scale/multi-scale urban traffic control problems.

Coseismic Coulomb stress changes on intraplate faults in the western Quebec seismic zone following three major earthquakes in the past century

There is currently no active fault map for the intraplate western Quebec seismic zone (WQSZ) in eastern Canada, and consequently, no detailed finite-fault source models, which are critical for seismic hazard assessments in this region with a rapidly growing population. While previous numerical stress modelling studies have shown that mostly NNW–SSE to NW–SE-striking faults exhibit the highest potential for reactivation under the present-day tectonic stress field, such modelling is unable to take into account the interaction of faults and earthquakes. This study attempts to identify possible future rupture zones using Coulomb stress analysis. We explore the static stress transfer caused by the 1935 M W 6.1 Témiscaming, 1944 M W 5.8 Cornwall–Massena, and 2013 M W 4.7 Ladysmith earthquakes, which are proximal to faults in the WQSZ that exhibit a relatively high reactivation potential, to determine whether these faults have an increased potential for failure. The significance of Coulomb stress changes (ΔCFS) observed on the nearby “receiver” faults varied widely. Among the events analyzed in this study, only the 1935 M W 6.1 Témiscaming earthquake caused extensively positive ΔCFS (≥0.1 bar) on its receiver fault. The areal extent of the receiver fault that has been promoted to failure suggests that earthquakes with a comparable magnitude to the 1935 event can be triggered. This work is the first attempt to provide a physical basis for seismic hazard assessment input parameters in the WQSZ based on the results of numerical stress modelling.

Investigation of anchorage plate opening on bursting force

Three-dimensional (3D) Finite Element analyses were performed to investigate transverse stresses and bursting forces at the anchorage zone in post-tensioned concrete beams. The behavior of the concrete beam was investigated through linear elastic Finite Element analyses. To validate the Numerical modeling results, an experimental study was performed on a beam and results are compared in terms of ultimate strength. The effect of duct hole and eccentricity are investigated by varying diameter and eccentric distances on the concrete beam. Furthermore, the results are validated for bursting stress and bursting force. Altogether, the equation of bursting force in IS 1343:2012 hasn’t considered the effect of duct hole and eccentricity, the results are compared with IS 1343:2012, and an equation is presented considering the effect of duct and eccentricity.

STABILIZATION OF DISCRETE-TIME SYSTEMS WITH STATE-DELAYS AND SATURATING CONTROL INPUTS

Context. The presence of time delays occurs in many complex dynamical systems, particularly in the areas of modern communication and information technologies, such as the problem of stabilizing networked control systems and high-speed communication networks. In many cases, time-delays lead to a decrease in the efficiency of such systems and even to the loss of stability. In the last decade, many interesting solutions using the Lyapunov-Krasovskii functional have been proposed for stability analysis and synthesis of a stabilizing regulator for discrete-time dynamic systems with unknown but bounded state-delays. The presence of nonlinear constraints on the amplitude of controls such as saturation further complicates this problem and requires the development of new approaches and methods.
 Objective. The purpose of this study is to develop a procedure for calculating the control gain matrix of state feedback that ensures the asymptotic stability of the analyzed system, as well as a procedure for calculating the maximum permissible value of the state-delay under which the stability of the closed-loop system can be ensured for a given set of admissible initial conditions.
 Method. The paper uses the method of descriptor transformation of the model of a closed-loop system and extends the invariant ellipsoids method to systems with unknown but bounded state-delays. The application of the Lyapunov-Krasovskii functional and the technique of linear matrix inequalities made it possible to reduce the problem of calculating the control gain matrix to the problem of semi-definite programming, which can be solved numerically. An iterative algorithm for solving the bilinear matrix inequality is proposed for calculating the maximum permissible value of the time-delay.
 Results. The results of numerical modeling confirm the effectiveness of the proposed approach in the problems of stabilizing discrete-time systems under the conditions of state-delays and nonlinear constraints on controls, which allows to recommend the proposed method for practical use for the problem of stability analysis and synthesis of stabilizing regulator, as well as for calculating the maximum permissible value of time-delay.
 Conclusions. An approach is proposed that allows extending the invariant ellipsoids method to discrete-time dynamic systems with unknown but bounded state-delays for solving the problem of system stabilization using static state feedback based on the application of the Lyapunov-Krasovskii functional. The results of numerical modeling confirm the effectiveness of the proposed approach in the presence of the saturation type nonlinear constraints on the control signals.

Derivation of heterogeneous material distributions and their sensitivity to HM-coupled two-phase flow models exemplified with the LASGIT experiment

Advective gas transport in bentonite, a possible buffer material in repositories for radioactive materials, is difficult to simulate in numerical continuum models, partly due to the complicated microstructure of bentonite. To generate reliable models of repositories nevertheless, spatially distributed heterogeneous material properties can be used to allow localization of gas flow. In this study, a pore-size-dependent stochastic approach of the gas entry pressure is derived from Mercury Intrusion Porosimetry, which is used to replicate measurements from the LASGIT experiment. In addition, three benchmark tests are simulated to investigate the dependence of heterogeneous distributions of material properties on the mesh discretization, the temporal dependence, and the coupling between the processes influenced by the heterogeneous parameters. The numerical modeling results of the LASGIT experiment show that the onset of gas flow into the system and the subsequent increase in pressure and stress can be well reproduced using heterogeneous distributions. Compared to a model with homogeneous material properties, heterogeneous distributions may allow the generation of dilatancy-controlled microfractures—an important feature with regard to the advective gas flow in bentonites. However, it can be observed that the heterogeneous distributions in LASGIT are less significant, as technical gaps or differences in material types could have a greater impact.

Effectiveness of compensatory injection as a means of regulating the stresses in complex geotechnical conditions

Deep excavations in dense urban areas often result in additional settlement of buildings that fall within the influence area of new construction. The most sensitive to such settlements, according to the normative literature, are historic buildings. One of the ways to regulate additional building settlement in the influence area of deep excavations is compensatory injection. The hydraulic fracture network created in the foundation creates additional lateral compression of the soil and changes its physical and mechanical properties. The article deals with a real example of an emergency situation at one of the objects in Tyumen, where during excavation of a pit with the designed depth of 7 meters for the construction of a five-storey building with an extended two-level underground space significant excessive settlement of the adjacent historic building and damage to its bearing structures were revealed as a consequence of the additional settlement development. The purpose of the present study is to evaluate the effectiveness of compensatory injection using collar technology to reduce excessive additional settlement of buildings in the zone of influence of deep excavations. A comprehensive analysis of the impact of new construction on the surrounding environment has been carried out by the authors of the article by means of numerical simulation. Analyzing the results of numerical modelling, the authors of the article developed measures to eliminate the emergency situation and stabilise the foundation. Based on the results of geodetic monitoring, it was found that the installation of a geotechnical barrier by injecting soil using the collar technique not only stabilised the foundation but also brought the building into position to meet regulatory requirements.

Energy Transfer and Reverse Flow Characteristics in the Interaction Process between Non-Breaking Solitary Wave and a Steep Seawall: A Case Study

This study utilized a two-dimensional numerical viscous wave tank to simulate the run-up and run-down processes of non-breaking solitary waves on a steep seawall. The research aimed to investigate the transformation between wave potential energy and kinetic energy, the evolution mechanisms of the wave and flow fields, and the correlation between the dynamic pressure gradient and the reverse flow near the sloping bed. The numerical model results were consistent with laboratory measurements of free surface elevations and flow velocity profiles, demonstrating the accuracy of the numerical model. This study focused on a solitary wave with a wave-height-to-water-depth ratio of 0.15, propagating on a representative seawall with a steep slope of 1:3 along the western coast of Taiwan. The simulation results indicate that the maximum run-up height occurs when the potential energy is at its highest. Undertow is caused by the adverse pressure gradient within the flow field, and the dynamic pressure on the sloping bed is directly proportional to the free surface elevation. Therefore, by observing the spatial changes in the free surface elevation, we can indirectly determine the occurrence time of undertow.