Depth-averaged vs. Full 3-D SPH Models: A Comparison of Accuracy and Computational Speed for Simulating Dam Break Flash Flood

. Dam-break flow is a three-dimensional (3-D) phenomenon that is often numerically modeled using depth-averaged formulation, omitting the 3-D effect for computational efficiency. This study compares the performance and accuracy of depth-averaged and 3-D models, particularly using Smoothed Particles Hydrodynamics (SPH) method. Both models accurately predict flow evolution, with the 3-D model capturing detailed 3-D effects. However, the depth-averaged model shows significantly shorter computation time (by 20 times) and memory usage due to fewer particles used. However, it shows slower computational effort per particle due to the time-consuming Newton-Raphson iterative procedure. This study provides insights into the performance and accuracy of two commonly used models in simulating violent fluid dynamics, aiding model selection for specific applications.


Introduction
Flash floods can cause significant damage to infrastructure and threaten human life, making accurate predictions of flow dynamics critical for mitigating their impact.Dam breaks are one of the most common causes of flash floods, and simulating such events can aid in developing strategies to minimise their impact.Numerical simulations are an effective tool for predicting flow dynamics, and the Smoothed Particle Hydrodynamics (SPH) method has become increasingly popular for such simulations due to its ability to handle complex geometries and free surfaces.
The SPH method discretises fluid as a set of particles and represents the fluid properties (e.g., velocity, pressure, and density) as a continuous field through interpolation between the particles.Unlike traditional grid-based methods, SPH particles are free to move independently, and there is no need to impose a fixed grid structure.The method is also Lagrangian, meaning that the particles move with the fluid [1].
The origins of the SPH method can be traced back to the 1970s when it was initially introduced as a technique for simulating astrophysical phenomena [2,3].Over the years, significant progress has been made in the underlying theory and hardware acceleration utilising graphics processing units (GPUs), which has resulted in the SPH method becoming increasingly robust, accurate, and used across various engineering fields [4,5].
The shallow water equations (SWEs) have been widely used to model the behaviour of water in large horizontal domains.In recent years, researchers have explored the use of SPH to solve the SWEs.Ata and Soulaïmani [6] and Rodriguez-Paz and Bonet [7] were among the first researchers to apply the SPH method to the SWEs.Since then, various advancements have been made by different researchers in the field such as particle splitting [8] and coalescing [9] algorithm, modified virtual boundary condition [10], open boundary condition [11], single node central processing unit (CPU) parallelisation [12], GPU parallelisation [13], rainfall term [14], 1-D model with junctions [15], positivity preserving [16] and Eulerian scheme [17].These advancements have led to the development of robust and accurate SPH-based models that are capable of simulating shallow water flows with high fidelity.Despite these advancements, a comprehensive analysis of the performance of the depth-averaged SPH model in comparison to the 3-D model has yet to be conducted.
This paper aims to provide a detailed quantitative analysis of the performance of the depth-averaged SPH model compared to the full 3-D version, particularly in simulating dam break flow.The comparison will be based on accuracy and computational performance, taking into consideration factors such as the number of particles used, CPU and GPU memory requirements, and total computation time.The results of this study will provide insights into the advantages and limitations of each model and assist in selecting the appropriate model for specific applications.the detailed equations are not included in this paper and readers are referred to the original references.
The full 3-D simulation was performed using the DualSPHysics software, which is a free and open-source software for simulating free surface flows and multiphysics problems [5,18].In this method, the fluid body is initially discretised into particles that contain fluid properties such as velocity, pressure, and density.These particles are Lagrangian and can move freely by solving the governing equations.The fluid properties are interpolated from neighbouring particles using the smoothing kernel function.The solver is based on the weakly compressible SPH, where the fluid pressure and density are connected by Tait's equation of state [19], which is based on the numerical speed of sound.The interaction of the fluid with the solid boundary is solved using the novel modified dynamic boundary conditions (m-DBC) proposed by Aaron et al. [20].More recently, DesignSPHysics (https://design.sphysics.org/),a graphical user interface (GUI) plug-in for FreeCAD (https://www.freecad.org/),has made it easier to the workflow of running the simulation, including preprocessing, solving and post-processing.
The depth-averaged SPH model was run using a selfwritten code developed by converting the DualSPHysics code from 3-D to depth-averaged formulation, keeping its computational efficiency using the CPU and GPU parallelisation.A more complete explanation regarding the governing equations is available in Aslami et al. [21].However, the depth-averaged model is not available yet in the DesignSPHysics GUI.
The SWE-SPH formulation in this study is based on the variational approach for SPH with variable smoothing length [22], which was further developed for the depth-averaged model [7,8].Instead of artificial viscosity, the Lax-Friedrichs flux formulation proposed by Ata and Soulaïmani [6] is utilised for SPH stabilisation due to its simplicity and no need for coefficient tuning.The boundary condition is handled using dummy particles, as proposed by Adami et al. [23], with adaptations made for the depth-averaged SPH model with variable smoothing length [21].Table 1 provides a detailed presentation of the numerical parameters used in both the 3-D and depthaveraged SPH models.To ensure a fair comparison between the two models, the same initial inter-particle distance ( = 0.005 m) was utilised, which is sufficient for representing the wall thickness with five particles.Both models employed the symplectic time integrator with a CFL number of 0.2, and the 5 th -order Wendland kernel [24].The bed friction (Manning) coefficient used in the SWE-SPH model was determined based on experimental testing and set to 0.007 sm -1/3 .In contrast, the 3-D model did not incorporate the Manning formulation and instead treated fluid-boundary interaction by activating or deactivating the (artificial) viscosity [25] term in the force computation using the ViscoBoundFactor (1 or 0).In this simulation, the ViscoBoundFactor value was set to 0 due to the relatively low friction on the flume surface.The viscosity term in the SWE-SPH model was implemented using the Lax-Friedrichs flux formulation [6], which requires no tuning value.In the 3-D SPH model, the pressure field can be unstable due to the weakly compressible formulation.To address this issue, the density diffusion term [26] was used.The simulation was run for a duration of 5.5 s, during which the data was recorded at intervals of 0.02 s.

Results and discussions
The findings from the dam-break simulation using 3-D and depth-averaged flow models are presented in this section.Firstly, the flow characteristics are visually examined and phenomena that can and cannot be captured by the models are discussed.Following this, a detailed comparison is made between the time history of water depth measurements in the laboratory and the corresponding numerical results.Finally, the numerical performance of the model for shallow and violent flow is discussed.

Planar view
Fig. 2 illustrates the sequence of dam-break flow screenshots, comparing experimental photographs [27] taken from a planar view with the results obtained from 3-D and depth-averaged SPH simulations.The focus is primarily on the flow spreading pattern of the water flow in these figures.At = 0.4 s, the flow had just been released and started hitting the obstacle.The SWE-SPH model appears to be slightly lagging.At = 0.75 s, the flow surge had been separated by the obstacle towards the side walls.The hydraulic jump can be observed near the obstacle due to the transition from the supercritical to the subcritical flow regime.At the next time step ( = 1.45 s), the flow had been reflected by the side wall and the front surge had reached the end wall on the righthand side of the figures.Here the second hydraulic jump can be observed near the side walls.At = 2.16 s, the flow front had fully covered the end wall and was being reflected in the opposite direction.The reflected flow on the side wall had separated towards two opposing directions, which were initially flowing in only one direction.At = 2.86 s, the last hydraulic jump was gradually moving backwards due to the decreasing flow rate.From all sequence snapshots, both models show general agreement with the experimental photographs.These simulations demonstrate the capability of the model to reproduce complex, free-surface and violent flows, particularly in numerically capturing wet-dry interfaces for the SWE-SPH model.
One limitation of 3-D SPH is its inability to simulate very shallow flow when there are only one or two particles present at that depth.The flow tends to cluster and create voids due to inter-particle interaction forces, as seen at = 1.45 s.This non-physical behaviour is not observed in the SWE-SPH model, demonstrating its superiority.On the other hand, SWE-SPH cannot reproduce hydraulic jump phenomena like the 3-D SPH model since it is a natural characteristic of the depthaveraged model.Instead, the hydraulic jump is reproduced in the form of a shock wave.Shortly after water is released ( = 0.1 s), the water surface profile differs significantly due to the 3-D effect, which cannot be captured by the depth-averaged model.The SWE-SPH model assumes a uniform velocity profile along the water depth, causing the water column to move simultaneously as if it were sliding on the bottom surface while changing its shape from "tall and thin" to "short and fat" or vice versa according to the governing equations.In contrast, the 3-D SPH model shows a more realistic scenario where the bottom front of the water moves forward faster than the water at the top surface, which behaves more like a falling fluid.However, as the water spreads, the velocity profile along the water depth becomes more uniform, reducing the 3-D effect, and making the depth-averaged assumption more accurate for water flow simulation.This is supported by the more similar water surface profiles shown by both models beyond = 0.1 s.At = 0.4 s, the obstacle was hit by the flow front and the water started creeping up the wall.Subsequently, at = 0.78 s, the water flow was reflected in the form of a plunging-type breaking wave which evolved into a dynamic and violent hydraulic jump.The hydraulic jump persisted until the end of the simulation (only shown up to = 4.26 s in the figure), during which the water depth gradually decreased following the water level in the reservoir tank.Since this phenomenon is inherently three-dimensional, it cannot be accurately simulated by the depth-averaged model, which can only represent it as a shock wave.An advanced shockcapturing method could be implemented to produce a sharper shock capture, but it was not utilised in this simulation.Nonetheless, the depth-averaged model reasonably predicts the water surface profile within the subcritical region.The overall water surface profile, including the last reflections at the end wall and the back of the obstacle ( = 2.86 s and = 4.26 s), is accurately reproduced by the depth-averaged model.Additionally, the wet-dry interfaces are more clearly depicted from this perspective with comparable accuracy.

Water depth accuracy
Fig. 4 presents a plot of the time history of water depth measurements at the wave gauge location (164.7,2.2 cm) for both experimental and numerical simulations.The results of the 3-D and depth-averaged SPH models are generally in agreement with the experimental data.However, the initial peak in the experimental data, which may be due to the splashing reflection of the water flow, is not captured by the depth-averaged model, resulting in an underestimation of the water depth by approximately 21.6%.In contrast, the 3-D SPH model provides a closer match to the experimental data.As the flow becomes less splashy, both numerical models become more accurate in predicting the water depth.
To assess the overall accuracy of the model, two statistical parameters, namely amplitude ( ) and phase ( ), as defined in Gómez-Gesteira et al. [28], are analysed.The parameters are computed as follows: where represents the time step sequence, is the total number of time series data, is the numerical water depth at time , and , is the experimental depth.Perfect agreement between the experimental and numerical results gives = 1 and = 0.The data presented in Fig. 4 results in = 1.076 and = 0.142 for the SWE-SPH model, and = 1.064 and = 0.179 for the 3-D SPH model.These results demonstrate that the SWE-SPH model yields reasonably similar accuracy to the 3-D model for the dam-break simulation with much less computational effort.A detailed performance analysis is provided in the following section.

Computational performance
In this section, the computational performance of depthaveraged and 3-D SPH models was evaluated concerning the number of particles, CPU and GPU memory requirements, and total computation time.The depth-averaged model employed 38,557 fluid particles and required 35.76 Megabytes (MB) of CPU memory and 76.08 MB of GPU memory.The total computation time was approximately 3.35 minutes, which is 20 times faster than the 3-D SPH model.However, the computational effort per particle is slower in the depthaveraged model due to the Newton-Raphson iterative procedure required for each time step.
In contrast, the 3-D SPH model utilised 1.1 million particles, with a CPU memory requirement of 200 MB and a GPU memory requirement of 420 MB.The 3-D SPH model required more memory than the depthaveraged model due to the larger number of particles, which was five times more in this case.The total computation time for the 3-D SPH model was 67 minutes to simulate 5.5 seconds of actual time.
The results of the comparison demonstrate that the depth-averaged model is computationally efficient and requires less CPU and GPU memory usage while maintaining a similar accuracy for shallow water flow problems.On the other hand, the 3-D SPH model can provide a detailed 3-D effect of the flow dynamics, but it requires more computational resources.The selection of the appropriate model depends on the specific simulation requirements, including the desired accuracy, available computational resources, and time constraints.
Table 2. Computational performance of the 3-D and depth-averaged SPH models.

Conclusions
In this study, we compared the performance and accuracy of the depth-averaged and 3-D SPH models in simulating a dam-break flow over a dry bed.Both models were found to accurately predict the flow evolution, with the 3-D SPH model being able to capture the detailed 3-D effect of the flow dynamics.However, the depth-averaged model showed a significantly shorter computation time (20 times faster) and lesser both CPU and GPU memory usage (one-fifth of the 3-D model).This is due to the smaller number of particles used in the simulation.Nonetheless, the depthaveraged model showed a slower computational effort per particle compared to the 3-D model due to the timeconsuming Newton-Raphson iterative procedure for each particle at each time step.
The choice between the two models will ultimately depend on the specific needs of the simulation, including the desired accuracy, available computational resources, and time constraints.If computational efficiency is a priority and the 3-D effect of the flow dynamics is not essential, the depth-averaged model is a suitable choice.On the other hand, if high accuracy and detailed 3-D effect of the flow dynamics are crucial, the 3-D SPH model is a better option, although it requires more computational resources and longer computation time.This is a piece of common knowledge among CFD experts; however, this study demonstrates the quantitative analysis of the argument, especially for the SPH method in simulating violent expanding flows.
Overall, the findings of this study provide insights into the performance and accuracy of two commonly used models in simulating fluid dynamics, which can assist in selecting an appropriate model for specific applications.

Fig. 1 .
Fig. 1.The layout and dimensions of the experimental facility at the Hydraulic Laboratory of Parma University.

Table 1 .
Numerical parameters used in the simulations.The initial setup for the simulation is based on the physical dam break experiments conducted by Aureli et al.[27] at the Hydraulic Laboratory of Parma University.The flume configuration is shown in detail in Fig.1and includes surrounding and dividing walls, a sluice gate, and a rectangular prism as a fixed obstacle.The flume dimensions are 1.2 m by 2.6 m, with surrounding walls that are 2.5 cm thick.The dividing wall is located at = 0.8 m and has a thickness of 2.5 cm.The left part of the flume serves as a reservoir with an initial water depth of 15 cm, while the other part is initially dry and can be flooded.The obstacle, measuring 15 × 30 cm, is located at the front middle of the flood front on the right side of the flume to generate a strong flow perturbation.Once the sluice gate is quickly opened, the time history of the water depth at the gauge location (164.7,2.2 cm) is recorded for numerical model verification.

Fig. 2 .
Fig. 2. The screenshots from the planar view of the dam-break flows at different instants (t = 0.4, 0.75, 1.45, 2.16 and 2.86 s) comparing the (a) experimental photographs (source: Aureli et al., 2008 [27]) with the results of (b) 3-D SPH and (c) SWE-SPH simulations.Note that the size of SWE-SPH particles is following the base area of the water column.

Fig. 3 .
Fig. 3.The lateral view of the dam-break flow along the cross section at the middle of the tank ( = 0.6 m), comparing the results of 3-D and SWE-SPH simulations from the initial state to = 4.26 s.The SWE-SPH particle represents the top elevation of the water column.

Fig. 3
Fig. 3 displays the flow sequences from a lateral view, where particles are clipped between = 0.55 to = 0.65 m, showing only the middle section of the flume.Due to the difficulty in obtaining corresponding experimental photos for this view, it demonstrates the advantages and flexibility of numerical methods compared to physical experiments.The solid boundary is represented by blue particles, while grey and red particles represent the 3-D SPH fluid and the depth-averaged fluid, respectively.It should be noted that the SWE-SPH particles represent a column of water that "slides" on the bottom surface involving friction, whereas, in this plot, the red dots represent the top elevation of the water column.Shortly after water is released ( = 0.1 s), the water surface profile differs significantly due to the 3-D effect, which cannot be captured by the depth-averaged model.The SWE-SPH model assumes a uniform velocity profile along the water depth, causing the water column to move simultaneously as if it were sliding on the bottom surface while changing its shape from "tall and thin" to "short and fat" or vice versa according to the governing equations.In contrast, the 3-D SPH model shows a more realistic scenario where the bottom front of the water moves forward faster than the water at the top surface, which behaves more like a falling fluid.However, as the water spreads, the velocity profile along the water depth becomes more uniform, reducing the 3-D effect, and making the depth-averaged assumption more accurate for water flow simulation.This is

Fig. 4 .
Fig. 4. The time history of the water depth at the measurement point (164.7,2.2 cm).