A Hydrodynamic-Based Robust Numerical Model for Debris Hazard and Risk Assessment

sustainability

Article A Hydrodynamic-Based Robust Numerical Model for Debris Hazard and Risk Assessment

Yongde Kang, Jingming Hou *, Yu Tong and Baoshan Shi

State Key Laboratory of Eco-Hydraulics in Northwest Arid Region of China, School of Water Resources and Hydroelectric Engineering, Xi’an University of Technology, Xi’an 710048, China; [email protected] (Y.K.); [email protected] (Y.T.); [email protected] (B.S.) * Correspondence: [email protected]

Abstract: Debris flow simulations are important in practical engineering. In this study, a graphics processing unit (GPU)-based numerical model that couples hydrodynamic and morphological processes was developed to simulate debris flow, transport, and morphological changes. To accurately predict the debris flow sediment transport and sediment scouring processes, a GPU-based parallel computing technique was used to accelerate the calculation. This model was created in the framework of a Godunov-type finite volume scheme and discretized into algebraic equations by the finite volume method. The mass and momentum fluxes were computed using the Harten, Lax, and van Leer Contact (HLLC) approximate Riemann solver, and the friction source terms were calculated using the proposed splitting point-implicit method. These values were evaluated using a novel 2D edge-based MUSCL scheme. The code was programmed using C++ and CUDA, which can run on GPUs to substantially accelerate the computation. After verification, the model was applied to the simulation of the debris flow process of an idealized example. The results of the new scheme better reflect the characteristics of the discontinuity of its movement and the actual law of the evolution Citation: Kang, Y.; Hou, J.; Tong, Y.; of erosion and deposition over time. The research results provide guidance and a reference for the Shi, B. A Hydrodynamic-Based in-depth study of debris flow processes and disaster prevention and mitigation. Robust Numerical Model for Debris Hazard and Risk Assessment. Keywords: debris flow; Godunov-type scheme; numerical model; graphics processing unit (GPU) ac- Sustainability 2021, 13, 7955. https:// celeration doi.org/10.3390/su13147955

Academic Editors: Ashraf Dewan and Jui-Sheng (Rayson) Chou 1. Introduction Debris ﬂows have occurred in many provinces and municipalities in China, and they Received: 29 May 2021 can be disastrous, often destroying nearly everything in their path and threatening lives, Accepted: 8 July 2021 Published: 16 July 2021 property, and infrastructure [1–3]. It is generally believed that a debris ﬂow is a series of processes between the movement of blocks such as collapse and landslide and sediment-

Publisher’s Note: MDPI stays neutral bearing water flow, which is formed by the interaction and development of solid-liquid with regard to jurisdictional claims in two-phase materials on mountain slopes or channels. published maps and institutional affil- At present, most debris flow models can be clarified as dynamic models and numeri- iations. cal models. The dynamic model of debris flow can be divided into continuous medium, discrete medium, and mixed medium models from the perspective of describing the composition and movement. Continuous medium models can be divided into single-fluid [4,5] and multifluid models [6–10]. The two-phase model is attractive because it can explicitly reveal the relative motions and interactions between the fluid and solid phases. Accord- Copyright: © 2021 by the authors. ingly, however, the increase in computing costs and also the demand for extra relationships Licensee MDPI, Basel, Switzerland. This article is an open access article to close the governing equations constrain its applications. Moreover, it is still unclear distributed under the terms and if they can perform considerably better than traditional single-phase models in terms of conditions of the Creative Commons modelling accuracy [11]. Single fluid models are generally applicable to debris flows with a Attribution (CC BY) license (https:// large bulk density or a small velocity difference between two phases [12]. Non-Newtonian creativecommons.org/licenses/by/ models include the Bingham body [13], the Bagnold expansion body [14], and the viscoelas- 4.0/). tic body [15]. The multifluid model mainly refers to the two-component hydrodynamic

Sustainability 2021, 13, 7955. https://doi.org/10.3390/su13147955 https://www.mdpi.com/journal/sustainability Sustainability 2021, 13, 7955 2 of 19

equations established by considering the momentum exchange between solid and liquid phases. These models can explicitly reveal the relative motion and interaction between the fluid and the solid phase [16]. For example, in the Coulomb mixed flow model, considering the constitutive relationship of granular matter and the interaction force between two phases [17–19], Pitman and Le [20] proposed a two-fluid model considering the effect of liquid buoyancy. The discrete medium model simplifies the debris flow into a system composed of a large number of material particles of a certain size. In recent years, with the development of thermal dynamics theory and turbulence theory, the traditional Boltzmann equation has been extended to the study of turbulence and particle flow [21], which provides a method different from the traditional Navier–Stokes (N-S) equation for the theoretical and numerical study of debris flows. The mixed medium model uses a continuous medium model and a discrete medium model to describe the movement of the liquid and solid parts of debris flows. In practice, debris flows are composed of two phases; only certain special types of debris flows, such as mud flows, can be simplified as a phase flow. Although the multifluid model describes the liquid phase and the solid phase with different equations, the equation describing the motion of solid particles is still a continuum model. Generally, the liquid phase of debris flows is a slurry liquid composed of fine particles and water that is smaller than a certain size, which can be regarded as a continuous medium. The solid phase of large particles above a certain size should be described by a discrete medium model. The liquid phase of debris flows is described by a non-Newtonian continuous medium, and the motion equation adopts the two-dimensional depth-averaged shallow water wave equation. The numerical calculation model of debris flow is closely related to the research problems. Due to the complexity of the debris flow process, a number of models were developed to simulate the flow behavior. The first numerical calculations of debris flows were performed using a one-dimensional homogeneous single-phase continuum model [22–24]. With the development of computational fluid dynamics, numerical calculation methods, computer technology, and debris flow theory, numerical approaches have been gradually extended to include multidimensional, heterogeneous, and multiphase models [25–33]. It is well known that numerical simulations are an important tool for studying debris flow disaster process. Although full three-dimensional models [34] can increase the precision of high-resolution simulations and may help very detailed resolution of the phenomena, when calculating large practical cases, this method is impractical due to the high calculation cost. In contrast, using the principles of the conservation of mass and momentum, depth-averaged models provide a reasonable balance of completeness and theoretical application and have therefore been broadly applied. Future science and engineering breakthroughs depend on computing. In recent years, through the implementation of parallelization techniques, there has been a reliable way to significantly reduce computational effort, such as multiprocessing (open MP) and message passing interface (MPI), which allow simulations to be run on cluster machines [35]. Graph- ics processing units (GPUs) are a new computation engine for high-resolution modeling as computation burdens become increasingly heavy. To resolve the challenge of further accelerating up computations, modern high-performance computer systems increasingly employ GPUs and other accelerators, such as open MP and MPI. Their disadvantages are related hardware cost and energy processor requirements, which usually create a limitation on their practical usage. On the contrary, hardware accelerators, such as GPU, have become a low-cost option, because they can be used on simple personal computers. Previous studies have developed strategies for implementing the pure shallow water equations on graphics processors [36,37]. However, less research has focused on high-resolution terrain using GPU acceleration technology. GPU computing is expected to become more common in future systems. Therefore, programs must be adapted to the use of graphics processors, because graphics processors provide high performance with low costs and power requirements, and therefore they have become the main computing resources of many of the largest supercomputers. Under the framework of a shallow sediment-geomorphology dynamic Sustainability 2021, 13, 7955 3 of 19

model, this study focused on a two-dimensional depth-averaged quasi-multiphase mixing model with non-uniform sediment transport to simulate the evolution of debris ﬂows on inclined bed slopes. Namely, by using the GPU Accelerated Surface Water Flow and Transport Model (GAST) model, which can accurately predict debris sediment transport and the debris sediment scouring process, GPU techniques were applied in a numerical model, making it possible to simulate the sediment transport and bed evolution in a high- resolution but efﬁcient way. This method resolves the realistic features of debris sediment transport and uses a GPU-based parallel computing technique to accelerate calculations. The proposed model was validated against experimental benchmark tests. Finally, brief conclusions are drawn.

2. Model Description Debris flows consist of a multiphase body including solid particles, water, and gas. However, the solid and liquid parts are fully mixed in motion, showing holistic motion. Therefore, debris flows can be regarded as a single fluid that follows the N-S equation of fluid motion. In the process of debris flow movement, the vertical scale is much smaller than the horizontal and vertical scales, so the vertical change in debris flow movement can be ignored. The Saint-Venant equation describing the two-dimensional motion of debris flow can be obtained by the vertical integration of the N-S equation. The vector form of the two-dimensional nonlinear shallow water equation is as follows [38]:

∂q ∂F ∂G + + = S (1) ∂t ∂x ∂y

where,      η  qx qy 2 2 q  kgh   + kgh   x   uqx + 2   vqy 2   q       y   uqy   vqy  q =  F =  G =   (2)  hcs   qxCs   qyCs         hcb   βqxCb   βqyCb  zb 0 0   0  S + S   bx f x   +   Sby S f y  S = (3)  ω0(Csae − Csa)   ∗   (qxcb−βq )   − L   h ∗ i  1 qxcb−q 1−p α βL + ω0(1 − α)(Csa − Csae) where t is time; x and y are the two-dimensional Cartesian coordinates; h is the flow depth; F and G are the flux vectors of the conserved variables in the x and y directions, respectively; g is the gravitational acceleration with a value of 9.81 m·s−2; u and v are the depth-averaged velocities in the and directions, where qx = uh and qy = vh; hCs and hCb are the conserved concentrations of the debris flow suspended load and the bed load, respectively; zb is the debris flow bed elevation; D and E denote the deposition and entrainment rates, respectively; and the subscripts s and b denote the debris flow suspended load and the bed load, respectively. √ 2 2+ 2 S = τ + κηu + gn u u v f x γm 8hγm h√1/3 2 2+ 2 S = τ + κηv + gn v u v f y γm 8hγm h1/3

where τ represents the yield stress, γm represents the unit weight of the debris ﬂow, and η represents the viscosity coefﬁcient. Sustainability 2021, 13, 7955 4 of 19

k is the lateral earth pressure coefﬁcient, which depends on the strain rate of the moving material columns [39] and can be expressed as follows:

2 q k = × 1 ∓ 1 − (1 + tan2 δ) cos ϕ − 1 (4) cos2 ϕ

The conservation Equations (1)–(3) are solved by the GAST model, and the finite volume method in the center format is used to calculate the interface flux. The HLLC approximate Riemann solution of the approximate Godunov method is used. The function of this format is better than that of other formats when dealing with dry elements. This format can better deal with the alternating dry and wet changes in complex terrains and has a strong shock capture ability, considering that the approximate Riemann solution based on HLLC is used to solve the interface flux with only first-order accuracy in space to increase the spatial accuracy of the numerical solution to the second order. At the same time, to avoid the phenomenon of numerical oscillations where the water surface gradient is large after the numerical reconstruction, the model adopts the TVD (Total Variation Diminishing)-MUSCL (Monotonic Upstream-Centered Scheme for Conservation Laws) method for numerical reconstruction. In addition, to ensure that the numerical solution is improved to the second-order accuracy as a whole while maintaining the stability of the numerical solution, the two-order Runge–Kutta format is used for the time step to obtain the second-order accuracy. For dry and wet dynamic boundary treatment, the water depth tolerance is 0.000001 m to distinguish between dry and wet grid cells. The calculation of the friction resistance term uses the split point implicit method to improve stability. In the calculation process, Courant is 0.5, and the CFL (Courant, Friedrichs, Lewy) criterion is used to predict the new time step of the next iteration [40–42]. In this work, the proposed numerical scheme was implemented on a GPU using the NVIDIA CUDA toolkit. CUDA (Compute Unified Device Architecture) is a CPU + GPU hybrid programming framework proposed by NVIDIA. The CUAD-based code used a CPU as the Host and the GPU as the co-processor or Device. In the proposed numerical model, the CPU and GPU work together. The CPU is responsible for processing the data transaction and performing serial calculations, whereas the GPU is performing parallel processing tasks. The CUDA parallel computing function running on a GPU is called a kernel function. The C++ programming language is applied on the Host, while the kernel function of the Device must follow the syntax rules and API interface provided by the CUDA. The kernel function can be executed on the GPU after being compiled by the NVCC compiler. As shown in Figure1, when performing the parallel computing task, the data are first read into the Host memory through the Host terminal program, and the corresponding GPU device space is allocated for the variable information, such as the grid, initial and boundary conditions, and hydrological and hydraulic parameters. Before calling the kernel function, data must be copied to the device memory. The kernel function is then executed on the GPU for loops. Finally, the computed results are copied back from the Device to the Host for saving and visualization. If the data exchange between the Host and Device memory is too frequent, the data transfer can reduce the performance of the model. Sustainability 2021, 13, x FOR PEER REVIEW 5 of 20

GPU device space is allocated for the variable information, such as the grid, initial and boundary conditions, and hydrological and hydraulic parameters. Before calling the kernel function, data must be copied to the device memory. The kernel function is then executed on the GPU for loops. Finally, the computed results are copied back from the Device Sustainability 2021, 13, 7955 5 of 19 to the Host for saving and visualization. If the data exchange between the Host and Device memory is too frequent, the data transfer can reduce the performance of the model.

GPU(C++) GPU(CUDA)

Data transportation from Yes Initialise and load Output required GPU copy CPU domain data

Advance simulation time Compile program for Write output device files to disk Apply friction effects Calculate work No Simulation Force time progression dimensions complete? No Source terms approximation Yes Copy domain data to Calculate new queue device buffers addition size Release computes Numerical fluxes calcuation device resources

Queue scheme kernels Limited gradient calculation

FigFigureure 1.1. FlowchartFlowchart of of the the computing computing process process using using GPUs GPUs for the for model. the model.

3.3. Model Model V Verificationerification ToTo verify the abilityability ofof thethe model model to to capture capture the the debris debris flow flow dynamics dynamics on on non-corrosive non-corro- siveand and erosive erosi channelve channel beds, beds, a numerical a numerical comparison comparison was was carried carried out usingout using the flumethe flume test testresults results of the of USGS. the USGS. In this In experiment, this experiment, two debris two debris flows flows were considered: were considered: debris debris flows flowson rough on rough channel channel beds andbeds debris and debris flows flows on rough on rough channel channel bed channel bed channel beds beds covered covered with witherosive erosi sediments.ve sediments. The experimentalThe experimental data data for the for coarse the coarse bed debrisbed debris flow cameflow came from from eight eightrepeated repeated USGS USGS flume flume experiments experiments [43,44 ].[43 The,44]. experiment The experiment recorded recorded unstable unstable and uneven and unevenerosive erosive channel channel bed debris bed flows debris from flows the from beginning the beginning to the end. to the The end. flow The front, flow the front, flow thesurface flowperpendicular surface perpendicular to the bed, to andthe bed, the bedand deformationthe bed deformation were recorded, were recorded, which basically which basicallyprovides provides a unique anda unique systematic and systematic set of observational set of observational data for testing data for mathematical testing mathemat- models icalof debris models flows of debris that can flows erode that channel can erode beds. channel In allexperiments, beds. In all experiments, debris flows debris were caused flows wereby the caused sudden by releasethe sudden of large release amounts of large of amounts water-sediment of water mixtures.-sediment Themixtures. experimental The ex- perimentaldevice, data device, processing, data processing, and material and characteristics material characteristics of the details of canthe details be found can in be [43 found] and inthe [43] specific and the parameters specific parameters are shown inare Table shown1. in Table 1.

TableTable 1. SimulatiSimulationon parameter attributes. ◦ ◦ ◦ ◦ ρs ρω ρ cρa ϕ1( )  ϕ2( )  δ( )  ϕvoe( )  cz λz  s   ca 1 () 2 ()  () voe () cz z Rough Bare Bed 2700 1000 2020 0 40 40 40 15 12 / Rough Bare Bed 2700 1000 2020 0 40 40 40 15 12 / Erodible Bed 2700 1000 2020 0 40 40 40 15 12 0.8 ExperimentalErodible Value Bed 27002700 10001000 2020 2010 <4000 3940 3940 40.740 /15 /12 /0.8 Experimental Value 2700 1000 2010 <400 39 39 40.7 / / /

Figures2 and3 show a comparison between the numerical solutions and the experi- Figures 2 and 3 show a comparison between the numerical solutions and the experimental results of the flow height–time curves of rough bare bed and erodible sediments. mental results of the flow height–time curves of rough bare bed and erodible sediments. The experimental curve in Figure3 corresponds to experiment C by Iverson [ 45]. Addition- The experimental curve in Figure 3 corresponds to experiment C by Iverson [45]. Addi- ally, the results are close to those of the simulation in [38]. The simulation curve of the water tionally, the results are close to those of the simulation in [38]. The simulation curve of the flow height with time is basically consistent with the experimental results. For the erodible water flow height with time is basically consistent with the experimental results. For the channel bed, the time to reach two positions is slightly earlier than the experimental results. erodible channel bed, the time to reach two positions is slightly earlier than the experi- Therefore, this simple model with a constant pore pressure coefficient can capture complex mentaldebonding results. flow Therefore, dynamics this [45 ].simple The results model show with a that constant the model pore establishedpressure coefficient in this paper can capturecan be used complex for debris debonding flow simulations. flow dynamics [45]. The results show that the model established in this paper can be used for debris flow simulations. Sustainability 2021, 13, 7955x FOR PEER REVIEW 6 6of of 20 19

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4. Numerical Tests 4. Numerical Tests 4.1. Numerical SimulationSimulation of Dam-BreakingDam-Breaking DebrisDebris LowLow The general process of the dam-breakingdam-breaking debris flowflow is that the waterwater flowflow isis fullyfully topped, then the the dam dam body body is is flattened, flattened, and and finally, finally, the the entire entire body body is is broken broken to to form form a adebris debris flow. flow. To Toverify verify the the rationality rationality of the of theproposed proposed GAST GAST model, model, the experiment the experiment per- performedformed by byKomatina Komatina in in1997 1997 [46] [46 was] was selected selected as as an an example. example. This This experiment experiment has been used by many scholars to verify numerical models that simulate dam dam-break-break debris flows. flows. The length of of the the water water tank tank in in the the calculation calculation example example was was 4.5 4.5 m, m, and and the the width width was was 1.5 1m..5 mThe. The fluid fluid was was a mixture a mixture of ofwater water and and kaolinite kaolinite clay clay with with solid solid particles particles of differentdifferent concentrations. The The average diameter of the particles was 0.0060.006 mm.mm. The proportionproportion ofof clay material was 2.65. The dam dam-breaking-breaking debris flowflow was simulated at different moments in the smoothsmooth flume.flume. The measured slurry was placed inin a container with aa lengthlength ofof 22 mm and a width of 1.5 m at thethe toptop ofof thethe waterwater tank,tank, withwith anan initialinitial heightheight ofof 11 m,m, andand thethe slurry was blocked by a movable baffle.baffle. Dam-breakDam-break flows flows are caused by a mixturemixture beingbeing releasedreleased byby thethe reservoirreservoir locatedlocated upstream of the flume.flume. The discharge caused by the suddensudden removalremoval ofof gatesgates betweenbetween reservoirs and and rivers rivers is is actually actually instantaneous. instantaneous. The The propagating propagating positive positive waves waves were were cap- capturedtured by bycameras. cameras. The The velocity velocity of ofthe the wave wave and and the the form form of of the the wave wave array array were were digitally recorded.recorded. In each experimental process, the levellevel waswas alsoalso continuouslycontinuously monitored.monitored. TheThe initial reservoir depthdepth waswas 10–3010–30 cm, the channel bed slope was 0.0–0.1%,0.0–0.1%, andand the volume concentration waswas different. The channel slope was 0.1%. The wavefront propagation timetime increased significantlysignificantly asas thethe concentrationconcentration increased.increased. ForFor differentdifferent concentrations,concentrations, thethe simulated and observed values are basically consistent, and it is clearclear thatthat thethe modelmodel isis reliable (Figure(Figure4 4).).

4.0 Simulation Value (C=4.0%) Measure Value (C=4.0%) Simulation Value (water) Measure Value (water) Simulation Value (C=8.6%) Measure Value (C=8.6%) 3.5

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0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 x/H Figure 4.4. SimulatedSimulated andand observed observed values values of of particle particle concentration concentration on waveon wave front front propagation propagation velocity. velocity. At the beginning of the test, the movable baffle was pulled out to allow the slurry to flowAt freely, the andbeginning the deformation of the test, ofthe the movable mud front baffle edge was and pulled the movement out to allow distance the slurry of the to frontflow freely, edge were and the recorded deformation at each of time the mud node. front The edge simulation and the time movement was 10 distance s, five typical of the momentsfront edge of were t = 0recorded s, t = 1 s, at t =each 5 s, time t = 7 s,node. and The t = 10simulation s were selected, time was and 10 the s, five movement typical distancemoments of of the t = mud 0 s, t front = 1 s, obtained t = 5 s, t by = 7 numerical s, and t = simulation 10 s were changedselected, withand the time. movement Figure5 Sustainability 2021, 13, x FOR PEER REVIEW 9 of 20

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distance of the mud front obtained by numerical simulation changed with time. Figure 5 shows that the destruction process of the dam-breaking debris flow can be divided into shows that the destruction process of the dam-breaking debris flow can be divided into three stages: when t = 0 s, the flow was in a static state; when t = 1 s, with the continuous three stages: when t = 0 s, the flow was in a static state; when t = 1 s, with the continuous infiltration of water, the shear surface developed and reached the middle of the tank; and infiltration of water, the shear surface developed and reached the middle of the tank; and when t = 5 s, the debris flow moved quickly and started to move to the bottom of the tank. when t = 5 s, the debris flow moved quickly and started to move to the bottom of the tank. In addition, the color in the flume was red at first and gradually transitioned to a green In addition, the color in the flume was red at first and gradually transitioned to a green and blue distribution, indicating that the material sources in the flume began to erode and blue distribution, indicating that the material sources in the flume began to erode after after a certain period of time, forming a complete debris flow process. a certain period of time, forming a complete debris flow process.

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Figure 5. Numerical simulation of morphological changes in debris flows at different times. Figure 5. Numerical simulation of morphological changes in debris flows at different times. Figure 5. Numerical simulation of morphological changes in debris flows at different times. A horizontal distance simulation overover timetime inin differentdifferent stagesstages inin thethe drainagedrainage troughtrough A horizontal distanceisis shownshown simulation inin FigureFigure over6 6.. TheThe time maximummaximum in differ horizontalenthorizontal stages in distancedistance the drainage reachedreached trough fromfrom 0–150–15 ss waswas 120120 cm.cm. is shown in Figure 6AtAt. The 00 s,s, maximum thethe debrisdebris horizontal flowflow depositsdeposits distance rapidlyrapidly reached accumulated,accumulated, from 0–15 butbut s was thethe 120 sinksink cm. waswas straightstraight andand stillstill At 0 s, the debris flowhad deposits a certain rapidly capacitycapacity accumulated, forfor entrainment.entrainment. but Asthe asink result,result, was thethe straight debrisdebris and flowflow still curvecurve displaysdisplays aa slightslight had a certain capacitydownward for entrainment. trend afterafter As a rapidrapid result, sedimentation.sedimentation. the debris flow At curve 5 s andand displays 77 s,s, thethe a horizontal horizontalslight distancedistance ofof thethe downward trend afterdebris rapid flowsflows sedimentation. firstfirst increasedincreased At 5 rapidlyrapidly s and 7 andand s, the thenthen horizontal increasedincreased distance graduallygradually of the fromfrom 68–10568–105 cm.cm. TheThe debris flows first increasedinitialinitial rapidrapid rapidly increaseincrease and waswasthen causedcaused increased byby thethe gradually rapidrapid sedimentationsedimentation from 68–105 ofofcm. thethe The debrisdebris flowflow material.material. initial rapid increase was caused by the rapid sedimentation of the debris flow material. 1.4 T=5s FLO-2D Modle GAST Modle 1.4 T=5s FLO-2D Modle GAST Modle 1.2 1.2

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0 20 40 60 80 100 120 L(cm) Figure 6. Debris ﬂowflow simulation depth at 5 s, 7 s,s, and 10 s.

In FigureFigure7 7,, a a resting resting velocity velocity of of 0 0 m/s m/s cancan bebe observedobserved at at t t = = 00 s.s. AsAs thethe slidingsliding speedspeed increased, the velocityvelocity accumulationaccumulation peaked at tt == 55 s,s, withwith thethe speedspeed reachingreaching 1212 cm/s,cm/s, and the corresponding horizontal distance was 40 cm. As the debris flowflow movedmoved inin thethe horizontal direction, the the mud mud depth depth showed showed a atend tendencyency to to fluctuate fluctuate and and thicken thicken—that—that is, is,there there was was a possibility a possibility of accumulation of accumulation—additionally,—additionally, the thearea area of the of thedebris debris flow flow was was ap- approximatelyproximately 330 330 m2 m. At2. Att = t15 = s, 15 the s, thedebris debris flow flow eroded eroded the thechannel channel bed bed and and moved moved to the to end of the flume. The slope of the breach continued to slide, which made the breach wider Sustainability 2021, 13, 7955 13 of 19

the end of the flume. The slope of the breach continued to slide, which made the breach wider than it was at t = 7 s. The velocity was mostly between 10 cm/s and 5.5 cm/s, and the maximum velocity was 17.5 cm/s. In summary, the reasons for this phenomenon may be attributed to the following points. Some researchers think that more comparisons of fixed-bed and entrainment models must be conducted to judge the reliability of the suggested enhancement of the model calibration process [47]. Although the two models used to simulate the propagation are based on different rheological laws, the results obtained in terms of debris flow paths and, more in general, of inundation depth and velocity, are very similar [48]. Bed entrainment Sustainability 2021, 13, x FOR PEER REVIEW 13 of 20 plays a major role in determining the whole propagation pattern; however, there can be different aspects of erosion in nature [49]. Because of the addition of material from the ground surface to the moving mass, the reduction in kinetic energy of the system might bethan greater it was than at t the= 7 increases. The velocity in its potential was mostly energy. between However, 10 cm/s those and complex 5.5 cm/s, behaviors and the alsomaximum depend velocity on which was erosion 17.5 cm/s. mechanism takes place, which flow theologies are involved, and howIn summary, the mass the and reasons momentum for this productions phenomenon are consideredmay be attributed in the dynamical to the following model equations.points. Some This researcher requiress further think that research. more comparisons of fixed-bed and entrainment models mustThe be FLO-2D conducted model to isjudge a tool the adopted reliability worldwide of the suggested for the enhancement simulation of of debris the model flow phenomenacalibration process and hydraulic [47]. Although flooding the of urbantwo models areas [ 47used,48]. to At simulate present, the the propagation FLO-2D model are hasbase possiblyd on different been mostrheological widely laws, applied the results to natural obtained debris in flowsterms orof debris compared flow with paths other and, modelsmore in [general,49–52]. Inof orderinundation to verify depth the and simulation velocity results, are very of similar the GAST [48]. model, Bed entrainment this paper comparesplays a major the simulation role in determining results with the those whole of thepropagation FLO-2D model pattern by; h simulatingowever, the there slidingcan be phenomenondifferent aspects at different of erosion stages. in nature It was [49]. found Because that the of simulatedthe addition horizontal of material distance from and the debrisground flow surface velocity to the were moving basically mass, consistent the reduction with the in actual kinetic physical energyprocess. of the system This verifies might thebe greater rationality than of the the increase GAST numerical in its potential model inenergy. this paper. However, those complex behaviors also dependTo objectively on which compare erosion the mechanism accuracy of takes the model, place, common which flow statistical theologies indicators are involved, includ- ingand the how root the mean mass square and momentum error (RMSE productions), coefficient ofare determination considered in (R the2), anddynamical mean relative model errorequations. (MRE This) were requires adopted, further and theresearch. index calculation formula is shown in Equation (5).

14 T=5s FLO-2D Modle GAST Modle

V (cm/s) V 6

0 10 20 30 40 50 60 70 80 L(cm) Figure 7. Cont. SustainabilitySustainability2021 2021,,13 13,, 7955 x FOR PEER REVIEW 1414 of 1920

14 T=7s FLO-2D Modle GAST Modle

6 V (cm/s)

0 20 40 60 80 100 120 L(cm)

12 T=10s FLO-2D Modle GAST Modle

V (cm/s) 4

0 20 40 60 80 100 120 L(cm) FigureFigure 7.7. DebrisDebris ﬂowflow simulationsimulation velocityvelocity atat 55 s,s, 77 s,s, andand 1010 s.s.

AnTheRMSE FLO-2Dvalue model of zero is a indicatestool adopted a perfect worldwide ﬁt Equation for the (5). simulation If the value of isdebris less thanflow halfphenomena of the standard and hydraulic deviation flooding of the observations,of urban areas the [47 model,48]. A performancet present, the is FLO considered-2D model to behas good. possibly been most widely applied to natural debris flows or compared with other s 2 models [49–52]. In order to verify the simulation∑n (M results− S ) of the GAST model, this paper RMSE = i=1 i i (5) compares the simulation results with those of the FLON -2D model by simulating the sliding phenomenon at different stages. It was found that the simulated horizontal distance and debris flow velocity were basically consistent with the actual physical process. This verifies the rationality of the GAST numerical model in this paper. Sustainability 2021, 13, 7955 15 of 19

The coefficient of determination R2 reflects the degree of coincidence of the simulated and observed final bed and water levels (Equation (6)). The closer R2 is to 1, the higher the degree of coincidence with the simulated runoff process is. R2 can be calculated as follows:

2 nd ∑i=1 M − M S − S R2 = (6) nd 2 nd 2 ∑I=1 Mi − M ∑I=1 Si − S

The efﬁciencies of the simulated and observed processes were evaluated by the coefﬁ- cient of relative error (MRE), calculated as follows (Equation (7)):

1 n |S − M | MRE = ∑ i i × 100% (7) n i=1 Mi

where M is an observed value, S is a simulated value, M is a mean observed value, S is a mean simulated value, and N is the sample number. The calculation results for the three indicators are compared in Table2.

Table 2. Performance comparison for different model for the depth and velocity level simulations.

Content Time RMSE R2 (%) MRE 5 s 0.1822 95.8 1.325 Depth-L 7 s 0.0208 97.2 1.289 10 s 0.0459 96.4 1.244 Average 0.2489 96.4 1.286 5 s 0.035 90.2 1.412 Velocity-L 7 s 0.056 97.6 1.366 10 s 0.062 94.5 1.402 Average 0.051 94.1 1.393

As shown in Table2 and Figures6 and7, the proposed GAST model was reliable and could achieve good predictive results for morphological processes. The comparative analysis showed that the accuracy of the simulated results could also be reﬂected by R2, RMSE, and MRE. Figures6 and7 show the correlation between the measured and simulated depth and velocity values based on the corresponding coefﬁcient of determination. Therefore, the predictive model proposed in this paper shows strong generalizability. Meanwhile, the model performance was acceptable.

4.2. GPU and CPU Runtime Several tests were performed to prove the validity of our GAST model. All tests were conducted using the same computational devices; the GPU was an NVIDIA GeForce GTX 1080 Ti with 16 G memory, and the CPU was an Intel (R) Core (TM) i7-6700 K CPU @ 4.00 GHz. The debris-flow-prone area in Pusa Village, Guizhou Province, China, covers 2.68 km2. The 12 m resolution, high-precision terrain was simulated using the GAST model for 3 h. The number of calculation grids for the 12 m resolution terrain was 2,592,280, the CPU calculation time was 5151.9 s, and the GPU time was 352.21 s. The hazard CPU and GPU simulation times and the results of the debris-flow-prone area were compared using the single-core Intel CoreTM i7-7700 as a comparison standard, and the speedup ratio was set to 1. Compared with the Intel CoreTM i7-7700 (single-core CPU), the mudslide simulation calculation performed using the NVIDIA GeForce GTX 1080 (GPU) was up to 14.62 times more efficient. Therefore, the GPU computing power far exceeded the CPU computing power (Table3). Sustainability 2021, 13, 7955 16 of 19

Table 3. CPU and GPU simulation performance comparison for the execution time and speed up for a removable debris ﬂow.

CPU (Four Cores) GPU

t t sup 5151.9 s 352.21 s 14.62

To compare the computational costs of the current complete GAST model [53,54] and the two-phase model, Table4 shows the CPU run time and GPU time of the two models in the FB and EB cases. The results show that the erosive bed required more processing time than the fixed bed because the former involved the bed evolution equation, which reflects the mass exchange between the flow and the bed.

Table 4. Relative CPU and GPU runtimes of the GAST model and the two-phase model.

Model FB (Fixed Bed) EB (Erodible Bed) Two-phase model (Li et al. 2017a, b) 1.22 1.31 0.75 0.98 GAST (GPU Accelerated Surface Water Flow 0.80 1.10 and Transport Model) 0.79 1.04 1.00 0.97 Speed-up (Four averages) 18–35% 16–25%

However, the calculation time of the GAST model was 16–35% faster than that of the two-phase model [53,54]. In addition, if the number of equations increases, the computation time is bound to increase [53,54]. Therefore, it can be predicted that the central processor run time of the two-phase model will inevitably increase compared with that of the GAST model, especially in the simulation of large-scale debris ﬂows.

5. Discussion Some researchers think that more comparisons of fixed bed and entrainment models must be conducted to judge the reliability of the suggested enhancement of the model calibration process [53]. Although the two models used to simulate the propagation are based on different rheological laws, the results obtained in terms of debris flow paths and, more in general, of inundation depth and velocity, are very similar [54]. Bed entrainment plays a major role in determining the whole propagation pattern; however, there can be different aspects of erosion in nature [55]. Because of the addition of material from the ground surface to the moving mass, the reduction in kinetic energy of the system might be greater than the increase in its potential energy. However, those complex behaviors also depend on which erosion mechanism takes place, which flow theologies are involved, and how the mass and momentum productions are considered in the dynamical model equations. This can probably be the situation for flows in moderate to low slope angles. Nevertheless, the erosion related mobility can be site- and material-specific. This requires further research. Regarding the effect of density change for debris flow, some research finds that for debris flow in the jump to the fast-moving incoming flow, compared to the predicted runup height, the difference in the predicted impact load appears to be even negligible [56]. Com- pressibility in natural debris flow is highly variable in its temporal and spatial distribution. Especially for nearly liquefied debris flows for where state of material varies de-pending on the barrier deformation pattern, the active state of downstream-jump debris material due to barrier deflection facilitates to further reduce the impact load. The contribution of jump volume resistance to the load attenuation obviously depends on the jump length, which in turn is controlled by the incoming flow properties. The compressibility (density change) does not significantly contribute to the load attenuation. Sustainability 2021, 13, 7955 17 of 19

At present, because of the complicated solid–fluid interaction within the two-phase debris flow [9,34,47], possible scale-dependent effects [17–19] cannot be explicitly considered. Some researchers conclude that the peak flow depth being out of sync with the peak velocity (a tapered flow front) causes of this discrepancy. Characterization of the debris properties is more important than selection of debris impact model. However, parameter values are constrained to differing degrees in different types of flows. Perhaps the greatest limitation on the model’s predictive capability results from the assumption that flows maintain constant masses as they move down slope. Mass change is an important feature of some debris flow sand avalanches, and although mass change terms may be appended to the model equations with little difficulty, the magnitude of such terms depends on external forces, which are poorly constrained in most instances [57]. In the future, the application of the novel erosion rate model to experimental and complex natural events of debris flow and avalanche mixtures would require substan- tial additional work and corresponding parameter estimates, either derived from field measurements or back calculations, involving observation data, which, therefore, must be deferred to some future contributions [9]. At the same time, debris flow hazard is viewed as a threat that has the potential to overwhelm people, property, and the environment. It is a pre-existing condition that can turn into a catastrophe depending on the influence of exogenous and endogenous factors. We should use multiple models for comprehensive research, such as the Pressure and Release (PAR) Model and Access Model, the Hazards-of- Place (HOP) Model, the Regions of Risk Model [58], and so on. These models techniques can significantly contribute to the deeper understanding of debris flow hazard, risk, and vulnerability, providing better representation and visualization of human–environment interaction [59,60].

6. Conclusions Based on the theory of shallow water deposition morphology dynamics, a GAST model suitable for debris flow was developed. The model was established under the framework of a Godunov-type finite volume scheme, and the control equation was solved by a completely conservative numerical algorithm. Compared with previous experiments and actual debris flows, the new model can reflect the dynamic process of debris flows. Although there are other two-phase models that perform better than the GAST model, the GAST model is still attractive because GPU acceleration technology greatly improves the computational time of the model. The experimental results in this paper prove that GPU acceleration technology can improve the performance by 14.62 times and increase the computational efficiency by 16–35%, especially in applications that require a large amount of high-performance computing. The general-purpose computing of GPUs will address areas and problems that require considerable manual computing in a low-cost and high-efficiency manner.

Author Contributions: Conceptualization, methodology, formal analysis, writing—original draft preparation, writing—review and editing, visualization, Y.K. and J.H.; formal analysis, Y.T. and B.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (52079106, 52009104, 51609199) and the National Key Research Program (2016YFC0402704). Acknowledgments: This research was funded by the National Natural Science Foundation of China (52079106, 52009104, 51609199) and the National Key Research Program (2016YFC0402704). We also thank the anonymous reviewers and editors for their constructive comments that have improved the manuscript. Conﬂicts of Interest: The authors declare no conﬂict of interest. Sustainability 2021, 13, 7955 18 of 19

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