Navigable flow condition simulation based on two-dimensional hydrodynamic parallel model ＊

Shang-hong Zhang (張尚弘), Yu Wu (吳昱), Zhu Jing (荊柱), Yu-jun Yi (易雨君)

1. Renewable Energy School, North China Electric Power University, Beijing 102206, China

2. State Key Laboratory of Water Environment Simulation and Pollution Control, School of Environment, Beijing Normal University, Beijing 100875, China

Abstract: Navigable flow condition simulations can provide detailed information on water depth and velocity distribution, simulation speed is one of the key factors which influence real-time navigation. In this paper, a navigable flow condition simulation system is developed to provide useful information for waterway management and shipping safety. To improve the simulation speed of 2-D hydrodynamic model, an explicit finite volume method and OpenMP are used to realize parallel computing. Two mesh schemes and two computing platforms are adopted to study the parallel model?s performance in the Yangtze River, China. The results show that the parallel model achieves dramatic acceleration, with a maximum speedup ratio of 34.94×. The parallel model can determine the flow state of the navigable channel in about 4 min, efficiency is further improved by a flow simulation scheme database. The developed system can provide early warning information for shipping safety, allowing ships to choose better routes and navigation areas according to real-time navigable flow conditions.

Key words: 2-D hydrodynamic model, finite volume method, parallel computation, OpenMP, navigable flow condition

Introduction

Inland river shipping has the advantages of low cost, low energy consumption, light pollution, and large capacity. Its importance in promoting regional economic development means that countries around the world attach great significance to the development of inland river shipping. However, the increase in the number of ships leads to management problems, and extreme hydrological conditions cause ship navigation accidents, resulting in casualties and loss of property[1].Therefore, safety is increasingly the focus of attention in the shipping community.

There are many causes of shipping accidents.One of the primary factors affecting the safety of shipping is the complex flow conditions caused by extreme hydrological conditions. For example, the accident of cargo ship “Yang Shi No. 8” in 1998 was caused by too high flow velocity in the waterway[2].On July 10, 2011, the passenger liner “Bu Jiaer”encountered a rainstorm in the Volga River that caused a big surge of river water to pour onto the right side of the ship, which led to its capsizing. Therefore,it is very important to obtain detailed and accurate information of the navigable flow conditions and supply an early warning, if necessary, to ensure the safety of the ship.

Currently, inland navigation relies mainly on navigation marks and electronic navigation charts.There are two problems in real applications. One is that electronic navigation charts provide only the distribution of water depth. However, research has revealed that velocity, gradient, and flow state are also important factors in the safety of navigation[3]. In an emergency, it is very important to obtain detailed,real-time navigable flow conditions to aid navigation and decision-making. The second problem is that navigation marks provide the navigable range for all ships, but each ship has different dynamic conditions and loading capacities. Thus, their suitable navigable flow conditions are not the same, and navigation marks cannot reflect this difference. Hence, providing real-time information about the current flow field distribution is important for the safety of navigation,because different ships can choose an appropriate route according to their own circumstances. In addition, detailed flow conditions can serve as an effective supplement to navigation marks, making full use of the waterway resources and improving the utilization rate of the waterway.

Previous studies on navigable flow conditions have mostly focused on approach channel, outlet area of ship lock, or parts of waterway[4-5]. Little research has considered real-time navigable flow simulations on long waterways. In navigable flow condition simulations, a 2-D hydrodynamic model can effectively simulate the flow field distribution and change processes[6-8], supplying enough flow information to the ship?s driver. The computational load of the 2-D hydrodynamic model is somewhat large, so obtaining the simulation results can take a lot of time. Although this is acceptable in waterway planning and engineering, it is not suitable for real-time navigable flow condition simulations, especially in emergency scenarios. Thus, the key point in using the 2-D hydrodynamic model for real-time navigable flow condition simulations is to improve the simulation speed and accuracy of the model. To solve the 2-D hydrodynamic model, the explicit finite volume method is widely used[9-10]. Because the discretization of friction sources takes an explicit form, the calculation of boundary fluxes uses only previous values, and there is no correlation between grid calculations in the same time step. This makes the model algorithm simpler and more effective[11].

To reduce the computation time and improve the calculation efficiency of the model, parallel computing can be applied to the 2-D hydrodynamic model[12-14]. The parallel computation of mathematical models is mainly applied in three ways: generalpurpose computing on graphics processing units, via a message processing interface, or using a shared memory architecture (OpenMP)[15-17]. In a shared memory model, communication between threads is achieved by reading and writing directly to the shared memory. Models developed in accordance with the OpenMP standard tend to be simple, extendable, and portable[18], and have obtained many good results in flood prediction and sediment transport simulations[19-20].

In this paper, a navigable flow condition simulation system to support inland river shipping is developed based on a 2-D hydrodynamic model and parallel computing technology. The main objectives of this study are to: (1) build a 2-D hydrodynamic model and study the implementation method and key technologies of 2-D hydrodynamic parallel computing based on OpenMP, (2) explore the acceleration effect of the 2-D parallel model for different meshes and hardware configurations, and (3) discuss the applications and advantages of the 2-D parallel model for inland river shipping.

1. Study area and system framework

The Yangtze River is the longest river in China,having a vast watershed area and abundant water reserves. Therefore, its navigable conditions and ship-through capacity are extensive, and it is the main focus of the development of inland shipping. Since the impoundment of the Three Gorges reservoir, shipping conditions in the upper reaches of the Yangtze River have been improved, making this region a very busy shipping line. This paper studies the waterway from Jiangjin to Chongqing, in the upper reaches of the Yangtze River (Fig. 1), a stretch of approximately 78 km,and there are nine hydrological stations along this reach. The terrain of the waterway has a large slope,and some reaches are narrow and curved. In addition,the water flow is severely affected by the Jialing River tributary. Hence, the flow conditions are very complex,making navigation difficult and leading to the occurrence of shipping accidents in recent years.

Fig. 1 (Color online) Sketch of the study area

To determine the real-time navigable flow conditions, a navigable flow condition simulation system was developed. The functional modules of the system include data monitoring, flow calculation, database,and visualization modules. The structure of the system is illustrated in Fig. 2. The data monitoring module is used to obtain the upstream inflow and water level data, and provides the initial calculation parameters for the flow calculation module. These data are obtained from monitoring terminals installed in hydrological stations (see Fig. 1). In this study reach,the hydrological data is refreshed once per hour. The flow calculation module reads the terrain data from the database module and the boundary data acquired by the monitoring module, and outputs navigable flow condition information from the parallel simulation model. When the simulation has been finished, the results are stored in the database module. This module also stores the terrain data of the waterway, the boundary data from the data monitoring module, and the simulation results from the flow calculation module. The visualization module takes the flow simulation results from the database module, and feeds data back to the user in a vivid way, providing a reference for the navigation decision. The main focus of this paper is to study the implementation of the flow calculation module.

2. Method

2.1 2-D hydrodynamic model

2.1.1 Control equation

In modeling the waterway flow, the horizontal scale is larger than the vertical depth scale. Therefore,ignoring the impact of the Coriolis force and wind, the 2-D shallow water equation is used as the control equation. The conservation of the 2-D shallow water equation is as follows:the coordinate system used in the equation is the plane Cartesian coordinate system, where u and v represent the velocity components in the x and y directions, respectively, g represents gravitational acceleration, Sox=-?Zb/?x , Soy=-?Zb/?y represent the bottom slope term of the x and y directions,respectively, Zbrepresents the bottom elevation,represent the friction slope of the x and y directions,respectively, and n is the Manning coefficient.

To adapt to the irregular terrain boundary, the study area is divided into an unstructured quadrilateral mesh, and the finite volume method is used for the numerical discretization of this mesh (Fig. 3).

The integral of the mesh elements is calculated as

Fig. 2 Schematic overview of the navigable flow condition simulation system

Fig. 3 Schematic diagram of the finite volume discretization

Using Gauss? formula to convert the volume integral to a line integral along the boundary of the mesh cell, we obtain

where i is the edge of each mesh element, Δliis the edge length of the mesh element,iF is the numerical flux of edge i, niis the external normal unit vector of edge i, and ΔAkis the area of the mesh.

Roe?s scheme for the Riemann approximate solution is used to solve the interfacial numerical flux

where URand ULare the right and left conservative variables of the interface, respectively, and J is the Jacobian matrix of the Roe scheme.

To allow the model to adapt to the complicated terrain, the bottom slope source term is processed by the bottom slope characteristic decomposition method

In this paper, an explicit scheme is adopted for the solution of these equations. Thus, the final discrete equation on the mesh is as follows

2.1.2 Calculation process

Once the calculation has started, the model reads the mesh data and the flow boundary data, initializing the calculation parameters and providing initial values for solving the flux and source terms. The mesh equations are then solved with the explicit finite volume method, which enables the state of the current moment of each mesh to be associated with that of the previous moment of the adjacent meshes. Thus, the solution of the flux and source terms for each mesh is independent, and we can obtain the state information(i.e., the water depth and velocity components in the x and y directions) of each mesh at time T+Δt as long as the interface flux, bottom slope source term,and friction source term are solved according to the state information of each mesh at time T. When the calculation is finished, state information is obtained for each mesh at time Tmax. Finally, state information for each mesh at Tmaxis output and stored in the database. The calculation process of the model is illustrated in Fig. 4.

2.2 Parallel model implementation based on openmp

2.2.1 Basic concepts of openmp

OpenMP is a shared memory programming model rather than an independent parallel language. It can be used in Fortran77/90/95 and C/C++. Guidance notes are added to the source code, and the compiler automatically enables the program to be executed in parallel. Thus OpenMP has the advantages of simplicity, portability, and scalability. The execution model of OpenMP takes a fork-join form, where forks create a new thread or wake up existing threads and joins represent the merging of multiple threads. Initially,only the main thread is executed. This main thread enlists subthreads to perform tasks in parallel if it encounters a part of the code that allows parallel computing. After the parallel computing, the subthreads are blocked or terminated, and the model returns to the main thread.

Fig. 4 Flowchart of the numerical simulation in the model

2.2.2 Openmp parallel implementation

Because there is no correlation between the numerical calculations for each mesh unit in the same time step, the OpenMP parallel computing method can be used to implement parallel computation in the model. The calculation of different mesh units is allocated to different threads; i.e., the total mesh calculation is decomposed into several small groups,each group can be executed as one thread. Because all the threads can be run at the same time, this parallel architecture realizes considerable acceleration,reducing the total calculation time. The model was coded in Fortran, and the partial codes of the OpenMP parallel calculations are shown in Fig. 5.

In Fig. 5:

Fig. 5 Schematic diagram of partial OpenMP parallel computing codes in the model

(1) !$OMP is an OpenMP directive statement identifier.

(2) !$OMP PARALLEL and !$OMP END PARALLEL determine the parallel region, i.e., the part between these two statements is the parallel domain.

(3) NUM_THREADS (number) creates threads in the parallel domain. Generally, the number of threads that can be accommodated by the parallel domain is equal to the number of physical cores. For example, NUM_THREADS (8) calls eight threads.

(4) DEFAULT (PRIVATE) and SHARED (variables) define the variable properties in the parallel region. In a multithreaded environment, the variable properties can be shared or private. To ensure normal access to variables and the correctness of the model,the variable properties must be defined correctly, and this definition can significantly affect the acceleration of the model. In this paper, the definition of the variable properties in the parallel domain is divided into two steps. The first step uses DEFAULT(PRIVATE) to define all the variables in the parallel domain as private so that each thread has a private copy of the variable. The second step picks up variables that are accessible by all threads in a parallel computation and are not affected by the threads, and uses SHARED (variables) to define them as shared variables.

(5) !$OMP DO and !$OMP END DO are executable statements, i.e., the content within the DO loop is executed in a parallel computing method.

3. Navigable flow condition simulation

According to the upstream hydrological monito-ring data, the navigable flow condition simulation system should output real-time flow condition data.Even though the speed of the 2-D parallel model meets the system requirements, the computer performing the calculations should run at full load all the time. Considering that most of the flow discharge cases can be calculated in advance, a flow schemes database for flow simulation results is established to store simulation results for different flow conditions.The flow scheme database uses an interpolation method to obtain the current flow field distribution according to the current inflow data and the inflow schemes in the database. Because the time required for interpolation is much less than the 2-D parallel model calculation time, the real-time performance of the water flow simulation is further improved.

The system first searches the simulation results in the flow scheme database according to the current inflow conditions, i.e., the discharge of upstream section and the water level of downstream section. If no result can be found in the database, the 2-D parallel model simulates the new flow scheme, and the result is added to the flow scheme database. Thus, the real-time simulation of water flow is ensured by the combination of the flow scheme database and the 2-D parallel model.

4. Results

4.1 Model verification

To test the accuracy of the hydrodynamic model,we compared three discharge levels with the corresponding measured water levels and velocity at nine different points (shown in Fig. 1). The verification results are presented in Table 1. In the table, QYangtzeis the inflow to the Yangtze River and QJialingis the inflow to the Jialing River. The three discharge levels represent small, medium, and large flow levels of the river reach. Table 1 indicates that the deviation in water depth is between -0.08 m and 0.08 m, the deviation in flow velocity is between -0.07 m/s and 0.06 m/s, and the relative deviation is between -3.2%and 4.3%. The simulation results are in good agreement with the measured data.

4.2 Parallel performance analysis

To test the parallel performance of the model,two different mesh sizes were used to divide the study area.

Scheme 1: mesh size 30 m, giving 131 855 cells and 136 948 nodes.

Scheme 2: mesh size 20 m, giving 253 161 cells and 260 579 nodes.

Table 1 Water depth and flow velocity verification

Table 2 Time (s), speedup, efficiency (%) and time-saving ratio (%) of different calculation schemes on platform 1

Table 3 Time (s), speedup, efficiency (%) and time-saving ratio (%) of different calculation schemes on platform 2

The parallel model was executed on two different computing platforms to test its performance. The main configuration parameters of the different platforms are as follows:

Platform 1: two Intel E5-2680 V2 2.80 GHz processors, 20 physical cores, 64 GB memory, and the Intel Visual Fortran Composer XE 2013 SP1 compiler.

Platform 2: 32 Intel Xeon E5-4620 eight-core processors at 2.20 GHz, 256 physical cores, shared memory of 4 TB, NUMA architecture, and the Intel Visual Fortran Composer XE 2013 SP1 compiler.

In the process of parallel computing, Platform 1 calls 2, 4, 8, 12, 16, and 20 threads, whereas Platform 2 calls 16, 32, 48, 64, 80, 96, 112, and 128 threads.We analyzed the speedup, efficiency, and time-saving ratio of the models. The results are given in Tables 2,3.

The speedup ratio S is calculated as follows

where TSis the serial computation time and TPis the parallel computation time.

Efficiency E is calculated as follows

where n is the number of threads.

The time-saving ratio δT is calculated as follows

4.3 Navigable flow simulation speed

From Tables 2, 3, we can see that the shortest flow condition calculation time is 2.2 min (132 s) in scheme 1 and 4.1 min (246 s) in scheme 2. Using the flow scheme database to obtain the flow conditions,approximately 10 s is required for interpolation, which fully satisfies the real-time requirements of the system.Thus, with the combination of the 2-D parallel model and the flow scheme database, the flow condition simulation system can supply detailed flow information for shipping when the changes in flow discharge are relatively slow. When extreme flow conditions occur in the upper reaches of the river system, such as during a flood or surge event, the speed of the model will enable an early warning to be supplied to ships downstream.

5. Discussion

5.1 Acceleration effect analysis

The simulation time of the proposed system is influenced by the computational load and the platform.As the number of grid cells increases, both the serial and parallel computing time increase, although the speedup and efficiency also increase.

For example, when platform 2 calls 128 threads,the speedup with mesh scheme 1 is 26.93 and the speedup with mesh scheme 2 is 34.94. When platform 1 calls 20 threads, the efficiency with mesh scheme 1 is 82% and the efficiency with mesh scheme 2 is 89%.It can be seen that the parallel acceleration effect is more obvious with higher computational loads.

Fig. 6 (Color online) The navigable flow condition simulation visualization

The model has achieved the purpose of parallel acceleration on two different platforms, but the calculation time, speedup, efficiency, and time-saving ratio are different. From Tables 2, 3, we can see that:(1) The minimum computation times with mesh schemes 1 and 2 are 132 s and 246 s, respectively, on platform 1, and 217 s and 359 s, respectively, on platform 2. This indicates that the model performance is better on platform 1 than on platform 2. (2) The maximum speedup ratios of mesh schemes 1 and 2 are 16.30 and 17.80, respectively, on platform 1, and 26.93 and 34.94, respectively, on platform 2, which shows that the number of physical cores is the main factor affecting the speedup ratio. (3) On platform 1,the model is generally very efficient, and only a small reduction in efficiency is apparent as the number of threads increases, demonstrating that platform 1 has a higher degree of resource utilization than platform 2.The computing speed and data transmission efficiency of platform 1 are very high, and the computation time is low when the calculation task is small. However,the small number of physical cores on platform 1 makes it difficult to maintain a fast computing speed as the computing task becomes more onerous.Platform 2 has many physical cores, so it has greater computing potential and can be used for large-scale computing tasks. However, the low computation speed and poor data transmission efficiency mean that additional time is needed as well as computational resources. Generally, besides the number of physical cores and dominant frequency, the connection and integration of the physical cores also affects the acceleration effect. Platforms with high levels of connection and integration will perform better than those with poor connection and integration in terms of parallel acceleration. Thus, platforms with fewer physical cores that are better connected and integrated(e.g., platform 1) are more suitable for parallel computing tasks with small computational loads. In contrast, platforms with more physical cores but a lower level of connection and integration (e.g.,platform 2) are better suited to larger computational loads.

5.2 System application

The navigable flow condition simulation system not only provides more detailed information about the water depth, but also provides the flow velocity distribution. When the flow condition changes suddenly, the system determines these flow changes through the simulation, providing sufficient reaction time for downstream ships to avoid accidents. The system can be combined with meteorological information to simulate the flood process. The parallel model will reduce the computation time, enabling the flood propagation process and corresponding propagation time to be determined in advance and providing a reference for the safety of downstream ships. For example, when the flow of the study area increases from 15 000 m3/s to 35 000 m3/s and the flood propagation time is about 4 h, the 2-D parallel model requires only 4 min to simulate the flow condition,providing plenty of time for downstream ships to react.

In addition, the system provides warning information to any ships with a targeted flow, achieving maximum use of waterway resources. For example,for a 1 000 t ship, the scope for navigation varies depending on whether the flow in a waterway is, e.g.,20 000 m3/s or 30 000 m3/s. Similarly, when the flow in a waterway is 30 000 m3/s , the scope for navigation will depend on whether the ship weighs,e.g., 1 000 t or 3 000 t. Traditional aids to navigation and electronic navigation charts cannot reflect changes in the scope of navigation for different ships in different flow conditions and areas that have very large flow velocities or a vortex. That is, they cannot divide the scope for navigation, leading to a waste of waterway resources. However, the navigable flow condition simulation system can obtain the distribution of water depth (h) and flow velocity (v) from the model (see Figs. 6(a), 6(b)), and according to the water depth and flow velocity threshold for each tonnage vessel, the system can show corresponding secure waterway. We can see from Figs. 6(c), 6(d)that different tonnage vessel has different secure scope,which is different with navigation mark system. In addition, for the same tonnage vessel, the secure scope is different in different river discharge (see Fig. 6(d)).With all these timely status information, the vessels can make full use of secure waterway resources that change according to river discharge.

6. Conclusions

The rapid simulation of navigable flow conditions can provide detailed and timely information for ship navigation, which is beneficial to improving the safety of navigation. In this paper, we have studied the flow simulation module of a navigable flow condition simulation system. A navigable flow simulation model based on 2-D hydrodynamic equations was established, and an explicit finite volume method was used to solve the model equations. To achieve parallel acceleration, OpenMP was adopted, as the calculations in each grid cell are not related in each time step,so they can be calculated separately on different computation cores. The following conclusions can be drawn.

Through a comparative analysis of the parallel effect of the model, we observed that, for the same computing platform, the speedup ratio increases with the number of threads as long as the thread number is less than the total number of physical cores. Moreover,larger computational loads lead to bigger speedup ratios and better acceleration performance. A maximum speedup ratio of 34.94 was achieved in the parallel performance test. A comparison of different computing platforms produced different acceleration effects. As well as the number of physical cores and dominant frequency, the platform with better connection and integration of the cores performed better than that with lower connection and integration levels in terms of parallel acceleration.

Using real-time monitoring of upstream inflow discharge, the 2-D parallel model was able to solve the flow state of the study channel in about 4 min.Combined with the construction of a flow simulation scheme database, the efficiency of the navigation flow simulation can be further improved. High-speed simulations of navigable flow can provide a timely warning for ship navigation in extreme hydrological conditions or emergency events, and detailed flow condition information can aid navigation. In addition,specific ships can choose appropriate safe areas according to the detailed flow conditions and ship characteristics, further enhancing the safety of navigation and greatly improving the utilization of the navigable channel.