Method and apparatus for simulating rapid sediment transport due to dam discharge

Abstract

The present invention provides a method and apparatus for simulating rapid sediment transport due to dam release. The method for simulating rapid sediment transport due to dam release includes the steps of: acquiring a dam discharge and a downstream water level; determining a superdiffusion coefficient of a one-dimensional spatial fractional-order advection-diffusion equation based on the discharge, the downstream water level, and a pre-established superdiffusion coefficient library, where the one-dimensional spatial fractional-order advection-diffusion equation includes a spatial fractional derivative diffusion term; and constructing a rapid sediment transport path after dam release based on the superdiffusion coefficient and the one-dimensional spatial fractional-order advection-diffusion equation, where the rapid transport path is intended to represent the rapid sediment transport process after dam release. According to the present invention, by simulating the sediment diffusion phenomenon using the spatial fractional derivative diffusion term, the rapid sediment transport process during dam release can be accurately described.

Description

[Technical Field]

[0001] FIELD OF THE INVENTION Embodiments of the present invention relate to the field of computers, and more particularly to a method and apparatus for simulating rapid sediment transport due to dam release. [Background technology]

[0002] Dam discharges cause dramatic changes in hydraulic parameters over time, resulting in rapid flood waves. The sudden turbulence of the water body caused by the rapid flood waves significantly affects the particle motion state within the water body, usually resulting in rapid transport phenomena. Currently, most dam groups use collaborative scheduling to improve overall sediment discharge levels and thereby extend the dam's lifespan. Accurate simulation of the sediment transport process is crucial for collaborative scheduling of dam groups.

[0003] In the prior art, the water flow and sedimentation process of a dam is usually simulated using a one-dimensional advection-diffusion equation, but this model cannot describe the rapid sediment transport process caused by the sudden occurrence of turbulent fluctuations, which is unfavorable for the joint scheduling of dam sediment. Summary of the Invention [Problem to be solved by the invention]

[0004] In order to accurately simulate the sediment rapid transport process of a dam, the present invention proposes a method and apparatus for simulating the sediment rapid transport caused by dam discharge. [Means for solving the problem]

[0005] In a first aspect, the present invention provides a method for producing a composition comprising: 1. A method for simulating rapid sediment transport due to dam discharge, comprising: obtaining a discharge flow rate and a downstream water level of the dam; determining a superdiffusion coefficient of a one-dimensional spatial fractional order advection-diffusion equation based on the discharge flow rate, the downstream water level, and a pre-established superdiffusion coefficient library, the one-dimensional spatial fractional order advection-diffusion equation including a spatial fractional order derivative diffusion term; and constructing a rapid transport path of sediment after dam release based on a hyperdiffusion coefficient and a one-dimensional spatial fractional-order advection-diffusion equation, the rapid transport path being intended to represent the rapid transport process of sediment after dam release.

[0006] The above method utilizes the memory and inheritance properties of the spatial fractional derivative diffusion term to describe the superdiffusion phenomenon of sediment movement that changes with time and space, simulates the sediment concentration at different positions during dam discharge and the time it takes for the sand peak to reach a predetermined position, accurately describes the rapid sediment transport process during dam discharge, and provides a basis for dam sediment scheduling. In addition, by adopting a one-dimensional advection-diffusion equation, the complexity of the simulation calculation is reduced and the simulation efficiency is improved.

[0007] In a first embodiment of the first aspect, the hyperdiffusion coefficient library is established based on a historical discharge flow rate and a historical downstream water level under each operating condition, and a one-dimensional spatial fractional-order advection-diffusion equation; Obtaining a historical discharge flow rate and a historical downstream water level under each operating condition; Deriving a hyperdiffusion coefficient for each operating condition based on the historical discharge flow rate and the historical downstream water level for each operating condition, and a one-dimensional spatial fractional-order advection-diffusion equation; and establishing a superdiffusion coefficient library based on the superdiffusion coefficients at each operating condition.

[0008] In a second embodiment of the first aspect, in combination with the first aspect or the first embodiment of the first aspect, Based on the superdiffusion coefficient and the one-dimensional spatial fractional advection-diffusion equation, the steps to construct the rapid transport path of sediment after dam discharge are as follows: determining sediment concentrations at different times at a given location after dam discharge based on a hyperdiffusion coefficient and a one-dimensional spatial fractional-order advection-diffusion equation; and constructing a rapid transport route for sediment after dam release based on the sediment concentration.

[0009] Combining the first aspect, in a third embodiment of the first aspect, the one-dimensional spatial fractional-order advection-diffusion equation is as follows:

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[0010] Combining the third embodiment of the first aspect, in a fourth embodiment of the first aspect, the spatial fractional derivative diffusion term is:

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[0011] In a second aspect, the present invention provides an apparatus for simulating rapid sediment transport due to dam discharge, the apparatus comprising: an acquisition module for acquiring the discharge flow rate and downstream water level of the dam; a determination module for determining a superdiffusion coefficient of a one-dimensional spatial fractional order advection-diffusion equation based on the discharge flow rate, the downstream water level, and a pre-established superdiffusion coefficient library, the one-dimensional spatial fractional order advection-diffusion equation including a spatial fractional order derivative diffusion term; The present invention further provides an apparatus including: a construction module for constructing a rapid transport path of sediment after dam release based on a superdiffusion coefficient and a one-dimensional spatial fractional-order advection-diffusion equation, wherein the rapid transport path is for representing the rapid transport process of sediment after dam release.

[0012] According to the above device, the memory and inheritance characteristics of the spatial fractional derivative diffusion term are utilized to describe the super-diffusion phenomenon of sediment movement that changes with time and space, and the sediment concentration at different positions during dam discharge and the time when the sand peak reaches a predetermined position are simulated, which accurately describes the rapid sediment transport process during dam discharge and provides a basis for dam sediment scheduling. In addition, by adopting a one-dimensional advection-diffusion equation, the complexity of the simulation calculation is reduced and the simulation efficiency is improved.

[0013] In combination with the second aspect, in a first embodiment of the second aspect, the hyperdiffusion coefficient library of the determination module is established based on the historical discharge flow rate and the historical downstream water level under each operating condition, and a one-dimensional spatial fractional-order advection-diffusion equation; an acquisition sub-module for acquiring historical discharge flow rate and historical downstream water level under each operating condition; a derivation submodule for deriving the hyperdiffusion coefficient at each operating condition based on the historical discharge flow rate and the historical downstream water level at each operating condition, and a one-dimensional spatial fractional-order advection-diffusion equation; and an establishment sub-module for establishing a superdiffusion coefficient library based on the superdiffusion coefficients at each operating condition.

[0014] In combination with the second aspect or the first embodiment of the second aspect, in a second embodiment of the second aspect, the construction module a determination submodule for determining sediment concentrations at different times at a given location after dam discharge based on a hyperdiffusion coefficient and a one-dimensional spatial fractional-order advection-diffusion equation; and a construction sub-module for constructing a rapid transport route for sediment after dam release based on the sediment concentration.

[0015] In a third aspect, the present invention further provides a computer apparatus including a memory and a processor, the memory and the processor being communicatively connected to each other, the memory having computer instructions stored therein, the processor executing the computer instructions to perform the steps of the method for simulating rapid sediment transport due to dam release in the first aspect or any embodiment of the first aspect.

[0016] In a fourth aspect, the present invention further provides a computer readable storage medium having stored thereon a computer program which, when executed by a processor, implements the steps of the method for simulating rapid sediment transport due to dam release of the first aspect or any embodiment of the first aspect. [Brief explanation of the drawings]

[0017] In order to more clearly describe the specific embodiments of the present invention or the technical solutions in the prior art, the following will briefly describe the drawings that need to be used in the description of the specific embodiments or the prior art. It should be apparent that the drawings in the following description are some embodiments of the present invention, and those skilled in the art can further obtain other drawings based on these drawings without exerting creative efforts. [Figure 1] 1 is a flowchart of a proposed method for simulating rapid sediment transport due to dam release according to one example embodiment. [Figure 2] In one example, a diagram comparing the simulation results of a method for simulating rapid sediment transport due to dam discharge proposed in an embodiment of the present invention with a simulation method without a superdiffusion term is shown. [Figure 3] 1 is a structural schematic diagram of a proposed device for simulating rapid sediment transport caused by dam discharge according to one exemplary embodiment; [Figure 4] 1 is a schematic diagram of a proposed hardware configuration of a computer device according to one exemplary embodiment; DETAILED DESCRIPTION OF THE INVENTION

[0018] The technical solutions of the present invention will be clearly and completely described below with reference to the drawings.

[0019] The technical features according to the embodiments of the present invention described below may be combined with one another as long as they are not inconsistent with one another.

[0020] Many diffusion phenomena exist in nature, such as the movement of contaminants in soil, groundwater permeation, and turbulent flow. These diffusion phenomena do not satisfy the classical Fickian law of gradient diffusion and are called "anomalous" diffusion. Anomalous diffusion processes are essentially processes with temporal memory and spatial nonlocality. Because the definition of the integer-order derivative limit is local, integer-order diffusion equations cannot accurately describe this type of anomalous diffusion process. However, it has been proven that spatial fractional derivatives can accurately describe physical processes with memory, inheritance, and path-dependence.

[0021] In order to accurately simulate the rapid transport process of sediment from a dam, the present invention proposes a method, an apparatus, a computer device and a medium for simulating the rapid transport of sediment caused by dam discharge.

[0022] 1 is a flowchart of a proposed method for simulating rapid sediment transport due to dam discharge according to an exemplary embodiment. As shown in FIG. 1, the method includes the following steps S101 to S103.

[0023] Step S101: The discharge amount and downstream water level of the dam are acquired.

[0024] Specifically, the discharge flow rate includes the discharge flow velocity and sand content upstream of the dam.

[0025] Step S102: Determine the superdiffusion coefficient of a one-dimensional spatial fractional order advection-diffusion equation based on the discharge flow rate, downstream water level, and a pre-established superdiffusion coefficient library, where the one-dimensional spatial fractional order advection-diffusion equation includes a spatial fractional order derivative diffusion term.

[0026] Specifically, when the dam discharge and downstream water level are different, the corresponding superdiffusion coefficients of the one-dimensional spatial fractional-order advection-diffusion equation are different. The superdiffusion coefficient library consists of superdiffusion coefficients corresponding to different discharge and downstream water levels. Depending on the current dam discharge and downstream water level conditions, it is possible to find the same or similar superdiffusion coefficients in the superdiffusion coefficient library.

[0027] Step S103: Based on the superdiffusion coefficient and the one-dimensional spatial fractional-order advection-diffusion equation, a rapid transport path of sediment after dam release is constructed, and the rapid transport path is used to represent the rapid transport process of sediment after dam release.

[0028] For example, the finite difference method can be adopted to solve a one-dimensional spatial fractional order advection-diffusion equation to simulate the rapid sediment transport process under this condition.

[0029] In the rapid transport channel, the sudden flood wave after dam release causes rapid sediment movement. The sediment movement process after dam release is a superdiffusion phenomenon, and the classical one-dimensional advection-diffusion equation cannot accurately describe this superdiffusion phenomenon, i.e., it cannot accurately describe the rapid sediment transport process. According to the method provided in the embodiment of the present invention, the memory and inheritance properties of the spatial fractional derivative diffusion term are utilized to describe the superdiffusion phenomenon of sediment movement that changes over time and space. By simulating the sediment concentration at different locations during dam release and the time when the sand peak reaches a predetermined position, the rapid sediment transport process during dam release can be accurately described, providing a basis for dam sediment scheduling. Furthermore, the adoption of the one-dimensional advection-diffusion equation reduces the complexity of the simulation calculation and improves simulation efficiency.

[0030] In one example, in the above step S102, the superdiffusion coefficient library is established based on the historical discharge flow rate and historical downstream water level at each operating condition, and a one-dimensional spatial fractional-order advection-diffusion equation, specifically including the following steps:

[0031] First, obtain the historical discharge flow rate and historical downstream water level under each operating condition. Next, the hyperdiffusion coefficient for each operating condition is derived based on the historical discharge flow rate and historical downstream water level under each operating condition, as well as the one-dimensional spatial fractional-order advection-diffusion equation. Finally, a superdiffusion coefficient library is established based on the superdiffusion coefficients at each operating condition.

[0032] In one example, in the above step S103, the steps for constructing a rapid transport route for sediment after dam release are as follows.

[0033] First, the sediment concentration at a given location after dam discharge at different times is determined based on the hyperdiffusion coefficient and a one-dimensional spatial fractional-order advection-diffusion equation.

[0034] Specifically, the predetermined position can be set as needed, and may be a cross section near the downstream dam, or may be any notable position from the upstream to the center downstream of the dam.

[0035] Next, based on the sediment concentration, a rapid transport route for sediment after dam release is constructed.

[0036] Specifically, by obtaining the sediment concentration at a specific location at different times, it is possible to obtain the change in sediment concentration from upstream to downstream of the dam when the dam discharges, i.e., the rapid sediment transport process when the dam discharges.

[0037] In one example, a one-dimensional spatial fractional order advection-diffusion equation is expressed as follows:

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[0038] In one possible implementation, the spatial fractional derivative diffusion term is:

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[0039] In one example, a dam area free from the operating conditions of lateral sand-laden flow and riverbank collapse is selected, and multiple monitoring stations are installed near the upstream dam at the tail of the dam to monitor the discharge flow rate and sand content data of the upstream dam in real time and collect historical related data.

[0040] Considering that the sediment scheduling time is short and the erosion and deposition of the dam are in a relatively balanced state, the net exchange between suspended sediment and bed load sediment is ignored, and factors such as lateral sand-laden flow and riverbank collapse are not taken into account. A one-dimensional model of sediment transport under the corresponding discharge operating conditions is constructed. The control equation is as follows:

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[0041] Figure 2 compares the simulation results of the method proposed in the embodiment of the present invention for simulating rapid sediment transport due to dam discharge with those of a simulation method without a superdiffusion term. At the initial time of dam discharge, sediment exhibits rapid transport, i.e., sediment concentration changes rapidly at the initial time of dam discharge. As shown in Figure 2, the simulation method proposed in the embodiment of the present invention can accurately capture the rapid sediment transport process near the initial time of upstream dam discharge. The sediment concentration changes rapidly, i.e., a small amount of sediment reaches a predetermined location downstream near the initial time. However, the simulation results obtained using a diffusion model without a superdiffusion term show that the sediment concentration is zero near the initial time. However, in the actual process, a small amount of sediment exists near the initial time of dam discharge. Therefore, the diffusion model without a superdiffusion term cannot describe the rapid sediment transport at the initial time. On the other hand, the width of the sand peak simulated using this method is wider than the width of the sand peak simulated using a diffusion model without a superdiffusion term, and the width of the sand peak is intended to represent the duration of sediment concentration higher than a predetermined sediment concentration. During sediment scheduling for a downstream dam, the duration of sediment scheduling is usually determined by the width of the sand peak of the sediment, i.e., the duration of sediment scheduling corresponds to this sand peak process. Therefore, when performing sediment scheduling for the sand peak simulated according to an embodiment of the present invention, the scheduling time is relatively long, and sediment can be discharged as completely as possible.

[0042] Based on the same inventive concept, an embodiment of the present invention further provides an apparatus for simulating rapid transport of sediment due to dam release, and as shown in FIG. 3, the apparatus includes an acquisition module 301, a determination module 302, and a construction module 303.

[0043] The acquisition module 301 is for acquiring the discharge amount and downstream water level of the dam. For details, please refer to the description of step S101 in the above embodiment, and the description will not be repeated here.

[0044] The determination module 302 is for determining the super-diffusion coefficient of the one-dimensional spatial fractional-order advection-diffusion equation based on the discharge flow rate, the downstream water level, and a pre-established super-diffusion coefficient library, where the one-dimensional spatial fractional-order advection-diffusion equation includes a spatial fractional-order derivative diffusion term. For details, please refer to the description of step S102 in the above embodiment, and the description will not be repeated here.

[0045] The construction module 303 is for constructing a rapid transport path of sediment after dam discharge based on the hyperdiffusion coefficient and the one-dimensional spatial fractional advection-diffusion equation, and the rapid transport path is for representing the rapid transport process of sediment after dam discharge. For details, please refer to the description of step S103 in the above embodiment, and the description will not be repeated here.

[0046] In one example, the hyperdiffusion coefficient library of the determination module 302 is established based on historical discharge flow rates and historical downstream water levels at each operating condition, and a one-dimensional spatial fractional-order advection-diffusion equation, and the determination module 302 includes an acquisition submodule, a derivation submodule, and an establishment submodule.

[0047] The acquisition sub-module is for acquiring the historical discharge flow rate and the historical downstream water level under each operating condition, for details of which please refer to the description of the above embodiment and will not be described again here.

[0048] The acquisition submodule is for deriving the hyperdiffusion coefficient for each operating condition based on the historical discharge flow rate and the historical downstream water level for each operating condition, and a one-dimensional spatial fractional order advection-diffusion equation. For details, please refer to the description of the above embodiment, and the description will not be repeated here.

[0049] The establishment sub-module is for establishing a superdiffusion coefficient library based on the superdiffusion coefficients at each operating condition, for details of which please refer to the description of the above examples and will not be repeated here.

[0050] In one example, the construction module 303 includes a determination sub-module and a construction sub-module.

[0051] The determination submodule is for determining the sediment concentration at a predetermined location at different times after dam discharge based on the hyperdiffusion coefficient and a one-dimensional spatial fractional advection-diffusion equation. For details, please refer to the description of the above embodiment, and the description will not be repeated here.

[0052] The construction sub-module is for constructing a rapid transport route for sediment after dam discharge based on sediment concentration, for details, please refer to the description of the above embodiment, and will not be described again here.

[0053] In one example, the one-dimensional spatial fractional order advection-diffusion equation for the device is:

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[0054] In a further example, the spatial fractional derivative diffusion term of the device is:

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[0055] The specific limitations and beneficial effects of the above device may be referenced to the limitations of the above method for simulating rapid sediment transport due to dam discharge, and will not be described again here. Each of the above modules can be realized in whole or in part by software, hardware, or a combination thereof. Each of the above modules may be integrated into or independent of the processor of a computer device in the form of hardware, or may be stored in the memory of a computer device in the form of software so that the processor can easily call and execute the operations corresponding to each of the above modules.

[0056] Figure 4 is a schematic diagram of a hardware configuration of a proposed computer device according to one exemplary embodiment. As shown in Figure 4, the device includes one or more processors 410 and memory 420, including persistent memory, volatile memory, and a hard disk, with one processor 410 illustrated in Figure 4. The device may further include an input device 430 and an output device 440.

[0057] The processor 410, memory 420, input device 430, and output device 440 may be connected via a bus or in other ways, and FIG. 4 illustrates the connection via a bus.

[0058] The processor 410 may be a central processing unit (CPU). The processor 410 may also be other general-purpose processors, chips such as digital signal processors (DSPs), application specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, or a combination of the above. The general-purpose processor may be a microprocessor, or may be any conventional processor, etc.

[0059] The memory 420 may be a non-transitory computer-readable storage medium, including persistent memory, volatile memory, and a hard disk, and may be used to store non-transitory software programs, non-transitory computer-executable programs, and modules, such as program instructions / modules corresponding to the method for simulating rapid sediment transport during dam release in the embodiments of the present application. The processor 410 executes the non-transitory software programs, instructions, and modules stored in the memory 420 to perform various functional applications and data processing of the server, i.e., to realize any of the methods for simulating rapid sediment transport during dam release.

[0060] Memory 420 may include a program storage area capable of storing an operating system, an application program required for at least one function, and a data storage area capable of storing data used as needed. Additionally, memory 420 may include high-speed random access memory, as well as non-transitory memory, such as at least one magnetic disk memory device, flash memory device, or other non-transitory solid-state memory device. In some embodiments, memory 420 optionally includes memory located remotely from processor 410, which may be connected to the data processing device via a network. Examples of such networks include, but are not limited to, the Internet, an intranet, a local area network, a mobile communications network, and combinations thereof.

[0061] The input device 430 can receive input numeric or textual information and generate signal inputs for user settings and function control. The output device 440 can include a display device such as a display.

[0062] The one or more modules, when stored in memory 420 and executed by one or more processors 410, perform the method shown in FIG.

[0063] The above-mentioned product can execute the method according to the embodiment of the present invention, and has corresponding functional modules and beneficial effects. For technical details not described in detail in this embodiment, please refer to the relevant description in the embodiment shown in FIG.

[0064] An embodiment of the present invention further provides a non-transitory computer storage medium having stored thereon computer-executable instructions capable of performing the simulation method of any of the above method embodiments, wherein the storage medium may be a magnetic disk, an optical disk, a read-only memory (ROM), a random access memory (RAM), a flash memory, a hard disk drive (HDD), or a solid-state drive (SSD), or the storage medium may further include a combination of the above types of memory.

Claims

1. 1. A method for simulating rapid sediment transport due to dam discharge, comprising: obtaining a discharge flow rate and a downstream water level of the dam; determining a superdiffusion coefficient of a one-dimensional spatial fractional-order advection-diffusion equation based on the discharge flow rate, the downstream water level, and a pre-established superdiffusion coefficient library, the one-dimensional spatial fractional-order advection-diffusion equation including a spatial fractional-order derivative diffusion term; A method comprising: constructing a rapid transport path of sediment after dam release based on the superdiffusion coefficient and the one-dimensional spatial fractional-order advection-diffusion equation, wherein the rapid transport path is intended to represent the rapid transport process of sediment after dam release.

2. the hyperdiffusion coefficient library is established based on the historical discharge flow rate and the historical downstream water level under each operating condition, and the one-dimensional spatial fractional-order advection-diffusion equation; Obtaining a historical discharge flow rate and a historical downstream water level under each operating condition; Deriving a hyperdiffusion coefficient for each operating condition based on the historical discharge flow rate and the historical downstream water level for each operating condition, and the one-dimensional spatial fractional-order advection-diffusion equation; and establishing the superdiffusion coefficient library based on the superdiffusion coefficients at each operating condition.

3. The step of constructing a rapid transport path for sediment after dam release based on the hyperdiffusion coefficient and the one-dimensional spatial fractional-order advection-diffusion equation includes: determining sediment concentrations at different times at a predetermined location after dam discharge based on the hyperdiffusion coefficient and the one-dimensional spatial fractional-order advection-diffusion equation; The method according to claim 1 or 2, further comprising the step of constructing a rapid transport route for sediment after dam release based on the sediment concentration.

4. The one-dimensional spatial fractional order advection-diffusion equation is as follows: [Equation 1] where u is the cross-sectional average flow velocity, C is the sediment concentration in the water body, t is time, x is the position along the direction of water flow, D is the hyperdiffusion coefficient, v is the lateral sediment inflow per unit time, and w is the net exchange rate between suspended sediment and bedload sediment. [Equation 2] 2. The method of claim 1, wherein is the spatial fractional derivative diffusion term, α is the spatial fractional order, b is the location of the outlet dam, and RL represents the Riemann-Liouville integral.

5. The spatial fractional derivative diffusion term is: [Equation 3] 5. The method of claim 4, wherein b is the location of the tailrace dam, x is the location along the direction of water flow movement, α is the fractional order, t is time, and RL represents the Riemann-Liouville integral.

6. A device for simulating rapid transport of sediment due to dam discharge, an acquisition module for acquiring the discharge flow rate and downstream water level of the dam; a determination module for determining a superdiffusion coefficient of a one-dimensional spatial fractional order advection-diffusion equation based on the discharge flow rate, the downstream water level, and a pre-established superdiffusion coefficient library, the one-dimensional spatial fractional order advection-diffusion equation including a spatial fractional order derivative diffusion term; and a construction module for constructing a rapid transport path of sediment after dam release based on the superdiffusion coefficient and the one-dimensional spatial fractional-order advection-diffusion equation, wherein the rapid transport path is for representing the rapid transport process of sediment after dam release.

7. The hyperdiffusion coefficient library of the determination module is established based on the historical discharge flow rate and the historical downstream water level under each operating condition, and the one-dimensional spatial fractional-order advection-diffusion equation; an acquisition sub-module for acquiring historical discharge flow rate and historical downstream water level under each operating condition; a derivation submodule for deriving a hyperdiffusion coefficient at each operating condition based on the historical discharge flow rate and the historical downstream water level at each operating condition, and the one-dimensional spatial fractional-order advection-diffusion equation; and an establishment sub-module for establishing the superdiffusion coefficient library based on the superdiffusion coefficients at each operating condition.

8. The construction module comprises: a determination submodule for determining sediment concentrations at different times at a predetermined location after dam discharge based on the hyperdiffusion coefficient and the one-dimensional spatial fractional-order advection-diffusion equation; The apparatus according to claim 6 or 7, further comprising a construction sub-module for constructing a rapid transport route for sediment after dam release based on the sediment concentration.

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