Real-time dynamic simulation method and system for temperature field in continuous casting process

Abstract

The invention discloses a real-time dynamic simulation method and system for a temperature field in a continuous casting process. The method comprises the following specific steps that continuous casting equipment with various sensors and executing mechanisms is built; performing a continuous casting experiment to obtain unsteady-state real-time data under the condition of variable process parameters; a temperature field dynamic simulation model based on a mechanism is constructed, and data is used for training and optimization; then replacing a temperature field solving part which is long in time consumption in the mechanism-based temperature field dynamic simulation model with a generative neural network; therefore, a mechanism-neural network coupled continuous casting process temperature field real-time dynamic simulation model is constructed, and rapid and accurate calculation of the temperature field is realized.

Description

Technical Field

[0001] The present invention belongs to the field of metal material preparation and processing, and in particular relates to a real-time dynamic simulation method and system for a continuous casting process. Background Art

[0002] Continuous casting (CC) is a widely used key technology for metal material preparation. During the continuous casting process, molten metal continuously flows into a mold, where it cools and solidifies under the action of the crystallizer. It is then continuously pulled out by the dummy bar, resulting in a metal billet of a certain size, providing raw material for subsequent deformation processes (such as rolling).

[0003] The temperature field during the continuous casting process determines the initial microstructure and properties of the metal material, significantly impacting subsequent processing and the final performance of the product. Currently, the primary method for analyzing the temperature field during the continuous casting process is finite element simulation, which then uses the trial-and-error results of the simulation to propose preset optimized process parameters. For example, Chinese patent CN116738518B proposes a numerical simulation method for the location of light-reduction cracks during continuous casting, as well as an internal quality control method. Through simulation and experimental verification, this method guides the setting of process parameters to reduce continuous casting microcracks.

[0004] However, finite element method-based temperature field simulations during continuous casting suffer from large numbers of grid nodes and strong nonlinearity, resulting in computational times of several hours. This makes the simulations suitable only for pre-design and post-analysis, and incapable of real-time simulation. Furthermore, traditional simulations are only suitable for steady-state temperature field analysis under fixed process parameters and are unable to simulate the dynamic changes in the unsteady-state temperature field during continuous casting. However, in actual production, only real-time dynamic simulation of the temperature field, tailored to the ever-changing continuous casting conditions, can achieve online intelligent control of the continuous casting process.

[0005] In order to solve the above problems, it is urgent to develop a real-time dynamic simulation method and related devices for the temperature field of the continuous casting process to realize the real-time dynamic calculation of the temperature field of the continuous casting process. Summary of the Invention

[0006] The present invention discloses a real-time dynamic simulation method and system for a temperature field in a continuous casting process, so as to solve any of the above and other potential problems in the prior art.

[0007] In order to solve the above technical problems, the technical solution of the present invention is: a real-time dynamic simulation method of the temperature field of a continuous casting process, the method specifically comprising the following steps:

[0008] S1) connecting a data acquisition unit to a continuous casting device and using real-time temperature field data;

[0009] S2) constructing a dynamic simulation model of the continuous casting temperature field based on the mechanism;

[0010] S3) using the non-steady-state real-time data of S1) as input, optimizing the dynamic simulation model of the continuous casting temperature field of the mechanism, and obtaining an optimized dynamic simulation model of the continuous casting temperature field of the mechanism;

[0011] S4) replacing the time-consuming temperature field solution portion of the mechanism-based temperature field dynamic simulation model optimized in S3) with a generative neural network to obtain a mechanism-neural network coupled temperature field dynamic simulation model;

[0012] S5) inputting the real-time collected process data into the temperature field dynamic simulation model coupled with the mechanism-neural network to obtain real-time dynamic simulation data.

[0013] Furthermore, the temperature field real-time data includes non-steady-state real-time data and steady-state real-time data; the non-steady-state real-time data includes: continuous or jump changes in melt temperature, cooling water flow, cooling water temperature and casting speed during continuous casting.

[0014] Furthermore, the specific contents of S2) include:

[0015] S2.1) Use the convection heat transfer equation to calculate the boundary heat transfer state between the cooling water and the water-cooled copper jacket as the boundary condition for the three-dimensional heat conduction differential equation;

[0016] S2.2) Use the three-dimensional heat conduction differential equation to calculate the heat conduction process between and within the mold, the copper alloy liquid phase, and the copper alloy solid phase;

[0017] S2.3) Calculate the heat release during the solidification of the copper alloy using the phase transformation kinetics equation as input to the three-dimensional heat conduction differential equation;

[0018] S2.4) The thermophysical parameters of the substance are expressed as a function of temperature. The thermal conductivity is expressed as: k = K(T); the specific heat capacity is expressed as: C p =P(T); density is denoted as ρ=ρ(T); thermal expansion coefficient is denoted as α=α(T);

[0019] S2.5) integrates the contents of S2.1) to S2.4) to obtain a dynamic simulation model of the continuous casting temperature field based on the mechanism; the molten copper alloy inlet temperature T g , cooling water temperature T w , continuous casting speed V w , cooling water flow U w , and the temperature field T at the current moment i As the input of the dynamic simulation model of continuous casting temperature field, the output is the temperature field T at the next moment i+1 , and there is the following relationship: T i+1 =f(Ti , T g , T w , V w , U w ).

[0020] Further, the specific steps of S3) are:

[0021] S3.1) Add correction terms to the cooling water-copper jacket convection heat transfer equation, the interfacial thermal conductivity between different materials, the phase change kinetics equation, and the material thermophysical property calculation formulas;

[0022] As shown below:

[0023]

[0024]

[0025] k=K(T)+ω 20 ; C p =P(T)+ω 21 ; ρ=ρ(T)+ω 22 ; α=α(T)+ω 23 ;

[0026] All correction terms are denoted as ω, and the following relationship is obtained: T i+1 =f(ω,T i , T g , T w , V w , U w ), part ω is calculated by the neural network with temperature and process parameters as input and becomes a variable associated with the cooling state, that is: ω = ω(T, x), where x represents the process parameters;

[0027] S3.2) Compare the actual cooling water outlet temperature T i jk , expressed as the temperature field T i The jth row, kth column element) and the calculated temperature The difference is: Calculate the actual solidus position x i and calculate the solidus position The difference is: Then, the gradient descent algorithm is used to train and optimize the correction parameter ω of the temperature simulation model to obtain the optimized mechanism-based temperature field dynamic calculation model.

[0028] Furthermore, the specific steps of S4) are: replacing the time-consuming temperature field solution process with a generative neural network to obtain a mechanism-neural network coupled temperature field dynamic simulation model, which takes the temperature field, thermophysical property parameter matrix and current process at the current moment as input, and the temperature field at the next moment as output, and its trained temperature field data comes from the calculation results of the optimized mechanism-based temperature field dynamic simulation model.

[0029] Another object of the present invention is to provide a system for realizing a real-time dynamic simulation method of a temperature field in a continuous casting process, the system comprising:

[0030] A data acquisition unit is used to collect real-time data of the temperature field of the continuous casting equipment at different times;

[0031] The temperature field calculation unit is used to calculate the temperature field at different times according to the collected real-time temperature field data to perform simulation and obtain real-time dynamic simulation data.

[0032] Furthermore, the data acquisition unit includes a water inlet flow acquisition unit, a water inlet flow control acquisition unit, a water inlet temperature acquisition unit, a water outlet temperature acquisition unit, a melt temperature acquisition unit and a crystallizer temperature acquisition unit.

[0033] Furthermore, the system also includes: a data visualization unit: used for performing data mapping to intuitively display the changes or distribution characteristics of process data and temperature data.

[0034] A computer-readable storage medium includes instructions, which, when running on a computer, enable the computer to execute the above-mentioned real-time dynamic simulation method of the continuous casting process.

[0035] The beneficial effects brought about by the technical solution provided by the present invention include at least:

[0036] 1. This paper proposes a modeling method that couples a mechanistic model with a neural network model. This method replaces the unclear and computationally time-consuming parts of the mechanism with a neural network, significantly improving the model's accuracy and speed. Furthermore, embedding the clear mechanistic formula into the neural network reduces the learning difficulty of the neural network, enabling good results to be achieved even with less training data.

[0037] 2. Compared with existing simulation methods, the present invention can realize real-time dynamic calculation of the temperature field during the continuous casting process, thereby laying the foundation for online intelligent control of the continuous casting process. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for describing the embodiments. The drawings described below are only used to illustrate the present invention and do not constitute a limitation of the present invention. In the drawings:

[0039] Figure 1 This is a flow chart of a method for real-time dynamic simulation of temperature field in a continuous casting process according to the present invention;

[0040] Figure 2 This is a logic block diagram of a real-time dynamic simulation system for temperature field in a continuous casting process according to the present invention;

[0041] Figure 3 This is a dynamic real-time simulation effect diagram of the temperature field of the continuous casting process in an embodiment of the present invention. DETAILED DESCRIPTION

[0042] In order to make the purpose, technical solutions and advantages of the present invention more clear, the technical solutions of the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that the embodiments are only used to illustrate the present invention, and the scope of protection of the present invention is not limited to the embodiments. The scope of protection of the present invention shall be based on the scope of protection of the claims.

[0043] The following describes the present invention by taking its application in copper alloy continuous casting production as an example.

[0044] like Figure 1 As shown, the present invention provides a real-time dynamic simulation method for the temperature field of a continuous casting process, which specifically includes the following steps:

[0045] S1) connecting a data acquisition unit to a continuous casting device and using real-time temperature field data;

[0046] S2) constructing a dynamic simulation model of the continuous casting temperature field based on the mechanism;

[0047] S3) using the non-steady-state real-time data of S1) as input, optimizing the dynamic simulation model of the continuous casting temperature field of the mechanism, and obtaining an optimized dynamic simulation model of the continuous casting temperature field of the mechanism;

[0048] S4) replacing the time-consuming temperature field solution portion of the mechanism-based temperature field dynamic simulation model optimized in S3) with a generative neural network, thereby obtaining a mechanism-neural network coupled temperature field dynamic simulation model, wherein the temperature field trained by the neural network is derived from the temperature field calculated by the mechanism-based dynamic simulation model of the continuous casting temperature field;

[0049] S5) inputting the real-time collected process data into the temperature field dynamic simulation model coupled with the mechanism-neural network to obtain real-time dynamic simulation data.

[0050] As a further explanation of the present invention, in said S1, the necessary data acquisition unit includes sensors and control devices including: water inlet flow sensor, water inlet flow control valve, water inlet temperature sensor, water outlet temperature sensor, melt temperature sensor, crystallizer temperature sensor, programmable controller, etc.

[0051] As a further illustration of the present invention, step S3 specifically includes:

[0052] S301: Continuously or jumpily change the melt temperature, cooling water flow, cooling water temperature, and casting speed during continuous casting, and obtain non-steady-state real-time data through the sensors in S1;

[0053] S302: Based on the non-steady-state real-time temperature data, an optimization algorithm is used to modify the dynamic temperature field simulation model to improve its calculation accuracy.

[0054] As a further illustration of the present invention, in step S5, network communication technology is used to transmit the data collected by the sensor to the digital model in real time for calculation to obtain the real-time temperature field of the continuous casting process.

[0055] like Figure 2 As shown, the present invention provides a real-time dynamic simulation system for the temperature field of a continuous casting process, the system comprising:

[0056] A data acquisition unit is used to collect real-time data of the temperature field of the continuous casting equipment at different times;

[0057] The temperature field calculation unit is used to calculate the temperature field at different times according to the collected real-time temperature field data to perform simulation and obtain real-time dynamic simulation data.

[0058] The data acquisition unit includes a water inlet flow acquisition unit, a water inlet flow control acquisition unit, a water inlet temperature acquisition unit, a water outlet temperature acquisition unit, a melt temperature acquisition unit and a crystallizer temperature acquisition unit.

[0059] The system further comprises: a data visualization unit: used for performing data mapping to intuitively display the changes or distribution characteristics of process data and temperature data.

[0060] Example:

[0061] The real-time dynamic simulation method of the temperature field in the copper alloy continuous casting process is as follows:

[0062] 1) Install the water inlet flow sensor, water inlet flow control valve, water inlet temperature sensor, water outlet temperature sensor, melt temperature sensor, and mold temperature sensor to the corresponding positions of the continuous casting equipment.

[0063] 2) Use the three-dimensional heat conduction differential equation to calculate the heat conduction process between the mold, copper alloy liquid phase, and copper alloy solid phase, as well as within each other. The differential equation is as follows:

[0064]

[0065] Where: ρ is the density (kg / m 3 ) is a function of temperature; Cp is the specific heat capacity (J / (kg·K)) as a function of temperature; T is the temperature (K); V w is the continuous casting speed (m / s); k is the thermal conductivity (W / (m·K)). The thermal conductivity inside the object is directly calculated, and the thermal conductivity of the contact surface of the object is corrected by the thermal resistance; t is the time (s); r and z are the radial and axial coordinates (m), respectively. is the internal heat source. Here it is represented by the solidification heat of the metal:

[0066] 3) The convection heat transfer equation is used to calculate the boundary heat transfer state of the cooling water and the water-cooled copper sleeve as the input of the three-dimensional heat conduction differential equation.

[0067] The convective heat transfer calculation formula is as follows:

[0068] Non-boiling zone:

[0069]

[0070] f≈1 / (-1.8log 10 ((ε / 3.7) 1.11 +6.9 / Re)) 2

[0071] Where: η is the dynamic viscosity, the subscripts f and w represent the average temperature of the fluid and the wall temperature, respectively, the Pr subscript is Pr at the average temperature and the wall temperature, l is the tube length; d is the equivalent diameter; f is the resistance coefficient of turbulent flow in the tube; d2 and d1 are the outer diameter and inner diameter, respectively.

[0072] Boiling area:

[0073] The comprehensive heat transfer coefficient is:

[0074] h tp =Fh f +Sh nb

[0075] where h f : Heat transfer coefficient when all is liquid, the calculation is consistent with the non-boiling zone; h nb Corrected pool boiling heat transfer coefficient.

[0076] Usually F = 1; S is calculated by the following formula:

[0077]

[0078] h nb Calculated by the following formula:

[0079]

[0080] Where: σ is the surface tension N / m; Hfg is the latent heat of vaporization, J / Kg; μ l is the dynamic viscosity of the liquid kg / (ms); ρ g is the gas density kg / m 3 ;λ l is the liquid thermal conductivity W / (m*K); c p，l is the liquid specific heat capacity J / kg / K; ρ l is the liquid density kg / m 3 .P sat (T w ) is the saturated water vapor pressure pa at the wall temperature; P sat (T sat ) is the boiling temperature, the saturated water vapor pressure MPa. Temperature unit ℃

[0081] On the water boundary The temperature gradient at the water flow boundary of the three-dimensional heat conduction differential equation can be obtained.

[0082] 4) The heat release during the solidification process of the copper alloy is calculated using the phase transformation kinetics equation as input to the three-dimensional heat conduction differential equation. This is described by the following formula:

[0083] F=1-exp(-(K(T)t n ) / d m )

[0084]

[0085] 5) The thermophysical properties of a substance are obtained from experiments or literature and are functions of temperature.

[0086] Thermal conductivity is recorded as: k = K (T); specific heat capacity is recorded as: C p =P(T); density is denoted as ρ=ρ(T); thermal expansion coefficient is denoted as α=α(T).

[0087] 6) The above four parts are integrated to obtain the dynamic simulation model of the continuous casting stability field. The input of the model is the inlet temperature of the molten copper alloy T g , cooling water temperature T w , continuous casting speed V w , cooling water flow U w , and the temperature field T at the current moment i , the output is the temperature field T at the next moment i+1 , and there is the following relationship: T i+1 =f(T i , T g , T w , V w , U w ).

[0088] 7) Conduct continuous casting experiments, collect real-time data of continuous casting under process fluctuation conditions, and obtain the actual temperature of the temperature measuring points during the continuous casting process.

[0089] 8) Add correction terms to the cooling water-copper sleeve convection heat transfer equation, the interface thermal conductivity between different materials, the phase change kinetics equation, and the material thermal property calculation formula.

[0090] As shown below:

[0091]

[0092] k=K(T)+ω 20 ; C p =P(T)+ω 21 ; ρ=ρ(T)+ω 22 ; α=α(T)+ω 23 .

[0093] All correction terms are denoted as ω, and the following relationship is obtained: T i+1 =f(ω,T i , T g , T w , V w , U w The part ω is calculated by a neural network with temperature and process parameters as input and becomes a variable associated with the cooling state, that is: ω = ω(T, x), where x represents the process parameters.

[0094] 9) By comparing the actual cooling water outlet temperature T i jk (expressed as temperature field T i The jth row, kth column element) and the calculated temperature The difference is: Calculate the actual solidus position x i and calculate the solidus position The difference is: Then, the gradient descent algorithm is used to train and optimize the correction parameter ω of the temperature field dynamic simulation model, and a dynamic simulation model of the continuous casting temperature field based on the mechanism is obtained;

[0095] 10) The time-consuming temperature field solution process is replaced by a generative neural network to obtain a mechanism-neural network coupled temperature field dynamic simulation model, which takes the current temperature field, thermophysical parameter matrix and current process as input and the temperature field at the next moment as output.

[0096] The model can be expressed as: T i+1 =g(T i , T g , T w , V w , Uw ,ψ(T i )), ψ established the temperature field T i and the functional relationship between thermal properties; the temperature field trained by the neural network is derived from the temperature field calculated by the dynamic simulation model of the temperature field based on the mechanism;

[0097] 11) The process data collected by the continuous casting equipment sensors are transmitted in real time to the temperature field dynamic simulation model of the mechanism-neural network coupling constructed above, and the real-time dynamic temperature field is calculated.

[0098] This embodiment achieves a fast and high-precision solution of the temperature field in the copper alloy continuous casting process, provides necessary temperature field data for microstructure and performance prediction, and lays the foundation for the dynamic control of the continuous casting process.

[0099] Those skilled in the art will appreciate that embodiments of the present invention may be provided as methods, systems, or computer program products. The present invention may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to CD-ROMs, disk storage devices, etc.) containing computer-usable program code.

[0100] The above describes in detail a method and system for real-time dynamic simulation of a continuous casting process provided by the embodiments of this application. The description of the above embodiments is intended only to facilitate understanding of the method and core concept of this application. Furthermore, those skilled in the art will appreciate that variations in the specific implementation and scope of application may occur based on the concepts of this application. In summary, this specification should not be construed as limiting this application.

[0101] For example, certain words are used in the specification and claims to refer to specific components. Those skilled in the art should understand that hardware manufacturers may use different nouns to refer to the same component. This specification and claims do not use differences in names as a way to distinguish components, but use differences in the functions of components as the criteria for distinction. For example, "including" and "comprising" mentioned throughout the specification and claims are open-ended terms, so they should be interpreted as "including / including but not limited to". "Approximately" means that within an acceptable error range, those skilled in the art can solve the technical problems within a certain error range and basically achieve the technical effects. The subsequent description in the specification is a preferred embodiment of the present application, but the description is for the purpose of illustrating the general principles of the present application, and is not used to limit the scope of the present application. The scope of protection of the present application shall be as defined in the attached claims.

[0102] It should also be noted that the terms "include," "comprises," or any other variations thereof are intended to encompass non-exclusive inclusion, such that a product or system comprising a series of elements includes not only those elements but also other elements not explicitly listed, or elements inherent to such product or system. In the absence of further limitations, an element defined by the phrase "comprises a..." does not exclude the presence of other identical elements in the product or system comprising the element.

[0103] It should be understood that the term "and / or" as used herein is merely a description of the relationship between associated objects, indicating that three possible relationships exist. For example, "A and / or B" can represent: A exists alone, A and B exist simultaneously, or B exists alone. Furthermore, the character " / " in this document generally indicates that the associated objects are in an "or" relationship.

[0104] The above description shows and describes several preferred embodiments of the present application. However, as previously mentioned, it should be understood that the present application is not limited to the form disclosed herein and should not be construed as excluding other embodiments. Instead, the present application can be used in various other combinations, modifications, and environments and can be modified within the scope of the application concept described herein through the above teachings or technology or knowledge in the relevant field. Modifications and variations made by those skilled in the art that do not depart from the spirit and scope of the present application should be protected by the claims appended hereto.

Claims

1. A real-time dynamic simulation method for temperature field of continuous casting process, characterized in that: The method specifically comprises the following steps: S1) connecting a data acquisition unit to a continuous casting device and using real-time temperature field data; S2) constructing a dynamic simulation model of the continuous casting temperature field based on the mechanism; S3) using the non-steady-state real-time data of S1) as input, optimizing the dynamic simulation model of the continuous casting temperature field of the mechanism, and obtaining an optimized dynamic simulation model of the continuous casting temperature field of the mechanism; S4) replacing the time-consuming temperature field solution portion of the mechanism-based temperature field dynamic simulation model optimized in S3) with a generative neural network to obtain a mechanism-neural network coupled temperature field dynamic simulation model; S5) inputting the real-time collected process data into the temperature field dynamic simulation model coupled with the mechanism-neural network to obtain real-time dynamic simulation data.

2. The method according to claim 1, characterized in that The temperature field real-time data includes non-steady-state real-time data and steady-state real-time data; the non-steady-state real-time data includes: continuous or jump changes in melt temperature, cooling water flow, cooling water temperature and casting speed during continuous casting.

3. The method according to claim 1, characterized in that The specific contents of S2) include: S2.1) Use the convection heat transfer equation to calculate the boundary heat transfer state between the cooling water and the water-cooled copper jacket as the boundary condition for the three-dimensional heat conduction differential equation; S2.2) Use the three-dimensional heat conduction differential equation to calculate the heat conduction process between and within the mold, the copper alloy liquid phase, and the copper alloy solid phase; S2.3) Calculate the heat release during the solidification of the copper alloy using the phase transformation kinetics equation as input to the three-dimensional heat conduction differential equation; S2.4) The thermophysical parameters of the substance are expressed as a function of temperature. The thermal conductivity is expressed as: k = K(T); the specific heat capacity is expressed as: C p =P(T); density is denoted as ρ=ρ(T); thermal expansion coefficient is denoted as α=α(T); S2.5) Integrate the contents of S2.1) to S2.4) to obtain a dynamic simulation model of the continuous casting temperature field based on the mechanism.

4. The method according to claim 1, wherein The specific steps of S3) are: S3.1) Add correction terms to the cooling water-copper jacket convection heat transfer equation, the interfacial thermal conductivity between different materials, the phase change kinetics equation, and the material thermophysical property calculation formulas; As shown below: K=k(T)+ω 20 ;C p =P(T)+ω 21 ;ρ=ρ(T)+ω 22 ;α=α(T)+ω 23 ; All correction terms are denoted as ω, and the following relationship is obtained: T i+1 =f(ω,T i , T g , T w , V w , U w ), part ω is calculated by the neural network with temperature and process parameters as input and becomes a variable associated with the cooling state, that is: ω = ω(T, x), where x represents the process parameters; S3.2) Compare the actual cooling water outlet temperature T j jk , expressed as the temperature field T i The jth row, kth column element) and the calculated temperature The difference is: Calculate the actual solidus position x i and calculate the solidus position The difference is: Then, the gradient descent algorithm is used to train and optimize the correction parameter ω of the temperature simulation model to obtain the optimized mechanism-based temperature field dynamic calculation model.

5. The method according to claim 1, wherein The specific step of S4) is to replace the time-consuming temperature field solution portion of the mechanism-based temperature field dynamic simulation model optimized in S3) with a generative neural network, thereby obtaining a mechanism-neural network coupled temperature field dynamic simulation model. This model takes the current temperature field, thermophysical property parameter matrix, and current process as inputs and outputs the temperature field at the next moment. The temperature field data used for training is derived from the calculation results of the optimized mechanism-based temperature field dynamic simulation model.

6. A real-time dynamic simulation system for temperature field of continuous casting process, characterized in that: The system comprises: A data acquisition unit is used to collect real-time data of the temperature field of the continuous casting equipment at different times; The temperature field calculation unit is used to calculate the temperature field at different times according to the collected real-time temperature field data to perform simulation and obtain real-time dynamic simulation data.

7. The system according to claim 6, characterized in that The data acquisition unit includes a water inlet flow acquisition unit, a water inlet flow control acquisition unit, a water inlet temperature acquisition unit, a water outlet temperature acquisition unit, a melt temperature acquisition unit and a crystallizer temperature acquisition unit.

8. The system according to claim 6, wherein: The system further comprises: a data visualization unit: used for performing data mapping to intuitively display the changes or distribution characteristics of process data and temperature data.

9. A computer-readable storage medium comprising instructions, which, when executed on a computer, enables the computer to execute the real-time dynamic simulation method for a continuous casting process according to any one of claims 1 to 5.

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