Information processing apparatus, information processing method, and program

Abstract

To suppress both an increase in the time required for deriving a state of molten metal in a mold of a continuous casting machine and a decrease in derivation accuracy of the state of molten metal.SOLUTION: An information processing device 100 derives a snapshot matrix that stores a value of a melt flow velocity at each calculation grid point and each time for each axis component. The information processing device 100 performs singular value decomposition on each of the snapshot matrices to derive base vectors of the flow velocities for each of the snapshot matrices. The information processing device 100 derives, for each axis component, a coefficient used when linearly combining the base vectors of the flow velocities. The information processing device 100 derives the flow velocity of the molten steel for each axial component by linearly combining the base vectors of the flow velocity by using the derived coefficients.SELECTED DRAWING: Figure 1

Description

Technical Field

[0001] The present invention relates to an information processing apparatus, an information processing method, and a program, and is particularly suitable for use in estimating the state of molten steel in a mold of a continuous casting facility.

Background Art

[0002] FIG. 18 is a diagram showing an example of the schematic configuration of a continuous casting facility. In each figure, the l1 axis, l2 axis, and l3 axis indicate directions. The one with an × shown in the ○ indicates the direction from the front side to the back side of the paper surface. The l1 axis, l2 axis, and l3 axis are the axes of a three-dimensional orthogonal coordinate system.

[0003] The molten steel 3 supplied from the ladle 1 to the tundish 2 is poured into the mold 4. A sliding nozzle 5 is provided at the bottom of the tundish 2. A submerged nozzle 6 is provided below the sliding nozzle 5. The submerged nozzle 6 is arranged at the center of the horizontal cross-section (l1-l3 cross-section) of the mold 4. A pair of left and right discharge ports 7 (a pair of discharge ports 7 at both ends in the l1-axis direction) are formed on the side surface of the tip of the submerged nozzle 6. The tip (discharge port 7) of the submerged nozzle 6 is immersed in the molten steel 3 supplied into the mold 4 (the region surrounded by the mold 4). FIG. 19 is a diagram showing an example of the flow of the molten steel 3 in the mold 4.

[0004] The molten steel 3 supplied to the tundish 2 flows down through the submerged nozzle 6 via the sliding nozzle 5 and is poured into the mold 4 from the pair of left and right discharge ports 7. After the molten steel 3 discharged from the pair of left and right discharge ports 7 collides with the solidification shell 8, it is divided into an upward flow and a downward flow as represented by the arrows in FIG. 19.

[0005] As shown in Fig. 19(a), during the steady state, the amount of molten steel discharged from the pair of left and right discharge ports 7 is substantially equal. However, as shown in Fig. 19(b), the amount of molten steel discharged from the pair of left and right discharge ports 7 may become unequal. Such a situation where the amount of molten steel discharged from the pair of left and right discharge ports 7 becomes unequal is called uneven flow. When uneven flow occurs, a difference in the flow velocity of the molten steel 3 on both the left and right sides sandwiching the immersion nozzle 6 is generated.

[0006] The causes of such uneven flow include the adhesion of oxides such as alumina to the inner surface of the immersion nozzle 6, etc., and the melting loss of the discharge port 7 resulting in a distorted shape of the discharge port 7. Also, due to the structure of the sliding nozzle 5, the molten steel 3 does not flow down axially symmetrically inside the immersion nozzle 6 but is biased to either the left or right and flows down, which is also cited as a cause of uneven flow.

[0007] When uneven flow of the molten steel 3 occurs in the mold 4 for the above reasons, in the region on either the left or right side of the regions sandwiching the immersion nozzle 6 where the amount of molten steel 3 is large, the molten steel 3 will vigorously split and flow upward and downward along the inner surface of the solidification shell 8. The upward flow of the vigorously flowing molten steel 3 causes the rise of the molten steel surface (meniscus). As a result, the supply of the flux sprayed on the molten steel surface to the space between the inner wall surface of the mold 4 and the solidification shell 8 is inhibited. Then, the shape of the solidification shell 8 is likely to become non-uniform, which causes wrinkles, cracks, etc. in the slab cast by the continuous casting facility.

[0008] Also, the downward flow of the vigorously flowing molten steel 3 reaches deep into the molten steel 3 and hinders the floating of non-metallic inclusions. The entry of non-metallic inclusions deep into the strand causes non-metallic inclusion defects in the slab cast by the continuous casting facility, etc. The molten steel surface level (the position in the height direction (l2-axis direction) of the molten steel surface (meniscus) of the molten steel 3) is measured by the molten steel surface level gauge 9. As the molten steel surface level gauge 9, an eddy current sensor or the like is used.

[0009] On one hand, in the regions on both the left and right sides sandwiching the immersion nozzle 6, in the region where the amount of molten steel 3 is small, the momentum of the flow of the discharged molten steel 3 becomes weak. When the momentum of the flow of the discharged molten steel 3 is weak, stagnation is likely to occur in the flow of the molten steel 3 in the discharge port 7, and there is a risk that precipitates such as alumina will adhere to the inside of the immersion nozzle 6. The adhesion of precipitates such as alumina causes problems such as blocking the flow path inside the immersion nozzle 6.

[0010] As described above, when a flow deviation occurs in the mold 4, it not only hinders the continuous casting operation, but also causes a deterioration in the quality of the slab, which is not preferable. Therefore, when a flow deviation occurs, for example, measures such as changing the casting speed, changing the flux, or replacing the immersion nozzle 6 are necessary. In order to perform these measures appropriately and quickly, it is required to estimate the state of the molten steel 3 in the mold 4 by numerical simulation.

[0011] Patent Document 1 describes deriving numerical analysis data, which is data of a state vector that is a vector at the calculation grid points (calculation positions) of the flow velocity and temperature of the molten steel 3, by numerical simulation. However, in the technique described in Patent Document 1, the number of variables to be derived is the product of the number of calculation grid points and the physical variables (for example, several thousand to several hundred thousand). For this reason, the time required to derive the flow velocity and temperature of the molten steel 3 becomes long. Therefore, in order to calculate the state of the molten steel 3 online in real time, it is necessary to use a computer with extremely high processing power.

[0012] Therefore, as described in Non-Patent Document 1 and Patent Document 2, it is conceivable to perform fluid analysis by proper orthogonal decomposition. Proper orthogonal decomposition expresses (approximates) the data as a linear combination of the basis vectors of the r-dimensional mode in order to represent the data in the r-dimensional phase space coordinate system (r≪n) instead of the n-dimensional grid point vector in which the data of the physical quantity to be derived exists. Incidentally, proper orthogonal decomposition is also called principal component analysis, etc. in the field of statistics.

[0013] In Non-Patent Document 1, when deriving the flow around a flat plate wing by fluid analysis using proper orthogonal decomposition, the basis vectors are derived by performing singular value decomposition on a snapshot matrix created from time series data of the kinetic energy derived from the velocity vector, and each component of the fluid flow velocity is expressed using the basis vectors.

[0014] Patent Document 2 discloses that when deriving the air flow by fluid analysis using proper orthogonal decomposition, each component of the flow velocity is grouped into one snapshot matrix, and the basis vectors are derived by performing singular value decomposition on the snapshot matrix.

Prior Art Documents

Patent Documents

[0015]

Patent Document 1

Patent Document 2

Non-Patent Documents

[0016]

Non-Patent Document 1

Non-Patent Document 2

Non-Patent Document 3

Non-Patent Document 4

Non-Patent Document 5

Non-Patent Document 6

Summary of the Invention

Problems to be Solved by the Invention

[0017] However, as described above, in the mold 4 of the continuous casting facility, the molten steel 3 discharged from the discharge port 7 of the immersion nozzle 6 collides with the solidification shell 8 and then suddenly changes its direction (see the upward flow and downward flow represented by the arrows in Fig. 19). Therefore, the velocity distribution of the molten steel 3 varies greatly in the l1-axis direction (width direction), l2-axis direction (casting direction), and l3-axis direction (depth direction). For this reason, when expressing the velocity of the molten steel 3 using the base vectors based on kinetic energy as described in Non-Patent Document 1, it is not easy to accurately derive the velocity of the molten steel 3 unless the velocity of the molten steel 3 is expressed (approximated) including the base vectors of higher-order modes. On the other hand, when expressing (approximating) the velocity of the molten steel 3 including the base vectors of higher-order modes, the time required to derive the velocity of the molten steel 3 becomes longer.

[0018] Also, the snapshot matrix described in Patent Document 2 stores, for each column, the velocity values at each computational grid point for each coordinate axis component, arranging the aggregates in order. Therefore, the base vectors obtained by proper orthogonal decomposition from this snapshot matrix have no physical meaning. Since the content of this base vector can be divided for each coordinate axis component, the base vectors corresponding to each coordinate axis component can be extracted from it. However, the base vectors obtained here are not mutually orthonormalized. For this reason, when performing proper orthogonal decomposition, a simple reduced equation (reduced simulator) cannot be constructed by Galerkin expansion, and the reduced equation becomes complicated. Therefore, when deriving the velocity of the molten steel 3 using the technique described in Patent Document 2, the time required to derive the velocity of the molten steel 3 becomes longer.

[0019] The present invention has been made in view of the above problems, and an object thereof is to suppress both the increase in the time required to derive the velocity of the molten metal in the mold of the continuous casting facility and the decrease in the estimation accuracy of the velocity of the molten metal.

Means for Solving the Problems

[0020] The information processing apparatus of the present invention is an information processing apparatus that executes a process including deriving the state of molten metal injected into a mold of a continuous casting facility, and includes a matrix derivation means for deriving a matrix storing values of a first physical quantity that is a physical quantity indicating the state of the molten metal at each calculation position and each time, and by performing singular value decomposition on the matrix storing the values of the first physical quantity derived by the matrix derivation means, a basis vector derivation means for deriving basis vectors of the first to M-th modes of the first physical quantity by deriving basis vectors of the first to M-th modes, and a state derivation means for deriving a state quantity that is a quantity indicating the state of the molten metal, wherein the first physical quantity includes the flow velocity of the molten metal and the temperature of the molten metal, and the matrix derivation means derives a matrix storing the value of the flow velocity of the molten metal for each axial component of the flow velocity performing, and deriving a matrix storing the values of the temperature of the molten metal, and the basis vector derivation means performs singular value decomposition on the matrix storing the value of the flow velocity of the molten metal derived by the matrix derivation means to derive basis vectors of the first to M-th modes, and performs this for each matrix storing the value of the flow velocity of the molten metal derived by the matrix derivation means, thereby deriving basis vectors of the first to M-th modes of the flow velocity of the molten metal for each axial component of the flow velocity performing, and by performing singular value decomposition on the matrix storing the values of the temperature of the molten metal derived by the matrix derivation means, deriving basis vectors of the first to M-th modes, thereby deriving basis vectors of the first to M-th modes of the temperature of the molten metal, and the state derivation means has a first coefficient derivation means for deriving coefficients used when linearly combining the basis vectors of the first to M-th modes of the first physical quantity after the basis vectors of the first to M-th modes of the first physical quantity are derived by the basis vector derivation means, the state quantity is determined based on the coefficients derived by the first coefficient derivation means, and the first coefficient derivation means derives, for each axial component, at least the coefficients used when linearly combining the basis vectors of the first to M-th modes of the flow velocity of the molten metal as the coefficients used when linearly combining the basis vectors of the first to M-th modes of the first physical quantity performing, and after the basis vectors of the first to M-th modes of the temperature of the molten metal are derived by the basis vector derivation means, deriving coefficients used when linearly combining the basis vectors of the first to M-th modes of the temperature of the molten metal, wherein the value of M is an integer of 2 or more, and for each axial component of the flow velocity for each temperature is determined, and the flow velocity of the molten metal is represented by linearly combining the base vectors of the first to M-th modes of the flow velocity of the molten metal derived by the base vector derivation means using the coefficients derived by the first coefficient derivation means. the temperature of the molten metal is represented by linearly combining the basis vectors of the first to M-th modes of the temperature of the molten metal derived by the basis vector derivation means using the coefficients derived by the first coefficient derivation means, It is characterized by the above.

[0021] The information processing method of the present invention is an information processing method for executing a process including deriving the state of a molten metal injected into a mold of a continuous casting facility, and includes a matrix derivation step of deriving a matrix storing values of a first physical quantity that is a physical quantity indicating the state of the molten metal at each calculation position and each time, and performing a singular value decomposition on the matrix storing the values of the first physical quantity derived by the matrix derivation step to derive base vectors of the first to M-th modes, thereby a base vector derivation step of deriving base vectors of the first to M-th modes of the first physical quantity, and a state derivation step of deriving a state quantity that is a quantity indicating the state of the molten metal. The first physical quantity includes the flow velocity of the molten metal. and the temperature of the molten metal, The matrix derivation step derives a matrix storing the value of the flow velocity of the molten metal for each axial component of the flow velocity. performing, and deriving a matrix storing the values of the temperature of the molten metal, The base vector derivation step performs singular value decomposition on the matrix storing the value of the flow velocity of the molten metal derived by the matrix derivation step to derive base vectors of the first to M-th modes, and executes this for each of the matrices storing the value of the flow velocity of the molten metal derived by the matrix derivation step, thereby deriving base vectors of the first to M-th modes of the flow velocity of the molten metal for each axial component of the flow velocity. performing, and by performing singular value decomposition on the matrix storing the values of the temperature of the molten metal derived by the matrix derivation step, deriving basis vectors of the first to M-th modes, thereby deriving basis vectors of the first to M-th modes of the temperature of the molten metal, , after the basis vectors of the first to M-th modes of the first physical quantity are derived by the basis vector derivation step, the state derivation step includes a first coefficient derivation step of deriving coefficients used when linearly combining the basis vectors of the first to M-th modes of the first physical quantity, the state quantity is determined based on the coefficients derived by the first coefficient derivation step, and the first coefficient derivation step derives, as coefficients used when linearly combining the basis vectors of the first to M-th modes of the first physical quantity, at least the coefficients used when linearly combining the basis vectors of the first to M-th modes of the flow velocity of the molten metal for each axis component performing, and after the basis vectors of the first to M-th modes of the temperature of the molten metal are derived by the basis vector derivation step, deriving coefficients used when linearly combining the basis vectors of the first to M-th modes of the temperature of the molten metal, , the value of M is an integer of 2 or more, and for each axis component of the flow velocity for each temperature is determined, and the flow velocity of the molten metal is represented by linearly combining the basis vectors of the first to M-th modes of the flow velocity of the molten metal derived by the basis vector derivation step using the coefficients derived by the first coefficient derivation step the temperature of the molten metal is represented by linearly combining the basis vectors of the first to M-th modes of the temperature of the molten metal derived by the basis vector derivation step using the coefficients derived by the first coefficient derivation step, characterized in that.

[0022] The program of the present invention is for causing a computer to function as each means of the information processing apparatus.

Effect of the Invention

[0023] According to the present invention, it is possible to suppress both the increase in the time required to derive the flow velocity of the molten metal in the mold of the continuous casting facility and the decrease in the estimation accuracy of the flow velocity of the molten metal.

Brief Description of the Drawings

[0024]

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Figure 15A

Figure 15B

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Figure 19

Embodiments for Carrying Out the Invention

[0025] Hereinafter, embodiments of the present invention will be described with reference to the drawings. 〔First Embodiment〕 First, the first embodiment will be described. FIG. 1 is a diagram showing an example of a functional configuration of an information processing apparatus 100. The hardware of the information processing apparatus 100 is realized by, for example, an information processing apparatus having a CPU, ROM, RAM, HDD, and various interfaces, or by using dedicated hardware. The information processing apparatus 100 executes processing including deriving the flow and temperature of the molten steel 3 (molten metal) in the mold 4 of the continuous casting facility as the state of the molten steel 3. In the present embodiment, the case where the information processing apparatus 100 creates data for visualizing the flow and temperature of the molten steel 3 in the mold 4 of the continuous casting facility is exemplified. In the present embodiment, deriving the distribution of physical quantities (such as flow velocity and temperature) of the molten steel (in some cases, the solidified shell) in all or part of the region in the continuous casting mold is referred to as visualization, and the region for deriving the distribution of the physical quantities of the molten steel by visualization is referred to as the visualization target region. Also, in each embodiment, the physical quantity indicating the state of the molten steel to be visualized (estimated) is also referred to as the first physical quantity. Also, in each embodiment, the physical quantity at each observation position of the continuous casting facility measured by the sensor is also referred to as the second physical quantity.

[0026] In this embodiment, a case where the flow and temperature states of the molten steel 3 in the mold 4 of the continuous casting facility shown in FIG. 18 are visualized will be exemplified. A thermometer (thermocouple) is embedded in the mold 4. FIG. 2 is a diagram showing an example of the thermometers L1 to L12 and F1 to F12 embedded in the mold 4. As shown in FIGS. 2(a) to 2(c), at a certain height position (position in the casting direction (l2-axis direction)) of the mold 4, on one long-side surface (referred to as the "F surface") of the mold 4, a plurality of thermometers F1, F3, F5, F7, F9, F11 are embedded at positions that are point-symmetrical with the center of the width direction (l1-axis direction) of the long side of the mold 4 as the symmetry point. Also, at the same height position (position in the casting direction (l2-axis direction)) of the mold 4, on the other long-side surface (referred to as the "L surface") of the mold 4, a plurality of thermometers L1, L3, L5, L7, L9, L11 are embedded at positions that are point-symmetrical with the center of the width direction (l1-axis direction) of the long side of the mold 4 as the symmetry point.

[0027] Further, on the F surface of the mold 4, below the plurality of thermometers F1, F3, F5, F7, F9, F11, a plurality of thermometers F2, F4, F6, F8, F10, F12 are embedded. The positions of the plurality of thermometers F1 to F12 in the depth direction (l3-axis direction) of the mold 4 are the same. Also, the intervals in the l1-axis direction of the plurality of thermometers F1, F3, F5, F7, F9, F11 and the intervals in the l1-axis direction of the plurality of thermometers F2, F4, F6, F8, F10, F12 are the same. Also, the positions in the l1-axis direction of the thermometers F1, F2, the positions in the l1-axis direction of the thermometers F3, F4, the positions in the l1-axis direction of the thermometers F5, F6, the positions in the l1-axis direction of the thermometers F7, F8, the positions in the l1-axis direction of the thermometers F9, F10, and the positions in the l1-axis direction of the thermometers F11, F12 are the same.

[0028] Similarly, on the L surface of the mold 4, below a plurality of thermometers L1, L3, L5, L7, L9, L11, a plurality of thermometers L2, L4, L6, L8, L10, L12 are embedded. The positions of the plurality of thermometers L1 to L12 in the depth direction (l3-axis direction) of the mold 4 are the same. Also, the intervals in the l1-axis direction of the plurality of thermometers L1, L3, L5, L7, L9, L11 and the intervals in the l1-axis direction of the plurality of thermometers L2, L4, L6, L8, L10, L12 are the same. Further, the positions in the l1-axis direction of the thermometers L1, L2, the positions in the l1-axis direction of the thermometers L3, L4, the positions in the l1-axis direction of the thermometers L5, L6, the positions in the l1-axis direction of the thermometers L7, L8, the positions in the l1-axis direction of the thermometers L9, L10, and the positions in the l1-axis direction of the thermometers L11, L12 are the same.

[0029] Also, the positions in the l2-axis direction of the plurality of thermometers F1, F3, F5, F7, F9, F11 on the upper side of the F surface of the mold 4 and the positions in the l2-axis direction of the plurality of thermometers L1, L3, L5, L7, L9, L11 on the upper side of the L surface of the mold 4 are all the same. The positions in the l2-axis direction of the plurality of thermometers F2, F4, F6, F8, F10, F12 on the lower side of the F surface of the mold 4 and the positions in the l2-axis direction of the plurality of thermometers L2, L4, L6, L8, L10, L12 on the lower side of the L surface of the mold 4 are all the same.

[0030] Also, the positions in the l1-axis direction of the thermometers F1, L1, the positions in the l1-axis direction of the thermometers F3, L3, the positions in the l1-axis direction of the thermometers F5, L5, the positions in the l1-axis direction of the thermometers F7, L7, the positions in the l1-axis direction of the thermometers F9, L9, the positions in the l1-axis direction of the thermometers F11, L11, the positions in the l1-axis direction of the thermometers F2, L2, the positions in the l1-axis direction of the thermometers F4, L4, the positions in the l1-axis direction of the thermometers F6, L6, the positions in the l1-axis direction of the thermometers F8, L8, the positions in the l1-axis direction of the thermometers F10, L10, and the positions in the l1-axis direction of the thermometers F12, L12 are the same. Further, the distance in the l3-axis direction from the F surface (inner wall surface) of the thermometers F1, F2, F3, F4, F5, F6, F7, F8, F9, F10, F11, F12 is the same as the distance in the l3-axis direction from the L surface (inner wall surface) of the thermometers L1, L2, L3, L4, L5, L6, L7, L8, L9, L10, L11, L12. In the following description, the plurality of thermometers F1 to F12 and L1 to L12 are collectively referred to as thermometers F and L as necessary.

[0031] In the technology described in Patent Document 1, when visualizing the flow and temperature states of the molten steel 3 in the mold 4 of the continuous casting facility online and in real time, it is necessary to perform numerical simulation of the flow velocity and temperature of the molten steel 3 and processes such as data assimilation using the results of the numerical simulation at a processing speed comparable to the elapse of time assumed in the (numerical simulation). For this reason, it is required to execute the numerical simulation of the flow velocity and temperature of the molten steel 3 at high speed without significantly degrading the calculation accuracy.

[0032] Therefore, the inventors focused on the fact that the velocity distribution of the molten steel 3 in the mold 4 of the continuous casting facility is not isotropically turbulent like the velocity distribution around an aircraft or a car, and is significantly different in the l1-axis direction (width direction), l2-axis direction (casting direction), and l3-axis direction (depth direction). Then, the inventors found that by performing proper orthogonal decomposition of the velocity of the molten steel 3 for each of its axial components, it is possible to achieve both a reduction in the computational load and suppression of a decrease in computational accuracy when numerically simulating the velocity of the molten steel 3. Hereinafter, an example of the functions of the information processing apparatus 100 of the present embodiment will be described. In the present embodiment, similar to Patent Document 1, the case of visualizing the flow and temperature of the molten steel 3 in a two-dimensional plane (l1-l2 plane) determined by the width direction (l1-axis direction) and the casting direction (l2-axis direction) of the mold 4 at the center position in the depth direction (l3-axis direction) of the mold 4 will be exemplified. As shown in FIG. 2(a), the center positions in the width direction (l1-axis direction) and the depth direction (l3-axis direction) of the mold 4 correspond to the position of the axis of the immersion nozzle 6. In the following description, this two-dimensional plane will be referred to as the visualization target surface as needed. In the present embodiment, the visualization target surface is an example of the visualization target region.

[0033] In FIG. 1, the numerical simulation unit 101, the matrix derivation unit 102, the basis vector derivation unit 103, and the reduced simulator derivation unit 104 are executed offline. On the other hand, the state derivation unit 111 (the first coefficient derivation unit 111a and the physical quantity derivation unit 111b), the first probability density function derivation unit 112, the observation data acquisition unit 113, the likelihood function derivation unit 114, the data assimilation unit 115, the visualization data creation unit 116, and the output unit 117 are executed online. The online-executed process is a process executed while inputting time-series data (measurement values) measured in the continuous casting facility through a communication line during the actual operation of the continuous casting facility. This online-executed process is preferably performed in real time. The real-time executed process is a process in which the time lag from the time when the observation data is acquired to the time when the state of the molten steel at that time (i.e., at least one of the flow velocity and the temperature) is output is such that appropriate and prompt measures can be taken against an output indicating an abnormal state of the molten steel such as a drift. The offline-executed process is a process for preparing the online (preferably, online real-time) executed process, and is a process executed in a state where time-series data (measurement values) measured in the continuous casting facility is not input in advance before the online-executed process.

[0034] (Numerical simulation unit 101) The numerical simulation unit 101 executes, offline, a numerical simulation of unsteady flow and heat transfer in the continuous casting facility to be visualized, using the operating parameters of the continuous casting facility and the physical property values of the molten steel 3. The operating parameters of the continuous casting facility and the physical property values of the molten steel 3 are those stored in advance by the numerical simulation unit 101.

[0035] The numerical simulation of the flow and heat transfer in the molten steel 3 can be obtained by solving the Navier-Stokes equations (momentum conservation equations) together with the continuity equation (mass conservation equation) and the energy conservation equation. For example, the numerical simulation is executed by discretizing and solving each equation using the finite volume method. When performing numerical simulation, it is necessary to set initial conditions and boundary conditions. The boundary conditions for fluid flow and heat transfer are set at the inner wall surface of the mold 4, the molten metal surface of the molten steel 3, the outer wall surface of the immersion nozzle 6, the discharge port 7, the interface between the solidified shell 8 and the molten steel 3, and the lower end of the molten steel 3. The lower end of the molten steel 3 refers to the lowest position in the visualization target region (the position at the end in the positive direction of the l2 axis). The initial conditions for fluid flow and heat transfer are set in the molten steel 3. The boundary conditions for fluid flow and heat transfer may be set according to the operating conditions assumed in the continuous casting facility. In this embodiment, the numerical simulation unit 101 exemplifies the case of performing numerical simulation without using measured values.

[0036] The numerical simulation unit 101 derives the flow velocity and temperature of the molten steel 3 at the calculation grid points at time t. In this embodiment, the calculation grid points are an example of the calculation positions. In the following description, the flow velocity of the molten steel 3 and the temperature of the molten steel 3 are abbreviated as flow velocity and temperature as necessary. Also, the flow velocity has values of three components in the l1-axis direction, l2-axis direction, and l3-axis direction. As described above, in this embodiment, to exemplify the case of visualizing the fluid flow and temperature of the molten steel 3 in a two-dimensional plane (l1-l2 plane), in the following description, a two-dimensional model is assumed as a model for visualizing the fluid flow and temperature of the molten steel 3. The l1-axis component of the flow velocity is denoted as u as necessary, the l2-axis component of the flow velocity is denoted as v as necessary, and the temperature is denoted as T as necessary. Also, the calculation grid points (positions) are denoted as l as necessary. The coordinates of the calculation grid point l in the l1-l2 plane are (l1, l2).

[0037] In this embodiment, the numerical simulation unit 101 derives the flow velocities u(l, t), v(l, t) and the temperature T(l, t) at each calculation grid point l at each time t with a time interval of step width Δt1. The Navier-Stokes equations, continuity equations, energy conservation equations for the continuous casting facility, and the contents of the initial conditions and boundary conditions are described, for example, in Non-Patent Document 3, so the detailed description thereof is omitted here.

[0038] (Matrix Derivation Unit 102) The matrix derivation unit 102 derives a snapshot matrix MU of the flow velocity u, a snapshot matrix MV of the flow velocity v, and a snapshot matrix MT of the temperature T as shown in the following equations (1), (2), and (3) based on the flow velocity u(l, t), v(l, t), and temperature T(l, t) at each calculation grid point l derived at each time t at the time interval of the step width Δt1 by the numerical simulation unit 101.

[0039] [Number]

[0040] In equations (1) to (3), the row numbers 1, ···, l M represent the calculation grid point l. In equations (1) to (3), the case where the number of calculation grid points l is l M is exemplified. The column numbers 1, ···, t M represent the time t. Thus, in the snapshot matrix MU of the flow velocity u, the snapshot matrix MV of the flow velocity v, and the snapshot matrix MT of the temperature T, the components included in the same row are the components with the same calculation grid point l, and the components included in the same column are the components with the same time.

[0041] The matrix derivation unit 102 extracts the flow velocity u(l, t) at each time t at the time interval of the step width Δt2, which is longer than the step width Δt1, from the flow velocity u(l, t) at each calculation grid point l derived at each time t at the time interval of the step width Δt1 by the numerical simulation unit 101. Then, the matrix derivation unit 102 derives the snapshot matrix in which the extracted flow velocity u(l, t) is stored as a component as the snapshot matrix MU of the flow velocity u. Note that the matrix derivation unit 102 may interpolate the flow velocity u(l, t) at each calculation grid point l derived at each time t at the time interval of the step width Δt1 and then extract the flow velocity u(l, t) at each time t at the time interval of the step width Δt2. By performing such processing, even when the step width Δt2 is set so as not to be an integer multiple of the step width Δt1, the flow velocity u(l, t) at each time t at the time interval of the step width Δt2 can be extracted.

[0042] Further, the matrix derivation unit 102 extracts the flow velocity v(l, t) at each time t with a time interval of step width Δt2, which is longer than the step width Δt1, from the flow velocity v(l, t) at each computational grid point l derived at each time t with a time interval of step width Δt1 by the numerical simulation unit 101. Then, the matrix derivation unit 102 derives a snapshot matrix in which the extracted flow velocity v(l, t) is stored as a component as the snapshot matrix MV of the flow velocity v. Note that the matrix derivation unit 102 may interpolate the flow velocity v(l, t) at each computational grid point l derived at each time t with a time interval of step width Δt1 and then extract the flow velocity v(l, t) at each time t with a time interval of step width Δt2. By performing such processing, even when the step width Δt2 is set so as not to be an integer multiple of the step width Δt1, the flow velocity v(l, t) at each time t with a time interval of step width Δt2 can be extracted.

[0043] Further, the matrix derivation unit 102 extracts the temperature T(l, t) at each time t with a time interval of step width Δt2, which is longer than the step width Δt1, from the temperature T(l, t) at each computational grid point l derived at each time t with a time interval of step width Δt1 by the numerical simulation unit 101. Then, the matrix derivation unit 102 derives a snapshot matrix in which the extracted temperature T(l, t) is stored as a component as the snapshot matrix MT of the temperature T. Note that the matrix derivation unit 102 may interpolate the temperature T(l, t) at each computational grid point l derived at each time t with a time interval of step width Δt1 and then extract the temperature T(l, t) at each time t with a time interval of step width Δt2. By performing such processing, even when the step width Δt2 is set so as not to be an integer multiple of the step width Δt1, the flow velocity T(l, t) at each time t with a time interval of step width Δt2 can be extracted.

[0044] Thus, in this embodiment, from the flow velocities u(l, t), v(l, t), and temperature T(l, t) at each time t with a time interval of step width Δt1 derived by the numerical simulation unit 101, the flow velocities u(l, t), v(l, t), and temperature T(l, t) at each time t with a time interval of step width Δt2 longer than the step width Δt1 are extracted, and snapshot matrices MU, MV, and MT of the flow velocities u, v, and temperature T are derived. Therefore, the computational load in the basis vector derivation unit 103 described later can be reduced compared to the case of handling a snapshot matrix in which the flow velocities u(l, t), v(l, t), and temperature T(l, t) at each time t with a time interval of step width Δt1 are stored as components. From such a perspective, the snapshot matrix needs to contain sufficient data to capture the characteristics of the assumed changes in flow and heat transport. Generally, however, the step width Δt1 is a time interval for maintaining the stability of the numerical simulation and is too small for the purpose of capturing the characteristics of the changes in flow and heat transport. Also, if the step width Δt2 is too large, the characteristics of the changes in flow and heat transport cannot be sufficiently captured. From such a perspective, the step width Δt2 is preferably 10 times or more the step width Δt1 and smaller than the acquisition interval of the measurement data (observation data acquired by the observation data acquisition unit 113 described later) in the continuous casting facility targeted.

[0045] (Basis Vector Derivation Unit 103) As described above, in this embodiment, the proper orthogonal decomposition is individually performed for each of the flow velocities u(l, t), v(l, t), and temperature T(l, t). In this case, the flow velocities u(l, t), v(l, t), and temperature T(l, t) are represented by the following equations (4), (5), and (6), respectively.

[0046] [Number]

[0047] In equation (4), φ u j (l) is a basis vector of the flow velocity u. The basis vector φ of the flow velocity u uj (l) is the left singular vector obtained by performing singular value decomposition on the snapshot matrix MU of the flow velocity u. φ u j The u in (l) indicates that it is the basis vector of the flow velocity u, and j indicates the mode number of the proper orthogonal decomposition. Positive integers are given in ascending order from 1 as the mode number j in order of decreasing singular values. For example, the basis vector of the flow velocity u, which is the left singular vector derived together with the largest singular value, is φ u 1(l). The basis vectors φ u j (l) form an orthonormal system with each other.

[0048] As shown in equation (4), in proper orthogonal decomposition, the flow velocity u(l,t) is represented by a linear combination of the basis vectors φ u j (l) of the flow velocity u. a u j (t) is the coefficient used when linearly combining the basis vectors φ u j (l) of the flow velocity u. Specifically, a u j (t) is the coefficient multiplied by the basis vector φ u j (l) of the flow velocity u. When the flow velocity u in equation (4) is represented by the snapshot matrix MU, the coefficient a u j (t) is derived as the product of the singular value of mode number j obtained by performing singular value decomposition on the snapshot matrix MU of the flow velocity u and the right singular vector corresponding to the singular value of mode number j. Note that the meanings of u and j in the coefficient a u j (t) are the same as the meanings of u and j in φ u j (l), respectively.

[0049] Also, in equation (5), φ v j (l) is the basis vector of the flow velocity v. The basis vectors φ v j(l) is the left singular vector obtained by performing singular value decomposition on the snapshot matrix MV of the flow velocity v. φ v j The v in (l) indicates that it is the basis vector of the flow velocity v, and j indicates the mode number of the proper orthogonal decomposition. Positive integers are assigned in ascending order from 1 as the mode number j in order of decreasing singular values. The basis vector φ of the flow velocity v v j (l) forms an orthonormal system with each other. As shown in equation (5), in the proper orthogonal decomposition, the flow velocity v(l,t) is the basis vector φ of the flow velocity v v j (l) is represented by a linear combination. a v j (t) is the coefficient used when linearly combining the basis vectors φ of the flow velocity v v j (l). Specifically, a v j (t) is the coefficient multiplied by the basis vector φ of the flow velocity u v j (l). When the flow velocity v in equation (5) is represented by the snapshot matrix MV, the coefficient a v j (t) is derived as the product of the singular value of mode number j obtained by performing singular value decomposition on the snapshot matrix MV of the flow velocity v and the right singular vector corresponding to the singular value of mode number j. Note that the meanings of v and j in the coefficient a v j (t) are the same as the meanings of v and j in φ v j (l), respectively.

[0050] Also, in equation (6), φ T j (l) is the basis vector of the temperature T. The basis vector φ of the temperature T T j (l) is the left singular vector obtained by performing singular value decomposition on the snapshot matrix MT of the temperature T. φ T jThe T in (l) indicates that it is the basis vector of temperature T, and j indicates the mode number of the proper orthogonal decomposition. Positive integers are assigned in ascending order from 1 as the mode number j in order of decreasing singular values. The basis vector φ of temperature T T j (l) forms a mutually orthonormal system. As shown in equation (6), in the proper orthogonal decomposition, the temperature T(l,t) is the basis vector φ of temperature T T j represented by a linear combination of (l). a T j (t) is the coefficient used when linearly combining the basis vector φ of temperature T T j (l). Specifically, a T j (t) is the coefficient multiplied by the basis vector φ of temperature T T j (l). When the temperature T in equation (6) is represented by the snapshot matrix MT, the coefficient a T j (t) is derived as the product of the singular value of mode number j obtained by performing singular value decomposition on the snapshot matrix MT of temperature T and the right singular vector corresponding to the singular value of mode number j. Incidentally, the meanings of T and j of the coefficient a T j (t) are the same as the meanings of T and j of φ T j (l) respectively.

[0051] Here, the basis vectors φ of the flow velocities u, v, and temperature T u j (l), φ v j (l), φ T j (l) are functions only of the position (spatial coordinates) of the computational grid point l. On the other hand, the coefficients a u j (t), a v j (t), a T j (t) are functions only of time t. That is, as shown in equations (4) to (6), in the proper orthogonal decomposition, the first physical quantity (flow velocities u, v, temperature T) is separated into variables with respect to time and space. The coefficient au j (t), a v j (t), a T j (t) is the basis vector φ u j (l), φ v j (l), φ T (l) corresponds to the contribution degree, and the singular value is the coefficient a u j (t), a v j (t), a T j (t) corresponds to the amplitude of the time variation.

[0052] As described above, in this embodiment, the basis vector derivation unit 103 derives the left singular vector obtained by performing singular value decomposition on the snapshot matrix MU of the flow velocity u derived by the matrix derivation unit 102 as the basis vector φ u j (l) of the flow velocity u. Further, the basis vector derivation unit 103 derives the left singular vector obtained by performing singular value decomposition on the snapshot matrix MV of the flow velocity v derived by the matrix derivation unit 102 as the basis vector φ v j (l) of the flow velocity v. Further, the basis vector derivation unit 103 derives the left singular vector obtained by performing singular value decomposition on the snapshot matrix MT of the temperature T derived by the matrix derivation unit 102 as the basis vector φ T j (l) of the temperature T.

[0053] In equation (4), M u is the number of the basis vectors φ u j (l) of the flow velocity u used in the calculation of equation (4). That is, in equation (4), the basis vectors φ u of the flow velocity u in the 1st to M u th modes, φ u MuIt represents that the flow velocity u(l, t) is derived by linearly combining (l). In equation (5), M v is the number of basis vectors φ v j (l) of the flow velocity v used in the calculation of equation (5). That is, in equation (5), the basis vectors φ v of the flow velocity v in the 1st to M v th modes, φ v Mv (l) are linearly combined to represent that the flow velocity v(l, t) is derived. In equation (6), M T is the number of basis vectors φ T j (l) of the temperature T used in the calculation of equation (6). That is, in equation (6), the basis vectors φ T of the temperature T in the 1st to M T th modes, φ T MT (l) are linearly combined to represent that the temperature T(l, t) is derived.

[0054] Thus, in this embodiment, the basis vector derivation unit 103 uses, among the basis vectors φ u j (l), φ v j (l), φ T j (l) of the flow velocities u, v, and the temperature T, M u pieces, M v pieces, M T pieces, respectively, of the basis vectors φ u 1(l) to φ u Mu (l), φ v 1(l) to φ v Mv (l), φ T 1(l) to φ T MT (l) in the calculations of equations (4), (5), and (6). The basis vectors φ u j (l), φ v j (l), φ T j (l) of the flow velocities u, v, and the temperature T used in the calculations of equations (4), (5), and (6), the number Mu and M v and M T The maximum value that M can take (i.e., the maximum order) is the number of columns of the snapshot matrix MU of the flow velocity u, the number of columns of the snapshot matrix MV of the flow velocity v, and the number of columns of the snapshot matrix MT of the temperature T, respectively.

[0055] Here, the basis vectors φ of the flow velocities u, v, and the temperature T used in the calculations of equations (4), (5), and (6) u j (l), φ v j (l), φ T j The number M of u and M v and M T An example of a method for deriving is described. The basis vectors φ of the flow velocities u, v, and the temperature T u j (l), φ v j (l), φ T j (l), when arranged in descending order of the singular values, their magnitudes decrease exponentially. Therefore, compared with the basis vectors φ of the lower-order modes u j (l), φ v j (l), φ T j (l), the contribution of the basis vectors φ of the higher-order modes u j (l), φ v j (l), φ T j (l) to the flow velocities u, v, and the temperature T becomes lower. For this reason, even if the flow velocities u, v, and the temperature T are approximated by a linear combination of only the basis vectors φ of the lower-order modes u j (l), φ v j (l), φ T j (l), the accuracy of the flow velocities u, v, and the temperature T does not decrease significantly.

[0056] Therefore, in the present embodiment, the basis vector derivation unit 103 accumulates the singular values obtained by performing singular value decomposition on the snapshot matrices MU, MV, and MT of the flow velocities u, v, and the temperature T in order from the singular values with a small mode number j (i.e., the singular values with a large value). Then, the basis vector derivation unit 103 uses the maximum value of the mode number j when the integrated value first satisfies a predetermined condition as the basis vectors φ of the flow velocities u, v, and the temperature T used in the calculations of equations (4), (5), and (6). u j (l), φ v j (l), φ T j (l) with the number M u , M v , M T be set. The singular values obtained by performing singular value decomposition on the snapshot matrices MU, MV, and MT of the flow velocities u, v, and the temperature T are different from each other. Therefore, the basis vectors φ of the flow velocities u, v, and the temperature T used in the calculations of equations (4), (5), and (6). u j (l), φ v j (l), φ T j (l) with the number M u , M v , M T can also be different values.

[0057] As the predetermined condition, for example, it can be adopted that the ratio for all singular values is equal to or greater than a predetermined value. As the predetermined value, for example, 99.999% can be adopted. In such a case, while incrementing the mode number j one by one with 1 as the initial value, it is determined whether the value obtained by dividing the integrated value of the singular values of mode numbers 1 to j obtained by performing singular value decomposition on the snapshot matrix MU of the flow velocity u by the integrated value of all the singular values obtained by performing singular value decomposition on the snapshot matrix MU of the flow velocity u is 0.99999 or more. Then, the mode number j when it is first determined that the value obtained by dividing the integrated value of the singular values of mode numbers 1 to j obtained by performing singular value decomposition on the snapshot matrix MU of the flow velocity u by the integrated value of all the singular values obtained by performing singular value decomposition on the snapshot matrix MU of the flow velocity u is 0.99999 or more is set as M u Let it be v For M T as well, in the above description, it is derived by replacing the snapshot matrix MU of the flow velocity u with the snapshot matrix MV of the flow velocity v and the snapshot matrix MT of the temperature T

[0058] (Reduced simulator derivation unit 104) The reduced simulator derivation unit 104 projects the equation of motion describing the flow of the molten steel 3 onto the basis vectors φ u j (l) and φ v j (l) of each axis component, thereby deriving the time evolution equations (differential equations) for the coefficients a u j (t) and a v j (t). Further, the reduced simulator derivation unit 104 projects the heat conduction equation describing the heat conduction of the molten steel 3 onto the basis vector φ T j (l) of the temperature T, thereby deriving the time evolution equations (differential equations) for the coefficients a u j (t), a v j (t), and a T j (t)

[0059] For example, the equation of motion describing the flow of the l1-axis direction (width direction) component of the molten steel 3, the equation of motion describing the flow of the l2-axis direction (casting direction) component of the molten steel 3, and the heat conduction equation describing the heat conduction of the molten steel 3 are represented by the following equations (7), (8), and (9), respectively.

[0060]

Number

[0061] Here, p is the pressure, ρ is the density, ν is the viscosity coefficient, and k is the thermal conductivity. Note that the equation of motion corresponds to the Navier-Stokes equation used in the numerical simulation unit 101, and the heat conduction equation corresponds to the energy conservation equation used in the numerical simulation unit 101.

[0062] For both sides of the equation obtained by substituting equations (4) and (5) into equation (7), take the inner product with the basis vector φ u i (l) of the flow velocity u of the i-th mode, and rearrange so that the term of the time derivative comes to the left side, then the following equation (10) is obtained. Also, for both sides of the equation obtained by substituting equations (4) and (5) into equation (8), take the inner product with the basis vector φ v i (l) of the flow velocity v of the i-th mode, and rearrange so that the term of the time derivative comes to the left side, then the following equation (11) is obtained. Also, for both sides of the equation obtained by substituting equations (4) to (6) into equation (9), take the inner product with the basis vector φ T i (l) of the temperature T of the i-th mode, and rearrange so that the term of the time derivative comes to the left side, then the following equation (12) is obtained.

[0063]

Number

[0064] Note that the basis vector φ u j (l) of the flow velocity u, the basis vector φv j (l), the base vector φ of temperature T T j (l) forms an orthonormal system respectively. Therefore, <φ u m (l), φ u n (l)> = δ mn , <φ v m (l), φ v n (l)> = δ mn , <φ T m (l), φ T n (l)> = δ mn holds. Here, <a, b> represents the inner product of a and b. Also, δ mn is the Kronecker delta (δ mn = 1 when m = n, δ mn = 0 when m ≠ n).

[0065] Furthermore, in equation (10), u and i of A u i indicate that A u i is a constant used in the equation giving the time derivative of the coefficient a u i (t) for the i-th mode of the flow velocity u. u of uu and i of ij of B uu ij indicate that B uu ij is a constant used in the equation giving the time derivative of the coefficient a u i (t) for the i-th mode of the flow velocity u, and u of uv and j of ij of B uu ij indicate that B u j is a constant multiplied by the coefficient a uv ij of u and i of ij of B uv ij indicates that B u iIt shows that it is a constant used in the equation for giving the time derivative of (t). For v in uv and j in ij, B uv ij is the coefficient a for the j-th mode of the flow velocity v v j It shows that it is a constant multiplied by (t). C uu ijk For the first u in uu and i in ijk, C uu ijk is the coefficient a for the i-th mode of the flow velocity u u i It shows that it is a constant used in the equation for giving the time derivative of (t). For j in ijk, C uv ijk is the coefficient a for the j-th mode of the flow velocity u u j It shows that it is a constant multiplied by (t). For the second u in uu and k in ijk, C uu ijk is the coefficient a for the k-th mode of the flow velocity u u k It shows that it is a constant multiplied by (t). C uv ijk For u in uv and i in ijk, C uv ijk is the coefficient a for the i-th mode of the flow velocity u u i It shows that it is a constant used in the equation for giving the time derivative of (t). For j in ijk, C uv ijk is the coefficient a for the j-th mode of the flow velocity u u j It shows that it is a constant multiplied by (t). For v in uv and k in ijk, C uv ijk is the coefficient a for the k-th mode of the flow velocity v v k It shows that it is a constant multiplied by (t).

[0066] Also, in equation (11), A v i For v and i in A v i is the coefficient a for the i-th mode of the flow velocity v v iIt shows that it is a constant used in the equation giving the time derivative of (t). B vu ij For v of vu and i of ij, B vu ij is the coefficient a for the i-th mode of the flow velocity v v i It shows that it is a constant used in the equation giving the time derivative of (t), and for u of vu and j of ij, B vu ij is the coefficient a for the j-th mode of the flow velocity u u j It shows that it is a constant multiplied by (t). B vv ij For the first v of vv and i of ij, B vv ij is the coefficient a for the i-th mode of the flow velocity v v i It shows that it is a constant used in the equation giving the time derivative of (t), and for the second v of vv and j of ij, B vv ij is the coefficient a for the j-th mode of the flow velocity v v j It shows that it is a constant multiplied by (t). C vu ijk For v of vu and i of ijk, C vu ijk is the coefficient a for the i-th mode of the flow velocity v v i It shows that it is a constant used in the equation giving the time derivative of (t), and j of ijk is, C vu ijk is the coefficient a for the j-th mode of the flow velocity v v j It shows that it is a constant multiplied by (t), and for u of vu and k of ijk, C vu ijk is the coefficient a for the k-th mode of the flow velocity u u k It shows that it is a constant multiplied by (t). C vv ijk For the first v of vv and i of ijk, C vv ijk is the coefficient a for the i-th mode of the flow velocity v v iIt shows that it is a constant used in the equation giving the time derivative of (t), and for j in ijk, C vu ijk is the coefficient a for the j-th mode of the flow velocity v v j It shows that it is a constant multiplied by (t), and for the second v in vv and k in ijk, C vv ijk is the coefficient a for the k-th mode of the flow velocity v v k It shows that it is a constant multiplied by (t).

[0067] Also, in equation (12), for T and i in A T i A T i is the coefficient a for the i-th mode of the temperature T T i It shows that it is a constant used in the equation giving the time derivative of (t). For the first T in TT of B and i in ij, B TT ij B TT ij is the coefficient a for the i-th mode of the temperature T T i It shows that it is a constant used in the equation giving the time derivative of (t), and for the second T in TT and j in ij, B TT ij is the coefficient a for the j-th mode of the temperature T T j It shows that it is a constant multiplied by (t). For T in Tu of B and i in ij, B Tu ij B Tu ij is the coefficient a for the i-th mode of the temperature T T i It shows that it is a constant used in the equation giving the time derivative of (t), and for u in Tu and j in ij, B Tu ij is the coefficient a for the j-th mode of the flow velocity u u j It shows that it is a constant multiplied by (t). For T in Tv of B and i in ij, B Tv ij B Tv ijis the coefficient a for the i-th mode at temperature T T i is a constant used in the equation giving the time derivative of (t), where v in Tv and j in ij are B Tv ij is the coefficient a for the j-th mode at flow velocity v v j is a constant multiplied by (t). C Tu ijk where T in Tu and i in ijk are C Tu ijk is the coefficient a for the i-th mode at temperature T T i is a constant used in the equation giving the time derivative of (t), where u in Tu and j in ijk are C Tu ijk is the coefficient a for the j-th mode at flow velocity u u j is a constant multiplied by (t), and k in ijk is C Tu ijk is the coefficient a for the k-th mode at temperature T T k is a constant multiplied by (t). C Tv ijk where T in Tv and i in ijk are C Tv ijk is the coefficient a for the i-th mode at temperature T T i is a constant used in the equation giving the time derivative of (t), where v in Tv and j in ijk are C Tv ijk is the coefficient a for the j-th mode at flow velocity v v j is a constant multiplied by (t), and k in ijk is C Tv ijk is the coefficient a for the k-th mode at temperature T T k is a constant multiplied by (t).

[0068] As described above, in this embodiment, the reduced simulator derivation unit 104 uses the basis vectors φ of the flow velocities u, v, and temperature T derived by the basis vector derivation unit 103 u 1(l) to φu Mu (l), φ v 1(l) to φ v Mv (l), φ T 1(l) to φ T MT (l) and the number M of basis vectors u , M v , M T and, using equations (4) to (9), the constant A in equation (10) i u , B uu ij , B uv ij , C uu ijk , C uv ijk and the constant A in equation (11) i v , B vu ij , B vv ij , C vu ijk , C vv ijk and the constant A in equation (12) i T , B TT ij , B Tu ij , B Tv ij , C Tu ijk , C Tv ijk and are derived. In this embodiment, equations (10) to (12) with the constants thus derived set are the reduced simulator. The reduced simulator corresponds to the reduced model described in Non-Patent Document 1. Projecting the governing equation (basic equation) onto a basis to derive the reduced model itself is described in Non-Patent Documents 1 and 2, so the detailed description thereof is omitted here.

[0069] (State derivation unit 111) The state derivation unit 111 includes a state vector x including, as components, state quantities that are quantities indicating the state of the molten steel 3 tDerive. In this embodiment, the state derivation unit 111 derives the state quantities to be included in the state vector x when performing data assimilation by the data assimilation unit 115 described later. t In addition, in this embodiment, a case where the state quantity is the flow velocities u, v and temperature T of the molten steel 3 which is an example of the first physical quantity will be described as an example. The state derivation unit 111 includes a first coefficient derivation unit 111a and a physical quantity derivation unit 111b.

[0070] ((First Coefficient Derivation Unit 111a)) The first coefficient derivation unit 111a, at time t, for each of a plurality of cases in which at least one of the flow velocity of the molten steel 3 at the computational grid point l at time t−Δt, the temperature of the molten steel 3 at the computational grid point l at time t−Δt, and at least one of the boundary condition parameters (for example, flow velocity, pressure, heat flux) between time t−Δt and time t is different, the solutions (a u 1(t)~a u Mu (t), a v 1(t)~a v Mv (t), a T 1(t)~a T MT (t)) of the differential equations of equations (10) to (12) are derived.

[0071] The first coefficient derivation unit 111a uses a preset value as an initial value for a plurality of cases in which at least one of the flow velocity of the molten steel 3 at the first time t (=t0), the temperature of the molten steel 3 at the first time t (=t0), and at least one of the boundary condition parameters between time t0 and time t0+Δt is different. For times t after the second time t, based on the observation data (information) acquired by the observation data acquisition unit 113 described later and based on the results derived by the data assimilation unit 115 described later, the flow velocity and temperature of the molten steel 3 at time t, and the boundary condition parameters between time t and time t+Δt are derived. Details of this point will be described later.

[0072] ((Physical Quantity Derivation Unit 111b)) The physical quantity derivation unit 111b substitutes the coefficient a for the flow velocity u, derived by the first coefficient derivation unit 111a, u 1(t)~a u Mu (t), and the basis vector φ of the flow velocity u, derived by the basis vector derivation unit 103, u 1(l)~φ u Mu (l) into Equation (4) to derive the flow velocity u of the molten steel 3 at each computational grid point l at time t. Further, the physical quantity derivation unit 111b substitutes the coefficient a for the flow velocity v, derived by the first coefficient derivation unit 111a, v 1(t)~a v Mv (t), and the basis vector φ of the flow velocity v, derived by the basis vector derivation unit 103, v 1(l)~φ v Mv (l) into Equation (5) to derive the flow velocity v of the molten steel 3 at each computational grid point l at time t. Further, the physical quantity derivation unit 111b substitutes the coefficient a for the temperature T, derived by the first coefficient derivation unit 111a, T 1(t)~a T MT (t), and the basis vector φ of the temperature T, derived by the basis vector derivation unit 103, T 1(l)~φ T MT (l) into Equation (6) to derive the temperature T of the molten steel 3 at each computational grid point l at time t. The physical quantity derivation unit 111b uses the coefficient a for the above-mentioned multiple cases u 1(t)~a u Mu (t), a v 1(t)~a v Mv (t), a T 1(t)~a T MT (t) respectively to derive the flow velocities u, v, and the temperature T of the molten steel 3 at each computational grid point l at time t.

[0073] In the following description of the present embodiment, a vector composed of all of the flow velocities u, v, and temperature T of the molten steel 3 at the computational grid point l is referred to as a state vector as needed, and the flow velocities u, v, and temperature T of the molten steel 3 derived by the physical quantity derivation unit 111b are referred to as numerical analysis data as needed. The numerical analysis data is derived by the physical quantity derivation unit 111b for each of the plurality of cases described above.

[0074] (First probability density function derivation unit 112) The first probability density function derivation unit 112 estimates the state vector x at time t t0:t-Δt based on the observation data (information) y up to time t - Δt acquired by the observation data acquisition unit 113 described later. t The conditional probability density function p(x t |y t0:t-Δt ) is derived. The probability density function p(x t |y t0:t-Δt ) is the probability density function indicated by the plurality of numerical analysis data {x t (k)} k derived by the physical quantity derivation unit 111b. In the following description, this probability density function p(x t |y t0:t-Δt ) is also referred to as the first probability density function as needed. In the present embodiment, the first probability density function derivation unit 112 derives the first probability density function p(x t |y t |y t0:t-Δt ) of the state vector x at each time t at time intervals of the step width Δt.

[0075] The first probability density function derivation unit 112 derives a frequency distribution using the relative frequency (value obtained by dividing the frequency by the total number of data) of the plurality of numerical analysis data {x t|t-Δt (k)} k at time t derived by the physical quantity derivation unit 111b. This frequency distribution becomes the first probability density function p(x t |y t |y t0:t-Δt ) of the state vector x at time t. Also, the plurality of numerical analysis data {x t|t-Δt (k)} kDerive the frequency distribution of each component (each first physical quantity at each computational grid point l), and take the product of these to obtain a plurality of numerical analysis data {x t|t-Δt (k)}, k which may be used as the frequency distribution. Alternatively, an approximation of these frequency distributions using a predetermined function (e.g., a spline function) may be used as the first probability density function.

[0076] Also, the first probability density function derivation unit 112 can derive the probability density function p(x t|t-Δt (k) ) k of the state vector x at time t by performing ensemble approximation (Monte Carlo approximation) using the plurality of numerical analysis data {x t} t at time t derived by the physical quantity derivation unit 111b, given y t0:t-Δt . In addition, assuming that the first probability density function p(x t ) t of the state vector x at time t, given y t0:t-Δt , is a Gaussian distribution, the first probability density function derivation unit 112 may derive the mean value and the variance value from the plurality of numerical analysis data {x t (k)} k derived at time t by the physical quantity derivation unit 111b. Furthermore, the first probability density function derivation unit 112 may correct the first probability density function p(x t ) t of the state vector x at time t, given y t0:t-Δt , derived as described above, using noise having a probability density function such as a Gaussian distribution.

[0077] (Observation data acquisition unit 113) The observation data acquisition unit 113 acquires observation data at a predetermined period based on time series data (measurement values) measured in the continuous casting facility. In the present embodiment, the observation data acquisition unit 113 inputs measurement values including the temperature measured by the thermometers F and L (temperature inside the mold 4) and the molten metal level measured by the molten metal level gauge 9.

[0078] The temperatures measured by the thermometers F and L are the temperatures of the mold 4, not the molten steel 3. Therefore, in the present embodiment, for each of the total 12 pairs of two thermometers F and L facing each other in the depth direction (l3-axis direction) of the mold 4, the arithmetic mean value of the temperatures measured simultaneously by the two thermometers F and L is derived. Note that the pairs of the two thermometers F and L are the thermometer F1 and L1, the thermometer F2 and L2, and so on. As a result, 12 temperatures are obtained. These temperatures are the temperatures at the positions (coordinates (l1, l2)) on the l1-l2 plane where the two thermometers F and L serving as the derivation sources of the temperatures exist.

[0079] Assuming that the heat transfer state in the long side of the mold 4 is one-dimensional steady heat conduction in the depth direction (l3-axis direction), the observation data acquisition unit 113 extrapolates these 12 temperatures based on the temperature of the cooling water of the mold 4 and the thermal conductivity of the mold 4, and derives the temperature of the molten steel 3 at each position on the visualization target surface. In addition, the observation data acquisition unit 113 derives the flow velocity in the casting direction (l2-axis direction) at each position corresponding to the molten steel surface of the visualization target surface from the change per unit time of a plurality of molten steel surface levels measured by the molten steel surface level meter 9.

[0080] As described above, in the present embodiment, the observation data acquisition unit 113 acquires, as observation data, the temperature of the molten steel 3 at each position on the two-dimensional plane to be visualized and the flow velocity in the casting direction at each position corresponding to the molten steel surface of the two-dimensional plane to be visualized. In the following description of the present embodiment, the position is referred to as an observation position as needed, and a vector composed of all of the temperature of the molten steel 3 and the flow velocity in the casting direction acquired as an example of the second physical quantity at the observation position is referred to as an observation vector. Note that in the present embodiment, the observation data acquisition unit 113 extracts data at the time corresponding to the time when the physical quantity derivation unit 111b performs numerical simulation from the time-series data measured in the continuous casting facility, and uses the extracted data to obtain, as observation data, the temperature of the molten steel 3 at each observation position on the two-dimensional plane to be visualized and the flow velocity in the casting direction at each observation position corresponding to the molten steel surface of the two-dimensional plane to be visualized.

[0081] (Likelihood Function Derivation Unit 114) The likelihood function derivation unit 114 evaluates the observation noise of the observation data y (where y is a vector composed of all the data of the temperature of the molten steel 3 and the flow velocity in the casting direction obtained at the observation position) at time t obtained by the observation data acquisition unit 113 from the past observation data of the same kind, and thereby determines the conditional probability density function p(y | x) of the observation vector y when the state vector x is obtained. The likelihood function L(x | y) is derived. The likelihood function L(x | y) is regarded as a function of the state vector x of the conditional probability density function p(y | x), and is determined as a function of the state vector x by substituting the observation data y into the observation vector y. In this embodiment, the likelihood function derivation unit 114 derives the likelihood function L(x | y) of the state vector x at each time t at time intervals of step width Δt. t (0) (y t (0) is a vector composed of all the data of the temperature of the molten steel 3 and the flow velocity in the casting direction obtained at the observation position) of the observation noise from the past observation data of the same kind, and thereby determines the conditional probability density function p(y t when the state vector x t is obtained. The likelihood function L(x t |x t ) of the state vector x t is derived. The likelihood function L(x t |y t ) is derived. The likelihood function L(x t |y t ) is the conditional probability density function p(y t |x t ) regarded as a function of the state vector x t , and is determined as a function of the state vector x by substituting the observation data y into the observation vector y t . In this embodiment, the likelihood function derivation unit 114 derives the likelihood function L(x t (0) |y t ) of the state vector x at each time t at time intervals of step width Δt. t The likelihood function L(x t |y t ) is derived.

[0082] The observation noise is assumed to follow, for example, a Gaussian distribution with a mean vector of 0 (zero vector), and the variance-covariance matrix of the observation data can be used as the covariance matrix of the observation noise. In this case, the variance-covariance matrix values of the past observation data of the same kind are calculated in advance for all types of observation data and stored by the likelihood function derivation unit 114. That is, the likelihood function derivation unit 114 stores the variance-covariance matrix values for each type of observation data. Also, the variance-covariance matrix value of the observation data may be the variance value of each component (value at each observation position) of the observation data, and the covariance value may be 0 (zero).

[0083] The likelihood function derivation unit 114 derives the likelihood function L(x t (0) |y t at time t based on the observation data y t at time t and the variance-covariance matrix value of the observation noise. t )

[0084] The observation noise is not limited to following a Gaussian distribution. For example, the likelihood function derivation unit 114 derives the average vector m(y t (0) at time t based on the observation data y t (0) at time t and the observation data y t-nΔt (0) in a predetermined period before time t, and derives the relative frequency (the value obtained by dividing the frequency by the total number of data) of the plurality of data {y t (0)},···,y t (0) ) of these plurality of data {y t-nΔt (0) -m(y t (0) ),···,y t (0) -m(y t (0) )} to derive a frequency distribution. Then, the likelihood function derivation unit 114 derives the frequency distribution as the probability density function of the observation noise at time t. Also, the likelihood function derivation unit 114 derives the frequency distribution of each component (each data at each observation position α) of the plurality of data {y t-nΔt (0) -m(y t (0) ),···,y t (0) -m(y t (0) )} and takes the product of these to obtain the plurality of data {y t-nΔt (0) -m(y t (0) ),···,y t (0) -m(y t (0))} may be used as the frequency distribution. Further, the likelihood function derivation unit 114 may use, as the probability density function of the observation noise, an approximation of this frequency distribution by a predetermined function (for example, a spline function).

[0085] (Data assimilation unit 115) The data assimilation unit 115 uses the first probability density function p(x t |y t ) of the state vector x at time t derived by the first probability density function derivation unit 112, and the likelihood function L(x 0:t-Δt |y t ) of the state vector x at time t when the observation data at time t derived by the likelihood function derivation unit 114 is obtained. By providing these to the algorithm of a filter that performs data assimilation by Bayesian statistical modeling based on Bayes' theorem, the conditional probability density function p(x t |y t ) of the state vector x at time t when the observation data up to time t is obtained is derived. In the following description, this probability density function p(x t |y t ) is also referred to as the second probability density function as necessary. In the present embodiment, the data assimilation unit 115 derives the second probability density function p(x t0:t |y t ) of the state vector x at each time t at time intervals of step width Δt. t0:t ) of the state vector x at each time t at time intervals of step width Δt. t |y t ) of the state vector x at each time t at time intervals of step width Δt. 0:t ) of the state vector x at each time t at time intervals of step width Δt.

[0086] When performing data assimilation, it is necessary to define a system equation (state equation) and an observation equation. The system equation is an equation that defines the relationship between the state vectors x t+ Δ t , x t at the previous and subsequent times at time intervals of step width Δt. In the present embodiment, this relationship is determined by a plurality of numerical simulations by the physical quantity derivation unit 111b. Further, the observation equation shows the relational expression between the observation vector y t (observed quantity) and the state vector x t (state quantity), and is given by equation (13). yt =H t x t +w t ···(13) Here, H t is the observation matrix, and w t is the observation noise. For example, if the α-th component of the observation vector is the same as the physical quantity of the β-th component of the state vector, the observation matrix H t is a matrix whose (α,β) component is 1 and the other components in the α-th row are 0 (zero). Note that the same physical quantity here means the same including the axis components. Therefore, the flow velocities u and v are different physical quantities. Also, if the observation position α does not coincide with any of the computational grid points l, the observation matrix H t can also be determined to interpolate with the state vector.

[0087] As a filter for performing data assimilation by Bayesian statistical modeling, for example, an ensemble Kalman filter can be used. In this case, the data assimilation unit 115 uses the first probability density function p(x t |y t ) of the state vector x at time t derived by the first probability density function derivation unit 112, and the likelihood function L(x 0:t-Δt |y t ) of the state vector x at the time t derived by the likelihood function derivation unit 114, and performs ensemble approximation to derive ensemble members (particles). Each ensemble member x t |y t ) at time t-Δt is updated based on the system equation, and the ensemble members x t-Δt (k) of the prediction distribution at the next time step (time t) are derived by the first probability density function derivation unit 112. Then, the data assimilation unit 115 derives the ensemble members x t|t-Δt (k) of the filter distribution from the ensemble members x t|t-Δt (k) of the prediction distribution at the next time step. By deriving the ensemble members of such a filter distribution, the state vector x at time t t (k) t ​The second probability density function is derived.

[0088] In the continuous casting facility, the flow velocity of the molten steel 3 cannot be measured at all positions. However, in the system equation, since the temperature and the flow velocity of the molten steel 3 are coupled, the probability density function of the state vector includes the probability density function of the flow velocity of the molten steel 3. In the present embodiment, since the flow velocity in the casting direction at the molten metal surface of the molten steel 3 is measured, data assimilation is also performed on the flow velocity in the casting direction at the molten metal surface of the molten steel 3, whereby the accuracy of the probability density function of the flow velocity of the molten steel 3 can be improved.

[0089] The filter that performs data assimilation by Bayesian statistical modeling is not limited to the ensemble Kalman filter. For example, a particle filter may be used. The ensemble Kalman filter and the particle filter are described in Non-Patent Documents 4 and 5. Further, specific calculation algorithms for the ensemble Kalman filter and the particle filter are described in Non-Patent Document 6. Therefore, detailed descriptions thereof are omitted.

[0090] The data assimilation unit 115 derives the mode x t of the second probability density function p(x t | y t ) at time t as an estimated value of the state vector x m at time t. Further, for each component of the state vector x t (each first physical quantity at each computational grid point l), the marginal probability density function of the second probability density function p(x t | y t | y t ) is derived, and a vector composed of the modes of the respective components of the marginal probability density function is used as the mode x t of the second probability density function p(x t | y m ) may be used. As described above, in the present embodiment, the data assimilation unit 115 uses the first probability density function p(x t of the state vector x t | y0:t-Δt ), and the state vector x at the time t derived by the likelihood function derivation unit 114. t The likelihood function L(x t |y t ) to obtain the most reasonable state vector x at time t. t The second probability density function p(x t |y t ) and calculate its most frequent value x m Let x be the state vector at time t. t is derived as an estimate of

[0091] In addition, the data assimilation unit 115 calculates the state vector x t The second probability density function p(x t |y t ) to the first coefficient derivation unit 111a. The first coefficient derivation unit 111a outputs the state vector x t The second probability density function p(x t |y t ), a first coefficient derivation unit 111a derives a plurality of pairs of the flow velocity u(l,t), v(l,t) and temperature T(l,t) of the molten steel 3 at each computational grid point l at time t, and parameters of the boundary conditions between time t and time t+Δt. u 1(l)~φ u Mu (l), φ v 1(l)~φ v Mv (l) and the basis vector φ for temperature T T 1(l)~φ T MT Based on (l), the coefficient a at time t is calculated using the following equations (14) to (16). u 1(t)~a u Mu (t), a v 1(t)~a v Mv (t), a T 1(t)~a T MT (t) is derived. In addition, the basis vectors φ of the flow velocities u and v areu 1(l) to φ u Mu (l), φ v 1(l) to φ v Mv (l) and the basis vector φ of temperature T T 1(l) to φ T MT (l) is derived by the basis vector derivation unit 103. Also, the first coefficient derivation unit 111a is the coefficient a at the time t derived in this way u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t) and the parameters of the boundary conditions between time t and time t + Δt, based on equations (10) to (12), the coefficient a u 1(t + Δt) to a u Mu (t + Δt), a v 1(t + Δt) to a v Mv (t + Δt), a T 1(t + Δt) to a T MT (t + Δt) is derived. The coefficient a u 1(t + Δt), a v 1(t + Δt), a T The derivation of 1(t + Δt) is executed for each of the above-mentioned multiple cases. Therefore, the coefficient a for the above-mentioned multiple cases u 1(t + Δt) to a u Mu (t + Δt), a v 1(t + Δt) to a v Mv (t + Δt), a T 1(t + Δt) to a T MT The ensemble members of (t + Δt) are derived.

[0092]

Number

[0093] (Visualization data creation unit 116) The visualization data creation unit 116 creates display data of at least one of the temperature T, flow velocities u and v of the molten steel 3 at each position (each computational grid point l) constituting the estimated value of the state vector at time t derived by the data assimilation unit 115. The visualization data creation unit 116 may create any display data as long as the temperature and flow velocity of the molten steel 3 at each position at each time can be recognized by the user of the information processing apparatus 100. However, it is preferable that the visualization data creation unit 116 creates display data that is intuitively easy for the user of the information processing apparatus 100 to understand.

[0094] For example, when the temperature range assumed as the temperature of the molten steel 3 is divided into a plurality of regions, the visualization data creation unit 116 stores in advance different display modes for each region. As the display mode, for example, at least one of color, pattern, and density can be adopted. The visualization data creation unit 116 specifies the temperature at each position from the temperature of the molten steel 3 at each position at time t derived by the data assimilation unit 115, and creates display data so that the image of each position on the visualization target surface is displayed in the display mode corresponding to the specified temperature.

[0095] In addition, when the flow velocity range assumed as the flow velocity of the molten steel 3 is divided into a plurality of regions, the visualization data creation unit 116 stores in advance different display modes for each region. As the display mode, for example, at least one of color, pattern, and density can be adopted. The visualization data creation unit 116 specifies the flow velocity at each position from the flow velocity of the molten steel 3 at each position at time t derived by the data assimilation unit 115, and creates display data so that the image of each position on the visualization target surface is displayed in the display mode corresponding to the specified flow velocity. Further, the visualization data creation unit 116 derives a representative direction of the molten steel 3 for each region preset for the visualization target surface, and creates display data so that arrow lines facing the direction of the direction are superimposed on the image of the region on the visualization target surface.

[0096] (Output unit 117) The output unit 117 displays the display data created by the visualization data creation unit 116 on a computer display. Further, instead of or in addition to such display data, the output unit 117 can output data on the flow velocity and temperature of the molten steel 3 at each position constituting the estimated value of the state vector at each time derived by the data assimilation unit 115. Examples of the output mode include transmission to a storage medium inside or outside the information processing apparatus 100 and transmission to an external device. In this way, display data for displaying the flow velocity and temperature of the molten steel 3 at each position at each time can be created separately later.

[0097] Note that the first probability density function derivation unit 112, the observation data acquisition unit 113, the likelihood function derivation unit 114, the data assimilation unit 115, the visualization data creation unit 116, and the output unit 117 can be realized in the same manner as those described in Patent Document 1.

[0098] (Operation flowchart) Next, an example of the offline processing of the information processing apparatus 100 will be described with reference to the flowchart of FIG. 3. First, in step S301, the numerical simulation unit 101 derives the flow velocities u(l, t), v(l, t) and the temperature T(l, t) at each calculation grid point l at each time t at a time interval of step width Δt1.

[0099] Next, in step S302, the matrix derivation unit 102 extracts the flow velocities u(l, t), v(l, t), and the temperature T(l, t) at each time t at a time interval of step width Δt2 that is longer than the step width Δt1 from the flow velocities u(l, t), v(l, t), and the temperature T(l, t) at each calculation grid point l derived at each time t at a time interval of step width Δt1 in step S301. Then, the matrix derivation unit 102 derives a snapshot matrix MU of the flow velocity u, a snapshot matrix MV of the flow velocity v, and a snapshot matrix MT of the temperature T in which the extracted flow velocities u(l, t), v(l, t), and the temperature T(l, t) are stored as components, respectively.

[0100] Next, in step S303, the basis vector derivation unit 103 uses the left singular vectors obtained by performing singular value decomposition on the snapshot matrices MU, MV, and MT of the flow velocities u and v and the temperature T derived in step S302 as the basis vectors φ u j (l) of the flow velocity u, the basis vector φ v j (l) of the flow velocity v, and the basis vector φ T j (l) of the temperature T. Further, the basis vector derivation unit 103 uses the basis vectors φ u j (l), φ v j (l), φ T j (l) of the flow velocities u and v and the temperature T for the calculations of equations (4), (5), and (6), and derives the numbers M u , M v , M T .

[0101] Next, in step S304, the reduced simulator derivation unit 104 derives and stores a reduced simulator based on the basis vectors φ u 1(l) to φ u Mu (l), φ v 1(l) to φ v Mv (l), φ T 1(l) to φ T MT (l) of the flow velocities u and v and the temperature T and the numbers M u , M v , M T . The reduced simulator has constants A i u , B uu ij , B uv ij , C uu ijk , C uv ijk , A i v , B vu ij , Bvv ij , C vu ijk , C vv ijk , A i T , B TT ij , B Tu ij , B Tv ij , C Tu ijk , C Tv ijk , M u , M v , M T is set. When the process of step S304 ends, the process according to the flowchart of FIG. 3 ends.

[0102] Next, with reference to the flowchart of FIG. 4, an example of the processing in the online of the information processing apparatus 100 will be described. First, in step S401, the first coefficient derivation unit 111a sets the time t to the initial value (t0). Next, in step S402, the first coefficient derivation unit 111a sets a plurality of sets of initial values as the initial values of the set of the flow velocity and temperature of the molten steel 3 at time t and the parameter of the boundary condition between time t and time t + Δt.

[0103] Next, in step S403, the first coefficient derivation unit 111a updates the time t to time t + Δt, and the process proceeds to step S404. Next, in step S404, the first coefficient derivation unit 111a calculates the flow velocities u(l, t - Δt), v(l, t - Δt), and temperature T(l, t - Δt) of the molten steel 3 at each computational grid point l at time t - Δt, and the basis vectors φ u 1(l) to φ u Mu (l), φ v 1(l) to φ v Mv (l), φ T 1(l) to φ T MTBy substituting (l) into equations (14), (15), and (16) respectively, the coefficients a for the flow velocities u, v, and temperature T u 1(t - Δt) to a u Mu (t - Δt), a v 1(t - Δt) to a v Mv (t - Δt), a T 1(t - Δt) to a T MT (t - Δt) is derived. And the first coefficient derivation unit 111a derives the solutions (a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t)) of the differential equations of equations (10) to (12) at time t. The coefficient a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t) is derived for each set of the flow velocities u(l, t - Δt), v(l, t - Δt) and temperature T(l, t - Δt) of the molten steel 3 at each computational grid point l at time t - Δt, and the parameters of the boundary conditions between time t - Δt and time t.

[0104] Next, in step S405, the physical quantity derivation unit 111b uses the coefficient a for the flow velocities u, v, and temperature T u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t) and the basis vectors φ of the flow velocities u, v, and temperature T derived in step S303 of FIG. 3 u 1(l) to φ u Mu (l), φv 1(l) to φ v Mv (l), φ T 1(l) to φ T MT By substituting (l) and φ into equations (4), (5), and (6) respectively, the flow velocities u(l,t), v(l,t) and temperature T(l,t) of the molten steel 3 at each computational grid point l at time t are derived. The flow velocities u(l,t), v(l,t) and temperature T(l,t) of the molten steel 3 at each computational grid point l at time t are derived for each set of the flow velocities u(l,t−Δt), v(l,t−Δt) and temperature T(l,t−Δt) of the molten steel 3 at each computational grid point l at time t−Δt and the parameters of the boundary conditions between time t−Δt and time t.

[0105] Next, in step S406, the first probability density function derivation unit 112 calculates the first probability density function p(x t (flow velocity and temperature of the molten steel 3 at each computational grid point l)) of the state vector x at time t t |y t-Δt ) at time t. Next, in step S407, the observation data acquisition unit 113 acquires the observation data y t (0) (temperature of the molten steel 3 and flow velocity in the casting direction (l2-axis direction) at each observation position α) at time t.

[0106] Next, in step S408, the likelihood function derivation unit 114 calculates the likelihood function L(x t (0) ) of the state vector x at time t based on the observation data y t t |y t ) at time t. Next, in step S409, the data assimilation unit 115 combines the first probability density function p(x t |y t ) of the state vector x at time t with the likelihood function L(x 0:t-Δt |y t ) of the state vector x at time t t |y tBy providing the above to the algorithm of the filter that performs data assimilation by Bayesian statistical modeling, the state vector x at time t t The second probability density function p(x t |y t0:t ) is derived.

[0107] Next, in step S410, the data assimilation unit 115 uses the second probability density function p(x t |y t |y t ) of the state vector x at time t to derive the mode value as the estimated value of the state vector at time t. Next, in step S411, the visualization data creation unit 116 creates display data of the flow velocity and temperature of the molten steel 3 at each position constituting the estimated value of the state vector at time t derived in step S410.

[0108] Next, in step S412, the output unit 117 determines whether the time t has reached a preset time t e If the result of this determination is that the time t has not reached the preset time t e , the process proceeds to step S413. In step S413, the first coefficient derivation unit 111a uses the flow velocity and temperature of the molten steel 3 at time t, which are the data used when deriving the solutions (a u 1(t + Δt) to a u Mu (t + Δt), a v 1(t + Δt) to a v Mv (t + Δt), a T 1(t + Δt) to a T MT (t + Δt)) of the differential equations of equations (10) to (12), and a set of boundary condition parameters between time t and time t + Δt, and derives a plurality of them based on the second probability density function p(x t |y t |y t0:t ) of the state vector x at time t derived by the data assimilation unit 115 in step S409.

[0109] Then, the process proceeds to step S403, and the processings of steps S403 to S413 are repeatedly executed until the time t reaches a preset time t e e . In step S404, a set of the flow velocity and temperature of the molten steel 3 derived in step S413 immediately before step S404 and the parameters of the boundary conditions is used. As described above, in step S412, when the time t reaches a preset time t e e , the process proceeds to step S414. In step S414, the output unit 117 outputs visualization data (the display data created in step S411 and the data of the flow velocity and temperature of the molten steel 3 at each position constituting the estimated value of the state vector at each time derived in step S410). Note that the display data may be displayed on a computer display each time the display data is created in step S411. When the process of step S414 ends, the process according to the flowchart of FIG. 4 ends.

[0110] (Example 1) Next, Example 1 will be described, but the present invention is not limited to the following Example 1. ((Inventive Example 1)) First, Inventive Example 1 will be described. In Inventive Example 1, visualization was performed by the information processing apparatus 100 on a continuous steel casting machine having the configuration shown in FIG. 18. The length of the long side (inner dimension) of the mold 4 is 1500 mm, and the length of the short side (inner dimension) is 250 mm.

[0111] In Inventive Example 1, the numerical simulation unit 101 calculated the flow velocity and temperature of the molten steel 3 at each calculation grid point l at each time t on the visualization target surface by the finite volume method without considering the generation of the solidification shell 8 (that is, assuming that the solidification shell 8 is not generated). The width (length in the l1-axis direction) of the visualization target surface was about 1500 mm, and the depth (length in the l2-axis direction) was about 8000 mm. 2600 calculation grid points l were set in the region of this visualization target surface. Also, necessary physical property values such as the thermal conductivity, specific heat, density, and viscosity coefficient of the molten steel 3 were set.

[0112] Furthermore, the heat transfer boundary condition was set to provide a constant heat flux at the outer peripheral boundary of the molten steel 3. Also, the fluid flow boundary condition was set to provide the vertical component of the flow velocity (the component in the l2-axis direction) at the molten steel surface, and to provide all components of the flow velocity at the inner wall surface of the mold 4, the outer wall surface of the immersion nozzle 6, and the lower end of the molten steel 3, and to provide pressure at the discharge port 7. And, as the initial value of the vertical component of the flow velocity (the component in the l2-axis direction) at the molten steel surface, 0 (zero) was set (slip condition). Also, as the initial values of all components of the flow velocity at the inner wall surface of the mold 4 and the outer wall surface of the immersion nozzle 6, 0 (zero) was set. Also, as the initial value of the flow velocity at a position 8000 mm below (in the positive direction of the l2-axis) from the molten steel surface (the lower end of the molten steel 3), the l1-axis direction was set to 0 (zero) and the l2-axis direction was set to be the same as the casting speed (the pulling speed of the slab). Also, as the initial value of the pressure at the discharge port 7, although it has a value other than 0 (zero) at the boundary with the outer wall surface of the immersion nozzle 6, it was set so that their average value becomes 0 (zero).

[0113] In Invention Example 1, the numerical simulation unit 101 inputs the initial values of the flow velocities u, v and temperature T of the molten steel 3 at each computational grid point l, as well as the heat flux distribution at the outer peripheral boundary of the molten steel 3, the pressure at the discharge port 7, and the casting speed. Then, the numerical simulation unit 101 calculates the flow velocities u(l,t), v(l,t) and temperature T(l,t) of the molten steel 3 at each computational grid point l at each time t at time intervals of 0.025 seconds by numerical simulation.

[0114] The matrix derivation unit 102 extracts 400 time series data at time intervals of 0.25 seconds from the flow velocities u(l,t), v(l,t) and temperature T(l,t) of the molten steel 3 at each computational grid point l at each time t derived by the numerical simulation unit 101, and derives snapshot matrices MU, MV, MT of the flow velocities u, v and temperature T. The snapshot matrices MU, MV, MT of the flow velocities u, v and temperature T are matrices of 2600 rows and 400 columns, respectively.

[0115] The basis vector derivation unit 103 derives the basis vectors φ u j (l), φ v j (l), φ T j (l) of the flow velocities u, v, and the temperature T by performing singular value decomposition on the snapshot matrices MU, MV, MT. Also, as described in this embodiment, the basis vector derivation unit 103 uses the basis vectors φ u j (l), φ v j (l), φ T j (l) of the flow velocities u, v, and the temperature T for the calculations of the formulas (4), (5), and (6) to derive the numbers M u , M v , M T of them.

[0116] FIG. 5 is a diagram showing an example of the singular values of the mode number j (FIG. 5(a)) obtained by performing singular value decomposition on the snapshot matrices MU, MV, MT of the flow velocities u, v, and the temperature T of the molten steel 3, and the ratio of the integrated value of the singular values of the mode numbers 1 to j (FIG. 5(b)). The ratio of the integrated value of the singular values of the mode numbers 1 to j is the integrated value of the singular values of the mode numbers 1 to j obtained by performing singular value decomposition on the snapshot matrices MU, MV, MT of the flow velocities u, v, and the temperature T, divided by the integrated value of all the singular values obtained by performing singular value decomposition on the snapshot matrices MU, MV, MT of the flow velocities u, v, and the temperature T, expressed as a percentage. Note that since the number of columns of the snapshot matrices MU, MV, MT of the flow velocities u, v, and the temperature T is 400, singular values are derived for each of the mode numbers from 1 to 400. However, for the sake of notation, FIGS. 5(a) and 5(b) omit the illustration of the singular values and the ratio of the integrated values of the singular values of the higher-order modes. The minimum value of the mode number j at which the ratio of the integrated value of the singular values of the mode numbers 1 to j becomes 99.999% or more is the number M of the basis vectors φ u j (l), φ v j (l), φ T j (l) of the flow velocities u, v, and the temperature T used for the calculations of the formulas (4), (5), and (6).u , M v , M T shall be set as follows.

[0117] From the results shown in Fig. 5(b), in Invention Example 1, the base vectors φ of the flow velocities u, v, and the temperature T used in the calculations of equations (4), (5), and (6) u j (l), φ v j (l), φ T j (l) and the number M of u , M v , M T are set to 15, 16, and 20, respectively.

[0118] The reduced simulator derivation unit 104 calculates the constants A, B, and C in equations (10) to (12) based on the base vectors φ of the flow velocities u, v, and the temperature T u 1(l) to φ u 15 (l), φ v 1(l) to φ v 16 (l), φ T 1(l) to φ T 20 (l) and the number M of the base vectors u (= 15), M v (= 16), M T (= 20). i u , B uu ij , B uv ij , C uu ijk , C uv ijk , A i v , B vu ij , B vv ij , C vu ijk , C vv ijk , A i T , B TT ij , B Tu ij , B Tv ij, C Tu ijk , C Tv ijk are derived.

[0119] The base vectors φ of the flow velocities u, v and the temperature T u j (l), φ v j (l), φ T j (l) are derived at each computational grid point l. Fig. 6 shows the base vectors φ of the flow velocities u, v and the temperature T in Invention Example 1 u 2(l), φ v 2(l), φ T 2(l). Fig. 6 shows the distribution of the base vectors φ of the flow velocities u, v and the temperature T u 2(l), φ v 2(l), φ T 2(l). In Fig. 6, the magnitudes of the base vectors φ of the flow velocities u, v and the temperature T are shown by contour lines. Fig. 6 shows the base vectors φ of the flow velocities u, v and the temperature T when the mode number j is 2 u 2(l), φ v 2(l), φ T 2(l). As shown in Fig. 6, it can be seen that the distributions of the base vectors φ of the flow velocities u, v are very different. u 2(l), φ v 2(l) The offline processing ends here.

[0120] Thereafter, online, the first coefficient derivation unit 111a, for each of 100 cases where at least one of the flow velocity of the molten steel 3 at time t - Δt, the temperature of the molten steel 3 at time t - Δt, and the parameter of the boundary condition between time t - Δt and time t is different, derives the solutions (a u 1(t) to a u 15 (t), a v 1(t) to a v 16 (t), a T 1(t) to a T 20 (t)) of the differential equations of formulas (10) to (12).

[0121] The physical quantity derivation unit 111b uses the coefficients a u 1(t) to a u 15 (t), a v 1(t) to a v 16 (t), a T 1(t) to a T 20 (t) derived by the first coefficient derivation unit 111a at each time t for the flow velocities u, v, and the temperature T, and the basis vectors φ u 1(l) to φ u 15 , φ v 1(l) to φ v 16 , φ T 1(l) to φ T 20 (l) and substitutes them into equations (4), (5), and (6) respectively to derive the flow velocities u(l, t), v(l, t), and the temperature T(l, t) of the molten steel 3 at each calculation grid point l at time t. As a result, at each time t, a large number of calculation results (an ensemble {x t (k)}) k ) are obtained.

[0122] The first probability density function derivation unit 112 derives the first probability density function p(x t (k) ) of the state vector x k at each time t from a large number of calculation results (an ensemble {x t}) t |y t-Δt ).

[0123] In Invention Example 1, observation data was obtained using the temperatures measured by the thermometers F1 to F12 and L1 to L12 embedded in the mold 4. The thermometers F1 to F12 and L1 to L12 are arranged as shown in FIG. 2. Therefore, in order to correspond with the numerical simulation of the measurement target surface of the information processing apparatus 100, for each of the 12 pairs of two thermometers F and L facing each other in the depth direction (l3-axis direction) of the mold 4, the arithmetic mean value of the temperatures measured simultaneously by the two thermometers F and L is derived. Then, the observation data acquisition unit 113 sets this arithmetic mean value as the temperature at the position (coordinates (l1, l3)) on the l1-l3 plane where the two thermometers F and L that are the sources of the derived temperature exist, and obtains the time series data thereof. The observation data acquisition unit 113 extrapolates the time series data of the 12 temperatures obtained in this way based on the temperature of the cooling water of the mold 4 and the thermal conductivity of the mold 4, and determines the temperature of the molten steel 3 at each observation position α of the visualization target surface as the observation data y t (0) and derives it as such.

[0124] The likelihood function derivation unit 114 uses, as the observation noise of the observation vector y t a plurality of observation data {y t-nΔt (0) , ···, y t (0)} derived during a certain period immediately before the measurement (for example, 6 hours or 24 hours). In this embodiment, the likelihood function derivation unit 114 assumes that the observation noise at time t follows a Gaussian distribution with a mean vector of 0 (zero vector), and calculates its variance-covariance matrix value. Thereby, the likelihood function L(x t |y t ) of the state vector x t at each time t is obtained.

[0125] The data assimilation unit 115 performs a filter operation by providing the first probability density function p(x t |y t ) of the state vector x 0:t-Δt at each time t and the likelihood function L(x t |y t ) of the state vector x t at each time t to the algorithm of the ensemble Kalman filter. Thereby, the state vector x tThe second probability density function p(x t |y t0:t ) is obtained. In equation (1), the observation matrix H t is set to derive the temperature of the molten steel 3 at each observation position α from the state vector x t , and the observation noise w t has a mean vector of 0 (zero vector), and the variance value of the variance-covariance matrix is the variance value of the observed data y t evaluated at the positions of the respective thermometers F1 to F12 and L1 to L12 for the past 24 hours of variation, and the covariance value is 0 (zero).

[0126] The data assimilation unit 115 derives the mode value of the second probability density function p(x t |y t |y t0:t ) of the state vector x t at time t as the estimated value of the state vector x at time t. t Also, the first coefficient derivation unit 111a randomly derives 100 sets of combinations of the flow velocities u(l, t), v(l, t) and temperature T(l, t) of the molten steel 3 at each computational grid point l at time t and the parameters of the boundary conditions between time t and time t + Δt from the second probability density function p(x t |y t0:t ) of the state vector x u derived by the data assimilation unit 115. Then, the first coefficient derivation unit 111a, based on each of the 100 sets of the temperature T(l, t) and flow velocities u(l, t), v(l, t) of the molten steel 3 at each computational grid point l at time t and the basis vectors φ u 15 1(l) to φ v 1(l) to φ v 16 and the basis vectors φ T 1(l) to φ T 20 (l) of the temperature T, according to equations (14) to (16), the coefficients a u 1(t) to a u 15 (t), a v 1(t) to a v 16 (t), aT 1(t) to a T 20 Derive 100 sets of ensemble members of (t). Then, the first coefficient derivation unit 111a derives the coefficient a u 1(t) to a u 15 (t), a v 1(t) to a v 16 (t), a T 1(t) to a T 20 Based on the ensemble members of (t) and the parameters of the boundary conditions from time t to time t + Δt, according to equations (10) to (12), the coefficient a u 1(t + Δt) to a u 15 (t + Δt), a v 1(t + Δt) to a v 16 (t + Δt), a T 1(t + Δt) to a T 20 Derive 100 sets of ensemble members of (t + Δt). The new boundary condition corrects only the pressure distribution at the position of the discharge port 7 of the immersion nozzle 6 based on the second probability density function p(x t of the state vector x at time t t |y t0:t ). The other boundary conditions were fixed as they were at the initial stage of the change. The time interval Δt was set to 1 second.

[0127] The visualization data creation unit 116 creates visualization data based on the flow velocity and temperature of the molten steel 3 at each position that constitutes the estimated value of the state vector at time t, which is derived by the data assimilation unit 115. The output unit 117 displays the visualization data on the computer display each time the visualization data is created by the visualization data creation unit 116.

[0128] ((Inventive Example 2)) Next, Inventive Example 2 will be described. In Inventive Example 2, compared with Inventive Example 1, the number of computational grid points l set in the region of the visualization target surface was increased from 2600 to 3600. In Inventive Example 2, in the same manner as in Inventive Example 1, the basis vectors φ of the flow velocities u, v, and temperature Tu j (l), φ v j (l), φ T j The number M of (l) u , M v , M T As a result of the derivation, they were 14, 14, and 18 respectively. Other processes and configurations in Invention Example 2 are the same as those in Invention Example 1.

[0129] ((Comparative Example 1)) Next, Comparative Example 1 will be described. In Comparative Example 1, in Invention Example 1, instead of the flow velocities u and v of the molten steel 3, the kinetic energy E of the molten steel 3 (= (u 2 + v 2 ) ÷ 2) is used. Also, in Comparative Example 1, the flow velocities u(l, t), v(l, t), and temperature T(l, t) are represented by the following equations (17), (18), and (19) instead of equations (4), (5), and (6) respectively.

[0130]

Equation

[0131] Here, φ E j (l) is the base vector of the kinetic energy E. In Comparative Example 1, the kinetic energy E of the molten steel 3 at each calculation grid point l at each time t is derived from the flow velocities u and v of the molten steel 3 at each calculation grid point l at each time t by numerical simulation. For the temperature T, in the same manner as in Invention Example 1, the temperature T of the molten steel 3 at each calculation grid point l at each time t is derived by numerical simulation.

[0132] From the kinetic energy E and temperature T of the molten steel 3 at each calculation grid point l at each time t, 400 time-series data are extracted at time intervals of 0.25 seconds, and the snapshot matrices ME and MT of the kinetic energy E and temperature T are derived. The snapshot matrices ME and MT of the kinetic energy E and temperature T are matrices of 2600 rows and 400 columns respectively.

[0133] By performing singular value decomposition on the snapshot matrices ME and MT, the basis vectors φ of the kinetic energy E and the temperature T E j (l), φ T j (l) are derived. Also, the basis vectors φ of the kinetic energy E and the temperature T used in the calculations of equations (17), (18), and (19) E j (l), φ T j (l) number M E , M T are derived.

[0134] FIG. 7 is a diagram showing an example of the singular values of the mode number j (FIG. 7(a)) obtained by performing singular value decomposition on the snapshot matrix ME of the kinetic energy E, and the ratio of the integrated value of the singular values of the mode numbers 1 to j (FIG. 7(b)). The ratio of the integrated value of the singular values of the mode numbers 1 to j is the same as that described in FIG. 5, and is the value obtained by dividing the integrated value of the singular values of the mode numbers 1 to j obtained by performing singular value decomposition on the snapshot matrix ME of the kinetic energy E by the integrated value of all the singular values obtained by performing singular value decomposition on the snapshot matrix ME of the kinetic energy E, expressed as a percentage. Note that, similar to FIGS. 5(a) and 5(b), in FIGS. 7(a) and 7(b) as well, the illustration of the singular values and the ratio of the integrated values of the higher-order modes is omitted. Similar to Invention Example 1, the minimum value of the mode number j at which the ratio of the integrated value of the singular values of the mode numbers 1 to j becomes 99.999% or more is used as the basis vectors φ of the kinetic energy E and the temperature T in the calculations of equations (17), (18), and (19) E j (l), φ T j (l) number M E , M T is set.

[0135] From the results shown in FIG. 7(b), in Comparative Example 1, the basis vector φ of the kinetic energy E used in the calculations of equations (17) and (18) e j (l) number M Ewas set to 38. Since the snapshot matrix MT of the temperature T in Comparative Example 1 is the same as that in Invention Example 1, the number M of the basis vectors φ T j (l) T becomes 20 as in Invention Example 1.

[0136] For both sides of the equations obtained by substituting equations (17) and (18) into equations (7) and (8), take the inner product with the basis vector φ E i (l) of the kinetic energy E of the i-th mode, and rearrange so that the time derivative term comes to the left side, then the same equations as equations (10) and (11) are obtained respectively. Also, for both sides of the equation obtained by substituting equations (17) to (19) into equation (9), take the inner product with the basis vector φ T i (l) of the temperature T of the i-th mode, and rearrange so that the time derivative term comes to the left side, then the same equation as equation (12) is obtained.

[0137] In Comparative Example 1, the basis vectors φ E 1(l) to φ E 38 (l), φ T 1(l) to φ T 20 (l) of the kinetic energy E, temperature T and the number M of the basis vectors u (=M E = 38), M v (=M E = 38), M T (= 20), based on these, the constants A i u , B uu ij , B uv ij , C uu ijk , C uv ijk , A i v , B vu ij , B vv ij , C vu ijk , C vv ijk , Ai T , B TT ij , B TE ij , C TE ijk are derived.

[0138] Figure 8 is a diagram showing the distribution of the basis vector φ E 2(l) of the kinetic energy E in Comparative Example 1. In Figure 8, the magnitude of the basis vector φ E 2(l) of the kinetic energy E is shown by contour lines. In Figure 8, the distribution of the basis vector φ E 2(l) of the kinetic energy E when the mode number j is 2 is shown. Incidentally, the distribution of the basis vector φ T 2(l) of the temperature T is the same as that shown in Figure 6. As shown in Figure 8, the distribution of the basis vector φ E 2(l) of the kinetic energy E is significantly different from the distribution of the basis vectors φ u 2(l), φ v 2(l) of the flow velocities u, v, and by combining the distributions of the basis vectors φ u 2(l), φ v 2(l) of the flow velocities u, v is very different, and it can be seen that the characteristics of the distribution of the basis vectors φ u 2(l), φ v 2(l) of the flow velocities u, v are not reflected. The offline processing ends here.

[0139] After that, online, in the same manner as in Invention Example 1, at each time t, for each of the 100 cases, the solutions (a u 1(t) ~ a u 38 (t), a v 1(t) ~ a v 38 (t), a T 1(t) ~ a T 20 (t)) of the differential equations of equations (10) to (12) are derived. Then, at each time t, the coefficients a u 1(t) ~ a u 38 (t), a v1(t) to a v 38 (t), a T 1(t) to a T 20 (t) and the basis vectors φ of the kinetic energy E and temperature T derived offline E 1(l) to φ E 38 , φ T 1(l) to φ T 20 By substituting them into equations (17), (18), and (19) respectively, the flow velocities u(l, t), v(l, t), and temperature T(l, t) of the molten steel 3 at each computational grid point l at time t are derived. As a result, at each time t, a large number of calculation results (an ensemble {x t (k)}) k are obtained. Since the subsequent online processing is the same as that in Invention Example 1, a detailed description of the processing is omitted.

[0140] ((Comparative Example 2)) Next, Comparative Example 2 will be described. In Comparative Example 2, the flow velocity and temperature of the molten steel 3 were derived and visualization data was created in the same manner as in Patent Document 1. That is, without having the numerical simulation unit 101, matrix derivation unit 102, basis vector derivation unit 103, and reduced simulator derivation unit 104, by using the numerical simulation unit described in Patent Document 1 instead of the first coefficient derivation unit 111a and physical quantity derivation unit 111b, the flow velocity and temperature of the molten steel 3 were derived and visualization data was created. The number of computational grid points l was set to 2600, the same as in Invention Example 1 and Comparative Example 1.

[0141] ((Comparative Example 3)) Next, Comparative Example 3 will be described. Comparative Example 3 is obtained by increasing the number of computational grid points l set within the region of the visualization target surface from 2600 to 3600 with respect to Comparative Example 2. That is, Comparative Example 3 is obtained by increasing the number of computational grid points l when deriving the temperature and flow velocity of the molten steel 3 and creating visualization data in the same manner as in Patent Document 1, compared to Comparative Example 2.

[0142] ((Comparison between the Invention Example and the Comparative Examples)) In Comparative Examples 2 and 3, the calculation time is long but the calculation accuracy is high. Therefore, based on Comparative Example 2, the calculation accuracies of Invention Example 1 and Comparative Example 1 are compared. FIG. 9 is a diagram showing the relationship between the temperature and time of molten steel 3 in Invention Example 1. FIG. 10 is a diagram showing the relationship between the temperature and time of molten steel 3 in Comparative Example 1. Note that the temperature of molten steel 3 is the temperature of molten steel 3 used for creating the visualization data.

[0143] In FIG. 9, graph 901 shows the relationship between the temperature and time of molten steel 3 in Invention Example 1. Graph 902 shows the relationship between the temperature and time of molten steel 3 in Comparative Example 2. In FIG. 10, graph 1001 shows the relationship between the temperature and time of molten steel 3 in Comparative Example 1. Graph 1002 shows the relationship between the temperature and time of molten steel 3 in Comparative Example 2 and is the same as graph 902. All of the graphs 901, 902, 1001, and 1002 show the temperature of molten steel 3 at the same calculation grid point l.

[0144] As shown in FIG. 9, in graph 901 showing the relationship between the temperature and time of molten steel 3 in Invention Example 1, it changed in the same manner as graph 902 showing the relationship between the temperature and time of molten steel 3 in Comparative Example 2, and substantially the same results were obtained. On the other hand, as shown in FIG. 10, graph 1001 showing the relationship between the temperature and time of molten steel 3 in Comparative Example 1 is significantly different from graph 1002 showing the relationship between the temperature and time of molten steel 3 in Comparative Example 2. In particular, in graph 1001 showing the relationship between the temperature and time of molten steel 3 in Comparative Example 1, a high-temperature state in the time range from 20 seconds to 30 seconds and a low-temperature state after 80 seconds could not be detected.

[0145] The difference in calculation accuracy between Invention Example 1 and Comparative Example 1 as described above is considered to be due to the difference in the base vectors φ u j (l), φ v j (l), φ E j (l). Therefore, the base vectors φ of the flow velocities u and vu j (l), φ v j (l) is used to reproduce the flow velocity and temperature in Comparative Example 2, and the base vector φ of the kinetic energy E E j The difference in accuracy was investigated between the case where the flow velocity and temperature in Comparative Example 2 were reproduced using (l) and the base vector φ of the kinetic energy E

[0146] First, the flow velocities u(l, t), v(l, t) and temperature T(l, t) derived in Comparative Example 2, and the base vectors φ of the kinetic energy E and temperature T derived in Comparative Example 1 E j (l), φ T j (l) are used to derive the coefficients a for the flow velocities u, v and temperature T according to the following equations (20), (21) and (16) u j (t), a v j (t), a T j (t). Also, the flow velocities u(l, t), v(l, t) and temperature T(l, t) derived in Comparative Example 2, and the base vectors φ of the flow velocities u, v and temperature T derived in Invention Example 1 u j (l), φ v j (l), φ T j (l) are used to derive the coefficients a according to equations (14), (15) and (16) u j (t), a v j (t), a T j (t).

[0147] [Number]

[0148] Then, the coefficients a derived from equations (20), (21) and (16) u j (t), a vj (t), a T j Substitute (t) into equations (17), (18), and (19), respectively. From the equations (17), (18), and (19) obtained in this way, the flow velocities u(l, t), v(l, t), and temperature T(l, t) in Comparative Example 2 can be reproduced. Reproducing such flow velocities u(l, t), v(l, t), and temperature T(l, t) using the basis vectors φ E j of the number M E of the kinetic energy E and the basis vectors φ T j of the number M T of temperature T is performed with different values. In the following description, the flow velocities u(l, t), v(l, t), and temperature T(l, t) reproduced in this way are referred to as the flow velocities û(l, t), v̂(l, t), and temperature T̂(l, t) reproduced from the results of Comparative Example 1 as necessary.

[0149] Also, the coefficients a u j (t), a v j (t), a T j (t) are substituted into equations (4), (5), and (6). From the equations (4), (5), and (6) obtained in this way, the flow velocities u(l, t), v(l, t), and temperature T(l, t) in Comparative Example 2 can be reproduced. Reproducing such flow velocities u(l, t), v(l, t), and temperature T(l, t) using the basis vectors φ u j of the number M u of the flow velocity u, the basis vectors φ v j of the number M v of the flow velocity v, and the basis vectors φ T j of the number M T of temperature T is performed with different values. In the following description, the flow velocities u(l, t), v(l, t), and temperature T(l, t) reproduced in this way are referred to as the flow velocities û(l, t), v̂(l, t), and temperature T̂(l, t) reproduced from the results of Invention Example 1 as necessary.

[0150] Then, the root mean squared error (RMSE) of the flow velocities u(l,t), v(l,t) and temperature T(l,t) reproduced from the results of Comparative Example 1 with respect to the flow velocities u(l,t), v(l,t) and temperature T(l,t) in Comparative Example 2 was derived. Similarly, the root mean squared error of the flow velocities u(l,t), v(l,t) and temperature T(l,t) reproduced from the results of Invention Example 1 with respect to the flow velocities u(l,t), v(l,t) and temperature T(l,t) in Comparative Example 2 was derived. The root mean squared error was calculated for each number of basis vectors used (M E , M u , M v , M T The root mean square error of each physical variable (flow velocities u, v, temperature T) was normalized by dividing it by the root mean square value of the fluctuation of the physical variable (flow velocities u, v, temperature T) from the average value in Comparative Example 2.

[0151] Figure 11 shows the root mean square error of the flow velocity u and the basis vector φ u j (l), φ E j (l) Number M u , M E 11(a) shows the root mean square error (RMSE) of the flow velocity u^(l,t) reproduced from the results of Example 1 of the invention with respect to the flow velocity u(l,t) in Comparative Example 2. FIG. 11(b) shows the root mean square error (RMSE) of the flow velocity u^(l,t) reproduced from the results of Comparative Example 1 with respect to the flow velocity u(l,t) in Comparative Example 2.

[0152] Figure 12 shows the root mean square error of the flow velocity v and the basis vector φ v j (l), φ E j (l) Number M v , M EIt is a diagram showing an example of the relationship with. Fig. 12(a) shows the root mean square error (RMSE) of the reproduced flow velocity v^(l,t) from the result of Invention Example 1 with respect to the flow velocity v(l,t) in Comparative Example 2. Fig. 12(b) shows the root mean square error (RMSE) of the reproduced flow velocity v^(l,t) from the result of Comparative Example 1 with respect to the flow velocity v(l,t) in Comparative Example 2.

[0153] Fig. 13 is a diagram showing an example of the relationship between the root mean square error of the temperature T and the number M of the basis vectors φ T j (l). T It is a diagram showing an example of the relationship with. Note that the basis vector φ of the temperature T T j (l) is the same in Invention Example 1 and Comparative Example 1. Therefore, the relationship between the root mean square error of the temperature T and the number M of the basis vectors φ T j (l) is the same whether the reproduced temperature T^(l,t) from the result of Invention Example 1 is used or the reproduced temperature T^(l,t) from the result of Comparative Example 1 is used. T

[0154] In Fig. 11(a), when the number M of the basis vectors φ u j (l) of the flow velocity u used in Invention Example 1 is M(=15), the root mean square error of the reproduced flow velocity u^(l,t) from the result of Invention Example 1 with respect to the flow velocity u(l,t) in Comparative Example 2 is about 0.038 (3.8%). On the other hand, in Fig. 11(b), when the number M of the basis vectors φ u (l) of the kinetic energy E used in Comparative Example 1 is M(=38), the root mean square error of the reproduced flow velocity u^(l,t) from the result of Comparative Example 1 with respect to the flow velocity u(l,t) in Comparative Example 2 is about 0.65 (65%). E j (l) E

[0155] Also, in Fig. 12(a), when the number M of the basis vectors φ v j (l) of the flow velocity v used in Invention Example 1 is M vWhen it is (=16), the root mean square error of the flow velocity u^(l,t) reproduced from the results of Invention Example 1 with respect to the flow velocity u(l,t) in Comparative Example 2 is about 0.011 (1.1%). On the other hand, in FIG. 12(b), the basis vector φ E j of the kinetic energy E used in Comparative Example 1, the number M E (=38), the root mean square error of the flow velocity v^(l,t) reproduced from the results of Comparative Example 1 with respect to the flow velocity v(l,t) in Comparative Example 2 is about 0.50 (50%).

[0156] Thus, it can be seen that in Comparative Example 1 using the basis vector φ E j (l) of the kinetic energy E, the reproduction errors of the flow velocities u and v in Comparative Example 2 are as large as comparable to the fluctuation ranges of the original flow velocities u and v. It is considered that this is because in Comparative Example 1, the original flow phenomenon of the molten steel 3 is not correctly reflected, so the result of data assimilation is not good either. On the other hand, it can be seen that in Invention Example 1 using the basis vectors φ u j (l) and φ v j (l) of the flow velocities u and v individually, the flow velocities u and v in Comparative Example 2 can be reproduced well.

[0157] In addition, as shown in FIG. 13, when the number M T j of the basis vectors φ T (l) of the temperature T used in Invention Example 1 and Comparative Example 1 is (=20), the root mean square error of the temperature T^(l,t) reproduced from the results of Invention Example 1 and Comparative Example 1 with respect to the temperature T(l,t) in Comparative Example 2 is about 0.090 (9.0%). Therefore, it can be seen that in Invention Example 1, the temperature T in Comparative Example 2 can be reproduced well.

[0158] Next, the calculation times of each invention example and each comparative example are compared. Table 1 shows the calculation times of Invention Example 1, Invention Example 2, Comparative Example 1, Comparative Example 2, and Comparative Example 3. The calculation times of Invention Example 1 and Invention Example 2 are, online, the coefficients a u 1(t)~au Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT The time required to derive (t) (processing time in the first coefficient derivation unit 111a). The calculation time of Comparative Example 1 is the coefficient a E 1(t) to a E ME (t), a T 1(t) to a T MT The time required to derive (t). The calculation times of Comparative Example 2 and Comparative Example 3 are the time required for the numerical simulation executed online in Patent Document 1 (processing time in the numerical simulation unit described in Patent Document 1). Incidentally, the unit of the numerical values shown in Table 1 is seconds.

[0159]

Table 1

[0160] In Table 1, the assumed time is the time assumed in the simulation. The time required to execute the simulation for the time shown in the column of the assumed time in the actual operation is the calculation time described in each column of Invention Example 1, Invention Example 2, Comparative Example 1, Comparative Example 2, and Comparative Example 3. As shown in Table 1, in Invention Example 1, Invention Example 2, and Comparative Example 1, the time required to execute the simulation for 3600 seconds (the calculation time when the assumed time is 3600 seconds) is much less than 1800 seconds, which is half of it. Therefore, even if data assimilation is executed, visualization data can be created within 3600 seconds. Thus, visualization data can be created in real time. On the other hand, in Comparative Example 2, the time required to execute the simulation for 3600 seconds is 3059 seconds, and the time until data assimilation is executed and visualization data is created may exceed 3600 seconds. Also, in Comparative Example 3, the time required to execute the simulation for 3600 seconds is 6154 seconds, and the calculation time greatly exceeds 3600 seconds before data assimilation is executed. When the calculation time exceeds 3600 seconds, the time lag from the time when the observation data is acquired to the time when the visualization data is created accumulates with the passage of time, and visualization data cannot be created in real time.

[0161] Summarizing the results of Invention Example 1, Invention Example 2, Comparative Example 1, Comparative Example 2, and Comparative Example 3, it becomes as shown in Table 2.

[0162]

Table 2

[0163] The application of POD indicates the presence or absence of the application of proper orthogonal decomposition (〇 indicates the application of proper orthogonal decomposition, and × indicates the absence of the application of proper orthogonal decomposition). The number of POD bases is the total number (M) of the basis vectors φ u j (l), φ v j (l), φ T 1(l), φ E 1(l) u 、M v, M T , M E shows the total number). The resolution shows the total number of computational grid points l. The accuracy indicates the quality of the calculation accuracy of the flow velocities u and v and the temperature T of the molten steel 3 (〇 indicates good, and × indicates poor). The calculation speed indicates the quality of the online calculation speed of the flow velocities u and v and the temperature T of the molten steel 3 (〇 indicates that real-time processing is sufficiently possible, △ indicates that there is no margin of time for real-time processing, and × indicates that real-time processing cannot be performed).

[0164] As shown in Table 2, in Invention Example 1 and Invention Example 2, both the accuracy and the calculation speed are good. On the other hand, in Comparative Examples 1 to 3, it can be seen that either the accuracy or the calculation speed is inferior to that of Invention Example 1 and Invention Example 2.

[0165] (Summary) As described above, in the present embodiment, the information processing apparatus 100 derives, offline for each axial component of the flow velocity, a snapshot matrix that stores the values of the flow velocities u and v of the molten steel 3 at each computational grid point l and each time t, as the snapshot matrices MU and MV. Then, the information processing apparatus 100 executes singular value decomposition on the snapshot matrices MU and MV, thereby, offline for each of the snapshot matrices MU and MV, deriving the basis vectors φ u 1(l) to φ u Mu (l), φ v 1(l) to φ v Mv (l). After that, when the information processing apparatus 100 linearly combines the basis vectors φ u 1(l) to φ u Mu (l), φ v 1(l) to φ v Mv (l), the coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv(t) for each axis component online. Then, the information processing device 100 derives the basis vectors φ u 1(l)~φ u Mu (l), φ v 1(l)~φ v Mv (l) is coefficient a u 1(t)~a u Mu (t), a v 1(t)~a v Mv (t) to linearly combine the flow velocities u and v of the molten steel 3 online for each axial component. Therefore, the estimation accuracy of the flow velocities u and v of the molten steel 3 can be improved compared to the case of performing proper orthogonal decomposition using kinetic energy as described in Non-Patent Document 1. In addition, the flow velocities u and v of the molten steel can be derived without using a complex calculation formula as described in Non-Patent Document 2. Therefore, it is possible to suppress both an increase in the time required to derive the flow velocities u and v of the molten steel 3 in the mold of the continuous casting equipment 4 and a decrease in the estimation accuracy of the flow velocities u and v of the molten steel 3.

[0166] In this embodiment, the information processing device 100 calculates basis vectors φ u 1(l)~φ u Mu (l), φ v 1(l)~φ v Mv Coefficient a multiplied by (l) u 1(t)~a u Mu (t), a v 1(t)~a v Mv (t), the coefficient a is calculated by solving the differential equations for each axis component simultaneously online. u 1(t)~a u Mu (t), a v 1(t)~a v Mv(t) is derived online for each axis component. Each of the differential equations is an equation of motion that describes the flow of the molten steel 3, and the flow velocities u and v of the molten steel 3 are expressed as linear combinations of the basis vectors φ u 1(l)~φ u Mu (l), φ u 1(l)~φ v Mv (l). By taking the inner product of both sides of the equation of motion expressed as a linear combination with the basis vectors φ u 1(l)~φ u Mu (l), φ u 1(l)~φ v Mv (l), differential equations (the differential equations with i being 1, ···, M u , 1, ···, M v are obtained). Therefore, by utilizing the fact that the basis vectors φ u j (l), φ v j (l) form an orthonormal system respectively, a reduced simulator can be easily derived.

[0167] Also, in this embodiment, the information processing device 100 derives offline a snapshot matrix MT that stores the values of the temperature T of the molten steel 3 at each calculation grid point l at each time t. Then, the information processing device 100 performs singular value decomposition on the snapshot matrix MT to derive offline the basis vectors φ T 1(l)~φ T MT (l) of the temperature T. After that, the information processing device 100 derives online the coefficients a u 1(t)~a T MT (t) used when linearly combining the basis vectors φ T 1(t)~a T MT (t) of the temperature T. And the information processing device 100 linearly combines the basis vectors φ T 1(l)~φ T MT (l) with the coefficients aT 1(t) to a T MT By linearly combining using (t), the temperature T of the molten steel 3 is derived online. Therefore, the temperature T can also be derived together with the flow velocities u and v of the molten steel 3. Thus, it is possible to suppress both the increase in the time required to derive the flow velocities u and v and the temperature T of the molten steel 3 in the mold of the continuous casting facility 4, and the decrease in the estimation accuracy of the flow velocities u and v and the temperature T of the molten steel 3.

[0168] Also, in the present embodiment, the information processing device 100 is the basis vector φ of the temperature T T 1(l) to φ T MT The coefficient a multiplied by (l) T 1(t) to a T MT By solving the differential equations representing the time changes of (t) online simultaneously, the coefficient a T 1(t) to a T MT (t) is derived online. Each of the differential equations is a heat conduction equation describing the heat conduction of the molten steel 3, and the temperature T of the molten steel 3 is the basis vector φ of the temperature T T 1(l) to φ T MT For both sides of the motion equation expressed as a linear combination of (l), the inner product with the basis vector φ of the temperature T T 1(l) to φ T MT is taken to form a differential equation (the differential equation with i being 1, ···, M in Equation (12)) T Therefore, by utilizing the fact that the basis vectors φ of the temperature T T j (l) form an orthonormal system, a reduced simulator can be easily derived.

[0169] Also, in this embodiment, the information processing apparatus 100 derives the flow velocities u(l, t) and v(l, t) and the temperature T(l, t) of the molten steel 3 offline at time intervals of step width Δt1 by numerical simulation. Then, the information processing apparatus 100 extracts the flow velocities u(l, t), v(l, t), and the temperature T(l, t) at each time t with a time interval of step width Δt2, which is longer than the step width Δt1, from the flow velocities u(l, t), v(l, t), and the temperature T(l, t) at each computational grid point l derived at each time t with a time interval of step width Δt1, and derives snapshot matrices MU, MV, and MT in which the extracted flow velocities u(l, t), v(l, t), and the temperature T(l, t) are stored as components. Therefore, the computational load during singular value decomposition can be reduced.

[0170] Also, in this embodiment, the information processing apparatus 100 determines, based on the singular values obtained by performing singular value decomposition on the snapshot matrices MU, MV, and MT, a value of M that is smaller than the maximum order of the modes that can be derived from the snapshot matrices MU, MV, and MT. Therefore, even without using overly high-order basis vectors, the flow velocities u, v, and the temperature T of the molten steel 3 can be approximated by a linear combination of the basis vectors φ u M v M n (l), φ u j (l), φ v j (l) with a precision required in practice. T j (l).

[0171] Also, in this embodiment, the information processing apparatus 100 provides the first probability density function p(x t |y t ) of the state vector x 0:t-Δt at time t and the likelihood function L(x t |y t ) of the state vector x t at time t to the algorithm of a filter that performs data assimilation, thereby obtaining the second probability density function p(x t |y t ) of the state vector x 0:tto derive it. The information processing apparatus 100 uses the mode value of this second probability density function p(x t |y 0:t ) as the estimated value of the state vector x t at time t. Further, the information processing apparatus 100 derives a plurality of parameters including the flow velocity and temperature of the molten steel 3 and the boundary conditions for obtaining the numerical analysis data x t |y 0:t ) at time t + Δt based on this second probability density function p(x t . Here, the flow velocities u and v and the temperature T of the molten steel 3 included as components of the state vector x t are derived by linearly combining the basis vectors φ u j (l), φ v j (l), φ T j (l) of the flow velocities u and v and the temperature T of the molten steel 3 using the coefficients a u j (t), a v j (t), a T j (t). Therefore, it is possible to visualize the flow and temperature distribution of the molten metal inside the continuous casting machine as factual data based on the observation data and reasonable numerical data based on numerical simulation based on physical laws without imposing a large computational load. Thus, it is possible to know the occurrence state of abnormal phenomena such as the uneven flow of the molten steel 3 at every moment. As a result, it becomes possible to flexibly control the casting operation conditions, and to change and replace problematic members at an appropriate timing, and it becomes possible to manufacture high-quality steel materials with good yield and productivity.

[0172] (Modification example) In this embodiment, the case where the flow and temperature of the molten steel 3 in the two-dimensional plane (l1-l2 plane) determined by the width direction (l1-axis direction) and the casting direction (l2-axis direction) of the mold 4 are visualized is illustrated. However, the flow and temperature of the molten steel 3 in the three-dimensional space may be visualized. In this case, in addition to the l1-axis component and the l2-axis component (flow velocities u, v) of the flow velocity of the molten steel 3, the l3-axis component of the flow velocity of the molten steel 3 is also included in the axial components of the molten steel 3. The processing for the l3-axis component of the flow velocity of the molten steel 3 can be realized by replacing the l1-axis component and the l2-axis component with the l3-axis component in the description of the l1-axis component and the l2-axis component (flow velocities u, v) of the flow velocity of the molten steel 3 in this embodiment.

[0173] Also, in this embodiment, the information processing apparatus 100 derived the mode value of the second probability density function p(x t |y 0:t ) as the estimated value of the state vector x t at time t. However, the average value of the second probability density function p(x t |y 0:t ) may be derived as the estimated value of the state vector x t at time t, or the median value of the second probability density function p(x t |y 0:t ) may be derived as the estimated value of the state vector x t at time t.

[0174] Also, in this embodiment, the visualization target region is limited to the region where the molten steel 3 exists, but it may be the region where the molten steel 3 and the solidification shell 8 exist. In this case, a numerical simulation of unsteady flow and heat transfer accompanied by solidification may be performed using the physical property values of the molten steel 3, the physical property values of the solidification shell 8, and the latent heat of solidification.

[0175] Further, if data assimilation is performed as in this embodiment to visualize the flow velocity and temperature of the molten steel 3, it is preferable because the occurrence state of abnormal phenomena such as the drift of the molten steel 3 can be known moment by moment. However, it is not always necessary to do so. When doing so, for example, the likelihood function derivation unit 114, the data assimilation unit 115, and the visualization data creation unit 116 become unnecessary. In this case, the boundary conditions of the equations (10) to (12) may be determined based on, for example, the observation data acquired by the observation data acquisition unit 113. Further, the output unit 117 outputs information indicating the flow velocities u and v and the temperature T of the molten steel 3 at each calculation grid point l at each time t derived by the physical quantity derivation unit 111b. Also, it is not necessary to derive the temperature T of the molten steel 3.

[0176] Further, in this embodiment, the case where the numerical simulation unit 101, the matrix derivation unit 102, the basis vector derivation unit 103, and the reduced simulator derivation unit 104 are executed offline is exemplified. However, the processing executed offline may be executed online at an operation time before the start of the processing executed online (the processing in the first coefficient derivation unit 111a). Also, the processing executed online may be executed offline. For example, without performing data assimilation (the processing by the first probability density function derivation unit 112, the observation data acquisition unit 113, the likelihood function derivation unit 114, and the data assimilation unit 115), the flow velocity u and the temperature T of the molten steel 3 may be derived by the numerical simulation unit 101, the matrix derivation unit 102, the basis vector derivation unit 103, the reduced simulator derivation unit 104, the first coefficient derivation unit 111a, and the physical quantity derivation unit 111b. When doing so, the first coefficient derivation unit 111a and the physical quantity derivation unit 111b may be executed offline.

[0177] 〔Second Embodiment〕 Next, a second embodiment will be described. In the first embodiment, the flow velocities u(l,t), v(l,t), and temperature T(l,t) of the molten steel 3 at each computational grid point l are used as state variables, and the temperature T and flow velocity v of the molten steel 3 at each observation position α are used as observed variables. Therefore, in the first embodiment, the flow velocities u(l,t-Δt), v(l,t-Δt), and temperature T(l,t-Δt) of the molten steel 3 at each computational grid point l at time t-Δt, and the basis vectors φ u 1(l) to φ u Mu (l), φ v 1(l) to φ v Mv (l), φ T 1(l) to φ T MT (l) are used to derive the coefficients a u 1(t-Δt) to a u Mu (t-Δt), a v 1(t-Δt) to a v Mv (t-Δt), a T 1(t-Δt) to a T MT (t-Δt) for the flow velocities u, v, and temperature T (step S404), and the coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t), and the basis vectors φ u 1(l) to φ u Mu (l), φ v 1(l) to φ v Mv (l), φ T 1(l) to φ T MTBased on (l), it is necessary to repeatedly execute the derivation of the flow velocities u(l,t), v(l,t), and temperature T(l,t) of the molten steel 3 at each computational grid point l at time t (step S405) every time data assimilation is performed for all ensemble members. The computational load of this step S405 increases as the number of computational grid points l increases. Also, it is expected that the estimation accuracy of the flow velocities u, v, and temperature T of the molten steel 3 will improve as the number of ensemble members is set larger (for example, 100 sets). Therefore, when the number of computational grid points l is large, there is a risk that real-time processing cannot be performed if the process of step S405 is repeatedly executed.

[0178] Therefore, in this embodiment, coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t) are used as state variables to form a state vector x t and data assimilation is performed by constructing an observation vector y v using, as observation variables, coefficients a^ v Mv (t), a^ T 1(t) to a^ T MT (t) derived from the temperature T and flow velocity v of the molten steel 3 at each observation position α, to derive an estimated value of the state vector, and by inverse-transforming the coefficients constituting the estimated value of the state vector, the flow velocity and temperature of the molten steel 3 are derived. Therefore, every time data assimilation is performed, the flow velocities u, v, and temperature T are converted into coefficients a t 1(t - Δt) to a u 1(t - Δt) to a u Mu (t - Δt), a v 1(t - Δt) to a v Mv (t - Δt), a T 1(t - Δt) to a T MT (t - Δt) in the above-described calculation, and coefficients a u1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (Instead of performing the above calculations to inverse-transform (t) into flow velocities u, v and temperature T for all ensemble members, for the measured values of flow velocity v and temperature T only, perform the calculation (step S1706 described later) to transform them into (t), and coefficient a v 1(t) to a^ v Mv (t), a^ T 1(t) to a^ T MT (Perform the calculation (step S1706 described later) to transform them into (t) only for the measured values of flow velocity v and temperature T. Coefficient a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (Since the calculation (step S1712 described later) to inverse-transform (t) into flow velocities u, v and temperature T may be performed only for the estimated values of the state vector, the computational load can be reduced.)

[0179] As described above, in the first embodiment, the first physical quantity is used as the state quantity and the second physical quantity is used as the observed quantity. In contrast, in this embodiment, when the first physical quantity is represented as a linear combination of basis vectors, the coefficient multiplied by the basis vector is used as the state quantity, and when the second physical quantity is represented as a linear combination of basis vectors, the coefficient multiplied by the basis vector is used as the observed quantity. The configurations and processes of this embodiment and the first embodiment mainly differ in this regard. Therefore, in the description of this embodiment, for the same parts as those in the first embodiment, the same reference numerals as those in FIGS. 1 to 13, FIGS. 18 to 19 are used, and detailed descriptions thereof are omitted.)

[0180] FIG. 14 is a diagram showing an example of a functional configuration of the information processing apparatus 1400. The hardware of the information processing apparatus 1400 is realized by, for example, the same hardware as that of the information processing apparatus 100 of the first embodiment shown in FIG. 1.

[0181] In the information processing apparatus 100 of the first embodiment shown in FIG. 1, the numerical simulation unit 101, the matrix derivation unit 102, the basis vector derivation unit 103, and the reduced simulator derivation unit 104 are executed offline. On the other hand, the information processing apparatus 1400 of the present embodiment has, in addition to the numerical simulation unit 101, the matrix derivation unit 102, the basis vector derivation unit 103, and the reduced simulator derivation unit 104, a second coefficient derivation unit 1405 and an observation matrix derivation unit 1406, and the numerical simulation unit 101, the matrix derivation unit 102, the basis vector derivation unit 103, the reduced simulator derivation unit 104, the second coefficient derivation unit 1405, and the observation matrix derivation unit 1406 are executed offline.

[0182] In the information processing apparatus 100 of the first embodiment, the state derivation unit 111, the first probability density function derivation unit 112, the observation data acquisition unit 113, the likelihood function derivation unit 114, the data assimilation unit 115, the visualization data creation unit 116, and the output unit 117 are executed online. In contrast, the information processing apparatus 1400 of the present embodiment has an estimated physical quantity derivation unit 1418 in addition to the state derivation unit 1411, the first probability density function derivation unit 1412, the observation data acquisition unit 1413, the likelihood function derivation unit 1414, the data assimilation unit 1415, the visualization data creation unit 1416, and the output unit 1417, and the state derivation unit 1411, the first probability density function derivation unit 1412, the observation data acquisition unit 1413, the likelihood function derivation unit 1414, the data assimilation unit 1415, the estimated physical quantity derivation unit 1418, the visualization data creation unit 1416, and the output unit 1417 are executed online. Further, some of the functions of the state derivation unit 1411, the first probability density function derivation unit 1412, the observation data acquisition unit 1413, the likelihood function derivation unit 1414, the data assimilation unit 1415, the visualization data creation unit 1416, and the output unit 1417 are different from those of the state derivation unit 111, the first probability density function derivation unit 112, the observation data acquisition unit 113, the likelihood function derivation unit 114, the data assimilation unit 115, the visualization data creation unit 116, and the output unit 117 of the first embodiment.

[0183] (Second coefficient derivation unit 1405) As described in the first embodiment, the coefficient a shown in equation (4) u j (t) is derived as the product of the singular value of mode number j obtained by performing singular value decomposition on the snapshot matrix MU of the flow velocity u and the right singular vector with respect to the singular value of mode number j for the flow velocity u, v(l,t), and temperature T(l,t) at each calculation grid point l derived by the numerical simulation unit 101 at each time t with a time interval of step width Δt1. The coefficient a shown in equation (5) v j(t) is derived as the product of the singular value of mode number j and the right singular vector corresponding to the singular value of mode number j obtained by performing singular value decomposition on the snapshot matrix MV of the flow velocity v with respect to the flow velocities u(l,t), v(l,t), and temperature T(l,t) at each calculation grid point l derived at each time t with a time interval of step width Δt1 by the numerical simulation unit 101. The coefficient a shown in equation (6) T j (t) is derived as the product of the singular value of mode number j and the right singular vector corresponding to the singular value of mode number j obtained by performing singular value decomposition on the snapshot matrix MT of the temperature T with respect to the flow velocities u(l,t), v(l,t), and temperature T(l,t) at each calculation grid point l derived at each time t with a time interval of step width Δt1 by the numerical simulation unit 101.

[0184] The second coefficient derivation unit 1405 uses the product of the singular value of mode number j and the right singular vector corresponding to the singular value of mode number j obtained by performing singular value decomposition on the snapshot matrix MU of the flow velocity u derived by the matrix derivation unit 102 as the coefficient a u j (t) for derivation. Also, the second coefficient derivation unit 1405 uses the product of the singular value of mode number j and the right singular vector corresponding to the singular value of mode number j obtained by performing singular value decomposition on the snapshot matrix MV of the flow velocity v derived by the matrix derivation unit 102 as the coefficient a v j (t) for derivation. Also, the second coefficient derivation unit 1405 uses the product of the singular value of mode number j and the right singular vector corresponding to the singular value of mode number j obtained by performing singular value decomposition on the snapshot matrix MT of the temperature T derived by the matrix derivation unit 102 as the coefficient a T j (t) for derivation.

[0185] (Observation matrix derivation unit 1406) The observation matrix derivation unit 1406 derives the observation matrix H t As shown in equation (13), the observation matrix H t is the observation vector yt (Observed quantity) and the state vector x t It is a matrix showing the relationship with (state quantity).

[0186] In the first embodiment, a vector composed of all of the flow velocities u, v, and temperature T of the molten steel 3 at the computational grid point l is referred to as a state vector as needed, and the flow velocities u, v, and temperature T of the molten steel 3 derived by the physical quantity derivation unit 111b are referred to as numerical analysis data as needed. On the other hand, in the present embodiment, when the flow velocities u, v, and temperature T of the molten steel 3 are represented as a linear combination of basis vectors, the coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t) are used as state quantities, and a vector composed of all of these coefficients is referred to as a state vector as needed. Further, the coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t) are referred to as numerical analysis data as needed.

[0187] Thus, instead of the first physical quantities themselves such as the flow velocities u, v, and temperature T of the molten steel 3, the coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t) are used to construct the state vector x t to construct the observation vector yt is not the measured value of the second physical quantity (the temperature T of the molten steel 3 at each position of the visualization target surface derived from the temperatures measured by the thermometers F and L and the flow velocity v derived from the molten metal level measured by the molten metal level gauge 9), but the measured value of the second physical quantity is the flow velocity v and the temperature T derived from the base vectors φ v j (l), φ T j the coefficients a^ multiplied by the base vectors when expressed as a linear combination of (l) v j (t), a^ T j Let it be a vector composed of all of (t).

[0188] When the present inventors use the vector composed of coefficients as the state vector and the observation vector as described above, the coefficients a^ v j (t), a^ T j the coefficients a for different physical quantities of (t) u j (t), a v j (t), a T j (t) have a higher dependence than when using the vector composed of the physical quantities themselves as the state vector and the observation vector. Therefore, the present inventors have found that even when the coefficient of the α-th component of the observation vector y t and the coefficient of the β-th component of the state vector x t are coefficients for different physical quantities (flow velocities u, v and temperature T), it is preferable to determine the (α,β) component of the observation matrix H t according to the correlation of those coefficients. Here, the physical quantities being the same means that they are the same including the axis components. Therefore, the flow velocities u and v are different physical quantities.

[0189] Therefore, the observation matrix derivation unit 1406 singular value decomposes the snapshot matrix MU of the flow velocity u, the snapshot matrix MV of the flow velocity v, and the snapshot matrix MT of the temperature T, and the coefficients a derived by the second coefficient derivation unit 1405u j (t), a v j (t), a T j The value indicating the correlation with (t) is derived as a component of the observation matrix H t The observation matrix H t is represented by the following equation (22).

[0190] [Number]

[0191] Here, H uv is the observation vector y t The coefficient a^ v j (t) and the coefficient a t that is a component of the state vector x u 1(t) to a u Mu (t). The observation matrix H t is a submatrix that stores the components indicating the relationship with v M rows and M u column matrix. H vv is the observation vector y t The coefficient a^ v j (t) and the coefficient a t that is a component of the state vector x v 1(t) to a v Mv (t). The observation matrix H t is a submatrix that stores the components indicating the relationship with v M rows and M v column matrix. H Tv is the observation vector y t The coefficient a^ v j (t) and the coefficient a t of the β-th component of the state vector x T 1(t) to a T MT (t). It is a submatrix that stores the components indicating the relationship with v M rows and M T column matrix. H uTis the coefficient a^ t that is a component of the observation vector y T j (t), and the coefficient a t that is a component of the state vector x u 1(t) to a u Mu (t). The observation matrix H stores the components indicating the relationship between them t and is a submatrix of T an M u by M vT matrix. H t is the coefficient a^ T j (t) that is a component of the observation vector y t and the coefficient a u 1(t) to a u Mu (t) that is a component of the state vector x t and is a submatrix of T an M v by M TT matrix. H t is the coefficient a^ T j (t) that is a component of the observation vector y t and the coefficient a T 1(t) to a T MT (t) that is a component of the state vector x t and is a submatrix of T an M T by M matrix.

[0192] Specifically, the submatrices H uv , H vv , H Tv , H uT , H vT , H TT are represented by the following equations (23) to (28), respectively.

[0193]

Equation

[0194] FIG. 15A shows the observation matrix H tSubmatrix H uT is a diagram showing a specific example. FIG. 15B shows the observation matrix H t Submatrix H vT is a diagram showing a specific example. FIG. 15C shows the observation matrix H t Submatrix H TT is a diagram showing a specific example. Here, due to the notation space, for temperature T, eight submatrices H are shown in order from the ones with larger eigenvalues (smaller mode numbers j of the eigen - orthogonal decomposition), and for flow velocities u and v, six submatrices H are shown in order from the ones with larger eigenvalues (smaller mode numbers j of the eigen - orthogonal decomposition). uT H vT H TT are shown only.

[0195] In equations (10) to (12) that constitute the reduced simulator, the physical variables are not made dimensionless. Therefore, to correct the magnitude of the numerical values of the coefficient a T j (t) related to temperature T and the coefficients a u j (t), a v j (t) related to flow velocities u and v, the Prandtl number Pt is used, and the observation matrix H t of equation (22) becomes as follows in equation (29) below.

[0196]

Equation

[0197] In this embodiment, the case of using equation (29) as the observation matrix H t is exemplified. Note that in equations (10) to (12) that constitute the reduced simulator, when the physical variables are made dimensionless, equation (22) may be used as the observation matrix H t .

[0198] (State derivation unit 1411) The state derivation unit 1411 includes a state vector x tDerive it. The state derivation unit 111 of the first embodiment includes a first coefficient derivation unit 111a and a physical quantity derivation unit 111b, and derives a vector composed of all of the flow velocities u, v, and temperature T of the molten steel 3 at the calculation grid point l as the state vector x t In contrast, in the present embodiment, the state vector x t is a vector composed of all of the coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t), and is derived by the first coefficient derivation unit 1411a. Therefore, the state derivation unit 1411 of the present embodiment has the first coefficient derivation unit 1411a and does not have the physical quantity derivation unit 111b.

[0199] The first coefficient derivation unit 1411a executes the same processing as the first coefficient derivation unit 111a of the first embodiment. However, in the present embodiment, the first coefficient derivation unit 1411a uses a preset value as an initial value for at least one of the coefficients a u 1(t0) to a u Mu (t0), a v 1(t0) to a v Mv (t0), a T 1(t0) to a T MT (t0) and at least one of the boundary condition parameters from time t0 to time t0 + Δt for a plurality of cases in which at least any one is different. For times t after the second time t, based on the results derived by the data assimilation unit 1415 described later, the coefficients a u 1(t0) to a u Mu (t0), a v 1(t0) to a v Mv (t0), a T 1(t0) to a T MT(t0), and derive the parameters of the boundary conditions between time t and time t + Δt. In addition, as described above, in this embodiment, the flow velocities u, v, and temperature T, which are examples of the first physical quantity, are coefficients a u 1(t) ~ a u Mu (t), a v 1(t) ~ a v Mv (t), a T 1(t) ~ a T MT (t) is not calculated.

[0200] (First Probability Density Function Derivation Unit 1412) The first probability density function derivation unit 1412, similar to the first probability density function derivation unit 112 of the first embodiment, is the observation data (information) y up to time t - Δt acquired by the observation data acquisition unit 1413 described later t0:t-Δt Based on this, the conditional probability density function p(x t |y t |y t0:t-Δt ) of the state vector x at time t is derived. The first probability density function derivation unit 1412 is a plurality of numerical analysis data {x t|t-Δt (k)} k at time t derived by the first coefficient derivation unit 1411a. A frequency distribution using the relative frequency (value obtained by dividing the frequency by the total number of data) is derived. This frequency distribution is the first probability density function p(x t |y t |y t0:t-Δt ) of the state vector x at time t. In the first embodiment, the flow velocities u, v, and temperature T of the molten steel 3 derived by the physical quantity derivation unit 111b are numerical analysis data. In contrast, in this embodiment, the coefficients a u 1(t) ~ a u Mu (t), a v 1(t) ~ a v Mv (t), a T 1(t) ~ a T MT(t) becomes numerical analysis data. Thus, the first probability density function derivation unit 1412 of the present embodiment and the first probability density function derivation unit 112 of the first embodiment only differ in the content of the numerical analysis data, and their processing contents are the same.

[0201] (Observation data acquisition unit 1413) The observation data acquisition unit 1413 acquires observation data at a predetermined period based on time-series data (measurement values) measured in the continuous casting facility. In the first embodiment, the observation data y t (0) is a vector composed of all of the measurement values of the second physical quantity (for example, the temperature T at each position of the molten steel 3 on the visualization target surface derived from the temperatures measured by the thermometers F and L, and the flow velocity v derived from the molten metal surface level measured by the molten metal surface level meter 9). In contrast, in the present embodiment, the observation data y t (0) is the coefficient (for example, the flow velocity v, the temperature T as the flow velocity v, the temperature T multiplied by the basis vectors φ v j (l), φ T j when expressed as a linear combination of (l), the coefficient a^ v j (t), a^ T j (t)) of all the basis vectors multiplied by the basis vectors when expressed as a linear combination.

[0202] Therefore, similar to the observation data acquisition unit 113 in the first embodiment, the observation data acquisition unit 1413 inputs measured values including the temperature measured by the thermometers F and L (the temperature inside the mold 4) and the molten metal surface level measured by the molten metal surface level gauge 9. Then, similar to the observation data acquisition unit 113 in the first embodiment, assuming that the heat transfer state in the long side of the mold 4 is one-dimensional steady heat conduction in the depth direction (l3-axis direction), based on the temperature of the cooling water of the mold 4 and the thermal conductivity of the mold 4, the observation data acquisition unit 1413 extrapolates these 12 temperatures and derives the temperature of the molten steel 3 at each position on the visualization target surface. Also, similar to the observation data acquisition unit 113 in the first embodiment, the observation data acquisition unit 1413 derives the flow velocity in the casting direction (l2-axis direction) at each position corresponding to the molten metal surface of the molten steel 3 on the visualization target surface from the change per unit time of a plurality of molten metal surface levels measured by the molten metal surface level gauge 9. In this embodiment as well, similar to the first embodiment, the derivation positions of the temperature of the molten steel 3 (each position on the visualization target surface) and the derivation positions of the flow velocity v (each position corresponding to the molten metal surface of the molten steel 3 on the visualization target surface) derived as described above are set as the observation positions.

[0203] Then, within the range of the visualization target surface S, the observation data acquisition unit 1413 individually performs, for each mode number j, the operation of taking the inner product of the base vector φ T j (l) of the temperature T derived offline by the base vector derivation unit 103 and the temperature T(l,t) at the observation position, so as to represent the temperature T(l,t) at the observation position as a linear combination using the base vector φ T j (l) of the temperature T derived by the base vector derivation unit 103, and derives the coefficient a^ T j (t) multiplied by the base vector φ T j (t).

[0204] Also, within the range of the molten metal free surface C, the observation data acquisition unit 1413 performs, for each mode number j, the operation of taking the inner product of the base vector φ v jBy individually taking the inner product of (l) and the flow velocity v(l,t) at the observation position for each mode number j, the flow velocity v(l,t) at the observation position is expressed as a linear combination using the basis vectors φ v j (l) of the flow velocity v derived by the basis vector derivation unit 103, and the coefficient a^ v j (t) multiplied by φ v j (l) is derived. The molten metal free surface C is the boundary region corresponding to the surface of the molten steel 3 among the boundary regions of the visualization target surface S.

[0205] Specifically, the observation data acquisition unit 1413 calculates the following equations (30) and (31) to derive the coefficients a^ T j (t) and a^ v j (t).

[0206]

Equation

[0207] Note that the coefficients a^ T j (t) and a^ v j (t) respectively represent the symbols on the left side of equation (30) and the symbol on the left side of equation (31) (a symbol with a ^ on a, T and v attached to the upper right of a, and (t) attached to the right of the symbol with j attached to the lower right of T and v). In this embodiment, the vector composed of all of the coefficients a^ T j (t) and a^ v j (t) derived by the observation data acquisition unit 1413 as described above becomes the observation data y at time t. As shown in equations (30) and (31), the coefficients a^ t (0) (t) and a^ T j (t) and a^ v jSince the integration range for deriving (t) is the visualization target surface S and the melt free surface C, the increase in the computational load of the observation data acquisition unit 1413 with respect to the observation data acquisition unit 113 does not become excessive.

[0208] In addition, in the present embodiment, the flow velocity v and the temperature T, which are examples of the second physical quantity, need to be included in the flow velocity u, v, and the temperature T, which are examples of the first physical quantity. This is because, unlike the first embodiment, in the present embodiment, it is necessary to use the basis vectors of the second physical quantity in the observation data acquisition unit 1413. On the other hand, in the first embodiment, since the second physical quantity is used as the observed quantity, even if the first physical quantity does not include the second physical quantity, the observation equation shown in Equation (13) can be constructed. Therefore, in the first embodiment, it is not always necessary for the first physical quantity to include the second physical quantity.

[0209] (Likelihood function derivation unit 1414) Similar to the likelihood function derivation unit 114 of the first embodiment, the likelihood function derivation unit 1414 evaluates the observation noise of the observation data y at time t acquired by the observation data acquisition unit 1413 from the past observation data, and thereby determines the likelihood function L(x t (0) |y t ) of the state vector x t when the observation vector y t is obtained. As described in the first embodiment, the likelihood function L(x t |y t ) is regarded as a function of the state vector x t for the conditional probability density function p(y t |x t ), and is determined as a function of the state vector x t by substituting the observation data y t into the observation vector y t . t t t (0) t

[0210] In the first embodiment, the observation vector y t ) is a vector composed of all of the temperature T of the molten steel 3 and the flow velocities u and v in the casting direction, which are acquired at the observation position, and the observation data y t which is the data of the observation vector y t (0) is acquired by the observation data acquisition unit 113. On the other hand, in the present embodiment, the observation vector y t is a vector composed of all of the coefficients a^ T j (t), a^ v j (t), and the observation data y t which is the data of the observation vector y t (0) is acquired by the observation data acquisition unit 1413. Thus, the likelihood function derivation unit 1414 of the present embodiment and the likelihood function derivation unit 114 of the first embodiment differ only in the content of the observation vector y t and the observation data y t (0) , and their processing contents are the same.

[0211] (Data assimilation unit 1415) The data assimilation unit 1415, similar to the data assimilation unit 115 of the first embodiment, uses the first probability density function p(x t |y t ) of the state vector x at time t derived by the first probability density function derivation unit 1412, and the likelihood function L(x 0:t-Δt ) of the state vector x at time t when the observation data at time t derived by the likelihood function derivation unit 1414 is obtained. By providing these to the algorithm of a filter that performs data assimilation by Bayesian statistical modeling based on Bayes' theorem, the conditional probability density function p(x t |y t ) of the state vector x at time t when the observation data up to time t is obtained t ) t |y t |y t0:t) is derived as the second probability density function. The data assimilation unit 1415 in this embodiment and the data assimilation unit 115 in the first embodiment only differ in the content of the state vector x t , the first probability density function p(x t |y 0:t-Δt ) and the likelihood function L(x t |y t ), and their processing contents are the same.

[0212] In addition, the data assimilation unit 1415 outputs the second probability density function p(x t |y t ) of the state vector x t to the first coefficient derivation unit 1411a. The first coefficient derivation unit 1411a derives the coefficients a t 1(t0)~a t (t), a t 1(t)~a u u (t), a Mu (t), a v 1(t)~a v Mv (t), a T 1(t)~a T MT (t), and the boundary condition parameters from time t to time t+Δt based on the ensemble members of the second probability density function p(x u |y u Mu ) of the state vector x v |y v Mv ). That is, similar to the first embodiment, the first coefficient derivation unit 1411a, based on the flow velocities u(l,t), v(l,t) and temperature T(l,t) of the molten steel 3 at each computational grid point l at time t, and the basis vectors φ T 1(l)~φ T MT (l), φ u 1(l)~φ u Mu (l) of the flow velocities u, v, and the basis vectors φ v 1(l)~φ v Mv (l) of the temperature T, derives the coefficients a T1(t) to a T MT There is no need to derive (t), and the coefficient a at time t u 1(t0) to a u Mu (t0), a v 1(t0) to a v Mv (t0), a T 1(t0) to a T MT Derive (t0) from the ensemble members of the second probability density function p(x t of the state vector x t |y t ).

[0213] (Estimated physical quantity derivation unit 1418) The estimated physical quantity derivation unit 1418 is the coefficient a that constitutes the estimated value of the state vector at time t, which is derived as the estimated value of the state vector by the data assimilation unit 1415 u 1(t) to a u Mu (t) and the basis vector φ of the flow velocity u derived by the basis vector derivation unit 103 u 1(l) to φ u Mu (l) are substituted into equation (4) to derive the flow velocity u of the molten steel 3 at each calculation grid point l at each time t from time t0 to t e Also, the estimated physical quantity derivation unit 1418 is the coefficient a that constitutes the estimated value of the state vector at time t, which is derived as the estimated value of the state vector by the data assimilation unit 1415 v 1(t) to a v Mv (t) and the basis vector φ of the flow velocity v derived by the basis vector derivation unit 103 v 1(l) to φ v Mv (l) are substituted into equation (5) to derive the flow velocity v of the molten steel 3 at each calculation grid point l at each time t from time t0 to t e Also, the estimated physical quantity derivation unit 1418 is the coefficient a that constitutes the estimated value of the state vector at time t, which is derived as the estimated value of the state vector by the data assimilation unit 1415 T 1(t) to a TMT (t) and the basis vectors φ of the temperature T derived by the basis vector derivation unit 103 T 1(l) to φ T MT By substituting (l) and (6) into equation (6), the temperature T of the molten steel 3 at each calculation grid point l at each time t from time t0 to t e is derived. The time t e is a preset time and is the end time of the creation period of the visualization data. Thus, in this embodiment, the estimated physical quantity derivation unit 1418 derives the flow velocities u, v, and temperature T of the molten steel 3 at each calculation grid point l at each time t after the data assimilation by the data assimilation unit 115 is completed.

[0214] (Visualization data creation unit 1416) The visualization data creation unit 1416 creates display data of at least one of the flow velocities u, v, and temperature T of the molten steel 3 at each position (each calculation grid point l) at time t derived by the estimated physical quantity derivation unit 1418. In the visualization data creation unit 116 of the first embodiment, the estimated value of the state vector at time t is obtained from the data assimilation unit 115 as the temperature T and flow velocities u, v of the molten steel 3 at each position. In contrast, in the visualization data creation unit 1416 of this embodiment, the flow velocities u, v, and temperature T of the molten steel 3 at each position at each time t derived by the estimated physical quantity derivation unit 1418 are obtained. The visualization data creation unit 116 of the first embodiment and the visualization data creation unit 1416 of this embodiment differ only in this respect, and their processing contents are the same.

[0215] (Output unit 1417) The output unit 1417 displays the display data created by the visualization data creation unit 1416 on a computer display, similar to the output unit 117 of the first embodiment. Further, the output unit 1417 can output, instead of or in addition to such display data, data on the flow velocity and temperature of the molten steel 3 at each position (each computational grid point l) at each time derived by the estimated physical quantity derivation unit 1418. In the output unit 117 of the first embodiment, the estimated value of the state vector at time t is acquired from the data assimilation unit 115 as the temperature T, flow velocities u, and v of the molten steel 3 at each position. In contrast, in the output unit 117 of the present embodiment, the flow velocities u, v, and temperature T of the molten steel 3 at each position at each time t derived by the estimated physical quantity derivation unit 1418 are acquired. The output unit 117 of the first embodiment and the output unit 1417 of the present embodiment differ only in this regard, and the processing contents thereof are the same.

[0216] (Operation flowchart) Next, an example of the processing in the offline state of the information processing apparatus 1400 will be described with reference to the flowchart of FIG. 16. The processing of steps S301 to S303 and S304 in FIG. 16 is the same as the processing of steps S301 to S303 and S304 in the flowchart of FIG. 3.

[0217] That is, in step S301, the flow velocities u(l, t), v(l, t) and the temperature T(l, t) are derived. Next, in step S302, a snapshot matrix MU of the flow velocity u, a snapshot matrix MV of the flow velocity v, and a snapshot matrix MT of the temperature T are derived. Next, in step S303, the basis vector φ u j (l) of the flow velocity u, the basis vector φ v j (l) of the flow velocity v, the basis vector φ T j (l) of the temperature T, and the number M of the basis vectors φ u j (l), φ v j (l), φ T j (l) of the flow velocities u, v, and temperature T u 、Mv and M T are derived.

[0218] When the process of step S303 ends, the process proceeds to step S1601. In step S1601, the second coefficient derivation unit 1405 calculates the product of the singular value of mode number j and the right singular vector corresponding to the singular value of mode number j, which are obtained by performing singular value decomposition on the snapshot matrices MU, MV, and MT of the flow velocities u, v, and temperature T derived in step S302, as the coefficients a u j (t), a v j (t), a T j (t). Note that step S1601 may be executed before step S303.

[0219] When the process of step S1601 ends, the process proceeds to the process of step S304, and a reduced simulator is derived. As described in the first embodiment, in the reduced simulator, in equations (10) to (12), the constants A i u , B uu ij , B uv ij , C uu ijk , C uv ijk , A i v , B vu ij , B vv ij , C vu ijk , C vv ijk , A i T , B TT ij , B Tu ij , B Tv ij , C Tu ijk , C Tv ijk , M u , M v , M Tis set. When the process of step S304 ends, the process proceeds to step S1602.

[0220] In step S1602, the observation matrix derivation unit 1406 derives, as components of the observation matrix H, values indicating the correlations for the coefficients a u j (t), a v j (t), a T j (t) (see equations (22) to (28)), and derives the observation matrix H t . When the process of step S1602 ends, the process according to the flowchart of FIG. 16 ends. Note that step S1602 may be executed before step S304. In this case, when the process of step S304 ends, the process according to the flowchart of FIG. 16 ends. t

[0221] Next, an example of the online process of the information processing apparatus 1400 will be described with reference to the flowchart of FIG. 17. First, in step S1701, the first coefficient derivation unit 1411a sets the time t to the initial value (t0). Next, in step S1702, the first coefficient derivation unit 1411a sets a plurality of sets of initial values as the initial values of the sets of the coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t) and the parameters of the boundary conditions between time t and time t + Δt.

[0222] Next, in step S1703, the first coefficient derivation unit 1411a updates the time t to time t + Δt, and the process proceeds to step S1704. Next, in step S1704, the first coefficient derivation unit 1411a derives the solutions (a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t)) of the differential equations of equations (10) to (12) at time t. The coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t) are derived for each set of the flow velocities u(l, t - Δt), v(l, t - Δt) and the temperature T(l, t - Δt) of the molten steel 3 at each computational grid point l at time t - Δt and the parameters of the boundary conditions between time t - Δt and time t.

[0223] Next, in step S1705, the first probability density function derivation unit 1412 derives the first probability density function p(x t (the coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t)) of the state vector x at time t given y t |y t-Δt ).

[0224] Next, in step S1706, the observation data acquisition unit 1413 acquires measurement values including the temperatures measured by the thermometers F and L and the molten metal level measured by the molten metal level gauge 9, and based on the acquired measurement values, derives the temperature of the molten steel 3 and the flow velocity in the casting direction (l2-axis direction) at each observation position α. Then, within the range of the visualization target surface S, the observation data acquisition unit 1413 uses the basis vectors φ of the temperature T derived in step S303 of FIG. 16T j By taking the inner product of (l) and the temperature T(l, t) at the observation position α for each mode number j individually, the coefficient a^ T j (t) is derived (see equation (30)). Further, the observation data acquisition unit 1413, within the range of the molten metal free surface C, the base vector φ of the flow velocity v derived in step S303 of FIG. 16 v j By taking the inner product of (l) and the flow velocity v(l, t) at the observation position α for each mode number j individually, the coefficient a^ v j (t) is derived (see equation (31)). Then, the observation data acquisition unit 1413 uses the coefficients a^ T j (t), a^ v j (t) to form a vector which is derived as the observation data y t (0) at time t.

[0225] Next, in step S1707, the likelihood function derivation unit 1414 derives the likelihood function L(x t (0) |y t of the state vector x t at time t based on the observation data y t ). Next, in step S1708, the data assimilation unit 1415 inputs the first probability density function p(x t |y t ) of the state vector x 0:t-Δt at time t and the likelihood function L(x t |y t ) of the state vector x t at time t into the algorithm of a filter that performs data assimilation by Bayesian statistical modeling, and thereby derives the second probability density function p(x t |y t ) of the state vector x t0:t at time t.

[0226] Next, in step S1709, the data assimilation unit 1415 determines the state vector x at time t t of the second probability density function p(x t |y t ) and derives the mode value as the estimated value of the state vector at time t. In step S410 of FIG. 4, the flow velocities u, v, and temperature T of the molten steel 3 are derived as the estimated values of the state vector. In contrast, in step S1709, the coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (t) are derived as the estimated values of the state vector.

[0227] Next, in step S1710, the data assimilation unit 1415 determines whether time t has reached a preset time t e . If the result of this determination is that time t has not reached the preset time t e , the process proceeds to step S1711.

[0228] In step S1711, the first coefficient derivation unit 1411a determines the coefficients a at time t that are used when deriving the solutions (a u 1(t + Δt) to a u Mu (t + Δt), a v 1(t + Δt) to a v Mv (t + Δt), a T 1(t + Δt) to a T MT (t + Δt)) of the differential equations of equations (10) to (12). The coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MTA pair of (t) and the parameters of the boundary conditions from time t to time t+Δt is derived by the data assimilation unit 1415 at step S1708, and is the state vector x at time t t from the second probability density function p(x t |y t0:t ). A plurality of them are derived based on

[0229] Then, the process proceeds to step S1703, and the processes of steps S1703 to S1711 are repeatedly executed until the time t reaches a preset time t e . Note that in step S1704, a pair of the coefficient derived in step S1711 immediately before step S1704 and the parameters of the boundary conditions is used. As described above, in step S1710, when the time t reaches a preset time t e , the process proceeds to step S1712. In step S1712, the estimated physical quantity derivation unit 1418 determines the coefficients a for the flow velocities u, v, and temperature T u 1(t)~a u Mu (t), a v 1(t)~a v Mv (t), a T 1(t)~a T MT (t), and the basis vectors φ of the flow velocities u, v, and temperature T derived in step S303 of FIG. 16 u 1(l)~φ u Mu , φ v 1(l)~φ v Mv , φ T 1(l)~φ T MT are respectively substituted into equations (4), (5), and (6) to derive the flow velocities u(l,t), v(l,t), and temperature T(l,t) of the molten steel 3 at each calculation grid point l at each time t from time t0 to t e .

[0230] Further, in step S410 of FIG. 4, the flow velocities u(l, t) and v(l, t) and the temperature T(l, t) of the molten steel 3 at each computational grid point l at time t are derived for each set of the flow velocities u(l, t−Δt) and v(l, t−Δt) and the temperature T(l, t−Δt) of the molten steel 3 at each computational grid point l at time t−Δt and the parameters of the boundary conditions between time t−Δt and time t. In contrast, in step S1712, the flow velocities u(l, t) and v(l, t) and the temperature T(l, t) of the molten steel 3 at each computational grid point l at each time t from time t0 to t e are derived.

[0231] Next, in step S1713, the visualization data creation unit 1416 creates display data of the flow velocity and temperature of the molten steel 3 at each position at each time t from time t0 to t e derived in step S1713. In step S411 of FIG. 4, display data of the flow velocity and temperature of the molten steel 3 at each position at one time is created. In contrast, in step S1714, the display data of the flow velocity and temperature of the molten steel 3 at each position at each time t from time t0 to t e is created in a batch.

[0232] Next, in step S1714, the output unit 1417 outputs the visualization data (data of the flow velocity and temperature of the molten steel 3 at each time and each position derived in step S1713 and the display data derived in step S1714). When the process of step S1714 ends, the process according to the flowchart of FIG. 17 ends.

[0233] (Example 2) Next, Example 2 will be described, but the present invention is not limited to the following Example 2. In Example 2, the second embodiment is compared with the first embodiment.

[0234] In Invention Examples 1 and 2 of Example 1, the number of computational grid points l was 2,600 points and 3,600 points, respectively. In contrast, in Example 2, the number of computational grid points l was set to 26,000 points so that a more accurate numerical simulation would be executed.

[0235] Also, in the actual system, it takes about 0.5 seconds to acquire the measurement data at all observation positions (the temperature T of the molten steel 3 at each position on the visualization target surface, and the flow velocity v at each position corresponding to the surface level of the molten steel 3 on the visualization target surface) and transfer it to the information processing devices 100 and 1300, which are computer systems for data assimilation. From this, after one step of data assimilation is completed, the time until the next measurement data is acquired is set to 0.5 seconds. In the flowchart of FIG. 4 of the first embodiment, this time is the time from the end of the process in step S409 to the end of the process in the next step S407. In the flowchart of FIG. 17 of the second embodiment, this time is the time from the end of the process in step S1709 to the acquisition of the measured values (the temperature measured by the thermometers F and L and the surface level measured by the surface level meter 9) in the next step S1706. Also, in Example 2, the number M of the basis vectors φ u j (l), φ v j (l), φ T j (l) is 8, 8, and 10 respectively. u 、M v 、M T was set to 8, 8, and 10 respectively.

[0236] The calculation conditions of this other Example 2 are the same as those of Invention Example 1 of Example 1. Using the same measurement data as the measurement data used in Example 1, visualization data was created in each of the methods described in the first embodiment and the second embodiment. Hereinafter, the method described in the second embodiment is referred to as Invention Example 3, and the method described in the first embodiment is referred to as Reference Example 1.

[0237] As a result, in Invention Example 3, the calculation accuracy of the flow velocity and temperature of the molten steel 3 at each position became equivalent to those of Invention Example 1 and Invention Example 2 of Example 1. Next, the calculation times of Invention Example 3 and Reference Example 1 are compared. Table 3 shows the calculation time of Invention Example 3. Table 4 shows the calculation time of Reference Example 1. The calculation time of Invention Example 3 is the time required to execute the processing of the flowchart in FIG. 17, and the calculation time of Reference Example 1 is the time required to execute the processing of the flowchart in FIG. 4. The units of the numerical values shown in Tables 3 and 4 are seconds.

[0238]

Table 3

[0239]

Table 4

[0240] In Tables 3 and 4, the assumed time has the same meaning as the assumed time shown in Table 1 of Example 1, and is the time assumed in the simulation. The time required to execute the simulation for the time shown in the column of the assumed time in actual operation is the time described in the column of the total calculation time of Invention Example 3 in Table 3 and the column of the total calculation time of Reference Example 1 in Table 4.

[0241] In Tables 3 and 4, the measurement value collection is the time required to acquire measurement values (the temperature T of the molten steel 3 at each position of the visualization target surface, the flow velocity v at each position corresponding to the molten steel surface of the molten steel 3 on the visualization target surface) during the period from the start to the end of the processing of the flowcharts in FIGS. 17 and 4. As described above, in this embodiment, after one step of data assimilation is completed, the time until the next measurement data is acquired is set to 0.5 seconds. Therefore, the value obtained by multiplying this time (=0.5 seconds) by the assumed time becomes the value in the column of measurement value collection.

[0242] PODSim is, online, the coefficient a u 1(t)~a u Mu (t), a v 1(t)~a v Mv (t), a T 1(t)~aT MT It is the time required to derive (t) (the processing time in the first coefficient derivation units 111a and 1411a), and corresponds to the times shown in the columns of Invention Example 1 and Invention Example 2 shown in Table 1 of Example 1. The others are other processing times online.

[0243] When comparing the times shown in the PODSim columns of Tables 3 and 4 with the times shown in the columns of Invention Example 1 and Invention Example 2 of Table 1, regardless of the number of computational grid points l (2600 points in Invention Example 1, 3600 points in Invention Example 2, 26000 points in Invention Example 3 and Reference Example 1), the time required for the operation of the reduced simulator (the processing time in the coefficient derivation units 111a and 1411a) is substantially the same.

[0244] The times shown in the total calculation time columns of Invention Example 3 in Table 3 are all below the assumed time, and it can be seen that in Invention Example 3 (the method of the second embodiment), visualization data can be created and output within the assumed time. On the other hand, the times shown in the total calculation time columns of Reference Example 1 in Table 4 are all above the assumed time, and it can be seen that in Reference Example 1 (the method of the first embodiment), when the number of computational grid points l increases, there is a possibility that visualization data cannot be created and output within the assumed time.

[0245] The results in Table 4 show that in Reference Example 1, a part of the processing in step S404 (based on the flow velocities u(l,t - Δt), v(l,t - Δt), and temperature T(l,t - Δt) of the molten steel 3 at each computational grid point l at time t, the coefficients a u 1(t - Δt) to a u Mu (t - Δt), a v 1(t - Δt) to a v Mv (t - Δt), a T 1(t - Δt) to a T MT (deriving (t - Δt)) and the processing in step S405 (the coefficients a u 1(t) to a u Mu (t), a v1(t) to a v Mv (t), a T 1(t) to a T MT (Based on (t), to derive the flow velocities u(l, t), v(l, t), and temperature T(l, t) of the molten steel 3 at each computational grid point l at time t) is repeatedly executed each time data assimilation is performed in step S409 for all ensemble members (each time the time t is updated). To cope with the fact that the computational time becomes long when the number of computational grid points l increases.

[0246] The results in Table 3 show that in Invention Example 3, a part of the process in step S1706 (Based on the temperature T and flow velocity v of the molten steel 3 at each observation position α at time t, the coefficient a^ for the temperature T and flow velocity v v 1(t) to a^ v Mv (t), a^ T 1(t) to a^ T MT (to derive (t)) is executed only for the measured values of the flow velocity v and temperature T, and the process in step S1713 (the coefficient a for the flow velocities u, v, and temperature T u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t), a T 1(t) to a T MT (Based on (t), to derive the flow velocities u(l, t), v(l, t), and temperature T(l, t) of the molten steel 3 at each computational grid point l at time t) is executed only for the estimated value of the state vector after all the data assimilation in step S1709 has ended (after time t e thereafter). Therefore, even when the number of computational grid points l increases, the computational time becomes shorter compared to Reference Example 1.

[0247] (Summary) As described above, in this embodiment, the information processing apparatus 1400 has the coefficient a u 1(t) to a u Mu (t), a v 1(t) to a vMv (t), a T 1(t) to a T MT A state vector composed of all of (t) is derived online as numerical analysis data {x t|t-Δt (k)} k at time t. Also, the information processing device 1400 derives the coefficient a^ T j (t), a^ v j An observation vector composed of all of (t) is derived online as observation data y t (0) at time t. The information processing device 1400 uses these numerical analysis data {x t|t-Δt (k)} k and the observation data y t (0) to perform data assimilation. Then, the information processing device 1400 uses the coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t) a T 1(t) to a T MT (t) to linearly combine the basis vectors φ u 1(l) to φ u Mu (l), φ v 1(l) to φ v Mv (l), φ T 1(l) to φ T MT (l) to derive the flow velocities u, v, and temperature T of the molten steel 3 at each computational grid point l at each time t. Therefore, each time data assimilation is performed for all ensemble members, the coefficients a u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t) a T 1(t) to a T MTIt becomes unnecessary to convert (t) into the flow velocities u, v and temperature T of the molten steel 3 at each computational grid point l. Therefore, in addition to the effects described in the first embodiment, an effect of being able to shorten the computational time online can be obtained.

[0248] Further, in the present embodiment, the information processing apparatus 1400 executes singular value decomposition on the snapshot matrices MU and MV, thereby obtaining the basis vectors φ u 1(l) to φ u Mu (l), φ v 1(l) to φ v Mv (l), φ T 1(l) to φ T MT When linearly combining (l), the coefficients a u 1(l) to φ u Mu (l), φ v 1(l) to φ v Mv (l), φ T 1(l) to φ T MT (l) multiplied by are derived, and the observation matrix H having components based on these correlations u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t)a T 1(t) to a T MT (t) are derived, and the observation matrix H having components based on these correlations t is derived. Therefore, the coefficients a^ v j (t), a^ T j (t) for different physical quantities of each are considered, and the observation matrix H u 1(t) to a u Mu (t), a v 1(t) to a v Mv (t)a T 1(t) to a T MT (t) is considered, and the observation matrix H tThe components can be derived. Therefore, the computational accuracy of data assimilation can be further improved. In addition, in this embodiment, various modifications described in the first embodiment may also be adopted.

[0249] In addition, the embodiments of the present invention described above can be realized by a computer executing a program. Also, a computer-readable recording medium recording the program and a computer program product such as the program can also be applied as embodiments of the present invention. As the recording medium, for example, a flexible disk, a hard disk, an optical disk, a magneto-optical disk, a CD-ROM, a magnetic tape, a non-volatile memory card, a ROM, etc. can be used. In addition, the embodiments of the present invention described above are merely examples of concretization in implementing the present invention, and the technical scope of the present invention should not be construed as being limited thereby. That is, the present invention can be implemented in various forms without departing from its technical idea or its main features.

Explanation of Reference Numerals

[0250] 1: ladle, 2: tundish, 3: molten steel, 4: mold, 5: sliding nozzle, 6: immersion nozzle, 7: discharge port, 8: solidified shell, 9: molten metal level gauge, 100, 1400: information processing device, 101: numerical simulation unit, 102: matrix derivation unit, 103: basis vector derivation unit, 104: reduced simulator derivation unit, 111: state derivation unit, 111a: first coefficient derivation unit, 111b: physical quantity derivation unit, 112: first probability density function derivation unit, 113: observation data acquisition unit, 114: likelihood function derivation unit, 115: data assimilation unit, 116: visualization data creation unit, 117: output unit, 1405: second coefficient derivation unit, 1406: observation matrix derivation unit, 1411: state derivation unit, 1411a: first coefficient derivation unit, 1412: first probability density function derivation unit, 1413: observation data acquisition unit, 1414: likelihood function derivation unit, 1415: data assimilation unit, 1416: visualization data creation unit, 1417: output unit, 1418: estimated physical quantity derivation unit 1418, F1 to F12, L1 to L12: thermometer

Claims

1. An information processing apparatus that executes a process including deriving the state of molten metal injected into a mold of a continuous casting facility, matrix derivation means for deriving a matrix storing values of a first physical quantity, which is a physical quantity indicating the state of the molten metal at each calculation position and each time; basis vector derivation means for deriving basis vectors of the first to M-th modes of the first physical quantity by performing singular value decomposition on the matrix storing the values of the first physical quantity derived by the matrix derivation means; state derivation means for deriving a state quantity that is a quantity indicating the state of the molten metal; comprising: the first physical quantity includes the flow velocity of the molten metal and the temperature of the molten metal; the matrix derivation means derives a matrix storing the value of the flow velocity of the molten metal for each axial component of the flow velocity and derives a matrix storing the value of the temperature of the molten metal; the basis vector derivation means performs singular value decomposition on the matrix storing the value of the flow velocity of the molten metal derived by the matrix derivation means to derive basis vectors of the first to M-th modes for each axial component of the flow velocity, and performs singular value decomposition on the matrix storing the value of the temperature of the molten metal derived by the matrix derivation means to derive basis vectors of the first to M-th modes of the temperature of the molten metal; the state derivation means has first coefficient derivation means for deriving coefficients used when linearly combining the basis vectors of the first to M-th modes of the first physical quantity after the basis vectors of the first to M-th modes of the first physical quantity are derived by the basis vector derivation means; the state quantity is determined based on the coefficients derived by the first coefficient derivation means. The first coefficient derivation means derives, for each axial component, at least the coefficients used when linearly combining the basis vectors of the first to M-th modes of the flow velocity of the molten metal, as the coefficients used when linearly combining the basis vectors of the first to M-th modes of the first physical quantity. After the basis vector derivation means derives the basis vectors of the first to M-th modes of the temperature of the molten metal, the first coefficient derivation means derives the coefficients used when linearly combining the basis vectors of the first to M-th modes of the temperature of the molten metal. The value of M is an integer of 2 or more, and is determined for each axial component of the flow velocity and for each temperature. The flow velocity of the molten metal is represented by linearly combining the basis vectors of the first to M-th modes of the flow velocity of the molten metal derived by the basis vector derivation means, using the coefficients derived by the first coefficient derivation means. The temperature of the molten metal is represented by linearly combining the basis vectors of the first to M-th modes of the temperature of the molten metal derived by the basis vector derivation means, using the coefficients derived by the first coefficient derivation means. An information processing apparatus characterized by this.

2. The first coefficient derivation means solves a system of first differential equations, which are differential equations representing the time change of the coefficients respectively multiplied by the basis vectors of the first to M-th modes of the flow velocity of the molten metal, and are differential equations configured for each axial component of the flow velocity, to derive, for each axial component of the flow velocity, the coefficients used when linearly combining the basis vectors of the first to M-th modes of the flow velocity of the molten metal. The first coefficient derivation means solves a system of second differential equations, which are differential equations representing the time change of the coefficients respectively multiplied by the basis vectors of the first to M-th modes of the temperature of the molten metal, to derive the coefficients used when linearly combining the basis vectors of the first to M-th modes of the temperature of the molten metal. Each of the first differential equations is a motion equation describing the flow of the molten metal, and is a differential equation configured by taking the inner product of both sides of the motion equation representing the flow velocity of the molten metal as a linear combination of the basis vectors of the first to M-th modes with the basis vectors of the first to M-th modes. Each of the second differential equations is a heat conduction equation that describes the heat conduction of the molten metal. By taking the inner product of both sides of the heat conduction equation, which represents the temperature of the molten metal as a linear combination of basis vectors of modes from the first to the M-th order, with the basis vectors of modes from the first to the M-th order, a differential equation is formed. The information processing apparatus according to claim 1 is characterized in that it is such a differential equation.

3. It further has numerical simulation means for deriving the first physical quantity at time intervals Δt1 by numerical simulation. The matrix derivation means derives, as the matrix, a snapshot matrix that stores the values of the first physical quantity at each time determined by a time interval Δt2 longer than the time interval Δt1, which is the first physical quantity derived by the numerical simulation means. The information processing apparatus according to claim 1 or 2 is characterized in that it is such.

4. Based on singular values obtained by performing singular value decomposition on the matrix that stores the values of the first physical quantity, which is derived by the matrix derivation means, the basis vector derivation means determines, as the value of M for the first physical quantity, a degree smaller than the maximum degree of the modes that can be derived from the matrix that stores the values of the first physical quantity. The information processing apparatus according to any one of claims 1 to 3 is characterized in that it is such.

5. The state quantity includes the first physical quantity. The state derivation means further has physical quantity derivation means for deriving the flow velocity of the molten metal as the first physical quantity for each axial component by linearly combining the basis vectors of modes from the first to the M-th order of the flow velocity of the molten metal, which are derived by the basis vector derivation means, using the coefficients derived by the first coefficient derivation means, and for further deriving the temperature of the molten metal as the first physical quantity by linearly combining the basis vectors of modes from the first to the M-th order of the temperature of the molten metal, which are derived by the basis vector derivation means, using the coefficients derived by the first coefficient derivation means. The information processing apparatus according to any one of claims 1 to 4 is characterized in that it is such.

6. The state derivation means derives, for each calculation position in the visualization target region of the molten metal injected into the mold of the continuous casting facility, numerical analysis data which is data of the state vector that is a vector composed of the state quantities at time t, at every time interval Δt. This is executed online for each of a plurality of cases in which at least one of at least one of the first physical quantities at time t−Δt, which is a time interval Δt before time t, and at least one of the parameters of the boundary conditions between time t−Δt and time t are different. The information processing apparatus includes first probability density function derivation means for deriving a first probability density function of the state vector at time t from the numerical analysis data at time t of the plurality of cases derived by the state derivation means. Observation data acquisition means for acquiring, for each observation position of the continuous casting facility measured by a sensor, observation data which is data of the observation vector that is a vector composed of observation quantities which are quantities based on a second physical quantity which is the physical quantity at each observation position, at every time interval Δt. Likelihood function derivation means for deriving a likelihood function of the state vector at time t from the observation data at time t acquired by the observation data acquisition means. Based on the first probability density function of the state vector at time t derived by the first probability density function derivation means and the likelihood function of the state vector at time t derived by the likelihood function derivation means, data assimilation means for deriving a second probability density function of the state vector at time t by a filter that performs data assimilation by Bayesian statistical modeling, and deriving an estimated value of the state vector at time t based on the second probability density function of the state vector. further comprises The first coefficient derivation means derives, based on the second probability density function of the state vector, the coefficient at time t online for each of the plurality of cases. The information processing apparatus according to any one of claims 1 to 5. **Claim 7** The state quantity includes the first physical quantity. The observation quantity is the second physical quantity. The state derivation means further includes a physical quantity derivation means for deriving the first physical quantity by linearly combining the basis vectors of the first to M-th modes of the first physical quantity derived by the basis vector derivation means using the coefficients derived by the first coefficient derivation means. The information processing apparatus according to claim 6, wherein the physical quantity derivation means derives the numerical analysis data at the time t for each of the plurality of cases online using the coefficients at the time t in each of the plurality of cases.

8. The first physical quantity includes the second physical quantity. The state quantity includes coefficients used when expressing the first physical quantity as a linear combination of basis vectors of the first to M-th modes. The observation quantity is a coefficient used when expressing the second physical quantity as a linear combination of basis vectors of the first to M-th modes. The apparatus further includes an estimated physical quantity derivation means for deriving the first physical quantity by linearly combining the basis vectors of the first to M-th modes of the first physical quantity derived by the basis vector derivation means using the coefficients included in the estimated value of the state vector derived by the data assimilation means. The observation data acquisition means derives the observation data based on the measured values of the second physical quantity at each observation position measured by a sensor and the basis vectors of the first to M-th modes of the second physical quantity derived by the basis vector derivation means. The information processing apparatus according to claim 6.

9. Second coefficient derivation means for deriving coefficients used when linearly combining the basis vectors of the first to M-th modes of the first physical quantity by performing the singular value decomposition on the matrix storing the values of the first physical quantity; Observation matrix derivation means for deriving an observation matrix showing the relationship between the coefficients used when linearly combining the basis vectors of the first to M-th modes of the first physical quantity, which are the state quantities, and the coefficients used when linearly combining the basis vectors of the first to M-th modes of the second physical quantity, which are the observation quantities. The observation matrix derivation means derives, as components of the observation matrix, a value indicating the correlation between the coefficient used when linearly combining the basis vectors of the first to M-th modes of the first physical quantity, which is the state quantity, based on the coefficient derived by the second coefficient derivation means, and the coefficient used when linearly combining the basis vectors of the first to M-th modes of the second physical quantity, which is the observed quantity. The information processing apparatus according to claim 8, characterized in that.

10. The information processing apparatus according to any one of claims 6 to 9, further comprising visualization data creation means for creating display data of the first physical quantity at each calculation position of the visualization target region at time t based on the estimated value of the state vector at time t derived by the data assimilation means.

11. An information processing method for executing a process including deriving the state of molten metal injected into a mold of a continuous casting facility, A matrix derivation step of deriving a matrix storing values of a first physical quantity, which is a physical quantity indicating the state of the molten metal at each calculation position and each time; A basis vector derivation step of deriving the basis vectors of the first to M-th modes of the first physical quantity by performing singular value decomposition on the matrix storing the values of the first physical quantity derived in the matrix derivation step, thereby deriving the basis vectors of the first to M-th modes of the first physical quantity; A state derivation step of deriving a state quantity, which is a quantity indicating the state of the molten metal; having The first physical quantity includes the flow velocity of the molten metal and the temperature of the molten metal. The matrix derivation step derives a matrix storing the values of the flow velocity of the molten metal for each axial component of the flow velocity, and derives a matrix storing the values of the temperature of the molten metal. The base vector derivation step derives base vectors of the first to M-th modes by performing singular value decomposition on the matrix storing the values of the flow velocity of the molten metal derived in the matrix derivation step, and this is performed for each matrix storing the values of the flow velocity of the molten metal derived in the matrix derivation step, thereby deriving base vectors of the first to M-th modes of the flow velocity of the molten metal for each axial component of the flow velocity. Also, by performing singular value decomposition on the matrix storing the values of the temperature of the molten metal derived in the matrix derivation step to derive base vectors of the first to M-th modes, base vectors of the first to M-th modes of the temperature of the molten metal are derived. After the base vectors of the first to M-th modes of the first physical quantity are derived in the base vector derivation step, the state derivation step includes a first coefficient derivation step of deriving coefficients used when linearly combining the base vectors of the first to M-th modes of the first physical quantity. The state quantity is determined based on the coefficients derived in the first coefficient derivation step. The first coefficient derivation step derives, for each axial component, the coefficients used when linearly combining at least the base vectors of the first to M-th modes of the flow velocity of the molten metal as the coefficients used when linearly combining the base vectors of the first to M-th modes of the first physical quantity. After the base vectors of the first to M-th modes of the temperature of the molten metal are derived in the base vector derivation step, the coefficients used when linearly combining the base vectors of the first to M-th modes of the temperature of the molten metal are derived. The value of M is an integer of 2 or more, and is determined for each axial component of the flow velocity and for each temperature. The flow velocity of the molten metal is represented by linearly combining the base vectors of the first to M-th modes of the flow velocity of the molten metal derived in the base vector derivation step using the coefficients derived in the first coefficient derivation step. The temperature of the molten metal is represented by linearly combining the base vectors of the first to M-th modes of the temperature of the molten metal derived in the base vector derivation step using the coefficients derived in the first coefficient derivation step. This is an information processing method.

12. A program for causing a computer to function as each means of the information processing apparatus according to any one of claims 1 to 10.

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