JP7747962B2 - Numerical analysis program and heat treatment device - Google Patents

Description

The present invention relates to a numerical analysis program and a heat treatment device for performing numerical analysis on objects with three-dimensional shapes.

Technologies for performing numerical analysis using computer simulations have been known for some time. For example, Patent Document 1 discloses a method for analyzing the mechanical properties of an object using the finite element method. More specifically, this document uses a technique for setting a coordinate system for each element, taking into account the anisotropy direction and/or the orientation of the curved surface of the object, so as to avoid complicating the construction of the characteristic value matrix.

Patent No. 6447454

In a typical finite element method, element division is performed according to the contours of the three-dimensional shape of an object. For example, by reducing the size of the elements, it is possible to represent more complex three-dimensional shapes. However, this results in a huge number of element divisions, and the positional relationships between the elements become more complex, which creates the problem of increased calculation time required for analysis.

The present invention was made in consideration of these problems, and its purpose is to provide a numerical analysis program and heat treatment device that can prevent the increase in the time required for analysis processing due to the complexity of the three-dimensional shape of the object being analyzed.

A numerical analysis program according to a first aspect of the present invention causes a computer to execute the following steps: an acquisition step for acquiring the three-dimensional shape of an object to be analyzed; a division step for dividing a three-dimensional space containing the object into a plurality of elements; a calculation step for determining the density of the object's volume within a unit space formed by the elements and calculating a density distribution indicating the density at each position of the elements; a construction step for determining a physical quantity correlated to the heat transfer characteristics of the elements using the calculated density distribution and constructing a heat transfer equation for each element including the physical quantity; and an analysis step for performing an analysis process on the object by solving the constructed heat transfer equation for each element, wherein the physical quantity is the amount of radiant heat exchanged between the object and the outside of the object; and in the construction step, the amount of change in the surface area of the object is calculated using the difference between a first density corresponding to a first element and a second density corresponding to a second element adjacent to the first element, and the amount of change is used to determine the amount of radiant heat exchanged.

In the numerical analysis program according to the second aspect of the present invention, the division step divides the three-dimensional space so that, when the density is expressed as a percentage, one or more of the calculated densities are intermediate values between 0% and 100%.

In the numerical analysis program according to the third aspect of the present invention, the division step defines a cylindrical coordinate system whose coordinate axes are the extension direction of the object, and divides the three-dimensional space into cylindrical elements coaxial with the coordinate axes. The calculation step calculates the density distribution indicating the density for each radial position and each axial position in the cylindrical coordinate system.

A fourth aspect of the present invention provides a heat treatment apparatus comprising an apparatus main body that applies heat treatment to a workpiece and a controller that controls the apparatus main body, wherein the apparatus main body comprises an insulating wall that is arranged to surround the object as the workpiece, and a heater that heats the inside of a heating chamber formed by the insulating wall, and the controller performs an acquisition step of acquiring the three-dimensional shape of the object, a division step of dividing a three-dimensional space including the object into a plurality of elements, a calculation step of determining the density of the volume of the object within a unit space formed by the elements and calculating a density distribution that indicates the density for each position of the elements, and a calculation step of calculating a density distribution using the calculated density distribution. The method executes a construction step of determining a physical quantity correlated with the heat transfer characteristics of the element and constructing a heat transfer equation including the physical quantity for each element, and a control step of performing an analysis process for the object by solving the constructed heat transfer equation for each element and controlling heating of the heater using the obtained analysis results, wherein the physical quantity is the amount of radiant heat exchanged between the object and the outside of the object, and in the construction step, the amount of change in surface area of the object is calculated using the difference between a first density corresponding to a first element and a second density corresponding to a second element adjacent to the first element, and the amount of change is used to determine the amount of radiant heat exchanged.

This invention makes it possible to prevent the time required for analysis processing from increasing due to the complexity of the three-dimensional shape of the object being analyzed.

1 is a schematic cross-sectional view of a vacuum carburizing furnace as a heat treatment apparatus according to an embodiment of the present invention. 2 is a flowchart showing an example of a numerical analysis operation performed by the controller of FIG. 1 . FIG. 1 is a diagram showing a three-dimensional shape of a part that is an example to be analyzed. FIG. 10 is a diagram illustrating a first example of a method for calculating a density distribution. FIG. 10 is a diagram illustrating a second example of a method for calculating a density distribution. FIG. 10 is a diagram showing the effect obtained by introducing a density distribution. FIG. 10 is a diagram illustrating an example of a method for determining a soaking time.

Embodiments of the present invention will now be described with reference to the accompanying drawings. To facilitate understanding, identical components in each drawing will be designated by the same reference numerals wherever possible, and duplicate descriptions will be omitted.

[Overall Configuration of Heat Treatment Apparatus 10]
1 is a schematic cross-sectional view of a heat treatment apparatus 10 according to one embodiment of the present invention. The heat treatment apparatus 10 includes a vacuum carburizing furnace 12 (corresponding to the "apparatus main body"), a controller 14 (corresponding to the "computer"), a heating power source 16, and temperature sensors 18 and 20.

The vacuum carburizing furnace 12 is a device for performing vacuum carburizing using a hydrocarbon gas at high temperature and reduced pressure. The vacuum carburizing furnace 12 comprises a furnace body 22 that houses the workpiece W (corresponding to the "heated object") and a frame 24 attached to the inside of the furnace body 22. Inside the frame 24, there are provided a relatively large, hollow, rectangular insulating member 26, a relatively small, hollow, rectangular insulating member 28, and multiple insulating members 30 for connecting the insulating member 26 to the insulating member 28. The insulating members 26, 28, and 30 are made of an insulating material such as ceramic fiberboard. The two insulating members 26, 28 are spaced apart to form a space 32 that functions as an insulating layer.

A carburizing chamber 36 for vacuum carburizing is formed within the furnace body 22, surrounded by a hollow insulating member 30. A hearth 38 is provided in the furnace body 22, extending from its lower part into the carburizing chamber 36. The workpiece W is placed on the upper end of the hearth 38.

Several heat radiation pipes 40 (corresponding to "heaters") are attached to the top of the furnace body 22, extending toward the inside of the carburizing chamber 36. Each heat radiation pipe 40 is positioned to cover a wide area of the carburizing chamber 36, from the top to the bottom. This allows the space defined by the carburizing chamber 36 to be heated approximately evenly. The interior of the carburizing chamber 36 is set to a high temperature, for example, exceeding 900°C.

Multiple gas introduction nozzles 44 are provided on the top and sides of the furnace body 22 so as to face the position of the workpiece W. Carburizing can be performed on the surface of the workpiece W by introducing hydrocarbon gas into the carburizing chamber 36 through the gas introduction nozzles 44 from a gas supply mechanism (not shown).

An exhaust pipe 46 is provided at the bottom of the furnace body 22, connecting the carburizing chamber 36 to the outside of the vacuum carburizing furnace 12. The pressure inside the carburizing chamber 36 can be adjusted by operating a vacuum pump (not shown) to evacuate air through the exhaust port 48.

The controller 14 is a computer that has a processor 50, memory 52, and operation panel 54 and controls the operation of each part of the vacuum carburizing furnace 12. By reading and executing programs and data stored in the memory 52, the controller 14 synchronously or in parallel performs the following operations: [1] supply control for supplying hydrocarbon gas into the carburizing chamber 36, [2] pressure control for adjusting the pressure inside the carburizing chamber 36, and [3] heating control for heating the carburizing chamber 36. To control these operations, a flow rate sensor, a pressure sensor (none shown), and temperature sensors 18 and 20 are connected to the controller 14. Here, the temperature sensor 18 is located in contact with the top of the furnace body 22, and the temperature sensor 20 is located on the periphery of the vacuum carburizing furnace 12.

The processor 50 acquires temperature signals from the temperature sensors 18 and 20 and uses the measured temperatures to calculate estimated values for the temperature inside the carburizing chamber 36 and the temperature of the workpiece W. The processor 50 determines the set conditions for the heat treatment of the workpiece W (hereinafter also referred to as "heat treatment conditions") based on the estimated values it has calculated. The processor 50 controls the heating of each heat sink tube 40 by controlling the thyristors of the heating power source 16.

Memory 52 stores setting conditions for the heat treatment of the workpiece W (i.e., heat treatment conditions) and information regarding a heat transfer model that describes heat transfer within the furnace body 22 (hereinafter also referred to as "model information"). The model information includes: [1] "shape information" regarding the three-dimensional shape of the object; [2] "arrangement information" regarding the definition of the three-dimensional coordinate system and the arrangement of elements; [3] "temperature information" regarding temperature constraints; and [4] "physical property information" regarding the physical properties of the object.

[Operation of controller 14]
<1. Temperature control operation>
The heat treatment apparatus 10 in this embodiment is configured as described above. Next, the first operation of the heat treatment apparatus 10, more specifically, the temperature control operation by the controller 14, will be outlined.

First, the controller 14 calculates the various control times included in the heat treatment conditions. Here, "control time" refers to the time involved in controlling each process that constitutes a series of heat treatments, and includes, for example, [1] the "heat-up time" required to reach the target temperature for carburizing, [2] the "soaking time" required to soak at the target temperature, or [3] the "heating time," which is the sum of the heat-up time and soaking time.

Next, the controller 14 sets the heat treatment conditions, including the calculated control time. After starting the heat treatment, the controller 14 performs a series of temperature control operations (e.g., feedback control or feedforward control of temperature) on the heat sink tube 40 in accordance with the set heat treatment conditions. By executing these control steps, the temperature increase process, soaking process, carburizing process, diffusion process, furnace cooling process, quenching and soaking process, and oil cooling process are carried out in sequence.

<2. Numerical analysis operation>
Next, the second operation of the heat treatment apparatus 10, more specifically, the numerical analysis operation by the controller 14, will be described with reference to the flowchart of FIG. 2 and FIGS.

(SP10: Acquisition step)
In step SP10 of FIG. 2, the controller 14 acquires model information including the three-dimensional shape of the object to be analyzed.

Figure 3 is a diagram showing the three-dimensional shape of part 60, which is the example to be analyzed. This part 60 is, for example, a gear shaft composed of a shaft portion 62 and a first gear portion 64 formed on the shaft portion 62. The shaft portion 62 comprises a shaft body 66 and a second gear portion 68 provided on the tip side of the shaft body 66. The first gear portion 64 has a plurality of teeth 70 (12 in this example) and is provided at an intermediate position on the shaft body 66. In this way, part 60 is approximately isotropic with respect to axis A, but has a relatively complex three-dimensional shape.

(SP12: division step)
2, the controller 14 divides the three-dimensional space 80, 90 containing the object (i.e., the part 60) into a plurality of elements 82, 92. A specific example of the division method will be described later with reference to FIGS.

(SP14: Calculation step)
In step SP14, the controller 14 calculates the density (i.e., density distributions 86, 96) for each position of the elements 82, 92 divided in step SP12. This "density" corresponds to the proportion of the volume of the object within the unit space formed by the elements 82, 92. The density can take various values within the range of [0, 1]. When expressing the density as a percentage, a value of "0" corresponds to 0% and a value of "1" corresponds to 100%. The density value may be a continuous value or a discrete value with three or more levels.

Figure 4 shows a first example of a method for calculating a density distribution 86. More specifically, Figure 4(A) shows an example of a method for dividing a three-dimensional space 80, and Figure 4(B) shows an example of the calculation results of a density distribution 86. In the first example, the three-dimensional space 80 is divided into multiple elements 82 based on a Cartesian coordinate system (x, y, z), and the density distribution 86 corresponds to a two-dimensional map of density with (y, z) as variables.

Figure 4(A) shows a schematic diagram of the cross-sectional shape 84 of the first gear portion 64 relative to the axis A shown in Figure 3. As can be seen from this figure, the three-dimensional space 80 is divided into multiple elements 82 according to a predetermined rule, regardless of the outline of the cross-sectional shape 84. Each element 82 is prismatic and extends in the X-axis direction.

In the density distribution 86 of Figure 4(B), the vertical axis of the graph indicates the position on the y-axis (unit: mm), and the horizontal axis of the graph indicates the density D (unit: dimensionless). As can be seen from this figure, the density distribution 86 changes in value depending on the position on the y-axis. Specifically, the density D increases as the position y approaches the axis A, while the density D decreases as the position y moves away from the axis A.

Figure 5 shows a second example of a method for calculating a density distribution 96. More specifically, Figure 5(A) shows an example of a method for dividing a three-dimensional space 90, and Figure 5(B) shows an example of the calculation results of a density distribution 96. In the second example, the three-dimensional space 90 is divided into multiple elements 92 based on a cylindrical coordinate system (r, θ, z), and the density distribution 96 corresponds to a two-dimensional map of density with (r, z) as variables.

Similar to FIG. 4(A), FIG. 5(A) schematically shows the cross-sectional shape 94 of the first gear portion 64 relative to the axis A shown in FIG. 3. As can be seen from this figure, the three-dimensional space 90 is divided into multiple elements 92 according to a predetermined rule, regardless of the outline of the cross-sectional shape 94. Each element 92 is cylindrical or columnar and is arranged coaxially.

In the density distribution 96 of Figure 5(B), the vertical axis of the graph represents the radial position r (unit: mm), and the horizontal axis of the graph represents the density D (unit: dimensionless). As can be seen from this figure, the density distribution 96 varies in value depending on the position r. Specifically, in the range of 0 < r < R1, the density D is constant (D = 1). In the range of r > R1, the density D decreases rapidly as the position r increases, and in the range of r > R2, the density D is constant (D = 0). Here, r = R1 corresponds to the bottom position of the tooth 70 (Figure 3), and r = R2 corresponds to the top position of the tooth 70 (Figure 3).

(SP16: Construction step)
2, the controller 14 uses the density distributions 86, 96 calculated in step SP14 to construct a heat transfer equation for each element 82, 92 that describes the heat transfer characteristics in the three-dimensional spaces 80, 90. For example, the heat transfer equation in the cylindrical coordinate system (r, θ, z) shown in FIG. 5 is expressed by the following equation (1):

Typically, when performing numerical calculations, the partial differential equation shown in equation (1) is approximated by differences and transformed into a difference equation. Examples of transformation methods include explicit methods, implicit methods, and ADI (Alternating Direction Implicit) methods. Here, by multiplying a physical quantity correlated with the heat transfer characteristics of elements 82 and 92 by density, the density distributions 86 and 96 can be reflected in the difference equation. Examples of such physical quantities include surface area, volume, thermal conductivity, and heat flux.

The change in surface area of the part 60 can also be calculated using density distribution 96, which uses (r, z) as a variable. For element 92 at position (r, z), assuming density D(r, z) and width (Δr, h), and assuming that the radiation distance between the part 60 and another object (radiation source) is sufficiently long, the change in surface area ΔS can be approximately calculated using the following equation (2):

By integrating this change ΔS along the radial or axial direction, the surface area of the part 60 can be calculated for each position without directly using the three-dimensional shape of the part 60. In other words, the effect of heat exchange by radiation can be reflected for each element 92. Furthermore, if the shape information of the part 60 includes the number, shape, or orientation of the teeth 70, the amount of heat exchange by radiation may be corrected according to this shape information. For example, in the case of radiation, the amount of heat exchange may be relatively increased as the angle between the heat incident angle and the tooth depth direction becomes smaller, while the amount of heat exchange may be relatively decreased as the angle becomes larger.

(SP18: Analysis step)
2, the controller 14 simultaneously or coupledly solves the heat transfer equations for each of the elements 82 and 92 constructed in step SP16, thereby obtaining a transient or steady-state solution for the temperature distribution in the three-dimensional spaces 80 and 90.

Figure 6 shows the effect achieved by introducing a density distribution. The horizontal axis of the graph represents elapsed time (unit: minutes), and the vertical axis of the graph represents temperature (unit: °C). The dotted curve represents the analysis results using a conventional division method, and corresponds to the "standard (3D-FEM)" calculation accuracy. Here, the element shape is a "tetrahedron," and the part is divided into multiple elements following the contour shape of part 60 (Figure 3). In this case, each value in the density distribution is binarized as 1 (all present) or 0 (absent).

The "Reference Example (2D-FEM)" shown by the two-dot chain line corresponds to the analysis results obtained by using the division method shown in Figure 5(A) while binarizing (1 or 0) the density distribution 96 shown in Figure 5(B) using a threshold value of D = 0.5. In other words, smoothing the three-dimensional shape of the part 60 results in the construction of a heat transfer model for a part with missing teeth 70. As a result, the curve for the "Comparative Example" shows a faster temperature rise and faster arrival at the target temperature To than the "Reference" curve, resulting in a larger deviation between the two (i.e., calculation error).

On the other hand, the "Example" shown by the solid line corresponds to the analysis results obtained using the division method shown in Figure 5(A) and the density distribution 96 shown in Figure 5(B). The curve for the "Example" roughly matches the curve for the "Reference" during the time period up to when the target temperature To is reached, demonstrating sufficient calculation accuracy.

(SP20: Decision step)
In step SP20 of FIG. 2, the controller 14 determines the heat treatment conditions from the analysis results obtained in step SP18.

Figure 7 shows an example of a method for determining the soaking time. The horizontal axis of the graph represents elapsed time (unit: minutes), and the vertical axis of the graph represents temperature (unit: °C). Three curves represent temperatures at different positions P1, P2, and P3 within the workpiece W. As can be seen from the contents of Figure 1, the shape of the curves differs because the degree of heat transfer varies depending on the position within the workpiece W. If the times to reach the target temperature at positions P1, P2, and P3 are defined as t1, t2, and t3, respectively, then the relationship t1 < t2 < t3 is satisfied. In this case, the maximum time difference (t3 - t1) can be determined as the soaking time.

[Summary of the embodiment]
As described above, in the numerical calculation program and method of this embodiment, one or more computers (here, the controller 14) execute an acquisition step (SP10 in FIG. 2 ) of acquiring the three-dimensional shape of the object to be analyzed (here, the part 60), a division step (SP12) of dividing the three-dimensional space 80, 90 containing the part 60 into a plurality of elements 82, 92, a calculation step (SP14) of determining the density of the volume of the part 60 within the unit space formed by the elements 82, 92 and calculating density distributions 86, 96 that indicate the density for each position of the elements 82, 92, and an analysis step (SP18, SP20) of performing an analysis process of the part 60 based on the finite element method using the calculated density distributions 86, 96.

In this way, by determining the density of the volume of part 60 within the unit space formed by elements 82, 92 and calculating density distributions 86, 96, it becomes possible to represent a pseudo-intermediate volume that is independent of the three-dimensional shape of part 60, increasing the design freedom for the shape or placement of elements 82, 92. For example, by dividing three-dimensional spaces 80, 90 to increase calculation efficiency, it is possible to prevent the increase in the time required for analysis processing due to the complexity of the three-dimensional shape of part 60.

Furthermore, in the division step, when the density is expressed as a percentage, the controller 14 may divide the three-dimensional spaces 80 and 90 so that one or more calculated densities are intermediate values between 0% and 100%. This makes it easier to express the outline of the part 60 more simply, and the number of elements can be reduced accordingly.

The controller 14 may also execute a construction step (SP16 in FIG. 2) in which the controller 14 uses the calculated density distribution to determine physical quantities correlated with the heat transfer characteristics of the elements 82, 92, and constructs heat transfer equations including the physical quantities for each element 82, 92. In this case, the controller 14 performs analysis processing of the part 60 in the analysis step by simultaneously or coupledly solving the constructed heat transfer equations for each element 80, 90.

Furthermore, if the physical quantity is the amount of radiant heat exchanged between the component 60 and the outside of the component 60, the controller 14 may calculate a change ΔS in the surface area of the component 60 in the construction step using the difference between a first density corresponding to a first element and a second density corresponding to a second element adjacent to the first element, and use the change ΔS to determine the amount of radiant heat exchanged.

In addition, in the division step, the controller 14 may define a cylindrical coordinate system whose coordinate axes are the extension direction of the part 60, and divide the three-dimensional space 90 into cylindrical elements 92 that are coaxial with the coordinate axes. In the calculation step, the controller 14 may calculate a density distribution 96 that indicates the density for each radial position (r) and each axial position (z) in the cylindrical coordinate system. By introducing a cylindrical coordinate system whose coordinate axes are the extension direction of the part 60, high-speed, high-precision analysis processing can be performed on objects that are isotropic or have high rotational symmetry (e.g., gear shafts).

The heat treatment apparatus 10 also includes an apparatus main body (here, a vacuum carburizing furnace 12) that performs heat treatment on the workpiece, and a controller 14 that controls the vacuum carburizing furnace 12. The vacuum carburizing furnace 12 includes insulating walls (here, insulating members 26, 28, 30) that surround the object (here, a component 60) serving as the workpiece, and a heater (here, a heat sink 40) that heats the interior of a heating chamber (here, a carburizing chamber 36) formed by the insulating walls.

The controller 14 then acquires the three-dimensional shape of the part 60, divides the three-dimensional space 80, 90 containing the part 60 into multiple elements 82, 92, determines the density of the volume of the part 60 within the unit space formed by the elements 82, 92, calculates density distributions 86, 96 that indicate the density for each position of the elements 82, 92, performs an analysis of the part 60 based on the finite element method using the calculated density distributions 86, 96, and controls the heating of the heat sink tube 40 using the obtained analysis results.

With this configuration, similar to the effects described above, the calculation time required for analysis can be reduced regardless of the complexity of the three-dimensional shape of the part 60. In particular, when heat treating a wide variety of parts 60, it becomes possible to execute heating control that is suited to various situations, thereby improving production efficiency.

[Modification]
The present invention is not limited to the above-described embodiment, and can be freely modified without departing from the spirit of the present invention. Alternatively, the respective configurations may be arbitrarily combined within the scope of the present invention without causing any technical contradiction.

In the above embodiment, the three-dimensional space 80, 90 is divided into rectangular parallelepiped elements 80 or cylindrical elements 90, but the method of dividing the elements is not limited to this. For example, the shape, size, arrangement, or combinations of these of the elements may be changed as needed.

In the above embodiment, the controller 14, which constitutes part of the heat treatment apparatus 10, executes the numerical analysis program, but the configuration of the apparatus is not limited to this. For example, a general-purpose computer independent of the vacuum carburizing furnace 12 may execute the heat calculation program for the purpose of computer simulation.

In the above embodiment, we have described a case where a computer performs thermal calculations, but the application of numerical calculations is not limited to this. This numerical calculation can be applied to any field problem described by partial differential equations, such as structural analysis, fluid analysis, and electromagnetic field analysis.

10...heat treatment device, 12...vacuum carburizing furnace, 14...controller (computer), 26, 28, 30...insulating member, 36...carburizing chamber, 50...processor, 52...memory, 60...part (object), 80, 90...3D space, 82, 92...elements, 86, 96...density distribution, W...workpiece (subject to be processed), SP10...acquisition step, SP12...division step, SP14...calculation step, SP16...construction step, SP18...analysis step

Claims (4)

an acquisition step of acquiring a three-dimensional shape of an object to be analyzed;
a division step of dividing a three-dimensional space including the object into a plurality of elements;
a calculation step of determining the density of the volume of the object within a unit space formed by the elements and calculating a density distribution indicating the density for each position of the elements;
a construction step of calculating a physical quantity correlated with the heat transfer characteristics in the element using the calculated density distribution, and constructing a heat transfer equation including the physical quantity for each element;
an analysis step of performing an analysis process of the object by solving the heat transfer equation for each of the constructed elements;
on the computer,
the physical quantity is the amount of radiant heat exchanged between the object and the outside of the object,
A numerical analysis program characterized in that in the construction step, a change in surface area of the object is calculated using the difference between a first density corresponding to a first element and a second density corresponding to a second element adjacent to the first element, and the amount of radiant heat exchange is determined using the change. The numerical analysis program described in claim 1, characterized in that in the division step, the three-dimensional space is divided so that, when the density is expressed as a percentage, one or more of the calculated densities are intermediate values between 0% and 100%. In the dividing step, a cylindrical coordinate system is defined in which the extension direction of the object is defined as a coordinate axis, and the three-dimensional space is divided into cylindrical elements coaxial with the coordinate axis;
3. The numerical analysis program according to claim 1, wherein the calculation step calculates the density distribution indicating the density for each radial position and each axial position in the cylindrical coordinate system. A heat treatment apparatus comprising: an apparatus main body that performs heat treatment on a workpiece; and a controller that controls the apparatus main body,
The device body includes:
a heat insulating wall provided to surround the object to be treated;
a heater for heating the inside of a heating chamber formed by the heat insulating wall;
Equipped with
The controller
acquiring a three-dimensional shape of the object;
a division step of dividing a three-dimensional space including the object into a plurality of elements;
a calculation step of determining the density of the volume of the object within a unit space formed by the elements and calculating a density distribution indicating the density for each position of the elements;
a construction step of calculating a physical quantity correlated with the heat transfer characteristics in the element using the calculated density distribution, and constructing a heat transfer equation including the physical quantity for each element;
a control step of performing an analysis process of the object by solving the heat transfer equation for each of the constructed elements, and performing heating control on the heater using the obtained analysis result;
Run
the physical quantity is the amount of radiant heat exchanged between the object and the outside of the object,
A heat treatment apparatus characterized in that in the construction step, the change in surface area of the object is calculated using the difference between a first density corresponding to a first element and a second density corresponding to a second element adjacent to the first element, and the amount of radiant heat exchanged is determined using the change.