Optimization of structural parameters and numerical simulation of stress ﬁ eld of composite crucible based on the indirect coupling method

: The research starts with the treatment of the multiscale transmission problem and establishes the electro-magnetic solidi ﬁ cation transmission coupling mathematical model based on the indirect coupling method. It uses the three-dimensional magnetic ﬁ eld ﬁ nite element theory to establish a three-dimensional crucible structure continuous casting model built on the electromagnetic solidi ﬁ - cation transmission coupling mathematical model. This model is used to optimize the parameters of the composite crucible structure and to simulate electromagnetic transmission and braking phenomena. The results show that the L-shaped static magnetic ﬁ eld has a more potent inhibition and a guidance e ﬀ ect on melt circulation. The braking e ﬀ ect of the actual magnetic ﬁ eld on the down-ward impact is worse. Under the in ﬂ uence of an L-shaped magnetic ﬁ eld, the ﬂ ow velocity of the melt is better, and the ﬂ ow state distribution is more smooth and uniform. The computational e ﬃ ciency test results show that the conversion calculation time of the method designed in this study is 18.03 min. The total calculation time is 680.48 min, which is superior to traditional methods. It proves that this model can accurately analyze the magnetic ﬁ eld coupling problem and at the same time ensure the superiority of its computing e ﬃ ciency.


Introduction
Modern manufacturing has developed rapidly. High requirements are increasing for the function and production efficiency of products. Driven by the strong demand of the external social environment, the continuous metal casting and casting technology under the crucible structure has gradually been developed [1][2][3]. Electromagnetic machining technology under the crucible structure is one of the main technologies. The main technical issue is how to achieve product surface performance and quality through heat and momentum transfer within the structure. During this process, the use of magnetic fields to control the liquid flow and other technologies can achieve quality control while reducing the possibility of production accidents [4][5][6]. Electromagnetic casting is not only a time-varying complex flow problem in space but also a strong coupling transmission process of multiclock field interaction. The macro and micro transmission behavior in the solidification process has an important impact on the solidification structure and properties of castings. The simulation experiment of energy field correlation and transmission phenomena in crucible structures requires a universal electromagnetic solidification transmission model for effective calculation. The goal of accurately controlling the transport behavior of different types of composite metals can be achieved. The indirect coupling method can simplify the transmission behavior problem and achieve the effect of an efficient solution [7][8][9]. Therefore, based on the mathematical model of electromagnetic solidification transmission coupling, this study established a mathematical model of electromagnetic solidification transmission coupling based on the indirect coupling method. A three-dimensional crucible structure continuous casting model was established using the three-dimensional magnetic field finite element theory. Finally, this model can be used to optimize the parameters of the composite crucible structure and simulate the electromagnetic transmission and electromagnetic braking phenomenon.

Related works
In recent years, the indirect coupling method has been applied more deeply. Mirehi and Heidari-Semiromi studied the formation of magnetization based on the interaction between electrons. In the research, different interaction intensity values are used to plot the energy values of electron interaction intensity. The different intensity levels of indirect coupling are determined by the changes in electron interaction intensity and the size of the nanosheet. The results show that it is mainly the electron interaction that changes the magnetization state at the edge of the nanosheet [10]. Kim et al. studied the sensor power conduction and acceleration feedback in the supersonic flight environment of aircraft by using the coupling analysis method, and proposed a filter feedback path control system on the basis of the coupling relationship [11]. This system can improve the stability margin characteristics of the aircraft control system through synchronous feedback, and eliminate the negative effects of structural coupling in the low-frequency range. Neves carried out environmental simulation of single-phase magnetic synchronous generator under transient state behavior through the finite element method [12]. Its simulation experiments can use indirect coupling methods to study the coupling interaction between the circuit environment and the magnetic circuit environment. The voltage distortion conditions formed by the terminal in the interactive simulation environment are mainly calculated based on Fourier transform. The research results show that this method can accurately simulate and predict the operation stability and operation fault status of synchronous generators. Ling et al. proposed a unified mathematical and physical model, which was difficult to describe the dynamic transmission characteristics of light between different resonators [13]. The indirect coupling scheme formed by this model can adjust the electronic density of plasma metamaterials, providing a theoretical basis for the formation of highperformance optoelectronic integrated devices. Starting from the perspective of mechanical assembly, Xu and Lu paid more attention to dynamic quality than traditional static quality selection [14]. They explored the impact of assembly coupling on the dynamic quality of mechanical assembly, and designed an assembly model with more dynamic stiffness using the indirect coupling method. The research results showed that this method is highly feasible. In this study, the indirect coupling method is applied to the structure of the composite crucible, and the parameter design is optimized.
On the other hand, the research on crucible structure has been deepened in recent years. Sawangboon et al. analyzed the performance of the composite crucible structure of alumina and quartz materials, mainly by means of burning, changing the density, and the number of constituent bonds [15]. The research results showed that the addition of alumina will have a significant impact on the crucible structure and its actual performance. Li et al. used the mechanically stirred crucible control to enhance the intensity of composite materials, and established a threedimensional flow field model [16]. The research results showed that the nonaxisymmetric pattern constructed by the square crucible can improve the turbulence state, and the preparation effect is stronger than that of the round crucible. Chen et al. analyzed the material structure and creep property of composite metal materials through cold crucible continuous casting technology, and explored the internal structure, tensile property, and creep state of composite metal from both micro and macro perspectives [8].
The research results showed that the directionally solidified structure can remarkably affect the creep performance of materials, and its impact on the ultimate tensile and tensile properties at room temperature is not sufficient. The local central concentrated stress effect generated by the internal colony boundary is conducive to promoting the cavity molding at the colony boundary. Palacz et al. used induction melting to design and study the cold crucible [17]. The research results showed that the characteristic sawtooth mainly produces the central part of the selfcooling crucible, and the research on the numerical model of simulation is helpful for the further design of the crucible. Xu et al. analyzed the cold crucible directional solidification technology, and characterized the composition of the solid-liquid interface mainly from the macro and micro perspectives [18]. The results showed that the columnar to equiaxed transition can be formed only when the solid-liquid interface after quenching meets the conditions of growth rate and heterogeneous nucleation.
The application of the indirect coupling method in the study of multifactor coupling interaction in complex environments is more in depth and extensive. It can analyze the influence path of elements through simulation, thus laying a theoretical foundation for solutions. In the research of crucible structure, the research on crucible performance and materials inside the crucible is basically focused on the analysis of the influence path of elements. Therefore, the advantage of the indirect coupling method in the analysis of multifactor coupling reaction is studied, and it is applied to the optimization design of the structural parameters of composite crucible. The mathematical model of electromagnetic solidification transmission coupling is established. On the ground of the three-dimensional continuous casting model, the electromagnetic transmission and electromagnetic braking phenomena are simulated using the electromagnetic solidification transmission coupling model. The research will establish a set of electromagnetic solidification transmission coupling mathematical models based on indirect coupling, and use it to optimize the structure parameters of the composite crucible and conduct numerical simulation of the stress field. In general, the basic method to deal with multiscale transport problems such as the composite alloy solidification process is to numerically simulate the transport behavior on the largest scale. Generally, the basic method for dealing with multiscale transport problems such as the solidification process of composite alloys is to numerically simulate the transport behavior at the maximum scale. Then, the solute is distributed in a small space, and its characteristic parameters are modeled. The modeled characteristic parameters are considered in the macroscale solute transport modeling equation [19][20][21]. Figure 1 shows the microplasma of the alloy solidification transport model. The unified model for the solidification transport phenomenon of composite alloy established in this study introduces the liquid phase volume fraction f L into all transport equations, and takes into account the solidification shrinkage effect and the influence of gravity on it. The same group of differential equations is used to express various transport laws occurring in the liquid phase region L, paste region + L S, and solid phase region S of binary alloy ingot solidified in the dendrite mode. To facilitate the study, it is assumed that external forces on the solidification system are only gravity and electromagnetic forces. There is no air hole in the area included in the calculation, that is, the casting space is continuous (formula (1)).
The solid phase is in a rigid state, and relative to the attachment, it exhibits a relatively static state during the solidification process. The composite alloy is binary or can be simplified into binary. The influence of the electromagnetic field on the solidification transfer behavior of composite alloys is mainly embodied in Joule heat and Lorentz force. Joule heat directly affects the solidification heat transfer process of composite alloys, and Lorentz force affects the flow behavior of liquid metal. The unified model of solidification and transport of composite alloys proposed in this article is combined with Maxwell's equations. A solidification and transport model of composite alloy continuous media for continuous casting slabs under a certain casting speed and electromagnetic field is derived, with the following heat transfer equation (formula (2)).
In formula (2), ρ indicates the charge density, m is solute, and t indicates a time variable. V 0 indicates the drawing speed and V represents the flow velocity. T is the freezing temperature, and q J is Joule heat. c P means the specific heat capacity of the object, and k is the equilibrium distribution coefficient, The mass transfer equation is as follows: In formula (3), D L and D S are the diffusion coefficients of paste solute and solid solute, respectively. C L and C S represent paste solute and solid solute concentration, respectively. The solidification characteristic function of composite alloy is expressed in the following equation: In formula (4), where C 0 represents the original component. The mass conservation equation is expressed as follows: x v x y z Figure 1: Alloy solidification transport model microplasma.
Optimization of structural parameters and numerical simulation  3 To quantitatively describe the solidification transmission process under the action of electromagnetic field, it is necessary to calculate the solidification temperature T , solid fraction f S , pressure P, solute concentration C L , flow velocity V , current density field J , and magnetic induction B. The seven basic physical field quantities are related to each other through the corresponding differential equations and various constitutive relations of solidification transport. The decisive factor is a pair of interactive coupling relations: The relationship between these field quantities is shown in Figure 2.
The momentum conservation equation describing the flow field is given as follows: In Eq. (7), ϕ represents u or v, I represents diffusion coefficient, and + S S ϕ c p represents the linearized source term. The calculation of electromagnetic solidification transmission coupling includes the analysis of the interaction between multiple engineering physical fields. Direct coupling and indirect coupling are two common methods for multifield coupling problems. In general, direct coupling only includes a set of coupling analysis, and the coupling functional unit it uses contains all unknown field quantities to be solved. The element matrix containing the required physical quantities is calculated and coupled. The indirect coupling includes two or more groups of coupling analysis, which couples two physical fields by applying the results of one analysis as loads to another analysis. Indirect coupling is more flexible and effective for multifield coupling with a low degree of nonlinearity. It requires more details to be defined. The transmission load needs to be set by oneself. The analysis between fields is relatively independent, and the coupling between them is bidirectional. One can transfer loads between different grids and analyses. To improve the melt flow in the crucible, harmonics are added in the electromagnetic casting process/traveling wave magnetic field or static magnetic field. A harmonic/traveling wave magnetic field has a relatively low frequency or wavelength λ. Much larger than the size of the crucible L. Therefore, the required static conditions are met. The magnetic Reynolds numberRe m is expressed in formula (8): In formula (8), V 0 and L 0 represent the characteristic speed and dimension of the system, respectively. v m represents the magnetic diffusion coefficient. At this time, the magnetic induction equation is given as follows: Formula (9) shows that the magnetic field is independent of the melt velocity. When calculating the numerical value of electromagnetic continuous casting, the original two-way coupling process of electromagnetic solidification transmission can be simplified as the unidirectional influence of electromagnetic field on the solidification transmission process. The magnetic field can be solved without considering the solidification transmission equation temporarily.

Optimization of structural parameters and numerical simulation of stress field
Throughout the electromagnetic continuous casting, the composite alloy melt flows out of the tundish and flows into the molten pool of the mold through the gap of the back wall of the mold under the action of gravity. The water-cooled copper strip moves continuously and uniformly at the speed of V 0 . The chilled solidified steel condensate shell is attached to the copper strip. The condensate shell and melt pass through the gap formed between the front of the mold and the copper strip under the combined action of the melt indenter, the copper strip drawing, and the adhesion force, and realize continuous rolling during the strip solidification cooling process. The length of crystallizer x is 0.2 m, and its sectional structure is shown in Figure 3.
To control the smooth flow of the melt, its impact on the condensate shell is reduced at the outlet and stable isokinetic supply of outflow is provided, and three static magnetic fields were used to brake the melt. They are horizontal strip magnetic field, vertical strip magnetic field, and horizontal and vertical L-shaped magnetic field. The horizontal and vertical strip magnetic field positions are shown in Figure 4.
The magnetic armature is close to the mold wall x. The direction length is 0.05 m. The static magnetic field controls the magnetic induction intensity by adjusting the turns and current of the DC coil around the armature. In the electromagnetic continuous casting, once the melt flows into the chilling zone, the heat of the melt will be transmitted from the melt surface and the copper strip to the outside through radiation and conduction. The position of the L-shaped magnetic field is shown in Figure 5.
Regardless of other factors affecting heat transfer, according to previous studies, the equivalent heat transfer coefficient is used for heat transfer of melt surface and substrate K s and K b . It is approximately assumed that the wall is partially adiabatic and the initial state is filled with melt with uniform temperature, and the velocity and temperature at the inlet are also uniform. When the local unidirectionality is assumed and the outlet interface is set at a place far away from the outlet where there is no return flow, the coefficient on the boundary of the discrete equation is equal to 0. When the velocity becomes a variable to be solved, the solution process does not need to know the outlet normal phase velocity. The study assumes that the relative rate of change of the normal phase velocity of the control volume on the outlet section is a constant, so  it is assumed that there is a relationship between different i points on the outlet section, as shown in Eq. (10).
Let v i M , 1 satisfy the overall mass conservation condition to determine the coefficient f , as shown in Eq. (12).
In Eq. (12), flow in indicates inflow quality and Δx Δz i k is the cross-sectional area, from which Eq. (13) is obtained: The calculation process of indirect coupling of electromagnetic solidification transmission also needs to make the following assumptions: the casting needs to be composed of solid and liquid phases, that is, the sum of the two phase volume fractions. The liquid phase needs to belong to the Newtonian fluid type and needs to belong to the laminar flow state. The solidified surface should be nonslip surface. The state of the melt is noncompressible. The thermophysical parameters are constant, and the solidification physical parameters need to exclude the effect of the magnetic field. The velocity formed by temperature and inflow must be in a uniform state, and the outlet is a unidirectional locally assumed outlet. At the same time, the tension effect on the melt surface should be ignored.
Convection has a strong directionality, and the processing of nonlinear convection terms involves the discrete format of the convection term, while the processing of the pressure gradient term in the momentum equation is related to the coupling relationship between pressure and velocity. From the perspective of the numerical calculation process and results, the dispersion of convection terms is related to three aspects of the characteristics of the solution process, including the accuracy, stability, and economy of the numerical solution. The study uses a first-order upwind scheme to discretize the control equation. To reduce the impact of false diffusion, a fine mesh discrete model is used. The continuous equation is discretized using an implicit scheme to obtain Eq. (14).
4 Numerical simulation of electromagnetic solidification transmission coupling model A unified numerical model for electromagnetic solidification transmission coupling has been established. Based on the staggered network, a method suitable for coupling calculation has been obtained built on the finite element volume method. In this section, the actual numerical value of electromagnetic solidification transmission coupling will be calculated and analyzed using this method.

Effect of different types of virtual magnetic fields on solution flow
When analyzing the effect of electromagnetic solidification transfer calculation, the influence of a three-dimensional model on solidification transfer behavior in steel continuous casting will be numerically simulated. In the simulation process, the study will analyze the flow state, flow velocity, and model calculation efficiency of the melt in the magnetic field. Then, judging whether the magnetic solidification transmission coupling model designed in the study can effectively simulate the numerical changes of electromagnetic casting. First, the melt flow in the magnetic field is simulated. The effect of different virtual magnetic field types on the braking force of melt flow is shown in Figure 6.
From Figure 6, in the horizontal strip virtual magnetic field, the electromagnetic force has greatly hindered the vertical up and down flow of the melt. Therefore, the overall melt flow is the most dense at the upper left, more dense at the lower right, and more sparse at other parts. The overall flow state is uneven. In this flow state, the supply of melt is difficult to rely on the convection supply without a magnetic field and can only rely on the overall sinking supply. In the virtual magnetic field of the vertical strip, the electromagnetic force has a great impact on the transverse flow of the melt. As a result, the overall melt flow is more concentrated on the left side and more concentrated on the edge than on the middle. The overall flow state is also uneven. In this flow state, the reflow zone of the melt is compressed to a greater extent and then forced to move toward the back wall. Under the influence of the L-shaped virtual magnetic field, compared with the horizontal strip virtual magnetic field and the vertical strip virtual magnetic field, the electromagnetic force has played a certain role in guiding the overall flow of the melt. Therefore, the overall melt flow presents a situation that the reflux area is significantly reduced and gradually moves backward, and the convection at the meniscus is increased. At the same time, the channel through which the melt flows out is relatively wide, so the overall flow state is relatively uniform. This state is particularly evident in the upper part. At the same time, the effect of the virtual strip magnetic field on the melt flow velocity was analyzed, as shown in Figure 7. the left side is greater than that on the right side. Although the overall melt flow state has improved, the flow state is still uneven, and fluency is also lacking. Under the influence of the L-shaped virtual magnetic field, because the electromagnetic force guides the overall flow of the melt to a certain extent, the flow velocity of the melt reflects a more uniform L-shaped flow channel, which is closer to the outside. The flow velocity of the melt in the channel is relatively high. The overall flow situation is more smooth and faster than the horizontal strip virtual magnetic field and the vertical strip virtual magnetic field.

Influence of melt on the impact velocity of condensate shell at the exit under magnetic field
The study further analyzed the influence of the melt on the impact velocity of the condensation shell at the outlet under the applied magnetic field, and the specific results are shown in Figure 8. From Figure 8, in a horizontal strip virtual magnetic field environment, only the magnetic field velocity component at 0 T affects the dotted line in a state of sharp decline.
The other magnetic field velocity components at 0.1, 0.2, and 0.4 T have no significant impact on the dotted line. In the true magnetic field environment of a horizontal bar, the magnetic field velocity components of various orders of magnitude have a significant impact on the dotted line, and the downward trend is consistent. Its downward trend is relatively uniform and has a high change synchronization rate. In a vertical strip virtual magnetic field environment, only the magnetic field velocity component at 0 T affects the dotted line in a state of sharp decline. The other magnetic field velocity components of 0.1, 0.2, and 0.4 T affect the dotted line. Although the decline is not significant, it is still a more significant downward change than the horizontal strip virtual magnetic field environment. In the vertical strip real magnetic field environment, the magnetic field velocity component of all orders of magnitude affects the decline of the broken line obviously. The decline trend is consistent, and the decline trend is more uniform and has a high change synchronization rate. However, the difference is that, with the enhancement of magnetic induction, the change gap between the broken lines of different orders of magnitude also appears. Compared with the virtual magnetic field, the braking effect of the real magnetic field on downward impact is poor. This is mainly because the existence of the electromagnetic force changes the distribution of the relative pressure, which leads to the inconsistency of the flow pattern and speed in the central area and the edge area, forming a negative pressure state at the outlet. This results in a velocity greater than the virtual magnetic field at the exit in the horizontal and downward impact directions. The overall flow correction effect is insufficient. The influence of the melt on the impact velocity of the condensation shell at the outlet under the L-shaped virtual magnetic field environment is shown in Figure 9. From Figure 9, under the L-shaped virtual magnetic field environment, the magnetic field velocity component influence polyline at 0 T is in a state of sharp decline. Although the magnetic field velocity component-influenced polylines at 0.1, 0.2, and 0.4 T are not significantly reduced, they are still significantly reduced compared with the horizontal strip virtual magnetic field environment and the vertical strip virtual magnetic field environment. In the vertical L-shaped real magnetic field environment, the decline of the magnetic field velocity component-influenced polyline of all orders of magnitude is very obvious, and the downward trend is consistent. The downward trend is relatively uniform and has a high change synchronization rate, similar to the vertical strip virtual magnetic field environment. With the enhancement of magnetic induction, the change gap between different orders of magnitude of polyline also appears, and the gap is more significant. It shows that the correction ability of melt velocity is relatively better in the L-shaped magnetic field environment. Because the coupling transportation of electromagnetic solidification and alkali hydrolysis is mainly carried out in the form that computers can only calculate, the calculation efficiency needs to be considered to some extent in the process of data simulation. The calculation environment selected for the calculation efficiency analysis is 2.1 GHz CPU main frequency and 2.0 GB memory. The comparison of operation efficiency is shown in Figure 10.
From Figure 10 in terms of file output, the number of single file outputs of the research and design method is higher when the file output frequency is consistent with that of traditional methods. The research and design method has certain performance advantages in terms of file output performance. In addition, the research and design method has a solidification calculation termination time of 40 s in terms of solidification transmission calculation termination time and electromagnetic field calculation time. The calculation time of electromagnetic field is 540 s, which is lower than that of traditional methods. So the research and design method has higher calculation efficiency and shorter calculation time under the same calculation effect. In addition, from the perspective of overall calculation efficiency, the conversion calculation time of the research and design method is 18.03 min, and the total calculation time is 680.48 min. Compared with the traditional method, the time of both methods is shorter. It means that the research and design model has higher operation efficiency, faster operation speed, and better performance from both the perspective of local operation and the perspective of the overall operation.

Conclusion
Aiming at the phenomenon of magnetic field solidification transmission, a coupling model of electromagnetic solidification transmission based on the indirect coupling method was established. Meanwhile, a three-dimensional continuous casting model was established using the three-position magnetic field finite element theory. The electromagnetic transmission and electromagnetic braking phenomena in the three-dimensional continuous casting model were simulated using the indirect coupling method. The original twoway coupling was transformed into a single coupling embodiment, and the indirect coupling analysis scheme was established. The results show that the L-shaped static magnetic field has stronger restraining and guiding effects on melt circulation under the virtual magnetic field and real magnetic field. The braking effect of the real magnetic field on the downward impact is worse than that of the virtual magnetic field. At the same time, in the L-shaped virtual magnetic field, the melt flow velocity reflects a more uniform L-shaped flow channel. The flow velocity of the melt in the channel is relatively high. The reflux area is significantly reduced, the convection condition at the meniscus is improved, and the channel through which the melt flows out is widened. The overall flow state is more uniform than the other two magnetic field states. In addition, regarding the calculation efficiency of the method, the single file output of the research and design method is 5. The output file frequency is 50. The solidification calculation termination time is 40 s. The electromagnetic field calculation time is 540 s. The conversion calculation time is 18.03 min. The total calculation time is 680.48 min. It is superior to the traditional method. All the aforementioend data indicate that the electromagnetic solidification transmission coupling model based on the indirect coupling method is more concise and efficient. It can effectively analyze the magnetic field coupling problem in the electromagnetic casting process of the composite crucible structure.
Funding information: Not applicable.

Conflict of interest:
The author declares that he has no competing interest.