A method for electromagneto-thermo- mechanical direct coupling analysis and its application to Tokamak structures

Abstract

In this work, a numerical method using a direct coupling algorithm is proposed for electromagneto-thermo-mechanical (EMTM) coupling analysis of structures in strong magnetic and thermal field. Both the electromagneto-mechanical (EMM) coupling effect and thermal effect are considered in the structural dynamical analysis. To realize this algorithm, a direct coupling element including degrees of freedom of displacement, temperature and electromagnetic fields is developed on the platform of the ANSYS user programmable features and verified by the comparison between the experimental data and simulation results of a typical EMTM problem. Finally, the EMTM dynamic response of a simplified model of the HL-2M vacuum vessel during plasma vertical disruption events is simulated. According to the simulation results, the EMTM coupling effect is found significant and necessary to be considered for the design and safety evaluation of Tokamak structures.

Keywords

Direct coupling algorithm electromagneto-thermo-mechanical coupling effect thermal effect numerical simulation Tokamak device

1. Introduction

Tokamak devices run in a high magnetic field with huge plasma current, high confined magnetic field and high temperature. Due to the instability of plasma, plasma disruptions (PDs) may happen and the plasma vertical displacement event (VDE) is one of major type of PDs. During VDEs, not only current quench but also heat quench will happen. On one hand, huge plasma current will decrease to zero in milliseconds and huge eddy currents will be induced in the in-vessel components of Tokamak device. This is called current quench. These eddy currents interact with the high confinement magnetic field, and huge electromagnetic (EM) forces will be generated on these components. On the other hand, plasma will move downward or upward more quickly than current quench and plasma of huge heat energy will strike on the surface of components as heat flux. This is called heat quench. Meanwhile, Joule heat will also be induced in the components due to the eddy currents. Under the heat flux and Joule heat, the temperature will rise and huge thermal stress may be acted on the in-vessel components too. Thus, such huge and complex EM loads and thermal loads will act on these structures at the same time which may directly lead to material failure and configuration instability. Therefore, the electromagneto-thermo-mechanical (EMTM) coupling effect is quite important for the design and safety evaluation of Tokamak structures. The key of EMTM coupling analysis is the treatment of the electromagneto-mechanical (EMM) coupling effect and the thermal effect. The EMM coupling effect means the influence to EM field due to structural movement or deformation. Since the Team Problem 12 and 16 were proposed, many studies have been done on the EMM coupling analysis [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The thermal effect is the variation of material constants such as the electric conductivity and the Young’s modulus due to the temperature change [14].

The EMTM coupling is a multi-fields coupling problem that needs to be solved by using sequential coupling method or direct coupling method. Series studies have been done about EMM and EMTM coupling through conventional sequential coupling method [15, 16, 17]. However, these studies did not consider the EMM coupling effect and the thermal effect at the same time and the methods are difficult to be applied to the simulation of complex structures such as in-vessel Tokamak components. The Lagrangian method, as a sequential coupling method, is based on the governing equations for eddy current analysis of deformable bodies in Lagrangian description and staggered load transfer for simulating coupling effect [8]. It can consider both the EMM coupling effect and the thermal effect at the same time but requires a lot of computing resources and time. On the other hand, there is almost no literature dealing with the EMTM coupling analysis with the direct coupling method. In this paper, the EMTM direct coupling method was proposed at first and a direct coupling element was then developed on the ANSYS user programmable features (UPFs) platform to realize the direct coupling. The proposed method and the developed element were verified through comparing the simulation results with the experimental data. At last, the algorithm was used to simulate the dynamic EMTM coupling responses of a simplified HL-2M vacuum vessel (VV) model during the plasma VDE.

2. Methods

The EMTM coupling analysis consists of the calculations of three fields, i.e., EM field, thermal field and mechanical displacement field. In EM field calculation, the additional motional eddy current is introduced into the governing equations of EM field to consider the EMM coupling effect, which can be described as

$\displaystyle\nabla\times\frac{1}{\mu}\nabla\times\vec{A}+\sigma\left({\frac{% \partial\vec{A}}{\partial t}+\nabla\varphi}\right)=\vec{J}_{v}$

(1) $\displaystyle\nabla\cdot\sigma\left({\frac{\partial\vec{A}}{\partial t}+\nabla% \varphi}\right)=\nabla\cdot\vec{J}_{v}$

where $\vec{A}$ and $\varphi$ are respectively the magnetic vector potential and electric scalar potential; $\sigma$ and $\mu$ are the electric conductivity and magnetic permeability; $\vec{J}_{v}$ is the motional eddy current induced by the structural movement in a strong magnetic field. The motional eddy current $\vec{J}_{v}$ is in form

$\displaystyle\vec{J}_{v}=\sigma\left[{\vec{v}\times\left({\vec{B}_{e}+\vec{B}_% {0}}\right)}\right]$ (2)

where $\vec{v}$ is the structural velocity, the induced magnetic flux intensity $\vec{B}_{e}=\nabla\times\vec{A}$ ; and $\vec{B}_{0}$ is the applied static magnetic field. In the subsequent discretization of Eq. (2), the time integral of electric scalar potential is employed to make the corresponding coefficient matrices symmetrical, and the Coulomb gauge is included to keep the magnetic vector potential $\vec{A}$ unique [18].

For the temperature field calculation, the governing equation and boundary condition are

$\displaystyle\rho c\frac{\partial T}{\partial t}=k\nabla^{2}T+q$ (3) $\displaystyle\frac{\partial T}{\partial n}=-\frac{q_{s}}{k}$ (4)

where $T$ respects the temperature; $\rho$ , $c$ and $k$ are the mass density, specific heat capacity and heat conductivity respectively; $q_{s}$ and $q$ are the heat flux density applied on the surface of conductor and the Joule heat density induced by eddy current respectively. The Joule heat is defined as

$\displaystyle q=\frac{\left|\vec{J}\right|^{2}}{\sigma}$ (5)

where $\vec{J}$ is the sum of eddy current density and is defined as

$\displaystyle\vec{J}=\sigma\left[{-\frac{\partial\vec{A}}{\partial t}-\nabla% \varphi+\vec{v}\times\left({\vec{B}_{e}+\vec{B}_{0}}\right)}\right]$ (6)

As for displacement field calculation, the dynamic equilibrium equation is as follows

$\displaystyle\frac{E}{2(1+\nu)}\nabla^{2}\vec{u}+\frac{E}{2(1+\nu)(1-2\nu)}% \nabla(\nabla\cdot\vec{u})+\vec{f}_{em}+\vec{f}_{T}=\rho\frac{\partial^{2}\vec% {u}}{\partial t^{2}}$ (7)

where $\vec{u}$ is the displacement vector; $E$ and $\nu$ respect the Young’s modulus and possion’s ratio respectively; $\vec{f}_{em}$ and $\vec{f}_{T}$ are respectively the thermal loads and EM forces of form

$\displaystyle\vec{f}_{em}=\vec{J}\times\left({\vec{B}_{e}+\vec{B}_{0}}\right)$ (8) $\displaystyle\vec{f}_{T}=-\frac{\alpha E}{1-2\nu}\nabla T$ (9)

where $\alpha$ is the coefficient of thermal expansion.

To consider the thermal effect, the variations of the Young’s modulus and the electric conductivity due to temperature are considered by using the following relations

$\displaystyle\sigma=\sigma_{0}/(1+\beta\cdot\Delta T)$ (10) $\displaystyle E=E_{0}\left({1-m\alpha_{0}\Delta T+0.5\left({m\alpha_{0}\Delta T% }\right)^{2}}\right)$ (11)

where $\Delta T$ respects the temperature change; $\sigma$ and $E$ respect the electric conductivity and Young’s modulus at temperature $T$ ; $\sigma_{0}$ , $E_{0}$ and $\alpha_{0}$ respect the electric conductivity, Young’s modulus and thermal expansion coefficient at reference temperature $T_{0}$ ; $\beta$ is the temperature coefficient and $m$ is a constant obtained through mathematical statistics for different materials. $m$ is usually about 24.66 for a metallic material.

Based on the theories referred above, the governing equations of three fields are discretized at the same time by using finite element method (FEM). The discrete system of linear governing equations for the direct coupling analysis which includes all the degrees of freedom (DOFs) of three fields is as follows

$\displaystyle[M^{M}]\left\{{{\begin{array}[]{{20}c}\ddot{u}\\ \ddot{T}\\ \ddot{A}\\ \ddot{\Phi}\\ \end{array}}}\right\}+\left[{{\begin{array}[]{{20}c}C^{M}\\ &C^{T}\\ &&C^{AA}&C^{VA}\\ &&C^{AV}&C^{VV}\\ \end{array}}}\right]\left\{{{\begin{array}[]{{20}c}\dot{u}\\ \dot{T}\\ \dot{A}\\ \dot{\Phi}\\ \end{array}}}\right\}+\left[{{\begin{array}[]{{20}c}K^{M}\\ &K^{T}\\ &&K^{AA}&0\\ &&&0\\ \end{array}}}\right]\left\{{{\begin{array}[]{{20}c}{u}\\ {T}\\ {A}\\ {\Phi}\\ \end{array}}}\right\}$ (12) $\displaystyle\quad=\left\{{{\begin{array}[]{{20}c}F^{EM}+F^{T}\\ Q^{EM}+Q^{S}\\ J^{vA}\\ J^{v\Phi}\\ \end{array}}}\right\}$

where $u$ , $T$ , $A$ and $\Phi$ are DOFs of displacement, temperature, magnetic vector potential and electric scalar potential respectively. [ $M$ ], [ $C$ ] and [ $K$ ] are the coefficient matrices related to the second, the first derivatives of the field variables, and the variables themselves (corresponding to mass, damping, stiffness matrices) and { $F$ }, { $Q$ }, { $J$ } are vectors of the excitation force, incident heat flux and excitation current density respectively.

Following the formulae above, a finite element including all the DOFs of displacement, temperature and EM field is developed on the ANSYS UPFs platform. The UserElem.F subroutine provided by ANSYS is rewritten and compiled based on the formulae above. After that, this subroutine is available to be called directly in ANSYS software as a user-defined element to fulfil the needs of the direct coupling method.

Table 1
Element information

Solving area Element type Node number Number of DOFs Element shape Source

of an element DOFs

Conductor area UserElem300 20 8 AX-AY-AZ- $\Phi$ - Hexahedron Developed by user

Temp-UX-UY-UZ

Air area Solid97 8 3 AX-AY-AZ Any Owned by ANSYS

The calculation areas in EMTM coupling analysis include conductor and air area. The new developed element has 8 DOFs for conductor area, and the original element has only 3 DOFs for air area in the EM field analysis. It is worth noting that element with 20 nodes has to be chosen for treating conductor area since 8-node element is not suitable for dynamic response analysis, while SOLID 97 with 8 nodes is the only node-based element available in ANSYS for EM field analysis. To correspond the field values of the DOFs between the 8-node element in the air area and the 20-node element in the conductor area, a scheme of hybrid element was proposed, i.e., shape functions of both the 8-node and the 20-node finite elements are included in the 20-node element in conductor area. The shape functions of 20-node element are used for the discretization of displacement and temperature field and those of the 8-node element are used for EM field. As for the correspondence of the values between the temperature, displacement with 20-node element shape function and those of the EM field with 8-node element shape function in conductor area, such as EM force, joule heat and motional eddy current, the value at the middle node of each edge was set as the average of the values of the two nodes at the ends of the same edge. The information of adopted elements is shown in Table 1. In addition, at each time step, the material constants are updated according to the new temperature distribution and the relative coefficient matrices are reformed to cope with the thermal effect. In this way, a direct coupling element was developed which is suitable to simulate the EMTM coupling dynamic response problem of a structure in transient magnetic field and pulsed heat flux on the ANSYS platform.
3. Verification

Solving area	Element type	Node number	Number of	DOFs	Element shape	Source
		of an element	DOFs
Conductor area	UserElem300	20	8	AX-AY-AZ- $\Phi$ -	Hexahedron	Developed by user
				Temp-UX-UY-UZ
Air area	Solid97	8	3	AX-AY-AZ	Any	Owned by ANSYS

The validity of the new element is verified through comparison of the experimental and the simulation results. As pointed above, the EMM coupling effect and thermal effect are the focus points of EMTM coupling analysis. The treatments of these two effects are verified respectively.

3.1 EMM coupling effect

Firstly the dynamic responses of the Team Problem 16 were simulated to validate the EMM coupling effect of the direct coupling element. Team Problem 16, as shown in Fig. 1, is a standard problem of simplified first wall of Tokamak device. A copper plate (0.3 mm $\times$ 40 mm $\times$ 115 mm) with its bottom constrained 10 mm was put in the static uniform magnetic field along the y axis (By $=$ 0.0, 0.1, 0.2, 0.3, 0.4 T). A 27-turn coil was lift 9.5 mm off one side of the plate with a pulse exciting current $I=800(exp(-500t)-exp(-6000t))$ A. The height, outer and inner diameters of this solenoid coil are 24.2 mm, 22.0 mm and 20.0 mm, respectively. The material parameters of copper are shown in Table 2. Under the pulse current exciting, eddy currents will be induced in the plate and interact with the static magnetic field. Pulse EM force will then be induced which leads to the rotation and vibration of the plate. In calculation, time steps for displacement, temperature and EM field were set to be the same and small enough until the results did not change significantly as time steps reduced. Displacements of point A and B (0.0 mm, $\pm$ 7.5 mm, 108 mm) were monitored.

Table 2
Material parameters of copper

Mass	Young’s	Possion	Relative	Electric	Heat	Heat	Coef.Ther.	Temperature
density	modulus	ratio	permeability	conductivity	conductivity	capacity	Expansion	coefficient
$\rho$ kg/m ${}^{3}$	$E_{0}$ GPa	$\upsilon$	$\mu_{r}$	$\sigma_{0}$ S/m	$k$ W/(m $\cdot$ K)	$c$ J/(kg $\cdot$ K)	$\alpha_{0}$ 1/K	$\beta$ ppm/K m
8610	110	0.34	1.0	5.81 $\times$ 10 ${}^{7}$	295	390	1.89 $\times$ 10 ${}^{-5}$	0.0039

Figure 1.

The model of team problem 16.

Figure 2.

Displacements at point A. (a) By $=$ 0.2 T; (b) By $=$ 0.4 T.

Figure 2 shows the results of point A including experiment [3], direct coupling method, Lagrangian method [8] and ANSYS without coupling with By $=$ 0.2 T and 0.4 T respectively. It can be seen that the direct coupling results meet well with the experimental results as well as the Lagrangian method results, but the ANSYS results are far away from the experimental results because the conventional commercial software can’t consider the EMM coupling effect easily. It can be concluded that the proposed direct coupling method and the developed element can treat EMM coupling effect properly. Additionally, although both the direct coupling method and Lagrangian method have the same accuracy, the former can save much calculating memory and reduce much computational burden to solve the EMM coupling problems because applied static magnetic field can be applied through setting the real constants of element in direct coupling method while helmholtz coils have to be modeled to produce the magnetic field in Lagrangian method and the coefficient matrixes need not to be reformed in direct coupling method while those of Lagrangian method have to be reformed to consider the EMM coupling effect in at each time step.

3.2 Thermal effect

As for the thermal effect, a simplified EMTM coupling experiment was carried out. As shown in Fig. 3, the experimental object is a copper plate (0.3 mm $\times$ 10 mm $\times$ 118 mm) with a 100-turn square coil which is 10 mm off the plate in the one side and a laser heat source whose maximum power is 100 W in the other side and the bottom of the plate is constrained. While the coil was excited by the pulse current generator, the eddy current and magnetic field would be induced in the plate. Meanwhile, when the laser source was triggered, the heat flux was irradiated onto the face and the temperature of the plate would change. The laser displacement sensor was adopted to monitor the displacement of point P (0.15 mm, 0.0 mm, 103 mm) on the plate.

Figure 3.

Experimental system for EMTM coupling.

Figure 4.

The monitored exciting current and heating power. (a) Exciting current; (b) Heating power.

Figure 5.

Experimental results with and without heating.

Figure 4 shows the monitored signals of exciting current monitored by the current transformer and the heating power monitored by the photoelectric sensor. As for that the heat power of laser source is 100 W and is not strong enough, the plate was preheated 2 seconds early by the heat source before the pulse current was applied to make the thermal effect more significant. Figure 5 shows the experiment results with and without heating respectively. From this figure, the displacement curve with heating has lower amplitude and longer vibration period compared with the result without heating. It can be seen that the thermal effect makes a big influence in this experiment. As for that there is no applied static magnetic field, the EMM coupling effect is not significant in this experiment.

Then the experiment progress was simulated using the developed element. Figure 6 shows the comparison of displacements obtained by experiment and simulation with and without heat respectively. Figure 6b also shows the temperature change at the centre of heating area and the maximum temperature is about 800 ${}^{\circ}$ C. Although the simulation accuracy tends to decline slightly due to the error accumulation as time increases, these two results meet well with each other both with and without heating, which reveals that the thermal effect was treated properly.

Above all, the comparison of the experiment and simulation results of EMM and EMTM coupling analysis proves the validity of the proposed method and developed element respectively.

4. Simulation of simplified VV of HL-2M Tokamak

To study the influence of EMTM coupling effect on safety evaluation of Tokamak device, the EMTM dynamic response of a simplified HL-2M vacuum vessel during plasma VDE with both current quench and thermal quench was simulated using the direct coupling element. Designed and under construction by the Southwestern Institute of Physics (SWIP) of China, HL-2M is a new experimental Tokamak device which can realize relative complicated divertor plasma configuration. The major design parameters of HL-2M are 2.5 MA plasma current, 2.2 T toroidal magnetic field, 5 s plasma duration time, 1.78 m major radius, 0.65 m minor radius, 14 Vs flux swing, 1.8–2.0 elongation and bigger than 0.5 triangularity [19, 20]. As shown in Fig. 7a, the VV of HL-2M is a D-shaped, double thin-wall structure consisting of inner shell, outer shell, reinforcing ribs and ports. The donuts shaped VV is 5.22 m in outer diameter, 3.02 m in height and the total wall thickness is 20 mm (the thickness of the outer shell, inner shell and the distance between the two shells are 5 mm, 5 mm and 10 mm respectively). Inconel 625 steel is chosen to be the material of the VV and material parameters of Inconel 625 steel are shown in Table 3. Besides, Fig. 2b shows the locations of coils of HL-2M including 16 poloidal field (PF) coils, 20 toroidal field (TF) coils and one centre solenoid (CS) and the initial plasma location.

Table 3
Material parameters of Inconel 625

Mass	Young’s	Possion	Relative	Electric	Heat	Heat	Coef.Ther.	Temperature
density	modulus	ratio	permeability	conductivity	conductivity ${}^{*}$	capacity ${}^{*}$	Expansion ${}^{*}$	coefficient ${}^{*}$
$\rho$ kg/m ${}^{3}$	$E_{0}$ GPa	$\upsilon$	$\mu_{r}$	$\sigma_{0}$ S/m	$k$ W/(m $\cdot$ K)	$c$ J/(kg $\cdot$ K)	$\alpha_{0}$ 1/K	$\beta$ ppm/K m
7860	189	0.3	1.0	1.4 $\times$ 10 ${}^{6}$	295	390	1.19 $\times$ 10 ${}^{-5}$	0.0039

Figure 6.

Experimental results with and without heating. (a) Without heating; (b) With heating.

Figure 7.

The VV of and locations of coils of HL-2M Tokamak device [21]. (a) Vacuum Vessel of the HL-2M; (b) Coil System of HL-2M.

Figure 8.

Simplified simulation model.

Figure 9.

Temperature distribution at time distant 120 ms.

Figure 10.

Displacement distribution at time distant 120 ms.

Figure 11.

Displacements with and without considering the EMM coupling effect while considering the thermal effect.

Figure 12.

Displacements with and without considering the thermal effect while considering the EMM coupling effect.

For simplification, a 5-degree model was set up consisting of VV, 16 PF coils, equivalent plasma current and air area (not shown) as shown in Fig. 8. The HL-2M VV with two-layer structures was simplified to one single layer structure which is 10 mm thick and the ports were ignored. To simulate the condition of plasma VDE, a total of 2.5 MA plasma current was set to decrease to zero in 20 milliseconds and a pulse heat flux 500 MW/m ${}^{2}$ was applied on the inner face of the bottom of the VV and was set to decrease to zero in 8 milliseconds as for that the heat quench is faster than the current quench. Besides, a static magnetic field of 2.2 T along the toroidal direction was applied to replace the field produced by TF coils.

As for the boundary condition, the cyclic symmetry boundary condition was considered on both sides of the 5 degree model. All DOFs of each pair of nodes on the 0 degree side and the 5 degree side were coupled by the “CPCYC” command of ANSYS. In addition, infinite elements were used in the outer layer of the model to simulate the infinite EM boundary. Besides, the nodes located in a small region at the middle outer surface of the VV (about 0.32 $\times$ 0.5 m ${}^{2}$ ) were restrained in both poloidal and toroidal directions to simulate the VV support.

Figure 9 shows the temperature distribution at 120 milliseconds. From this figure, it can be seen that the temperature in heating area is about 290 ${}^{\circ}$ C and is far higher than that in no heating area. Figure 10 shows the displacement distribution at 120 milliseconds. From Fig. 10, the VV moves and vibrates mainly in upright direction and the maximum displacement is about 7.3 mm. Figure 11 shows the displacements with and without considering the EMM coupling effect while considering the thermal effect. It can be seen that the vibration amplitude with considering the EMM coupling effect is smaller and decreases with the time going compared with that without considering the EMM coupling effect. Besides, the vibration period with considering the EMM coupling effect is longer. That is, both magnetic damping effect and magnetic stiffness effect are significant in this situation. Figure 12 shows the results respectively with and without considering the thermal effect while considering the EMM coupling effect and the temperature change of the heating area. It can be seen that the vibration amplitude is smaller but the vibration period changes little when considering the thermal effect. That is, the change of electric conductivity due to temperature variation made a great influence on structural vibration but the change of Young’s modulus not because the whole stiffness changes little as for that the heating area is partial. Therefore, the influence of thermal effect does make a great influence on the safety evaluation of Tokamak structures.

5. Conclusions

In this paper, a direct coupling method for EMTM coupling analysis with consideration of both the EMM coupling effect and the thermal effect at the same time is proposed and a new finite element is developed on the ANSYS UPFs platform for the direct coupling analysis. The validity of the proposed numerical method and developed element are verified through comparison between numerical simulations and experiments. The advantages of less memory requirements, less computational burden and the possibility to be applied to complex structures of this method are also demonstrated through comparison. Finally, the dynamic EMTM response of a simplified model of HL-2M VV under plasma VDE with both current quench and thermal quench is simulated by using the direct coupling method. Numerical results show that the magnetic damping effect, magnetic stiffness effect and thermal effect all have a significant influence on the vibration of HL-2M VV under plasma VDE, which reveals the importance to consider the EMTM coupling effect in the structure design. The proposed direct coupling method and developed element provide a feasible approach for the design and safety evaluation of Tokamak structures.

Footnotes

Acknowledgments

The authors would like to thank the National Magnetic Confinement Fusion Program of China (2013GB113005), National Natural Science Foundation (11502192, 51577139), and Postdoctoral Science Foundation Program of China (2015M570819) for funding this study in part.

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A method for electromagneto-thermo- mechanical direct coupling analysis and its application to Tokamak structures

Abstract

Keywords

1. Introduction

2. Methods

3.1 EMM coupling effect

Table 2 Material parameters of copper

Table 3 Material parameters of Inconel 625

Footnotes

Acknowledgments

References

Table 2
Material parameters of copper

Table 3
Material parameters of Inconel 625