CN114254528A - Finite element analysis method for superconducting magnet quench process - Google Patents

Finite element analysis method for superconducting magnet quench process Download PDF

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CN114254528A
CN114254528A CN202010993657.8A CN202010993657A CN114254528A CN 114254528 A CN114254528 A CN 114254528A CN 202010993657 A CN202010993657 A CN 202010993657A CN 114254528 A CN114254528 A CN 114254528A
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magnet
finite element
conductor
analysis method
eddy current
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平学伟
李黎
殷兴辉
李昌利
刘海韵
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Hohai University HHU
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Abstract

The invention discloses a finite element analysis method for a superconducting magnet quench process. The method comprises the steps of firstly obtaining the change of current in a coil along with time in the quenching process, then establishing an electromagnetic field control equation and a heat conduction equation, and solving the two equations in sequence to obtain the eddy current density, the electromagnetic force and the temperature distribution of each moment in a magnet. The method fully considers the coupling of the electromagnetic field and the temperature field, and can accurately simulate the change of the eddy current density, the electromagnetic force and the temperature of the magnet after quenching along with the time. The method has high simulation precision and can provide effective guidance for reasonably designing the magnet structure.

Description

Finite element analysis method for superconducting magnet quench process
Technical Field
The invention relates to a finite element analysis method, in particular to a finite element analysis method for superconducting magnet quench problem.
Background
The problem of superconducting magnet quench is a very important problem and is an inherent problem of the superconducting magnet. If the process of generating a superconducting magnet is not mature, the number of quenches per magnet may reach 5 or more. Even with mature processes, artificial quench is sometimes required to assess the stability of the magnet. During a quench, the electromagnetic energy stored in the coil is substantially released, thereby generating a large amount of heat that volatilizes the liquid helium, causing a significant loss. In addition, large electromagnetic force is generated on the metal wall inside the magnet, so that elastic strain is generated on the cold shield, the end cover, the framework and the like. When the magnet structure is not designed reasonably, the quenching can damage the components in the magnet, thereby affecting the stability of the magnet. In severe cases even plastic strain may be imparted to some metal parts. Therefore, it is very necessary to analyze the magnet quench process. The analysis of the quench process of the magnet coils is currently focused mainly on the magnet coils themselves, and few documents consider the effect of the quench on surrounding conductors such as cold shields and the like. From the foregoing, it is clearly essential to analyze the stress and temperature distribution of metal components such as the cold shield during the quench process. The parameters such as the conductivity of the conductor in the quenching process can change along with the rise of the temperature, which undoubtedly increases the difficulty of accurate simulation.
Disclosure of Invention
The purpose of the invention is as follows: in view of the above-mentioned deficiencies of the prior art, the present invention aims to provide an effective method for analyzing a quench process of a superconducting magnet by using time domain finite elements. By adopting the method, parameters such as an electric field and a magnetic field generated in the quenching process of the superconducting coil, eddy current density and electromagnetic force generated on the conductor and the like can be analyzed, so that guidance is provided for reasonably designing a magnet structure.
The technical scheme is as follows: in order to achieve the purpose, the finite element analysis method for the quench process of the superconducting magnet, provided by the invention, comprises the steps of firstly obtaining a time-varying relation curve I (t) of current in a magnet coil after the magnet quenches; then, I (t) is used as an excitation source, the coordinate of the quenching starting moment is set as t-0, the initial temperature of each part in the magnet before quenching is determined, and a time domain numerical simulation algorithm is adopted to calculate the temperature at t-t1The density of eddy currents momentarily generated on the conductor within the magnet; then any point on the conductor within the magnet system is calculated by the following formula
Figure BDA0002691666900000011
To the heat generated by the density of the eddy current
Figure BDA0002691666900000012
Figure BDA0002691666900000021
In the above formula, the first and second carbon atoms are,
Figure BDA0002691666900000022
as a coordinate point
Figure BDA0002691666900000023
The electrical conductivity of the (c) electrode,
Figure BDA0002691666900000024
is composed of
Figure BDA0002691666900000025
The modulus of the eddy current density;
then solving the following heat conduction control equation to obtain t ═ t1Temperature distribution in conductor at time:
Figure BDA0002691666900000026
in the above formula, T is temperature, rho is material density, c is specific heat of the material, x, y, z are coordinates of three directions in a Cartesian coordinate system, and k isx,ky,kzIs a coordinate point
Figure BDA0002691666900000027
Thermal conductivity in three directions of x, y and z;
and re-determining the conductivity of the conductor at each coordinate point according to the temperature distribution, and then repeating the process to calculate the eddy current density and the temperature at the next moment until the calculation is finished.
Further, the time-varying curve of the current after the coil quench in the step (1) is obtained by sampling the current signal through a magnet monitoring device, or is obtained through an ellipsoid diffusion model.
Further, numerical simulation is carried out on the eddy current density at each moment by adopting a finite element method, and finite element modeling is carried out by adopting the following electromagnetic field control equation:
in the conductor region:
Figure BDA0002691666900000028
in the non-conductor region:
Figure BDA0002691666900000029
in the above formula, A is the vector magnetic potential generated by the eddy current density, V is the scalar potential, V is the reciprocal of the magnetic permeability,
Figure BDA00026916669000000210
as a coordinate point
Figure BDA00026916669000000211
Electrical conductivity of the metal, JsIs the current density, sign, in the coil
Figure BDA00026916669000000212
The operation of the rotation degree is represented,
Figure BDA00026916669000000213
the operation of the divergence is represented by the operation of divergence,
Figure BDA00026916669000000214
representing a gradient operation.
Further, the eddy current density in the conductor is calculated by the following formula:
Figure BDA00026916669000000215
further, when the maximum in the conductorDensity of eddy current satisfies
Figure BDA00026916669000000216
When the simulation is finished, epsilon is the set threshold value.
Further, for the heat conduction control equation, a tetrahedral mesh is adopted for subdivision, and a node basis function is adopted as an interpolation function of the temperature field.
And further, dividing by adopting a tetrahedral mesh, and taking an edge basis function as an interpolation function of the magnetic vector bit.
Further, the lorentz force density at any point on the conductor within the magnet system is calculated by the following equation:
Figure BDA0002691666900000031
in the above formula, the first and second carbon atoms are,
Figure BDA0002691666900000032
as a coordinate point
Figure BDA0002691666900000033
The density of the eddy currents in the (d),
Figure BDA0002691666900000034
as a coordinate point
Figure BDA0002691666900000035
The magnetic field of (a).
Has the advantages that: the analytical method of the invention has the advantages that: the method accurately considers the coupling of an electromagnetic field and a thermal field, can accurately obtain the parameter changes of eddy current density, Lorentz force, heat and the like after the magnet is quenched, and has high simulation precision.
Drawings
FIG. 1 is a flow chart of a method of an embodiment of the present invention.
Fig. 2 is a graph of current versus time after a magnet quench in an embodiment of the present invention.
FIG. 3 is a graph showing the axial variation of the eddy current density in the cold shield after a quench of 1.2s in an embodiment of the present invention.
FIG. 4 is a graph of Lorentz force density on the cold shield after a quench of 1.2s in an embodiment of the present invention as a function of axial variation.
Detailed Description
The invention relates to a method for analyzing a superconducting magnet quench problem. The invention provides a scheme for analyzing a superconducting magnet quenching process by adopting a time domain finite element. The analytical procedure is described below:
as shown in fig. 1, in the finite element analysis method for a superconducting magnet quench process disclosed in the embodiment of the present invention, a time-dependent change curve i (t) of a current in a magnet coil after a magnet quench is obtained; then, I (t) is used as an excitation source, the coordinate of the quenching starting moment is set as t-0, the initial temperature of each part in the magnet before quenching is determined, and a time domain numerical simulation algorithm is adopted to calculate the temperature at t-t1The density of eddy currents momentarily generated on the conductor within the magnet; then any point on the conductor within the magnet system is calculated by the following formula
Figure BDA0002691666900000036
To the heat generated by the density of the eddy current
Figure BDA0002691666900000037
Figure BDA0002691666900000038
In the above formula, the first and second carbon atoms are,
Figure BDA0002691666900000039
as a coordinate point
Figure BDA00026916669000000310
The electrical conductivity of the (c) electrode,
Figure BDA00026916669000000311
is composed of
Figure BDA00026916669000000312
The modulus of the eddy current density.
Then solving the following heat conduction control equation to obtain t ═ t1Temperature distribution in conductor at time:
Figure BDA0002691666900000041
in the above formula, T is temperature, rho is material density, c is specific heat of the material, x, y, z are coordinates of three directions in a Cartesian coordinate system, and k isx,ky,kzIs a coordinate point
Figure BDA0002691666900000042
Thermal conductivity in three directions of x, y and z; the directions of the three coordinate axes can be determined according to the convenience of analyzing the problem. For superconducting magnet problems, the central axis of the magnet is generally defined as the z-axis, and any two orthogonal directions within a plane perpendicular to the z-axis are the x-axis and the y-axis.
And re-determining the conductivity of the conductor at each coordinate point according to the temperature distribution, and then repeating the process to calculate the eddy current density and the temperature at the next moment until the calculation is finished.
Before the method is adopted, a time-varying current curve of the coil after quenching needs to be obtained. For an already fabricated magnet, this curve can be obtained by collecting the current in the coil after the magnet has quenched. In the magnet design stage, an ellipsoid diffusion model can be adopted to calculate the current time-varying curve of the coil after quenching. Such a model is described in some documents, such as "design basis for superconducting magnets" in south and ceremony, and a specific calculation method is not given here.
After the current change in the coil is obtained, the eddy current density at each moment in the magnet is numerically simulated by using a finite element method. Finite element eddy current density simulations may use different governing equations. The finite element modeling is carried out by adopting the following electromagnetic field control equation:
in the conductor region:
Figure BDA0002691666900000043
in the non-conductor region:
Figure BDA0002691666900000044
in the above formula, A is the vector magnetic potential generated by the eddy current density, V is the scalar potential, V is the reciprocal of the magnetic permeability,
Figure BDA0002691666900000045
as a coordinate point
Figure BDA0002691666900000046
Electrical conductivity of the metal, JsIs the current density, sign, in the coil
Figure BDA0002691666900000047
The operation of the rotation degree is represented,
Figure BDA0002691666900000048
the operation of the divergence is represented by the operation of divergence,
Figure BDA0002691666900000049
representing a gradient operation.
It can be found by analysis that the above-mentioned governing equation has a unique solution after the boundary condition is determined. After the vector magnetic potential and the scalar potential in the magnet are obtained, the eddy current density in the conductor is obtained by the following formula:
Figure BDA0002691666900000051
after the current density is obtained, the Lorentz force density at any point on the conductor in the magnet system
Figure BDA0002691666900000052
Calculated by the following formula:
Figure BDA0002691666900000053
in the above formula, the first and second carbon atoms are,
Figure BDA0002691666900000054
as a coordinate point
Figure BDA0002691666900000055
The density of the eddy currents in the (d),
Figure BDA0002691666900000056
as a coordinate point
Figure BDA0002691666900000057
The magnetic field of (a).
Thus, by the simulation, the eddy current density, the electromagnetic force, and the temperature at any time in the magnet can be obtained. Because the conductivity of the conductor changes along with the temperature at low temperature, an electromagnetic field equation and a temperature equation need to be solved during simulation, and the conductivity value is updated after the solution, so that the simulation precision is ensured.
In the above algorithm, an exit criterion of time iteration needs to be set. We can use the value of the maximum eddy current density as the criterion when the maximum eddy current density value in the conductor satisfies
Figure BDA0002691666900000058
The simulation ends, where ε is an artificially set value.
For finite element simulation, a tetrahedral mesh is adopted to subdivide a heat conduction equation, and a node basis function is adopted as an interpolation function of a temperature field. For the electromagnetic field control equation, a tetrahedral mesh is also adopted for subdivision, and the edge basis function is adopted as the interpolation function of the magnetic vector bit. After the specific form, the grid and the basis function of the control equation are determined, the control equation can be discretized by adopting a Galerkin method. The steps of using the discrete governing equations of Galerkin's method are described in some documents, such as the electromagnetic field finite element method, by King Ming, and will not be described further herein.
A specific simulation example is given below. The change of the current amplitude over time obtained by monitoring after a superconducting magnet quenches is known as shown in fig. 2. In order to optimize the structure of the cold shield of the magnet, the quench process needs to be simulated. Since this is a simulation of an actual product, specific parameters such as coils and cold shields are not given here. A Cartesian coordinate system and a cylindrical coordinate system are established by taking the central axis of the magnet as a z-axis and the central point of the magnet as a coordinate origin. The coordinates in the Cartesian coordinate system are represented by (x, y, z), and the coordinates in the cylindrical coordinate system are represented by
Figure BDA0002691666900000059
And (4) showing. It is known from analysis that the force on the cold shield occurs at a certain moment in the beginning of the quench, because at this time the magnetic field generated by the coil is maximal. After the electromagnetic force reaches the peak at a certain time, the electromagnetic force gradually attenuates with the attenuation of the current. Fig. 3 is the simulated eddy current density distribution on the cold shield at t-1.2 s. Fig. 4 is a graph of the distribution of the lorentz force density f on the cold shield along the axial direction at t-1.2 s. Since f is a vector, the graph contains two curves. fz is the lorentz force density in the z direction, and f ρ is the lorentz force density in the ρ direction. Whether the stress of the cold shield is in a bearing range can be judged through the Lorentz force density change of different positions at different moments, so that guidance is provided for structural improvement.

Claims (8)

1. A finite element analysis method for a superconducting magnet quench process is characterized by comprising the following steps:
(1) obtaining a time-varying relation curve I (t) of current in a magnet coil after the magnet is quenched;
(2) taking I (t) as an excitation source, setting the coordinate of the quenching starting moment as t as 0, determining the initial temperature of each part in the magnet before quenching, and calculating the t as t by adopting a time domain numerical simulation algorithm1The density of eddy currents momentarily generated on the conductor within the magnet;
(3) calculating any coordinate on a conductor in a magnet system by the following formulaDot
Figure FDA0002691666890000011
To the heat generated by the density of the eddy current
Figure FDA0002691666890000012
Figure FDA0002691666890000013
In the above formula, the first and second carbon atoms are,
Figure FDA0002691666890000014
as a coordinate point
Figure FDA0002691666890000015
The electrical conductivity of the (c) electrode,
Figure FDA0002691666890000016
is composed of
Figure FDA0002691666890000017
The modulus of the eddy current density;
(4) solving the following equation of heat conduction control to obtain t ═ t1Temperature distribution in conductor at time:
Figure FDA0002691666890000018
in the above formula, T is temperature, rho is material density, c is specific heat of the material, x, y, z are coordinates of three directions in a Cartesian coordinate system, and k isx,ky,kzIs a coordinate point
Figure FDA0002691666890000019
Thermal conductivity in three directions of x, y and z;
(5) and (4) re-determining the conductivity of the conductor at each coordinate point according to the temperature distribution obtained in the step (4), and then repeating the steps (2) to (4) to calculate the eddy current density and the temperature at the next moment until the calculation is finished.
2. A finite element analysis method for a superconducting magnet quench process according to claim 1, wherein the time-varying curve of the current after the coil quench in step (1) is obtained by sampling the current signal with a magnet monitoring device or by an ellipsoid diffusion model.
3. A finite element analysis method for superconducting magnet quench process according to claim 1, wherein the finite element method is used in step (2) to numerically simulate the eddy current density at each time, and the finite element modeling is performed by using the following electromagnetic field control equation:
in the conductor region:
Figure FDA00026916668900000110
in the non-conductor region:
Figure FDA0002691666890000021
in the above formula, A is the vector magnetic potential generated by the eddy current density, V is the scalar potential, V is the reciprocal of the magnetic permeability,
Figure FDA0002691666890000022
as a coordinate point
Figure FDA0002691666890000023
Electrical conductivity of the metal, JsIs the current density, sign, in the coil
Figure FDA0002691666890000024
The operation of the rotation degree is represented,
Figure FDA0002691666890000025
the operation of the divergence is represented by the operation of divergence,
Figure FDA0002691666890000026
representing a gradient operation.
4. A finite element analysis method of a superconducting magnet quench process as claimed in claim 3, wherein the eddy current density in the conductor is obtained by the following formula:
Figure FDA0002691666890000027
5. a finite element analysis method for superconducting magnet quench processing according to claim 1, wherein in step (5), the maximum eddy current density in the conductor is satisfied
Figure FDA0002691666890000028
When the simulation is finished, epsilon is the set threshold value.
6. A finite element analysis method for a superconducting magnet quench process according to claim 1, wherein the heat conduction control equation in step (4) is subdivided by using a tetrahedral mesh and using a node basis function as an interpolation function of the temperature field.
7. A finite element analysis method for a superconducting magnet quench process as claimed in claim 3, wherein the tetrahedral mesh is used for subdivision, and the edge basis function is used as an interpolation function of the magnetic vector bits.
8. A finite element analysis method of a superconducting magnet quench process as claimed in claim 1, wherein any one coordinate point on a conductor in the magnet system
Figure FDA0002691666890000029
Lorentz force density of
Figure FDA00026916668900000210
Calculated by the following formula:
Figure FDA00026916668900000211
in the above formula, the first and second carbon atoms are,
Figure FDA00026916668900000212
as a coordinate point
Figure FDA00026916668900000213
The density of the eddy currents in the (d),
Figure FDA00026916668900000214
as a coordinate point
Figure FDA00026916668900000215
The magnetic field of (a).
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2024051618A1 (en) * 2022-09-05 2024-03-14 中车长春轨道客车股份有限公司 Method and apparatus for designing superconducting magnet
WO2024077861A1 (en) * 2022-10-12 2024-04-18 华中科技大学 Finite element-based electromagnetic, thermal and mechanics multi-field coupling simulation modeling method for superconducting magnet

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2024051618A1 (en) * 2022-09-05 2024-03-14 中车长春轨道客车股份有限公司 Method and apparatus for designing superconducting magnet
WO2024077861A1 (en) * 2022-10-12 2024-04-18 华中科技大学 Finite element-based electromagnetic, thermal and mechanics multi-field coupling simulation modeling method for superconducting magnet

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