A force-thermal coupling multi-scale analysis method for superconducting coils

Abstract

The application discloses a superconducting coil force-thermal coupling multi-scale analysis method, which is used for studying the force-thermal coupling response of the coil by constructing a sequential multi-scale analysis framework; specifically, a mesoscopic representative volume element (RVE) of NbTi strand is established, and homogenization treatment is carried out to obtain equivalent orthotropic material parameters, which are then applied to a macroscopic finite element model of a racetrack coil; in addition, by carrying out coupling analysis of different contact settings under the conditions of external pressure and electromagnetic force load, the coupling relationship between the local contact pressure and temperature change under different external load conditions is explored, and the critical contact pressure threshold is determined, so that the model optimization design is carried out by modifying the design parameters, so that the contact pressure of all regions of the model is lower than the critical contact pressure threshold, which provides effective guidance for evaluating the multi-scale force-thermal coupling behavior of the superconducting coil, and also provides theoretical support for the magnet design and stability optimization.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of superconductivity, and particularly relates to a force-thermal coupling multi-scale analysis method for a superconducting coil. BACKGROUND

[0002] Low-temperature superconductors, particularly niobium-titanium (NbTi) alloys, have been the dominant material in high-field magnet technology for decades. In recent years, racetrack-shaped NbTi superconducting coils have attracted much attention in the propulsion and levitation systems of high-speed maglev trains due to their excellent mechanical properties and good cost-effectiveness; NbTi superconductors need to be maintained at extremely low temperatures to maintain the superconducting state. In actual operation, NbTi coils bear significant multi-physical field loads, including stress caused by external impact, and strong electromagnetic force generated by the interaction of high operating current and strong magnetic field. These loads inevitably lead to complex contact behavior between the coil winding and its structural support (such as stainless steel rings), and trigger more complex contact interactions between superconducting strands inside the winding; the friction at these interfaces dissipates energy in the form of heat, causing local thermal disturbance. In the harsh low-temperature environment, such frictional heating poses a serious threat to the stability of the magnet, which may trigger a loss of superconductivity and cause operational failure. However, in the research on superconducting magnets, the key coupling between internal contact mechanics and thermal effects is often ignored. SUMMARY

[0003] The application aims to provide a force-thermal coupling multi-scale analysis method for a superconducting coil to solve the problems in the background art.

[0004] To achieve the above-mentioned purpose, the application provides the following technical scheme: a force-thermal coupling multi-scale analysis method for a superconducting coil, comprising the following steps:

[0005] S1: constructing an RVE model of a NbTi superconducting strand;

[0006] S2: performing finite element calculation on the mechanical response of the RVE model obtained in step S1 under uniaxial tension and pure shear load, obtaining the macroscopic stress-strain curve of the RVE model, and based on the calculated overall mechanical response, fitting and homogenizing the mechanical behavior of the RVE model using an equivalent orthotropic linear elastic constitutive relationship to obtain equivalent material parameters for macro-scale simulation, including Young's modulus, Poisson's ratio and shear modulus in three directions;

[0007] S3: Based on the homogenized material parameters obtained in step S2, a racetrack superconducting coil macroscopic finite element model containing internal NbTi windings and external stainless steel restraint structure is established, and the racetrack superconducting coil macroscopic finite element model is discretized by using coupled temperature-displacement analysis units. The boundary conditions are set as follows: symmetric constraints are applied on the symmetry plane, axial displacement constraints are applied on the bottom of the racetrack superconducting coil, and two kinds of contact settings are made, i.e. stainless steel-NbTi strand contact: defining the contact pair between stainless steel and NbTi strand, which contains friction and heat conduction properties; NbTi strand inter-contact: on the basis of the stainless steel-strand inter-contact, additional strand inter-contact pairs along the y-axis and z-axis directions are added, and the general contact and penalty friction formula is used for the contact setting formula, and the friction coefficient is 0.1;

[0008] S4: The model established in step S3 is respectively applied with external pressure load and electromagnetic force load. When the external pressure load is applied, a monotonically increasing pressure along the axial direction is applied on the upper surface of the racetrack superconducting coil to simulate the external impact or pre-tightening force. Then the electromagnetic force load is applied. First, the magnetic field distribution and Lorentz force body density generated by the coil under the given working current and number of turns are calculated by the electromagnetic field analysis software. Then the obtained magnetic field distribution and Lorentz force body density are applied to the racetrack superconducting coil as body load by the ABAQUS subprogram DLOAD, and the calculation equation of the electromagnetic force is as follows:

[0009] ; wherein F is the Lorentz force body density vector, the unit is N / M 3 , J is the current density vector, the unit is A / m 2 , J x , J y , J z are the components of the current density vector on the x, y, z coordinate axes respectively, B is the magnetic induction intensity vector, the unit is T, B x , B y , B z are the components of the electromagnetic force on the x, y, z coordinate axes respectively, is the unit vector in the x, y, z direction of the spatial rectangular coordinate system, the current density vector J distribution and the magnetic induction intensity vector B distribution are calculated by the electromagnetic field analysis software;

[0010] S5: Perform mechanical-thermal coupling analysis and critical threshold judgment on the racetrack superconducting coil subjected to external pressure load and electromagnetic force load in step S4. In the solving process, perform fully coupled mechanical-thermal analysis, convert all mechanical energy dissipated by the relative sliding friction of the contact interface into heat through the heat generation module of the finite element software, and participate in transient heat conduction calculation as internal heat source, so as to obtain the temperature field evolution of the racetrack superconducting coil caused by friction. By analyzing the simulation results under different loads and different contact settings, the distribution and evolution of key physical quantities are extracted, including Mises stress, contact pressure, and local temperature. The contact pressure-temperature coupling region that needs to be monitored is locked through the Mises stress cloud map, and the critical contact pressure threshold that causes the temperature to rise sharply to 14K is identified by monitoring the coupling change relationship between the contact pressure and the temperature of the strand contact interface.

[0011] S6: Optimize the design parameters of the racetrack superconducting coil, and perform simulation and simulation of steps S3-S5 again until the contact pressure of all regions in the established racetrack superconducting coil simulation result is lower than the critical contact pressure threshold.

[0012] Preferably, in step S3, when establishing the macro finite element model of the racetrack superconducting coil, because the establishment of the racetrack superconducting coil structure has symmetry, a quarter model is established to improve the calculation efficiency.

[0013] Preferably, when optimizing the design parameters of the racetrack superconducting coil, the design parameter optimization includes the structural stiffness of the stainless steel support, the friction coefficient of the strand contact interface, the filling material between the strands, and the strand layout.

[0014] Compared with the prior art, the beneficial effects of the present application are:

[0015] The present application constructs a sequential multi-scale analysis framework for studying the force-thermal coupling response of the coil; by establishing the mesoscopic representative volume element (RVE) of the NbTi strand and performing homogenization treatment to obtain equivalent orthotropic material parameters, which are then applied to the macro finite element model of the racetrack coil; in addition, by carrying out coupled analysis considering different contact settings under external pressure and electromagnetic force load, the coupling relationship between local contact pressure and temperature change under different external load conditions is explored, and the critical contact pressure threshold is determined, so that by modifying the design parameters, the contact pressure of all regions of the model is lower than the critical contact pressure threshold for model optimization design, which provides effective guidance for evaluating the multi-scale force-thermal coupling behavior of the superconducting coil, and also provides theoretical support for magnet design and stability optimization. BRIEF DESCRIPTION OF DRAWINGS

[0016] Figure 1The RVE model of the NbTi superconducting strand provided by the embodiment of the present application is a three-dimensional micro-representative volume element (RVE) model of the NbTi superconducting strand provided by the embodiment of the present application.

[0017] Figure 2 The macro stress-strain curve of the RVE model is provided by the embodiment of the present application.

[0018] Figure 3 The macro finite element model of the racetrack superconducting coil is provided by the embodiment of the present application.

[0019] Figure 4 The macro finite element model of the racetrack superconducting coil with different contact settings is provided by the embodiment of the present application.

[0020] Figure 5 The electromagnetic field analysis model of the racetrack superconducting coil is provided by the embodiment of the present application.

[0021] Figure 6 The stress distribution diagram of the racetrack superconducting coil with different contact settings under an external load scenario is provided by the embodiment of the present application.

[0022] Figure 7 The stress distribution diagram of the racetrack superconducting coil with different contact settings under an external electromagnetic force scenario is provided by the embodiment of the present application.

[0023] Figure 8 The contact pressure (a) and temperature field distribution (b) diagram of the NbTi strand under different contact settings is provided by the embodiment of the present application.

[0024] Figure 9 The contact pressure (a) and temperature field distribution (b) diagram of the NbTi strand under different contact settings is provided by the embodiment of the present application.

[0025] Figure 10 The contact pressure and temperature change with the electromagnetic force loading time step diagram. DETAILED DESCRIPTION

[0026] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the scope of protection of the present application.

[0027] Please refer to Figures 1-10 The present application provides a technical solution, which comprises the following steps:

[0028] S1: Construction and homogenization of the micro-representative volume element (RVE): first, a three-dimensional micro-representative volume element (RVE) model of the NbTi superconducting strand is constructed, as shown inFigure 1 As shown, the RVE model is a cubic structure which can reflect the real composite characteristics of the strand, and the specific structure is: copper in the inside, NbTi filament bundle in the middle layer, and epoxy resin in the outer layer; the copper material adopts an elastic-plastic constitutive model subject to the isotropic hardening rule, and the yield stress changes with the equivalent plastic strain; the NbTi filament and the epoxy resin are both simplified as isotropic linear elastic materials;

[0029] S2: Obtain the macroscopic stress-strain curve of the RVE model by performing finite element calculation on the mechanical response of the RVE model obtained in step S1 under uniaxial tension and pure shear load, as shown in Figure 2 Based on the calculated overall mechanical response, the mechanical behavior of the RVE model is fitted and homogenized using the equivalent orthotropic linear elastic constitutive relation, so as to obtain the equivalent material parameters for macro-scale simulation, including the Young's modulus, Poisson's ratio and shear modulus in three directions, and in this embodiment, the material parameters are shown in Table 1 as follows:

[0030] Table 1 Homogenized material parameters

[0031]

[0032] S3: Based on the homogenized material parameters obtained in step S2, a macroscopic finite element model of a racetrack type superconducting coil containing internal NbTi winding and external stainless steel constraint structure is established, because the establishment of the racetrack type superconducting coil structure has symmetry, a quarter model is established to improve the calculation efficiency, and the model is as shown in Figure 3 The racetrack type superconducting coil macroscopic finite element model is discretized using coupled temperature-displacement analysis elements, Figure 2 The internal part represents the NbTi strand, and the external part represents the stainless steel, 13440 C3D8T elements are used to discretize the internal NbTi strand, and 9088 C3D8T elements are used to discretize the outer layer stainless steel, the boundary conditions are set as: symmetrical constraints are applied to the outer surfaces along the x-axis and y-axis, displacement z=0 is applied to the bottom surface along the z-axis, and two kinds of contact settings are performed, as shown in Figure 4 The two kinds of contact settings are: stainless steel-NbTi strand contact: define the contact pair between stainless steel and NbTi strand, which contains friction and heat conduction properties; NbTi strand inter-contact: based on the contact between stainless steel and strand, additional strand inter-contact pairs along the y-axis and z-axis are added, the general contact and penalty friction formula is used for the contact setting formula, and the friction coefficient is 0.1;

[0033] S4: Apply external pressure load and electromagnetic force load to the model established in step S3 respectively, when applying external pressure load, apply monotone increasing pressure along the axial direction (z-axis) on the upper surface of the racetrack superconducting coil, the maximum pressure is set to 60 MPa, the loading step time step is simplified to 1, and the initial temperature is 4.2 K (liquid helium temperature) to simulate external impact or pre-tightening force; then apply electromagnetic force load, first calculate the magnetic field distribution and Lorentz force body density of the coil under the given working current (170 A in this embodiment) and the number of turns (1800 turns in this embodiment) by using electromagnetic field analysis software (the method of building a “uniform coil + air domain” model by using the MF module of COMSOL is adopted in this embodiment, and the model is as shown in Figure 5 The maximum magnetic field strength is 2.2 T, and the magnetic field distribution is symmetrical along the z-axis, then the obtained magnetic field distribution and Lorentz force body density are used to apply electromagnetic force as body load to the racetrack superconducting coil by using the subprogram DLOAD of ABAQUS, wherein the calculation equation of the electromagnetic force is as follows:

[0034] ; wherein F is the Lorentz force body density vector, the unit is N / M 3 , J is the current density vector, the unit is A / m 2 , J x 、 J y 、 J z are the components of the current density vector in the x, y, z three coordinate axes respectively, B is the magnetic induction intensity vector, the unit is T, B x 、 B y 、 B z are the components of the electromagnetic force in the x, y, z three coordinate axes respectively, is the unit vector in the x, y, z direction of the spatial rectangular coordinate system, the current density vector J distribution and the magnetic induction intensity vector B distribution are calculated by the electromagnetic field analysis software; the electromagnetic field analysis model is as shown in Figure 3 ;

[0035] The final simulation results are as follows:

[0036] External pressure load scenario:

[0037] The results of the stainless steel-NbTi strand contact model are as shown in Figure 6 The contact pressure is mainly distributed on the top surface and the bottom surface, and the contact pressure of other surfaces is zero, because the external pressure is applied along the z-axis direction, and the surface of the steel and the strand along the z-axis direction is more prone to extrusion than other directions, and further consideration is given to the setting of the NbTi strand contact (as shown inFigure 6 As shown in the lower model, the contact pressure not only appears on the top and bottom surfaces, but also on the contact surface between strands, and the values are similar; this result shows that the contact pressure between NbTi strands of the racetrack type superconducting coil model cannot be ignored, and the contact pressure between NbTi strands also needs to be optimized when optimizing the model contact pressure.

[0038] External electromagnetic force load scenario:

[0039] The macro stress distribution of the racetrack type NbTi coil under electromagnetic force loading is shown in Figure 7 The results of the two contact settings both show that the stress of the external stainless steel is higher, with a maximum value of 223.8 MPa, while the stress of the internal NbTi strand is about 60 MPa, indicating that the stainless steel bears most of the force, and different contact settings have little effect on the overall mechanical response.

[0040] S5: Perform mechanical-thermal coupling analysis and critical threshold judgment on the racetrack type superconducting coil subjected to external pressure load and electromagnetic force load in step S4, in the solving process, perform fully coupled mechanical-thermal analysis, through the heat generation module of the finite element software, all the mechanical energy dissipated by the relative sliding friction of the contact interface is converted into heat, which is used as an internal heat source to participate in transient heat conduction calculation, so as to obtain the local temperature field evolution caused by friction; and by analyzing the simulation results under different loads and different contact settings, the distribution and evolution of key physical quantities are extracted, including: Mises stress, contact pressure, and local temperature; by monitoring the coupling change relationship between the contact pressure and the temperature of the strand contact interface, the critical contact pressure threshold that causes the temperature to rise sharply and rise far above the critical temperature of NbTi (generally considered to be 14K) is identified;

[0041] In this embodiment, the simulation results under different load scenarios are as follows:

[0042] External pressure load scenario:

[0043] As shown in Figure 8 When considering "stainless steel-NbTi strand contact", as the contact pressure increases, the NbTi strand still maintains 4.2K in most areas, and the friction between the stainless steel and the strand cannot generate enough heat to significantly change the temperature of the strand; the results of the "NbTi strand contact" setting show that the friction between the strands can generate significant heat, causing the temperature of the local area to rise, and the contact pressure threshold at which the local temperature rises significantly is about 30 MPa, and as the contact pressure increases, the highest temperature reaches 26.5K, which is far above the critical temperature (4.2K) required for NbTi wire to maintain superconducting state, which may cause serious quenching and affect the stable operation of the NbTi coil.

[0044] External electromagnetic force load scenario:

[0045] As Figure 9 shown, the results of the two contact settings show that the contact pressure near the boundary area is higher; from Figure 9 unit A and unit B are selected from the above, the changes of local contact pressure and temperature under different electromagnetic force loading step time are studied, and the results are shown in Figure 10 shown, for unit A, the contact pressure under the two contact settings is lower than 5MPa, and the maximum temperature rise is within 0.2K, the contact pressure of unit B is obviously increased, the maximum value is about 32MPa, and the temperature rises from 4.2K to 4.95K~5.2K, which is smaller than the temperature change under the external pressure loading condition, and the critical contact pressure obtained is higher than the critical contact pressure obtained under the external pressure loading.

[0046] S6: the design parameters of the racetrack type superconducting coil are optimized, including the structural stiffness of the stainless steel support, the friction coefficient of the strand contact interface, the filling material between the strands, and the strand layout, and the simulation of steps S3~S5 is performed again until the contact pressure of all areas of the established racetrack type superconducting coil simulation result is lower than the critical contact pressure threshold (30MP) under the normal external impact in the application environment under the 2.2T magnetic field environment, in this embodiment, the design parameters of the racetrack type superconducting coil are that the Young's modulus of stainless steel is 210GPa, the Poisson's ratio is 0.28, the friction coefficient of the strand contact interface is 0.1, the filling material between the strands is epoxy resin, and the strand layout is straight line and array arrangement, through the above simulation analysis, the model of this embodiment meets the requirement of magnetic field condition, and only external impact experiment is needed.

[0047] Although the present application has been described in detail with reference to the foregoing embodiments, those skilled in the art can modify the technical solutions described in the foregoing embodiments, or make equivalent replacement to part of the technical features, any modification, equivalent replacement, improvement, etc. made within the spirit and principles of the present application shall be included in the protection scope of the present application.

Claims

1. A multi-scale analysis method for force-thermal coupling of superconducting coils, characterized in that, Includes the following steps: S1: Construct the RVE model for NbTi superconducting strands; S2: By performing finite element calculations on the mechanical response of the RVE model obtained in step S1 under uniaxial tension and pure shear loads, the macroscopic stress-strain curve of the RVE model is obtained. Based on the calculated overall mechanical response, the mechanical behavior of the RVE model is fitted and homogenized using an equivalent orthogonal anisotropic linear elastic constitutive relation, thereby obtaining equivalent material parameters for macroscopic simulation, including Young's modulus, Poisson's ratio and shear modulus in three directions. S3: Based on the homogenized material parameters obtained in step S2, a macroscopic finite element model of a racetrack-shaped superconducting coil, including an internal NbTi winding and an external stainless steel constraint structure, is established. The macroscopic finite element model of the racetrack-shaped superconducting coil is discretized using coupled temperature-displacement analysis elements. The boundary conditions are set as follows: symmetric constraints are applied on the symmetry plane, and axial displacement constraints are applied to the bottom of the racetrack-shaped superconducting coil. Two contact settings are implemented: Stainless steel-NbTi strand contact: a contact pair between stainless steel and NbTi strands is defined, which includes friction and heat conduction properties; NbTi strand contact: based on the stainless steel-NbTi strand contact, additional strand contact pairs are added along the y-axis and z-axis directions. The contact setting formula adopts the general contact and penalty friction formula, and the friction coefficient is taken as 0.

1. S4: Apply external pressure load and electromagnetic force load to the model established in step S3. When applying the external pressure load, apply a monotonically increasing pressure along the axial direction to the upper surface of the racetrack-shaped superconducting coil to simulate external impact or preload. Then apply the electromagnetic force load. First, calculate the magnetic field distribution and Lorentz force volume density generated by the coil under a given operating current and number of turns using electromagnetic field analysis software. Then, apply the obtained magnetic field distribution and Lorentz force volume density to the racetrack-shaped superconducting coil as a volume load through the ABAQUS subroutine DLOAD. The calculation equation for the electromagnetic force is as follows: Where F is the Lorentz force body density vector, with units of ; N / m 3 J is the current density vector, with units of 12000 m / s. A / m 2 , J x , J y , J z These are the components of the current density vector on the x, y, and z coordinate axes, respectively, and B is the magnetic flux density vector in tons (T). B x , B y , B z These are the components of the electromagnetic force on the x, y, and z coordinate axes. It is a unit vector in the x, y, and z directions in a spatial rectangular coordinate system. The current density vector J and the magnetic induction intensity vector B are calculated by electromagnetic field analysis software. S5: Perform mechanical-thermal coupling analysis and critical threshold determination on the racetrack-shaped superconducting coil subjected to external pressure load and electromagnetic force load in step S4. During the solution process, a fully coupled mechanical-thermal analysis is performed. Through the thermal generation module of the finite element software, all the mechanical energy dissipated at the contact interface due to relative sliding friction is converted into heat. This heat is used as an internal heat source to participate in the transient heat conduction calculation, thereby obtaining the temperature field evolution of the racetrack-shaped superconducting coil caused by friction. By analyzing the simulation results under different loads and contact settings, the distribution and evolution of key physical quantities are extracted, including: Mises stress, contact pressure, and local temperature. The contact pressure-temperature coupling region that needs to be monitored is identified by the Mises stress cloud map. By monitoring the coupling change relationship between contact pressure and temperature at the strand contact interface, the contact pressure that causes the temperature to rise sharply to 14K is identified as the critical contact pressure threshold. S6: Optimize the design parameters of the racetrack-shaped superconducting coil, and repeat the simulation steps S3 to S5 until the contact pressure in all regions of the established racetrack-shaped superconducting coil simulation results is lower than the critical contact pressure threshold.

2. The superconducting coil force-thermal coupling multi-scale analysis method according to claim 1, characterized in that, In step S3, when establishing the macroscopic finite element model of the racetrack-shaped superconducting coil, a quarter-model is established to improve computational efficiency because the structure of the racetrack-shaped superconducting coil has symmetry.

3. The superconducting coil force-thermal coupling multi-scale analysis method according to claim 1, characterized in that, When optimizing the design parameters of the racetrack-shaped superconducting coil, the optimization parameters include: the structural stiffness of the stainless steel support, the friction coefficient of the strand contact interface, the filling material between the strands, and the strand layout.

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