Multi-objective optimization method for non-planar coil of star simulator and star simulator magnet

Abstract

The invention provides a multi-objective optimization method for non-planar coils of a star simulator and a star simulator magnet. The optimization method comprises the following steps: obtaining initial structures of a plurality of non-planar superconducting coils according to magnetic field configuration data of target plasma; according to the target magnetic field parameter, the engineering constraint parameter and the material constraint parameter of the configuration data, an objective function is constructed, the engineering constraint parameter comprises the engineering limit geometric parameter of the non-planar superconducting coil, and the material constraint parameter comprises the limit current carrying of the non-planar superconducting coil material; carrying out minimization solution on the target function to obtain optimized coil data; and performing materialization processing on the plurality of non-planar superconducting coils according to the optimized coil data to obtain a plurality of optimized non-planar superconducting coil models. According to the optimization method, a magnetic field matching-engineering feasibility-material safety three-dimensional constraint system is constructed, and the problem that the engineering cannot be realized on the ground is avoided.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of superconductivity, and particularly relates to a stellarator coil optimization method and a stellarator magnet obtained by using the optimization method. BACKGROUND

[0002] Magnetic confinement nuclear fusion is an important direction for solving global energy crisis, which realizes controllable nuclear fusion reaction by restraining high-temperature plasma of millions of degrees in a specific geometric shape through a strong magnetic field generated by a coil. A stellarator is a kind of magnetic confinement nuclear fusion device, which generates field lines through a series of twisted coils to avoid saturation of induced current and realizes more stable long-term operation. The coil design not only needs to accurately generate a required magnetic field to confine the plasma, but also needs to take into account the engineering feasibility (such as coil manufacturing difficulty, mechanical stress, cost, etc.). The stellarator research field attempts to reduce the neoclassical transport level and high-energy particle loss level of the stellarator by optimizing the magnetic field shape.

[0003] The coil as the “skeleton” of the nuclear fusion device needs to meet the following requirements: (1) the generated magnetic field is highly accurately matched with the target plasma shape (ensuring the confinement performance); (2) the shape and size of the coil meet the manufacturing and installation requirements (ensuring the engineering feasibility); and (3) the current density does not exceed the material bearing limit (ensuring the operation safety). However, due to the complex three-dimensional coil system of the stellarator, the existing coil optimization technology of the stellarator has the following defects: single constraint system, the mainstream open source optimization framework only considers the magnetic field matching constraint, and the problem of “optimization result is physically feasible but engineering cannot be implemented” often occurs; poor initialization adaptability, the non-uniform distribution of the target shape magnetic field gradient is not considered; low optimization efficiency, when facing a complex objective function with multiple constraint couplings, iteration shock or slow convergence often occurs, and the optimization period is long. SUMMARY

[0004] The present application aims to solve the defects of imperfect constraint system, poor initialization adaptability and low optimization efficiency in the existing coil optimization method of the stellarator.

[0005] The present application provides a multi-objective optimization method for stellarator non-planar coils, comprising the following steps: S1: obtaining an initial structure of a plurality of non-planar superconducting coils according to magnetic field shape data of a target plasma, the magnetic field shape data comprising target magnetic field parameters; S2: constructing an objective function according to the target magnetic field parameters, engineering constraint parameters and material constraint parameters, wherein the engineering constraint parameters comprise engineering limit geometric parameters of the non-planar superconducting coils, and the material constraint parameters comprise limit current carrying of the non-planar superconducting coil materials; S3: minimizing the objective function to obtain optimized coil data; and S4: performing solidification processing on the plurality of non-planar superconducting coils according to the optimized coil data to obtain an optimized plurality of non-planar superconducting coil models.

[0006] The technical scheme is adopted, the magnetic field configuration of the pre-designed plasma is taken as a target reference, a target function including a core constraint term related to the magnetic field matching, an engineering constraint term and a material constraint term is constructed, the core constraint term is used to measure the matching degree of the optimized coil magnetic field and the target magnetic field, the engineering constraint term is used to constrain the manufacturing and installation feasibility of the coil, and the material constraint term is used to constrain the material bearing limit of the coil, a three-dimensional constraint system is formed, the optimization result is verified from three dimensions, and the problem that the engineering cannot be implemented is avoided.

[0007] According to another specific embodiment of the present application, the embodiment of the present application discloses a multi-objective optimization method for a non-planar coil of a stellarator, and the target magnetic field parameters include a target magnetic flux.

[0008] ,

[0009] In the formula, is a target function, is a deviation between the actual magnetic flux and the target magnetic flux, is a magnetic flux weight, is an engineering constraint parameter, is an engineering constraint weight, is a material constraint parameter, and is less than or equal to a preset current-carrying limit, is a material constraint weight.

[0010] The technical scheme is adopted, the deviation between the actual magnetic flux and the target magnetic flux is used to realize the quantification of the magnetic field matching error, and the weights of different constraint terms can be adjusted based on actual design requirements, so that the design method is more scientific, and the coil obtained by the design meets the requirements.

[0011] According to another specific embodiment of the present application, the embodiment of the present application discloses a multi-objective optimization method for a non-planar coil of a stellarator, and the engineering constraint parameter is calculated according to the following formula:

[0012] ,

[0013] In the formula, is a minimum coil constraint length, and is calculated according to the following formula:

[0014] ,

[0015] In the formula, L MIN is a preset minimum length of the coil, is an actual length of the i th non-planar superconducting coil, and q represents the number of non-planar superconducting coils.

[0016] is a maximum coil constraint length, and is calculated according to the following formula:

[0017] ,

[0018] where L MAX is the preset maximum length of the coil.

[0019] is the coil constraint spacing, calculated by:

[0020]

[0021] where d min is the preset minimum spacing between coils, is the actual distance between the i-th and j-th non-planar superconducting coils.

[0022] is the coil constraint distance, calculated by:

[0023]

[0024] where, is the minimum safe distance between the coil and the plasma magnetic surface, D is is the actual distance between the i-th non-planar superconducting coil and the magnetic surface.

[0025] is the maximum constraint curvature of the base half-period coil, calculated by:

[0026] ,

[0027] where k i is the actual maximum curvature of the i-th base half-period coil, k curv is the preset maximum curvature of the base half-period coil, and p is the number of base half-period coils.

[0028] is the square curvature of the base half-period coil, calculated by:

[0029]

[0030] where, is the actual average square curvature of the i-th base half-period coil, MSC y is the preset average square curvature of the base half-period coil.

[0031] According to another specific embodiment of the present invention, the multi-objective optimization method for nonplanar coils of a stellarator disclosed in the present invention includes step S1: S11: acquiring magnetic field configuration data, the magnetic field configuration data further including magnetic field gradient distribution parameters; S12: obtaining the initial structure of multiple nonplanar superconducting coils according to the magnetic field gradient distribution parameters; wherein, in the nonplanar superconducting coils in the region of large magnetic field gradient, a higher Fourier order is used to construct the initial profile of the nonplanar superconducting coil; in the region of small magnetic field gradient, a lower Fourier order is used to construct the initial profile.

[0032] By adopting the above technical solution, the initial structure of the multi-wire coil is set according to the magnetic field gradient distribution parameters and the principle of "gradient-fitting accuracy matching" to avoid problems such as low magnetic field control accuracy, accuracy redundancy, and poor initial structure adaptability.

[0033] According to another specific embodiment of the present invention, the multi-objective optimization method for nonplanar coils of stellarators disclosed in the embodiments of the present invention further includes step S12: allocating an initial current to the initial structure of each nonplanar superconducting coil, wherein the initial current of each nonplanar superconducting coil among the plurality of nonplanar superconducting coils is the same.

[0034] Using the above technical solution, the single-wire current of each non-planar superconducting coil is scaled according to the number of wires, so that the single-wire current is fixed to one coil current to avoid trivial solutions.

[0035] According to another specific embodiment of the present invention, the multi-objective optimization method for nonplanar coils of a stellarator disclosed in this embodiment further includes magnetic axis radius in the magnetic field configuration data; step S12 further includes: dividing the grid region according to the magnetic field gradient distribution parameters, refining the grid in the region with a large magnetic field gradient, and thinning the grid in the region with a small magnetic field gradient, generating a boundary point set; obtaining the initial contours of multiple nonplanar superconducting coils according to the magnetic axis radius and the boundary point set; wherein, the formula for the coil curve is:

[0036]

[0037] In the formula, Let θ be the parametric angle, x(θ) be the coordinate in the x-direction of the spatial coordinate system, y(θ) be the coordinate in the y-direction of the spatial coordinate system, z(θ) be the coordinate in the z-direction of the spatial coordinate system, and M be the Fourier order. These are the amplitudes of cos(mθ) and sin(mθ) in the Fourier representations of x(θ), y(θ), and z(θ), respectively.

[0038] According to another specific embodiment of the present application, the multi-objective optimization method for the non-planar coil of the stellarator disclosed by the embodiments of the present application comprises the following steps: in step S3, the L-BFGS-B algorithm is used to solve the minimum value of the objective function, and the iteration step and the convergence threshold are adjusted according to the gradient change rate of the objective function; wherein the calculation formula of the gradient change rate is:

[0039] ,

[0040] In the formula, Δη k is the gradient change value of the kth objective function, Δη k-1 is the gradient change value of the (k-1) th objective function.

[0041] If η is less than or equal to the preset adjustment threshold, the step is expanded; if η is greater than the preset adjustment threshold, the step is reduced and the convergence threshold is reduced; and when the gradient change value of the objective function is less than the adjusted convergence threshold, the iteration is stopped, and the optimized coil data is obtained.

[0042] By using the above technical solution, the improved L-BFGS-B algorithm is used to minimize the solution of the objective function, and the iteration step and the convergence threshold are dynamically adjusted according to the gradient change rate of the objective function, so that efficient optimization is realized.

[0043] According to another specific embodiment of the present application, the multi-objective optimization method for the non-planar coil of the stellarator disclosed by the embodiments of the present application comprises the following steps: in step S4, according to the coil data, the wire grouping, the peripheral wire positioning, the resampling mapping and the cross section splicing are performed, and the discrete wire is converted into a complete non-planar superconducting coil model.

[0044] According to another specific embodiment of the present application, the multi-objective optimization method for the non-planar coil of the stellarator disclosed by the embodiments of the present application comprises the following steps: in step S4, according to the coil data, the wire grouping, the peripheral wire positioning, the resampling mapping and the cross section splicing are performed, and the discrete wire is converted into a complete non-planar superconducting coil model.

[0045]

[0046]

[0047]

[0048] In the formula, N is the total number of vertices of the wire, respectively, are the three-dimensional coordinates of the i th vertex.

[0049] The expression of the within-group sum of squares objective function for wire grouping is:

[0050]

[0051] In the formula, r is the number of coil groups,​ is the center of the kth coil group, and x is the filament center point of the kth coil group. is the center of the kth coil group, and x is the filament center point of the kth coil group.

[0052] The peripheral filament positioning step includes obtaining the relative coordinates of each filament center point with the overall centroid of the plurality of filaments in each coil group as the origin.

[0053] The resampling mapping step includes mapping the total number of vertices of the filaments to the target number of points, and the mapping relationship function is:

[0054]

[0055] In the formula, is the original index value, N is the number of vertices of the filament, i is the index of the target point, is the target number of points.

[0056] The cross-section splicing step includes calculating the cross-section vertex identifier, positioning the vertices of adjacent cross-sections through identifier offset, and realizing the construction of the side surface quadrilateral element.

[0057] The application also provides a stellarator magnet, which comprises a plurality of non-planar superconducting coils arranged in an array, and the plurality of non-planar superconducting coils are obtained by using the multi-objective optimization method for stellarator non-planar coils. BRIEF DESCRIPTION OF DRAWINGS

[0058] Figure 1 is the flowchart of the multi-objective optimization method for stellarator non-planar coils provided by the application;

[0059] Figure 2 is the boundary profile boundary feature map of the target plasma preset in one specific embodiment of the multi-objective optimization method for stellarator non-planar coils provided by the application;

[0060] Figure 3 is a schematic diagram of a uniform multi-filament layout using a 2*2 multi-filament structure;

[0061] Figure 4 is a schematic diagram of a plurality of non-planar superconducting coil models obtained by one specific embodiment of the multi-objective optimization method for stellarator non-planar coils provided by the application. DETAILED DESCRIPTION

[0062] In order to make the purpose, technical scheme and advantages of the application clearer, the embodiments of the application will be further described in detail below with reference to the drawings.

[0063] Example 1

[0064] The application provides a multi-objective optimization method for a non-planar coil of a stellarator, wherein the non-planar superconducting coil is a coil made of superconducting cable and other materials, has a complex spatial structure, and presents a three-dimensional non-planar configuration. Figure 1 As shown in the figure, the optimization method comprises the following steps S1-S4.

[0065] S1: obtaining an initial structure of a plurality of non-planar superconducting coils according to position data, wherein the magnetic field position data comprises target magnetic field parameters.

[0066] Specifically, the application is used for the design of a stellarator magnet, the stellarator comprises a plurality of non-planar superconducting coils, and a complex magnetic field structure is generated by the plurality of coils to confine a plasma. The target plasma in the above steps refers to a preset target plasma, which can be preset by a VMEC program and form a magnetic field position input file. It should be noted that when the design scheme of the stellarator magnet is determined, the expected magnetic field position can be determined as long as the integral of the square of the normal magnetic field on the surface of the plasma is low enough. The magnetic field position data comprises the boundary poloidal / toroidal profile of the plasma, the magnetic axis radius (maximum radius), the magnetic field distribution parameter, the magnetic field gradient distribution parameter, the magnetic field period number, the target magnetic field parameter, and the like, wherein Figure 2 A boundary profile feature map of a preset target plasma in one specific embodiment is shown. The target magnetic field parameter is a parameter related to the magnetic field strength, which can be used to compare the matching of the obtained magnetic field and the target magnetic field of the plasma, and specifically comprises at least one of the following parameters: magnetic flux error, normalized magnetic flux error, and normal magnetic field component on the surface of the plasma.

[0067] In one specific embodiment, as shown in the figure, Figure 1 The step S1 specifically comprises the following steps S11 and S12.

[0068] S11: obtaining the magnetic field position data, specifically, a Surface RZ Fourier type optimization program can be used to load a magnetic field position file of a target plasma for design optimization, and the position file can be a VMEC format file, and then the magnetic field position data is obtained; the magnetic field position data comprises a magnetic field gradient distribution parameter for describing the spatial distribution rate of the magnetic field strength. Specifically, the magnetic axis radius, the boundary poloidal / toroidal profile, the magnetic field period number and other data are directly extracted from the position file, and then the magnetic field gradient distribution parameter is calculated by the finite difference method: ; wherein is the change amount of the magnetic field strength in unit distance, which can be expressed as = dB / dx (x-direction gradient), unit: T / m.

[0069] S12: Obtain the initial structure of the plurality of non-planar superconducting coils according to the magnetic field gradient distribution parameter G. Specifically, first determine the initial contour of the non-planar superconducting coil according to the configuration file, and then perform multi-filament layout based on the initial contour. Under the premise of keeping the total number of coil filaments fixed (such as a 2x2 multi-filament structure), adapt the magnetic field gradient distribution parameter by adjusting the Fourier order of the coil curve: in the region with large magnetic field gradient, a higher Fourier order is used to construct the initial contour of the non-planar superconducting coil, so as to improve the fitting accuracy of the coil curve to the complex magnetic field distribution; in the region with small magnetic field gradient, a lower Fourier order is used to construct the initial contour, which controls the calculation complexity and reduces the calculation cost while ensuring the basic fitting accuracy. Figure 3 A 2x2 multi-filament structure is shown. Adjust the Fourier order of the coil curve to adapt to the magnetic field gradient distribution parameter: in the region with large magnetic field gradient, a higher Fourier order is used to construct the initial contour of the non-planar superconducting coil, so as to improve the fitting accuracy of the coil curve to the complex magnetic field distribution; in the region with small magnetic field gradient, a lower Fourier order is used to construct the initial contour, which controls the calculation complexity and reduces the calculation cost while ensuring the basic fitting accuracy.

[0070] Further, in a specific embodiment, step S12 further comprises: after the above-mentioned layout of the superconducting coil filament arrangement in different regions, assigning an initial current to each initial structure of the non-planar superconducting coil, and the initial current of each non-planar superconducting coil in the plurality of non-planar superconducting coils is the same.

[0071] Specifically, the same initial current I0 is assigned to each non-planar superconducting coil with the filament arrangement laid out. The specific value of the initial current I0 can be determined according to the preset target, requirements, etc. of the designed stellarator, to ensure the matching accuracy of the magnetic field generated by the coil and the target magnetic surface of the plasma, which can be realized by flux error constraint (J_flux) optimization. In a specific embodiment, I0 can be 8.64x10 5 A. The initial current of each non-planar superconducting coil is the same, and the single-filament current is scaled according to the number of filaments (nfil = numfilaments n x numfilaments b) of each non-planar superconducting coil, i.e. the initial current of the non-planar superconducting coil is uniformly distributed to each single filament, and the current of each single filament is: total current / number of single filaments; this setting makes each single filament have a fixed coil current.

[0072] It should be noted that if each single filament does not have a fixed current, it will participate in optimization as a variable, and the objective function may converge to a trivial solution of all zeros due to the following reasons: the flux error is essentially the "sum of the squares of the differences between the coil magnetic field and the target magnetic field", and if the target magnetic field is locally zero, the all-zero current may make the error in that region zero, leading to the algorithm misjudging as the optimal solution; therefore, each single filament having a fixed coil current can avoid the trivial solution of "all-zero current". Moreover, fixed coil current is the simplest and most direct means to avoid the trivial solution of "all-zero current": by forcing the current to remain non-zero, it ensures that the optimization result meets the core physical requirement of the stellarator "using magnetic field to confine plasma", rather than degenerating into a meaningless zero magnetic field state.

[0073] Further, in a specific embodiment, the magnetic field configuration data in step S12 includes a magnetic axis radius , i.e., the radial reference coordinate value of the plasma magnetic axis (the center axis of magnetic field symmetry). It is the core geometric parameter of the magnetic axis, the average reference value of the magnetic axis three-dimensional curve in the R direction, and determines the overall radial position of the plasma and the reference scale of the coil design; step S12 further includes: dividing the grid area according to the magnetic field gradient distribution parameter G, and encrypting the grid in the area with large magnetic field gradient to improve the magnetic field matching calculation precision, and thinning the grid in the area with small magnetic field gradient to reduce the calculation cost, and finally generating a discrete boundary point set, whose expression is:

[0074] ,

[0075] wherein N is the total number of boundary points.

[0076] According to the magnetic axis radius (unit: m) and the boundary point set, the initial profile of a plurality of non-planar superconducting coils is obtained; specifically, the main radius of the initial coil is set, wherein is the safety distance between the coil and the plasma boundary, and the unit is m; then the Fourier order is used to describe the coil curve, and the formula of the coil curve is:

[0077]

[0078] wherein x(θ) is the coordinate in the x direction of the space coordinate system, y(θ) is the coordinate in the y direction of the space coordinate system, and z(θ) is the coordinate in the z direction of the space coordinate system; θ is the parameter angle, the unit is rad, and represents the parameterized angle variable of the coil curve, and the value range is [0, 2π) ; M is the Fourier order; and are the amplitudes of cos(mθ) and sin(mθ) in the Fourier representation of x(θ), y(θ), and z(θ), respectively.

[0079] It should be noted that the order of the above specific steps in step S12 is as follows.

[0080] S121: According to the magnetic field gradient distribution parameter, the grid area is divided, the grid is encrypted in the area with large magnetic field gradient, the grid is thinned in the area with small magnetic field gradient, and the boundary point set is generated.

[0081] S122: According to the magnetic axis radius and the boundary point set, the initial profile of a plurality of non-planar superconducting coils is obtained.

[0082] S123: According to the magnetic field gradient distribution parameter, the multi-wire is laid out to obtain the initial structure of the non-planar superconducting coil, and the magnetic field gradient distribution parameter is adapted by adjusting the Fourier order of the coil curve: in the area with large magnetic field gradient, higher Fourier order is used to construct the initial contour of the non-planar superconducting coil, so as to improve the fitting accuracy of the coil curve to the complex magnetic field distribution; in the area with small magnetic field gradient, lower Fourier order is used to construct the initial contour, which can control the calculation complexity and reduce the calculation cost while ensuring the basic fitting accuracy.

[0083] S124: Assign an initial current to each initial structure of the non-planar superconducting coil.

[0084] S2: Construct a target function according to the target magnetic field parameter, the engineering constraint parameter and the material constraint parameter, wherein the target magnetic field parameter can be at least one of the magnetic field strength and the magnetic flux of the target magnetic field; the engineering constraint parameter includes the engineering limit geometry parameter of the non-planar superconducting coil, and the material constraint parameter includes the limit current carrying capacity of the non-planar superconducting coil material; a multi-objective optimization target function is constructed based on the constraints of "magnetic field matching + engineering feasibility + material safety". The expression of the target function can be at least one of linear function, quadratic function, matrix form, etc.

[0085] In a specific embodiment, the target magnetic field parameter includes the target magnetic flux; the expression of the target function constructed by the linear function is:

[0086] ,

[0087] In the formula, is the target function, is the deviation of the actual magnetic flux from the target magnetic flux, the unit is Wb, and ω1 is the magnetic flux weight, is the engineering constraint parameter, ω2 is the engineering constraint weight, is the material constraint parameter, which is less than or equal to the preset limit current, the unit is A / m 2 , and ω3 is the material constraint weight. Wherein, ω1, ω2, ω3 are determined according to the importance of different parameters to the optimization design, and the specific values of ω1, ω2, ω3 can also be adjusted based on the actual design requirements, so that the coil obtained by the design meets the requirements.

[0088] Specifically, The calculation method of includes: first calculating the magnetic field strength generated by the initial structure of each non-planar superconducting coil (unit: T), which can be calculated by Biot-Savart law, and the square integral of the normal component of the target magnetic field is solved to obtain the deviation of the actual magnetic flux from the target magnetic flux, so as to realize the quantization of the magnetic field matching error, and the calculation formula is:

[0089]

[0090] wherein ds represents a micro-area element of the plasma surface, n represents a normal unit vector of the plasma, B target represents a magnetic field strength of the target plasma, with a unit of T.

[0091] In one specific embodiment, the calculation formula of the engineering constraint parameter is:

[0092] ,

[0093] wherein, is a minimum coil constraint length, and the calculation formula is:

[0094] ,

[0095] wherein, L MIN is a preset minimum length of the coil, is an actual length of the i-th non-planar superconducting coil, and q represents a number of non-planar superconducting coils, which can be specifically 12, 16, 20, 24, etc.

[0096] is a maximum coil constraint length, and the calculation formula is:

[0097] ,

[0098] wherein, L MAX is a preset maximum length of the coil.

[0099] is a coil constraint distance, and the calculation formula is:

[0100]

[0101] wherein, d min is a preset minimum distance between coils, is an actual distance between the i-th non-planar superconducting coil and the j-th non-planar superconducting coil.

[0102] is a coil constraint distance, and the calculation formula is:

[0103]

[0104] wherein, is a minimum safe distance between the coil and the plasma magnetic surface, D is is an actual distance between the i-th non-planar superconducting coil and the magnetic surface.

[0105] is a maximum constraint curvature of the basic half-period coil, and the calculation formula is:

[0106] ,

[0107] In the formula, k i Let k be the actual maximum curvature of the i-th fundamental half-cycle coil. curv p is the preset maximum curvature of the base half-cycle coil, and p is the number of base half-cycle coils.

[0108] The maximum curvature is obtained by first discretizing the parameterized space curve to calculate the curvature at each point, and then taking the maximum value of the discrete curvature set, with the unit being m. -1 The number of basic half-cycle coils is determined based on the total number of coils in the stellarator magnet, where the total number of coils is: p × 2 × number of cycles; as shown below. Figure 4 Taking the plasma structure shown as an example, the three-dimensional plasma structure includes four symmetrically arranged cycles. Multiple coils (four in the figure) in each cycle can be configured using a basic half-cycle coil. Figure 4 The basic half-cycle coil is generated by two coils.

[0109] It should be noted that each cycle of the plasma is symmetrical. Therefore, only the basic half-cycle coil needs to be set, and the other half-cycles will automatically generate the same coil symmetrically, with the only difference being the angle. Furthermore, after the coil of one cycle is completed, the remaining multiple cycles are exactly the same. Therefore, in this invention, only the basic half-cycle coil is calculated.

[0110] The square curvature of the basic half-cycle coil is an index describing the smoothness of the coil curve. The average square of the curvature at each discrete point of the coil is calculated to constrain the feasibility of the coil winding process; the calculation formula is:

[0111]

[0112] In the formula, MSC is the actual mean square curvature of the i-th fundamental half-cycle coil. y The preset mean square curvature of the base half-cycle coil.

[0113] In one specific implementation, the material constraint parameters are calculated according to the current density calculation formula, which is as follows: In the formula, S is the cross-sectional area of ​​the coil; and constraints are applied. Not exceeding the material's current carrying capacity Then the cross-sectional area of ​​the single wire in the coil must meet the following requirements. In the formula I fil This represents the current in a single wire.

[0114] The prior art open source optimization method only considers the magnetic field matching constraint, considers the coil length and coil spacing constraint on this basis, ignores key constraints such as coil curvature, average square curvature, coil-plasma distance and current density, and is prone to problems such as that the optimization result has physical feasibility but engineering cannot be implemented, for example, the problem that the optimized superconducting coil curvature is too large to cause winding difficulty and cannot be implemented. The application constructs a target function including a core constraint term related to magnetic field matching, an engineering constraint term and a material constraint term, measures the matching degree of the optimized coil magnetic field and the target magnetic field through the core constraint term, constrains the manufacturing and installation feasibility of the coil through the engineering constraint term, and constrains the material bearing limit of the coil through the material constraint term, forms a three-dimensional constraint system, realizes the verification of the optimization result from three dimensions, and can avoid the problem that engineering cannot be implemented.

[0115] S3: minimizing the target function to obtain the optimized coil data. Specifically, based on the target function constructed in the foregoing, the minimization solving can be realized by methods such as iteration, gradient descent and Newton method; and as shown in the following formula, in the optimization process, it is continuously judged whether the optimization result meets the preset requirement; if yes, the optimization is stopped, and the optimized coil data is output; if no, the minimization solving of the target function is continuously performed. Figure 1

[0116] In a specific embodiment, the L-BFGS-B (Limited memory Broyden Fletcher Goldfarb Shanno) algorithm is used to solve the minimum value of the target function. The L-BFGS-B is a Newton method optimization algorithm, which is used to solve the nonlinear optimization problem with upper and lower boundary constraints of variables. And the iteration step and convergence threshold are adjusted according to the gradient change rate of the target function. Wherein, the calculation formula of the gradient change rate is:

[0117] ,

[0118] In the formula, η is the gradient change value of the kth target function, is the gradient change value of the k-1th target function. Based on the gradient change rate, the iteration convergence trend is judged, and the algorithm parameters are dynamically adjusted, wherein if η is less than or equal to the preset adjustment threshold, it indicates that the convergence is relatively stable, and then the step is expanded to speed up the convergence; if η is greater than the preset adjustment threshold, it indicates that there is an iteration oscillation problem, and then the step is reduced and the convergence threshold is reduced. And real-time iteration monitoring is performed, and when the gradient change value of the target function is less than the dynamically adjusted convergence threshold

[0119] ​​​When the optimization result meets the preset requirements, the iteration stops, and the optimized coil data is obtained, including coil geometric parameters, current parameters, and magnetic field distribution data. The preset adjustment threshold can be set according to the actual target, for example, it can be set to 0.1.

[0120] Existing technologies employ optimization methods using the L-BFGS-B algorithm with fixed parameters. However, this approach is prone to iterative oscillations or slow convergence when dealing with complex objective functions with multiple coupled constraints, thus lengthening the optimization cycle. This invention utilizes an improved L-BFGS-B algorithm to minimize the objective function, dynamically adjusting the iteration step size and convergence threshold based on the gradient rate of change of the objective function, achieving efficient optimization.

[0121] S4: Based on the optimized coil data, the initial structures of multiple non-planar superconducting coils are materialized to obtain optimized models of multiple non-planar superconducting coils. Figure 4 A schematic diagram of multiple nonplanar superconducting coil models obtained after one optimization is shown, which have four periods. The four coils in different periods are the same. The coil labels in the figure indicate the period and position of the coil, such as "1_Coil3" indicating the third coil in the first period.

[0122] Specifically, such as Figure 1 As shown, during the solidification process, it is determined whether the processing result meets the preset requirements. If yes, multiple optimized non-planar superconducting coil models are output; if not, the solidification process continues. Specifically, the following methods can be used to determine whether the processing result meets the preset requirements: use tools such as Para View to confirm whether the processed coil is a closed structure without broken surfaces, whether it is generated along the original wire path and the cross-section is rectangular without deformation; determine whether a predetermined number of coils (e.g., 16) are output, whether a single coil contains a predetermined number of points (e.g., 400), a predetermined number of side elements (e.g., 398), and 2 end face elements, and whether all cross-sections of the same coil are basically consistent (specifically, the cross-sectional size fluctuation can be set ≤1%, and the length error with the original wire ≤0.5%). If all the above judgment results are yes, then the processing result can be judged to meet the preset requirements.

[0123] In one specific implementation, based on the coil data, the wires are grouped, the peripheral wires are located, resampling and mapping are performed, and the cross-sections are spliced ​​to transform the discrete wires into a complete non-planar superconducting coil model, thereby realizing the transformation of the coil from discrete data to a physical model.

[0124] Furthermore, in one specific embodiment, the step of grouping the threads includes: calculating the geometric center of all vertices of a single thread, using the following formula:

[0125]

[0126]

[0127]

[0128] wherein N is the total number of vertices of the wire, is the three-dimensional coordinate of the i-th vertex, respectively.

[0129] When grouping the wires, the grouping is achieved by minimizing the within-group sum of squares based on the Kmeans clustering algorithm, and the objective function expression of the within-group sum of squares is:

[0130]

[0131] wherein r is the number of coil groups, is the k-th coil group, is the center of the k-th coil group, and x is the wire center point of the k-th coil group. Wherein the coil group refers to a collection of a plurality of discrete wires aggregated by a specific rule (Kmeans clustering algorithm) to construct a single complete non-planar superconducting coil entity model, the center of the coil group refers to the three-dimensional coordinate mean value of the "wire center point" of all wires in a single coil group (i.e. the group centroid), and the wire center point refers to the three-dimensional geometric center of all vertices of a single wire, which is a key coordinate representing the spatial position of a single wire.

[0132] The peripheral wire positioning step includes: taking the overall centroid of a plurality of wires in each coil group as the origin to obtain the relative coordinates of each wire center point, and the calculation formula of the relative coordinates is:

[0133]

[0134] ,

[0135] wherein, is the centroid coordinate of the plurality of wire center points.

[0136] The resampling mapping step includes mapping the total number of vertices of the wire to the target number of points, and the mapping relationship function is:

[0137]

[0138] wherein, is the original index value, N is the number of vertices of the wire, and i is the index of the target point (such as 0~99), is the target number of points (such as 100 points).

[0139] The cross section splicing step includes: peripheral wire positioning, screening a plurality of peripheral wires by "relative coordinates with the mass center of the coil group as the origin", ensuring that the cross section is rectangular and meets the preset multi-wire structure; peripheral wire resampling, a plurality of peripheral wires are unified to the same target point number through "resampling mapping"; generating the vertices and ID of a single cross section, taking the i-th point of the plurality of peripheral wires as the four vertices of the current cross section for each target point index i; vertex ID assignment, sequentially adding the four vertices of the last step to the global point set of the solid coil to automatically generate continuous vertex IDs; constructing the vertices of the adjacent cross section through fixed ID offset, the vertex IDs of the connected cross sections exist fixed offset, and four edge elements are sequentially constructed according to the offset rule for each adjacent cross section; and finally realizing the construction of the side surface four edge element.

[0140] It should be noted that the present application adopts the principle of the above-mentioned cross section splicing step, which includes that the vertex IDs of adjacent cross sections exist fixed offset, wherein each cross section contains 4 vertices, and the offset expression is: .

[0141] In the formula, curr_pid is the ID of the vertex of the current cross section, is the ID of the corresponding vertex of the last cross section.

[0142] The design and optimization of the three-dimensional non-planar superconducting coil is the focus of the stellarator magnet design, and various constraints need to be considered to achieve the best balance. The multi-objective optimization method for the stellarator non-planar coil provided by the present application takes the magnetic field configuration of the pre-designed plasma as the target reference, constructs a three-dimensional constraint system of "magnetic field matching-engineering feasibility-material safety", verifies the optimization result from three dimensions, and avoids the problem that the engineering cannot be implemented. Further, the strategy of adopting the initial structure of the superconducting coil multi-wire layout adapted to the magnetic field gradient distribution and the improved algorithm of dynamically adjusting the algorithm parameters are adopted for optimization, so as to realize the global optimization of the optimized superconducting coil parameters and improve the optimization effect. The present application provides a multi-objective optimization method for quasi-isostatic non-planar superconducting coils, realizes multi-objective balance and efficient solution of coil design, improves the optimization convergence efficiency, and is suitable for coil design of various magnetic confinement nuclear fusion devices such as tokamak and stellarator, and provides key technical support for the engineering implementation of high-performance nuclear fusion devices.

[0143] Example 2

[0144] The present application provides a stellarator magnet, comprising a plurality of non-planar superconducting coils distributed in an array, wherein the plurality of non-planar superconducting coils are obtained by optimization using the multi-objective optimization method for the stellarator non-planar coil provided in example 1.

[0145] It is to be understood that the above description is intended to be illustrative and not restrictive. Many other embodiments will be apparent to those of skill in the art upon reading and understanding the above description. Although the application has been described with reference to particular means, materials and embodiments, from the foregoing description, one skilled in the art can effect a variety of changes and modifications. It is, therefore, desired to be protected in whatever field within the broadest interpretation of the application as set forth in the following claims.

[0146] While the application has been described with reference to certain implementations, it will be understood by those skilled in the art that various changes can be made and equivalents can be substituted for elements thereof without departing from the scope of the application. In addition, many modifications can be made to adapt a particular situation or material to the teachings of the application without departing from the central inventive concept described herein.

Claims

1. A multi-objective optimization method for nonplanar coils in stellarators, characterized in that, Includes the following steps: S1: Obtain the initial structure of multiple non-planar superconducting coils based on the magnetic field configuration data of the target plasma, wherein the magnetic field configuration data includes the target magnetic field parameters; S2: Construct an objective function based on the target magnetic field parameters, engineering constraint parameters, and material constraint parameters, wherein the engineering constraint parameters include the engineering limiting geometric parameters of the nonplanar superconducting coil, and the material constraint parameters include the limiting current carrying capacity of the nonplanar superconducting coil material; S3: Minimize the objective function to obtain the optimized coil data; S4: Based on the optimized coil data, the plurality of nonplanar superconducting coils are materialized to obtain optimized models of the plurality of nonplanar superconducting coils.

2. The multi-objective optimization method for nonplanar coils of stellarators as described in claim 1, characterized in that, The target magnetic field parameters include the target magnetic flux; in step S2, the expression for the constructed objective function is: ， In the formula, Let the objective function be... This represents the deviation between the actual magnetic flux and the target magnetic flux. For magnetic flux weighting, These are the engineering constraint parameters. For engineering constraint weights, The material constraint parameter is less than or equal to the preset limiting current carrying capacity. For material constraint weights.

3. The multi-objective optimization method for nonplanar coils of stellarators as described in claim 2, characterized in that, The calculation formula for the engineering constraint parameters is as follows: ， in, The minimum coil constraint length is calculated using the following formula: ， In the formula, L MIN Preset a minimum length for the coil. Let be the actual length of the i-th nonplanar superconducting coil, and q represent the number of nonplanar superconducting coils. The maximum coil constraint length is calculated using the following formula: ， In the formula, L MAX The maximum length is preset for the coil; The formula for calculating the coil constraint spacing is: In the formula, d min Preset a minimum spacing between coils. This represents the actual distance between the i-th and j-th nonplanar superconducting coils; The coil constraint distance is calculated using the following formula: In the formula, D is the minimum safe distance between the coil and the plasma magnetic surface. is Let be the actual distance between the i-th nonplanar superconducting coil and the magnetic surface; The maximum constraint curvature of the basic half-cycle coil is calculated using the following formula: ， In the formula, k i Let k be the actual maximum curvature of the i-th fundamental half-cycle coil. curv p is the preset maximum curvature of the base half-cycle coil; The square curvature of the basic half-cycle coil is calculated using the following formula: In the formula, Let be the actual mean square curvature of the i-th fundamental half-cycle coil. The preset mean square curvature of the base half-cycle coil.

4. The multi-objective optimization method for nonplanar coils of stellarators as described in claim 1, characterized in that, Step S1 includes: S11: Obtain the magnetic field configuration data, which further includes magnetic field gradient distribution parameters; S12: Obtain the initial structure of the plurality of nonplanar superconducting coils based on the magnetic field gradient distribution parameters; In the nonplanar superconducting coil, a higher Fourier order is used to construct the initial profile in the region with a large magnetic field gradient; in the region with a small magnetic field gradient, a lower Fourier order is used to construct the initial profile.

5. The multi-objective optimization method for nonplanar coils of stellarators as described in claim 4, characterized in that, Step S12 also includes: An initial current is assigned to the initial structure of each of the nonplanar superconducting coils, and the initial current of each of the plurality of nonplanar superconducting coils is the same.

6. The multi-objective optimization method for nonplanar coils of stellarators as described in claim 4, characterized in that, The magnetic field configuration data also includes the magnetic axis radius; step S12 further includes: The grid region is divided according to the magnetic field gradient distribution parameters. The grid is made denser in the region with large magnetic field gradient and thinner in the region with small magnetic field gradient, and a set of boundary points is generated. The initial contours of the plurality of nonplanar superconducting coils are obtained based on the magnetic axis radius and the boundary point set; The formula for the coil curve is: In the formula, x(θ) is the coordinate in the x-direction of the spatial coordinate system, y(θ) is the coordinate in the y-direction of the spatial coordinate system, and z(θ) is the coordinate in the z-direction of the spatial coordinate system. Let M be the parameter angle, and M be the Fourier order. These are the amplitudes of cos(mθ) and sin(mθ) in the Fourier representations of x(θ), y(θ), and z(θ), respectively.

7. The multi-objective optimization method for nonplanar coils of stellarators as described in claim 1, characterized in that, In step S3, the L-BFGS-B algorithm is used to solve for the minimum value of the objective function, and the iteration step size and convergence threshold are adjusted according to the gradient rate of change of the objective function; wherein, the formula for calculating the gradient rate of change is: ， In the formula, Let be the gradient change value of the objective function in the kth iteration. This represents the gradient change value of the objective function at the (k-1)th iteration; If η is less than or equal to the preset adjustment threshold, then increase the step size; If η is greater than the preset adjustment threshold, then the step size is reduced and the convergence threshold is decreased; Furthermore, when the gradient change value of the objective function is less than the adjusted convergence threshold, the iteration stops, and the optimized coil data is obtained.

8. The multi-objective optimization method for nonplanar coils of stellarators as described in any one of claims 1-7, characterized in that, In step S4, based on the coil data, the wires are grouped, the outer wires are located, resampling and mapping are performed, and the cross-sections are spliced ​​to transform the discrete wires into a complete non-planar superconducting coil model.

9. The multi-objective optimization method for nonplanar coils of stellarators as described in claim 8, characterized in that, The steps of grouping the silk threads include: The formula for calculating the geometric center of all vertices of a single thread is as follows: Where N is the total number of vertices of the thread. These are the three-dimensional coordinates of the i-th vertex; The objective function for grouping silk threads into groups and calculating the sum of squares within each group is expressed as follows: In the formula, r is the number of coil groups. For the k-th coil group, Let x be the center of the k-th coil group, and let x be the center point of the wire in the k-th coil group. The peripheral wire positioning step includes obtaining the relative coordinates of the center point of each wire, with the overall centroid of multiple wires within each coil group as the origin: The resampling mapping step includes mapping the total number of vertices of the thread to the target number of points, and the mapping relationship function is: In the formula, Here, N is the original index value, N is the number of vertices of the thread, and i is the index of the target point. The target number of points; The section splicing step includes: calculating the vertex identifier of the section, locating the vertex of adjacent sections by offsetting the identifier, and realizing the construction of the side quadrilateral unit.

10. A stellarator magnet, characterized in that, It includes multiple nonplanar superconducting coils arranged in an array, wherein the multiple nonplanar superconducting coils are obtained by optimization using the multi-objective optimization method for nonplanar coils of stellarators as described in any one of claims 1-9.

Images