Multi-objective optimization method for nonplanar coils in stellarators and stellarator magnets

Abstract

This invention provides a multi-objective optimization method for nonplanar coils in stellarators and a stellarator magnet. The optimization method includes: obtaining the initial structure of multiple nonplanar superconducting coils based on the magnetic field configuration data of the target plasma; constructing an objective function based on the target magnetic field parameters, engineering constraint parameters, and material constraint parameters of the configuration data, wherein the engineering constraint parameters include the engineering limiting geometric parameters of the nonplanar superconducting coils, and the material constraint parameters include the limiting current carrying capacity of the nonplanar superconducting coil material; minimizing the objective function to obtain optimized coil data; and solidifying the multiple nonplanar superconducting coils based on the optimized coil data to obtain optimized models of multiple nonplanar superconducting coils. The optimization method of this invention constructs a three-dimensional constraint system of "magnetic field matching - engineering feasibility - material safety," avoiding the problem of engineering implementation failure.

Description

Technical Field

[0001] This invention belongs to the field of superconducting technology, and specifically relates to an optimization method for stellarator coils and a stellarator magnet obtained by using the optimization method. Background Technology

[0002] Magnetic confinement fusion, which uses a strong magnetic field generated by coils to confine plasma at temperatures exceeding hundreds of millions of degrees Celsius within a specific geometric configuration, achieves controlled nuclear fusion reactions and represents a crucial direction for addressing the global energy crisis. Stellar reactors are a type of magnetic confinement fusion device. They utilize a series of twisted coils to generate field lines, avoiding induced current saturation and achieving more stable long-term operation. The coil design not only requires the precise generation of the necessary magnetic field to confine the plasma but also must consider engineering feasibility (such as coil manufacturing difficulty, mechanical stress, and cost). Stellar reactor research aims to reduce neoclassical transport levels and high-energy particle loss by optimizing the magnetic field configuration.

[0003] As the "skeleton" of a nuclear fusion device, the design of the coil must simultaneously meet the following requirements: (1) the generated magnetic field must be precisely matched with the target plasma configuration (to ensure confinement performance); (2) the shape and size of the coil must meet the manufacturing and installation requirements (to ensure engineering feasibility); and (3) the current density must not exceed the material's bearing capacity limit (to ensure operational safety). However, due to the complex three-dimensional coil system of the stellarator, the existing stellarator coil optimization technology has the following defects: the constraint system is singular, and the mainstream open-source optimization framework only considers magnetic field matching constraints, often resulting in the problem that "the optimization result is physically feasible but cannot be implemented in engineering"; the coil initialization adaptability is poor, and the non-uniform distribution of the magnetic field gradient of the target configuration is not considered; the optimization algorithm is inefficient, and when faced with complex objective functions with multiple constraints, it is easy to have iterative oscillations or slow convergence, resulting in a long optimization cycle. Summary of the Invention

[0004] The purpose of this invention is to solve the defects of existing stellarator coil optimization methods, such as imperfect constraint system, poor initialization adaptability, and low optimization efficiency.

[0005] This invention provides a multi-objective optimization method for nonplanar coils in stellarators, comprising the following steps: S1: Obtaining the initial structure of multiple nonplanar superconducting coils based on the magnetic field configuration data of the target plasma, wherein the magnetic field configuration data includes target magnetic field parameters; S2: Constructing 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 limit geometric parameters of the nonplanar superconducting coils, and the material constraint parameters include the limit current carrying capacity of the nonplanar superconducting coil material; S3: Minimizing the objective function to obtain optimized coil data; S4: Performing physical processing on the multiple nonplanar superconducting coils based on the optimized coil data to obtain optimized models of multiple nonplanar superconducting coils.

[0006] By adopting the above technical solution, and taking the pre-designed magnetic field configuration of the plasma as the target benchmark, an objective function is constructed that includes core constraints, engineering constraints, and material constraints related to magnetic field matching. The core constraints measure the matching degree between the optimized coil magnetic field and the target magnetic field, the engineering constraints constrain the manufacturing and installation feasibility of the coil, and the material constraints constrain the material bearing limit of the coil, forming a three-dimensional constraint system. This allows for the verification of the optimization results from three dimensions, avoiding the problem of engineering implementation failure.

[0007] According to another specific embodiment of the present invention, the multi-objective optimization method for nonplanar coils of a stellarator disclosed in this embodiment includes a target magnetic field parameter including a target magnetic flux; in step S2, the expression of the constructed objective function is:

[0008] ,

[0009] In the formula, Let be the objective function. This represents the deviation between the actual magnetic flux and the target magnetic flux. For magnetic flux weighting, For engineering constraint parameters, For engineering constraint weights, For material constraint parameters, less than or equal to the preset limit current carrying capacity. For material constraint weights.

[0010] By adopting the above technical solution, the magnetic field matching error can be quantified based on the deviation between the actual magnetic flux and the target magnetic flux; and the weights of different constraint terms can be adjusted based on actual design requirements, making the design method more scientific and the coil obtained by the design more in line with the requirements.

[0011] According to another specific embodiment of the present invention, the multi-objective optimization method for nonplanar coils of stellarators disclosed in this embodiment of the present invention uses the following formula for calculating the engineering constraint parameters:

[0012] ,

[0013] in, The minimum coil constraint length is calculated using the following formula:

[0014] ,

[0015] 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.

[0016] The maximum coil constraint length is calculated using the following formula:

[0017] ,

[0018] In the formula, L MAX Set the maximum length for the coil.

[0019] The formula for calculating the coil constraint spacing is:

[0020]

[0021] 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.

[0022] The coil constraint distance is calculated using the following formula:

[0023]

[0024] 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.

[0025] The maximum constraint curvature of the basic half-cycle coil is calculated using the following formula:

[0026] ,

[0027] 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.

[0028] The square curvature of the basic half-cycle coil is calculated using the following formula:

[0029]

[0030] 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.

[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 invention, the multi-objective optimization method for nonplanar coils of a stellarator disclosed in this embodiment of the present invention, in step S3, uses the L-BFGS-B algorithm to solve for the minimum value of the objective function, and adjusts the iteration step size and convergence threshold according to the gradient rate of change of the objective function; wherein, the formula for calculating the gradient rate of change is:

[0039] ,

[0040] In the formula, Let be the gradient change value of the objective function at the k-th iteration. This represents the gradient change value of the objective function at the (k-1)th iteration.

[0041] Specifically, if η is less than or equal to the preset adjustment threshold, the step size is increased; if η is greater than the preset adjustment threshold, the step size is decreased 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 adopting the above technical solution, the improved L-BFGS-B algorithm is used to minimize the objective function. The iteration step size and convergence threshold are dynamically adjusted according to the gradient change rate of the objective function to achieve efficient optimization.

[0043] According to another specific embodiment of the present invention, the multi-objective optimization method for nonplanar coils of stellarators disclosed in the embodiment of the present invention, in step S4, performs wire grouping, peripheral wire positioning, resampling mapping and cross-section splicing based on the coil data, and transforms the discrete wires into a complete nonplanar superconducting coil model.

[0044] According to another specific embodiment of the present invention, the multi-objective optimization method for nonplanar coils of stellarators disclosed in this embodiment includes the following step of grouping the wires: calculating the geometric center of all vertices of a single wire, using the following formula:

[0045]

[0046]

[0047]

[0048] Where N is the total number of vertices of the thread. These are the three-dimensional coordinates of the i-th vertex.

[0049] The objective function for grouping silk threads into groups and calculating the sum of squares within each group is expressed as follows:

[0050]

[0051] 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.

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

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

[0054]

[0055] 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.

[0056] The cross-section splicing steps include: calculating the vertex identifiers of the cross-sections, locating the vertices of adjacent cross-sections by offsetting the identifiers, and constructing the lateral quadrilateral units.

[0057] The present invention also provides a stellarator magnet, comprising a plurality of nonplanar superconducting coils arranged in an array, which are optimized using the multi-objective optimization method for nonplanar stellarator coils provided by the present invention. Attached Figure Description

[0058] Figure 1 A flowchart of a multi-objective optimization method for nonplanar coils of a stellarator provided by the present invention;

[0059] Figure 2 This is a boundary profile feature map of a preset target plasma in a specific embodiment of the multi-objective optimization method for nonplanar coils of a stellarator provided by the present invention.

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

[0061] Figure 4 This is a schematic diagram of multiple nonplanar superconducting coil models obtained in a specific embodiment of the multi-objective optimization method for nonplanar coils of stellarators provided by the present invention. Detailed Implementation

[0062] To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.

[0063] Example 1

[0064] This invention provides a multi-objective optimization method for nonplanar coils in stellarators, wherein the nonplanar superconducting coil is a coil made of materials such as superconducting cables, possessing a complex spatial structure and exhibiting a three-dimensional, nonplanar configuration. Figure 1 As shown, the optimization method includes the following steps S1-S4.

[0065] S1: Obtain the initial structure of multiple nonplanar superconducting coils based on the configuration data. The magnetic field configuration data includes the target magnetic field parameters.

[0066] Specifically, this invention is used for the design of stellarator magnets. The stellarator includes multiple non-planar superconducting coils, which generate a complex magnetic field structure to confine plasma. The target plasma in the above steps refers to a pre-defined target plasma, which can be pre-defined using the VMEC program to generate a magnetic field configuration input file. It should be noted that once the stellarator magnet design scheme is determined, the expected magnetic field configuration can be determined simply by ensuring that the integral of the square of the normal magnetic field over the plasma surface is sufficiently low. The magnetic field configuration data includes the plasma's boundary poloidal / circular profile, magnetic axis radius (maximum radius), magnetic field distribution parameters, magnetic field gradient distribution parameters, magnetic field period number, target magnetic field parameters, etc. Figure 2 The diagram illustrates the boundary contour features of a predefined target plasma in a specific embodiment. The target magnetic field parameters are parameters related to magnetic field strength. These parameters can be used to compare the matching between the designed magnetic field and the magnetic field of the target plasma. Specifically, they include at least one of the following parameters: magnetic flux error, normalized magnetic flux error, and the normal magnetic field component of the plasma surface.

[0067] In one specific implementation, such as Figure 1 As shown, step S1 specifically includes the following steps S11 and S12.

[0068] S11: Obtain magnetic field configuration data. Specifically, the magnetic field configuration file of the target plasma for design optimization can be loaded using a Surface RZ Fourier-type optimization program. This configuration file can be a VMEC format file, thereby obtaining the magnetic field configuration data. The magnetic field configuration data includes magnetic field gradient distribution parameters, which describe the rate of change of magnetic field intensity in space. Specifically, data such as magnetic axis radius, boundary poloidal / circumferential profile, and magnetic field period number are directly extracted from the configuration file, and then the magnetic field gradient distribution parameters are calculated using the finite difference method. ;in The change in magnetic field strength per unit distance can be expressed as: = dB / dx (gradient in the x-direction), with units of T / m.

[0069] S12: Based on the magnetic field gradient distribution parameter G, obtain the initial structure of multiple nonplanar superconducting coils. Specifically, first determine the initial outline of the nonplanar superconducting coils based on the configuration file, and then perform multi-filament layout based on the initial outline. According to the magnetic field gradient distribution parameter, while keeping the total number of coil filaments fixed (e.g., ...), the initial structure is determined. Figure 3 Under the premise of a 2×2 multi-wire structure, the magnetic field gradient distribution parameters are adapted by adjusting the Fourier order of the coil curve: in the region of large magnetic field gradient, a higher Fourier order is used to construct the initial profile of the non-planar superconducting coil to improve the fitting accuracy of the coil curve to the complex magnetic field distribution; in the region of small magnetic field gradient, a lower Fourier order is used to construct the initial profile, thereby controlling the computational complexity and reducing the computational cost while ensuring the basic fitting accuracy.

[0070] Furthermore, in one specific embodiment, step S12 further includes: after arranging the superconducting coil wires in different regions as described above, assigning an initial current to the initial structure of each non-planar superconducting coil, and the initial current of each non-planar superconducting coil among the multiple non-planar superconducting coils is the same.

[0071] Specifically, each pre-arranged non-planar superconducting coil is assigned the same initial current I0. The specific value of the initial current I0 can be determined based on the preset goals and requirements of the stellarator to be designed, ensuring the matching accuracy between the magnetic field generated by the coil and the magnetic surface of the plasma target. This accuracy can be optimized through magnetic flux error constraints (J_flux). In one specific implementation, I0 can be 8.64 × 10⁻⁶. 5 A. Each nonplanar superconducting coil is set with the same initial current. The single-wire current of each nonplanar superconducting coil is scaled according to the number of wires (nfil = numfilaments n×numfilaments b), that is, the initial current of the nonplanar superconducting coil is evenly distributed to each single wire. The current of each single wire is: total current / number of single wires. This setting makes each single wire have a fixed coil current.

[0072] It should be noted that if the current in each filament is not fixed, it will participate in the optimization as a variable. The objective function may converge to a zero-current solution for the following reasons: the magnetic flux error is essentially the "sum of squares of the difference between the coil's magnetic field and the target magnetic field." If the target magnetic field is locally zero, a zero-current may make the error in that region zero, causing the algorithm to misjudge it as the optimal solution. Therefore, having a fixed current in one coil for each filament avoids the trivial solution of "zero-current." Furthermore, fixing the coil current is the simplest and most direct way to avoid the trivial solution of "zero-current": by forcing the current to remain non-zero, it ensures that the optimization result meets the core physical requirement of the stellarator to "confine plasma with a magnetic field," rather than degenerating to a meaningless zero-magnetic-field state.

[0073] Furthermore, in one specific implementation, the magnetic field configuration data in step S12 includes the magnetic axis radius. , that is, the radial reference coordinate value of the plasma magnetic axis (the central axis of magnetic field symmetry). It is the core geometric parameter of the magnetic axis, the average reference value of the three-dimensional curve of the magnetic axis in the R direction, which determines the overall radial position of the plasma and the reference scale of the coil design; Step S12 also includes: dividing the grid region according to the magnetic field gradient distribution parameter G, refining the grid in the region of large magnetic field gradient to improve the accuracy of magnetic field matching calculation, and thinning the grid in the region of small magnetic field gradient to reduce the computational cost, finally generating a discretized boundary point set, the expression of which is:

[0074] ,

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

[0076] Based on the magnetic axis radius (Unit: m) The initial contours of multiple nonplanar superconducting coils are obtained from the boundary point set; specifically, the principal radii of the initial coils are set. In the formula, The safe distance between the coil and the plasma boundary is given in meters (m). Then, the coil curve is described using Fourier order, and the formula for the coil curve is:

[0077]

[0078] 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; θ is the parameter angle, in rad, representing the parameterized angle variable of the coil curve, with a value range of [0, 2π); M is 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.

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

[0080] S121: Divide the grid region according to the magnetic field gradient distribution parameters. The grid is refined in the region with large magnetic field gradient and thinned in the region with small magnetic field gradient, generating a boundary point set.

[0081] S122: Obtain the initial profiles of multiple nonplanar superconducting coils based on the magnetic axis radius and boundary point set.

[0082] S123: Based on the magnetic field gradient distribution parameters, lay out multiple wires to obtain the initial structure of the nonplanar superconducting coil. Adapt the magnetic field gradient distribution parameters by adjusting the Fourier order of the coil curve: In regions with large magnetic field gradients, use a higher Fourier order to construct the initial profile of the nonplanar superconducting coil to improve the fitting accuracy of the coil curve to complex magnetic field distributions; in regions with small magnetic field gradients, use a lower Fourier order to construct the initial profile, controlling computational complexity and reducing computational costs while ensuring basic fitting accuracy.

[0083] S124: Assign initial current to the initial structure of each nonplanar superconducting coil.

[0084] S2: Construct an objective function based on the target magnetic field parameters, engineering constraint parameters, and material constraint parameters. Specifically, the target magnetic field parameters can be at least one of the magnetic field strength and magnetic flux of the target magnetic field; 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. This constructs a multi-objective optimization objective function based on constraints of "magnetic field matching + engineering feasibility + material safety." The expression of the objective function can specifically adopt at least one of the following: linear function, quadratic function, matrix form, etc.

[0085] In one specific implementation, the target magnetic field parameters include the target magnetic flux; the expression for the target function constructed using a linear function is:

[0086] ,

[0087] In the formula, Let be the objective function. ω1 represents the deviation between the actual magnetic flux and the target magnetic flux, expressed in Wb, where ω1 is the magnetic flux weight. Here, ω2 represents the engineering constraint parameter, and ω2 represents the engineering constraint weight. These are material constraint parameters, less than or equal to the preset limiting current carrying capacity, in A / m. 2 ω3 represents the material constraint weight. ω1, ω2, and ω3 are determined based on the importance of different parameters to the optimization design. Furthermore, the specific values ​​of ω1, ω2, and ω3 can be adjusted based on actual design requirements to make the designed coil more compliant with the specifications.

[0088] Specifically, The calculation method includes: first calculating the magnetic field strength generated by the initial structure of each nonplanar superconducting coil. (Unit: T) Specifically, it can be calculated using Biot-Savart's law by solving the square integral of the magnetic flux with respect to the normal component of the target magnetic field. This yields the deviation between the actual magnetic flux and the target magnetic flux, thus quantifying the magnetic field matching error. The calculation formula is as follows:

[0089]

[0090] In the formula, ds represents the infinitesimal area element of the plasma surface, n represents the plasma normal unit vector, and B target This represents the magnetic field strength of the target plasma, measured in tons (T).

[0091] In one specific implementation, the formula for calculating the engineering constraint parameters is:

[0092] ,

[0093] in, The minimum coil constraint length is calculated using the following formula:

[0094] ,

[0095] In the formula, L MIN Preset a minimum length for the coil. Let q be the actual length of the i-th nonplanar superconducting coil, and q represent the number of nonplanar superconducting coils, which can be 12, 16, 20, 24, etc.

[0096] The maximum coil constraint length is calculated using the following formula:

[0097] ,

[0098] In the formula, L MAX Set the maximum length for the coil.

[0099] The formula for calculating the coil constraint spacing is:

[0100]

[0101] 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.

[0102] The coil constraint distance is calculated using the following formula:

[0103]

[0104] 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.

[0105] The maximum constraint curvature of the basic half-cycle coil is calculated using the following formula:

[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] Existing open-source optimization methods only consider magnetic field matching constraints, and on this basis, coil length and coil spacing constraints, neglecting key constraints such as coil curvature, mean square curvature, coil-plasma distance, and current density. This easily leads to problems where the optimized results are physically feasible but cannot be implemented in engineering. For example, the optimized superconducting coil may have excessive curvature, making winding difficult and impossible. This invention constructs an objective function that includes core constraints related to magnetic field matching, engineering constraints, and material constraints. The core constraints measure the degree of matching between the optimized coil's magnetic field and the target magnetic field, the engineering constraints constrain the feasibility of coil manufacturing and installation, and the material constraints constrain the coil's material load-bearing limit, forming a three-dimensional constraint system. This allows for verification of the optimization results from three dimensions, avoiding the problem of being unable to implement them in engineering.

[0115] S3: Minimize the objective function to obtain the optimized coil data. Specifically, based on the objective function constructed above, minimization can be achieved through methods such as iteration, gradient descent, and Newton's method; and, as... Figure 1 As shown, during the optimization process, it is continuously judged whether the optimization result meets the preset requirements; if yes, the optimization is stopped and the optimized coil data is output; if no, the objective function is minimized.

[0116] In one specific implementation, the L-BFGS-B (Limited Memory Broyden Fletcher Goldfarb Shanno) algorithm is used to solve for the minimum value of the objective function. L-BFGS-B is a Newton-Raphson optimization algorithm used to solve nonlinear optimization problems with upper and lower boundary constraints on variables. The iteration step size and convergence threshold are adjusted according to the gradient rate of change of the objective function. The formula for calculating the gradient rate of change is:

[0117] ,

[0118] In the formula, Let be the gradient change value of the objective function at the k-th iteration. This represents the gradient change value of the objective function at the (k-1)th iteration.

[0119] Based on the gradient change rate, the iterative convergence trend is determined, and the algorithm parameters are dynamically adjusted. If η is less than or equal to a preset adjustment threshold, it indicates relatively stable convergence, and the step size is increased to accelerate convergence. If η is greater than the preset adjustment threshold, it indicates iterative oscillation, and the step size and convergence threshold are decreased. Furthermore, real-time iterative monitoring is performed, and the gradient change value of the objective function is monitored. Less than the dynamically adjusted convergence threshold 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 solid 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] Where N is the total number of vertices of the thread. These are the three-dimensional coordinates of the i-th vertex.

[0129] When grouping the silk threads, the K-means clustering algorithm is used to achieve grouping by minimizing the sum of squares within each group. The objective function expression for the sum of squares within each group is:

[0130]

[0131] 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 x be the center point of the wire in the k-th coil group. Here, a coil group refers to a collection of discrete wires aggregated by specific rules (K-means clustering algorithm) to construct a single complete non-planar superconducting coil solid model. The center of a coil group refers to the mean of the three-dimensional coordinates of the "wire center points" of all wires in a single coil group (i.e., the centroid within the group). The wire center point refers to the three-dimensional geometric center of all vertices of a single wire, which is the key coordinate representing the spatial position of a single wire.

[0132] The peripheral wire positioning steps include: obtaining the relative coordinates of the center point of each wire, taking the overall centroid of multiple wires within each coil group as the origin. The formula for calculating the relative coordinates is:

[0133]

[0134] ,

[0135] in, The coordinates of the centroid of the center point of the multiple threads.

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

[0137]

[0138] In the formula, The original index value is N, where N is the number of vertices of the thread, and i is the index of the target point (e.g., 0~99). The target number of points (e.g., 100 points).

[0139] The cross-section splicing steps include: peripheral wire positioning, selecting multiple peripheral wires using "relative coordinates with the centroid of the coil group as the origin" to ensure the cross-section is rectangular and conforms to the preset multi-wire structure; peripheral wire resampling, unifying multiple peripheral wires to the same number of target points through "resampling mapping"; generating vertices and IDs for a single cross-section, indexing each target point i, and taking the i-th point of multiple peripheral wires as the four vertices of the current cross-section; vertex ID allocation, adding the four vertices from the previous step to the global point set of the solid coil in sequence, automatically generating consecutive vertex IDs; constructing and positioning vertices of adjacent cross-sections by using fixed ID offsets, with fixed offsets for the vertex IDs of connected cross-sections, and constructing quadrilateral units for each adjacent cross-section in sequence according to the offset rules; finally, the construction of the side quadrilateral units is achieved.

[0140] It should be noted that the principle of the above-mentioned cross-section splicing steps in this invention includes: the vertex IDs of adjacent cross-sections have a 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 current vertex of the cross section. This is the ID of the vertex corresponding to the previous section.

[0142] The design and optimization of three-dimensional nonplanar superconducting coils is a key aspect of stellarator magnet design, requiring consideration of various constraints to achieve the optimal balance of requirements. This invention provides a multi-objective optimization method for stellarator nonplanar coils, using a pre-designed magnetic field configuration of the plasma as the target benchmark. It constructs a three-dimensional constraint system encompassing "magnetic field matching, engineering feasibility, and material safety," enabling verification of optimization results from three dimensions and avoiding the problem of engineering implementation failure. Furthermore, it employs a strategy of using a multi-wire layout initial structure of the superconducting coil adapted to the magnetic field gradient distribution and an improved algorithm with dynamically adjusted parameters for optimization, achieving global optimization of the optimized superconducting coil parameters and improving optimization effectiveness. This invention provides a multi-objective optimization method for quasi-isomechanical nonplanar superconducting coils, achieving multi-objective balance and efficient solution in coil design, while improving optimization convergence efficiency. It is applicable to coil design in various magnetic confinement fusion devices such as tokamas and stellarators, providing key technical support for the engineering implementation of high-performance fusion devices.

[0143] Example 2

[0144] The present invention provides a stellarator magnet, comprising multiple non-planar superconducting coils arranged in an array. The multiple non-planar superconducting coils are obtained by optimization using the multi-objective optimization method for non-planar coils of stellarators provided in Example 1.

[0145] It should be noted that, in addition to the specific embodiments described above, those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. Although the description of the present invention is presented in conjunction with preferred embodiments, this does not mean that the features of the invention are limited to these embodiments. On the contrary, the purpose of describing the invention in conjunction with the embodiments is to cover other options or modifications that may be derived based on the claims of the present invention. To provide a deep understanding of the invention, many specific details are included in the above description, and the invention may also be implemented without using these details. Furthermore, to avoid confusion or obscuring the focus of the invention, some specific details will be omitted in the description. It should be noted that, unless otherwise specified, the embodiments and features in the embodiments of the present invention can be combined with each other.

[0146] While the present invention has been illustrated and described with reference to certain preferred embodiments, those skilled in the art should understand that the above description is a further detailed explanation of the invention in conjunction with specific embodiments, and should not be construed as limiting the specific implementation of the invention to these descriptions. Various changes in form and detail can be made by those skilled in the art, including several simple deductions or substitutions, without departing from the spirit and scope of the invention.

Claims

1. A multi-objective optimization method for a star imitator non-planar coil, characterized in that, The method comprises the following steps: S1: obtaining initial structures of a plurality of non-planar superconducting coils according to magnetic field configuration data of a target plasma, the magnetic field configuration data comprising target magnetic field parameters; S2: constructing a target function according to the target magnetic field parameters, engineering constraint parameters and material constraint parameters, wherein the target magnetic field parameters comprise target magnetic flux, the engineering constraint parameters comprise engineering limit geometric parameters of the non-planar superconducting coils, and the material constraint parameters comprise limit current-carrying capacity of non-planar superconducting coil materials; the expression of the constructed target function is: ， wherein, is the objective function, is the deviation of the actual magnetic flux from the target magnetic flux, is the magnetic flux weight, is the engineering constraint parameter, is the engineering constraint weight, is the material constraint parameter, less than or equal to a preset limit current-carrying capacity, is the material constraint weight; S3: minimizing the target function to obtain optimized coil data; wherein the L-BFGS-B algorithm is used to solve the minimum value of the target function, and the iteration step and the convergence threshold are adjusted according to the gradient change rate of the target function; wherein the calculation formula of the gradient change rate is: ， In the formula, is the gradient change value of the objective function at the kth time, is the gradient change value of the objective function at the k-1th time; if η is less than or equal to a preset adjustment threshold, the step is enlarged; 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 target function is less than the adjusted convergence threshold, the iteration is stopped, and the optimized coil data is obtained; 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; wherein according to the coil data, wire grouping, peripheral wire positioning, resampling mapping and cross-section splicing are performed to convert discrete wires into complete non-planar superconducting coil models.

2. The multi-objective optimization method for a star imitator non-planar coil of claim 1, wherein, The calculation formula of the engineering constraint parameter is: ， wherein is the minimum coil constraint length, calculated as: ， In the formula, L MIN preset minimum length for the coil, actual length of the i-th non-planar superconducting coil, and q represents the number of non-planar superconducting coils. For the maximum coil constraint length, the formula is: ， In the formula, L MAX is the maximum length preset for the coil; The calculation formula for the coil constraint distance is: In the formula, d min The preset minimum distance between coils is d The actual distance between the i-th and j-th non-planar superconducting coils is dij. The calculation formula for the coil constraint distance is: In the formula, Dmin is the minimum safe distance between the coil and the plasma magnetic surface, D is Dactual is the actual distance between the ith non-planar superconducting coil and the magnetic surface; To the maximum constraint curvature of the basic half-period coil, the calculation formula is: , in the formula, k i is the actual maximum curvature of the i-th basic half-period coil, k curv is the preset maximum curvature of the basic half-period coil, and p is the number of the basic half-period coils. For the square curvature of the base half-period coil, the calculation formula is: In the formula, is the actual average square curvature of the i-th base half-period coil, is the preset average square curvature of the base half-period coil.

3. The multi-objective optimization method for a star imitator non-planar coil of claim 1, wherein, Step S1 comprises: S11: obtaining the magnetic field configuration data, the magnetic field configuration data further comprising magnetic field gradient distribution parameters; S12: obtaining the initial structures of the plurality of non-planar superconducting coils according to the magnetic field gradient distribution parameters; wherein in the non-planar superconducting coils in the region with large magnetic field gradient, a higher Fourier order is used to construct the initial profile of the non-planar superconducting coils; in the region with small magnetic field gradient, a lower Fourier order is used to construct the initial profile.

4. The multi-objective optimization method for a star imitator non-planar coil of claim 3, wherein, Step S12 further comprises: an initial current is assigned to each of the initial structures of the non-planar superconducting coils, and the initial current of each of the plurality of non-planar superconducting coils is the same.

5. The multi-objective optimization method for a star imitator non-planar coil of claim 3, wherein, The magnetic field configuration data further comprises a magnetic axis radius; step S12 further comprises: dividing grid regions according to the magnetic field gradient distribution parameters, and densifying the grid in the region with large magnetic field gradient and thinning the grid in the region with small magnetic field gradient to generate a boundary point set; obtaining the initial profile of the plurality of non-planar superconducting coils according to the magnetic axis radius and the boundary point set; Wherein, the formula of the coil curve is: In the formula, x(θ) is the coordinate of x direction in the space coordinate system, y(θ) is the coordinate of y direction in the space coordinate system, z(θ) is the coordinate of z direction in the space coordinate system, is the parameter angle, M is the Fourier order, x(θ), y(θ), z(θ) are the amplitudes of cos(mθ), sin(mθ) in the Fourier display formula respectively.

6. The multi-objective optimization method for a star imitator non-planar coil of claim 1, wherein, The wire grouping step comprises: calculating the geometric center of all vertices of a single wire, and the calculation formula is: wherein N is the total number of vertices of the thread, are the three-dimensional coordinates of the i-th vertex, respectively. The expression of the intra-group sum of squares objective function for the wire grouping is: where r is the number of coil groups, is the kth coil group, is the center of the kth coil group, and x is the wire center point of the kth coil group. The peripheral wire positioning step comprises obtaining the relative coordinates of the center point of each wire with the overall centroid of the plurality of wires in each coil group as the origin: The resampling mapping step comprises mapping the total number of wire vertices to the target number of points, and the mapping relationship function is: In the formula, is the original index value, N is the number of vertices of the wire, i is the index of the target point, is the number of target points; The cross-section splicing step comprises: calculating the cross-section vertex identifier, positioning the vertices of adjacent cross-sections through identifier offset, and realizing the construction of a side four-edge element.

7. A stellarator magnet, characterized by A superconducting magnet comprising a plurality of non-planar superconducting coils arranged in an array, said plurality of non-planar superconducting coils being optimized using the multi-objective optimization method for a stellarator non-planar coil according to any one of claims 1-6.

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