Method and device for solving three-dimensional nonlinear coupling of poloidal spectral magnetohydrodynamics

Abstract

This invention discloses a method and apparatus for solving three-dimensional nonlinear coupled angular spectral magnetohydrodynamics, relating to the fields of three-dimensional perturbation simulation of fusion plasma and angular spectral mode discretization. The method includes: acquiring the angular spectral modal states on a substrate computational grid; transforming the modal fluid flux and modal magnetic flux to an angular node space or weak integral sampling space satisfying dealiasing conditions; calculating the three-dimensional nonlinear fluid update quantity and magnetic update driving quantity; back-projecting the calculation results into retained modes, and forming an update of modes to be accepted. By coupling the calculation of nonlinear flux, non-ideal terms, or internal source terms in the sampling space and back-projecting to retain modes, non-axisymmetric three-dimensional perturbations can be expressed at a lower modal cost while maintaining modal magnetic confinement.

Description

Technical Field

[0001] This invention relates to the fields of magnetohydrodynamic numerical calculation, fusion plasma three-dimensional perturbation simulation and angular spectrum mode discretization, and particularly to a three-dimensional nonlinear coupling solution method and apparatus for angular spectrum magnetohydrodynamics. Background Technology

[0002] Two-dimensional axisymmetric magnetohydrodynamic models are computationally efficient but struggle to describe tilt, rotation, and low-order non-axisymmetric perturbations. Complete three-dimensional finite volume meshes can express three-dimensional nonlinear effects, but they incur significant computational and storage requirements. Traditional Fourier-finite element or spectral methods, when preserving a small number of modes in the angular direction, are prone to aliasing errors if nonlinear terms are directly processed in the mode space; furthermore, if modal topological magnetic update constraints are not maintained, modal flux updates may violate discrete magnetic divergence constraints. Therefore, a solution method is needed that expresses complete nonlinear three-dimensional flux at finite angular modal costs while preserving local conservation updates and modal topological magnetic constraints. Summary of the Invention

[0003] The purpose of this invention is to provide a three-dimensional nonlinear coupling solution method and apparatus for angular spectrum magnetohydrodynamics, addressing the problems of difficulty in representing low-order non-axisymmetric perturbations using two-dimensional axisymmetric models, high computational cost of complete three-dimensional meshes, easy aliasing errors from directly processing nonlinear terms in truncated mode space, and easy destruction of magnetic constraints during mode flux updates. This invention forms a closed-loop processing chain through angular spectrum mode states, dealiasing angular node space or weak integral sampling space, three-dimensional nonlinear calculation of node space, mode back projection, mode topological magnetic update, and mode diagnostic acceptance. This closed-loop processing chain allows mode preservation to reduce the cost of angular degrees of freedom, allows the dealiasing angular node space or weak integral sampling space to couple the calculation of nonlinear products, geometric source terms, surface flux, non-ideal electric field terms, internal source term sampling quantities, and magnetic update driving quantities, and allows mode topological magnetic updates to maintain mode magnetic constraints.

[0004] The core constraints of this invention include: the angular spectrum modal state and the set of angular retained modes or the set of retained basis indices are determined on the basis computation grid; the angular spectrum modal state is transformed to an angular node space or weak integral sampling space that satisfies the dealiasing condition; the nonlinear fluid update quantity and the node space magnetic update driving quantity are formed under the same dealiasing angular node or weak integral sampling point, the same node weight or mass matrix, and the same geometric metric; the node space computation result is back-projected to the modal back projection operator or weak projection operator through the same node to become the update of the mode to be accepted on the retained mode; the modal fluid update and modal magnetic flux update are jointly accepted or jointly rejected in the same modal diagnostic acceptance. Each implementation revolves around the same angular spectrum modal state, the same sampling space, the same back projection link, and the same update of the mode to be accepted.

[0005] Based on the aforementioned core constraints, the technical objects of this invention include angular spectral modal states, dealiasing angular node space, weak integral sampling space, three-dimensional nonlinear calculation results of node space, magnetic update driving quantities of node space, modal back projection results, and modal topological magnetic update results. Different angular basis functions, radial or cross-sectional spectral bases, de Rham-compatible discrete basis forms, dealiasing rules, fluid flux construction methods, magnetic update methods, low-modal initialization methods, modal regularization methods, or node space boundary correction methods can be described as optional implementations, but all must ensure that the nonlinear fluid update quantity and the node space magnetic update driving quantity are formed based on the same sampling space, and that both return to the same angular preserved mode set or preserved basis index set through the same back projection link.

[0006] When the internal source terms include neutral-plasma exchange terms, anomalous transport terms, high-energy particle deposition terms, or externally heated deposition terms, the internal source terms are processed in the angular spectrum implementation according to the same mode transformation, dealiasing, back projection, and diagnostic acceptance rules; the internal source terms do not change the processing chain of this method, which is based on nodal space nonlinear calculation, mode back projection, and mode topological magnetic update. Neutral fluid state propagation, reaction collision rate formation, and particle, momentum, and energy exchange budget records are not considered as independent core inventions of this sub-project. The common limitation of extended magnetofluid terms, non-ideal electric field terms, and internal source terms is that they all form back projection contributions in the nodal space or weak integral sampling space, or form mode magnetic update residual terms to be accepted, and must not bypass the mode topological magnetic update module 105 to directly cover the modal surface magnetic flux.

[0007] To facilitate understanding of the technical solution of this invention, the relevant terms used in this invention will be explained below.

[0008] In this specification, angular spectral modes refer to a finite number of retained basis function coefficients used to represent the variation of a physical quantity along the angular, circumferential, field periodic direction, or radial / sectional direction coupled with the angular direction. These angular spectral modes include complex Fourier modes, real-valued sine / cosine modes, positive-negative mode pairings, field periodic modes, family of field periodic modes, spiral phase modes, rotating phase tracking modes, Fourier-Bessel radial spectral bases, Fourier-Zernike spectral bases, de Rham-compatible discrete form bases, vector mode combinations satisfying axis regularity, or basis function representations that are periodically compatible with the device and satisfy the same angular periodic boundary conditions. Different angular spectrum mode representations all receive modal fluid state, modal magnetic flux state, and angular retained mode set as inputs. They can all be transformed to angular node space or weak integral sampling space, and the update quantity on the retained mode can be obtained by backprojection of the nonlinear flux or magnetic update driving quantity in the node space. It does not include particle propulsion steps with particle position or velocity as the main propulsion object.

[0009] In this specification, the dealiasing angular node space refers to the set of angular sampling nodes that satisfy the preset dealiasing conditions; the weak integral sampling space refers to the quadrature nodes, quality matrix, and weak projection relation configured for non-orthogonal spectral bases, Fourier-Bessel spectral bases, Fourier-Zenick spectral bases, spectral element bases, or Delamin-compatible discrete form bases. The preset dealiasing conditions include the 2 / 3 sampling rule, the 2 / 3 truncation rule, the oversampling integral rule, the exact convolution sampling rule, the dealiasing filtering rule, or the adaptive node number rule. The common function of these rules is to reduce the aliasing error caused by the folding of the preserved modal nonlinear convolution terms back to lower-order modes, and to output the nodal space nonlinear fluid update quantity and the nodal space magnetic update driving quantity that can be projected back.

[0010] In this specification, a Fourier-Zernike spectral basis refers to a basis that represents physical or geometric quantities using Fourier basis functions in the circumferential or field-periodic directions and Zernike basis functions on the cross-sectional disk or normalized radial-angular plane. The Fourier-Zernike spectral basis takes as input a set of preserved basis indices, radial / sectional orders, angular orders, and circumferential orders, and outputs preserved basis function values, derivative values, and mass matrix terms that can be evaluated at cross-sectional quadrature nodes or in the weak integral sampling space. Common constraints include: the radial and angular orders satisfying axial regularity or parity relationships; and the node-to-basis back projection results still entering the same angular preserved mode set or preserved basis index set. The Fourier-Zernike spectral basis can be used for nested flux surface equilibrium initialization, cross-sectional spectrum representation, or local weak integral sampling, but this method does not require the assumption of a globally nested flux surface.

[0011] In this specification, the Fourier-Bessel radial spectral basis refers to a basis that represents cross-sectional perturbations using angular Fourier factors and Bessel functions or their discrete radial orders determined by radial boundary conditions. The Fourier-Bessel radial spectral basis takes angular mode indices, radial zeros or radial orders, and boundary conditions as inputs, and outputs radial-angular basis function values ​​that can be used for nodal space sampling or weak projection. Its common constraint is that the radial basis functions are compatible with the regularity and boundary conditions of the device axis, conductor walls, free boundaries, or local cross-sectional boundaries, and the back projection results still serve as modal fluid update quantities or modal magnetic update driving quantities within the retained basis set.

[0012] In this specification, de Rham-compatible discrete-form basis refers to discrete basis functions, sets of discrete degrees of freedom, or discrete exterior differential relations used to carry discrete 0-form, 1-form, 2-form, and 3-form degrees of freedom. The de Rham-compatible discrete-form basis takes as input a set of retained basis indices, discrete-form order, mesh, or weak-form geometric metric, and outputs discrete gradient, discrete curl, discrete divergence, mass matrix, or discrete Hodge metric mapping. Its implementation methods include finite element exterior differential space, hybrid spline finite element, polar coordinate spline, spectral element discrete form, finite volume point-edge-surface-volume relation, or hybrid discretization satisfying the same complex constraints. The common constraint of the above implementation methods is that the discrete gradient, discrete curl, and discrete divergence satisfy the relationship that curl-gradient composition is zero and divergence-curl composition is zero; edge quantities or 1-form degrees of freedom can be converted into surface magnetic flux or 2-form degrees of freedom through discrete curl; and surface magnetic flux can be converted into magnetic divergence diagnostic quantities through discrete divergence.

[0013] In this specification, the Hodge metric mapping refers to a discrete mapping that maps discrete-form degrees of freedom to dual space, weak-form space, or corresponding metric quantities based on mesh geometric metrics, coordinate Jacobian, area weights, volume weights, mass matrices, material coefficients, state equation quantities, or transport constitutive coefficients. In this sub-species, the Hodge metric mapping is used to derive magnetic field, magnetic field strength, current density, non-ideal electric field constitutive quantities, magnetic energy inner product, or diagnostic residual weights from modal surface magnetic flux. The Drum-compatible discrete-form basis guarantees the topological complex relationship between discrete gradients, discrete curls, and discrete divergences; the Hodge metric mapping guarantees the consistency in how geometry, material, constitutive, and energy inner products enter the discrete equations. The Hodge metric mapping does not change the zero-composite relationship of discrete gradients, discrete curls, and discrete divergences, nor does it serve as an entry point to directly cover modal surface magnetic flux, bypassing modal topological magnetic updates.

[0014] In this specification, "same dealiasing angular node sampling or weak integral sampling" refers to the formation of fluid surface flux, angular flux terms, cylindrical or curvilinear geometric source terms, non-ideal electric field terms, and nodal spatial magnetic renewal driving quantities under the same angular node or weak integral sampling point, the same node weight or mass matrix, and the same geometric metric. These quantities are then projected back or weakly onto the modal back projection or modal magnetic renewal driving quantities from the same node. This method avoids inconsistencies in fluid flux and magnetic renewal driving quantities based on different angular sampling states.

[0015] In this specification, nodal-space three-dimensional nonlinear computation refers to the process of recovering the original three-dimensional variables in the angular nodal space, constructing a reconstructed state, calculating radial, axial, and angular fluxes, and forming nonlinear fluid update quantities from the nodal-space state. The nonlinearity originates from the product of density, velocity, pressure, magnetic field, energy, geometric metrics, nonideal electric fields, or source term samples in the nodal space, flux functions, or Riemannian solutions, rather than being directly given by linearized fluxes in the preserved mode space. In this way, a small number of preserved modes can still express three-dimensional nonlinear interactions in the nodal space.

[0016] In this specification, the nodal space magnetic update driving quantity is used to represent the nodal space edge electric field circulation, the nodal space surface magnetic flux increment, the nodal space magnetic update residual term, or the magnetic update driving quantity that satisfies the discrete exterior differential relation. Modal topological magnetic update refers to the process of using the modal surface magnetic flux or the discrete state with the same modal discrete two-form degrees of freedom as the modal surface magnetic flux, which can be derived into a magnetic field, non-ideal electric field constitutive quantity, or magnetic energy inner product through modal Hodge metric mapping as the update object, and forming the magnetic flux update of the mode to be accepted through modal edge quantities, modal edge electric field circulation, modal surface magnetic flux increment, modal magnetic update residual term, or modal magnetic update driving quantity that satisfies the modal magnetic constraint residual acceptance condition; when using modal constraint transport (CT) to implement, the modal magnetic update driving quantity is the modal edge electric field circulation; modal de Rham update is an optional implementation. All of the above magnetic update methods output the magnetic flux update of the mode to be accepted, and all must satisfy the modal divergence diagnosis constraint or the corresponding modal magnetic constraint.

[0017] In this specification, the extended magnetofluid term refers to a processing object that takes density, velocity, pressure, electron pressure, flux-derived magnetic field, current density, collision frequency, transport coefficient, or closure configuration in the nodal space or weak integral sampling space as input, and outputs a nodal space non-ideal electric field term, a nodal space magnetic update driving quantity, or a modal magnetic update residual term. The extended magnetofluid term includes one or more of the following: Hall electric field term, electron pressure gradient term, electron inertia term, resistivity correction term, electron-neutral collision frequency correction term, anisotropic resistivity term, anisotropic transport correction term, or a non-ideal magnetic update residual term that meets the same magnetic constraint residual acceptance condition. The common constraint of the above extended magnetofluid terms is that they form a projection contribution to be reviewed in the angular nodal space or weak integral sampling space, or form a modal magnetic update residual term to be accepted, and enter the modal flux update to be accepted through the modal projection module 104 and the modal topological magnetic update module 105; it does not include processing of the modal surface flux, edge electric field circulation, or the flux state of the next time sub-stage.

[0018] In this specification, internal source term coupling refers to the way neutral-plasma exchange terms, anomalous transport terms, high-energy particle deposition terms, neutral particle injection deposition terms, radio frequency heating deposition terms, radiation loss terms, two-temperature energy exchange terms, or resistance heating terms are integrated into the angular spectrum solution chain. The internal source term coupling takes modal fluid state, node space state, state equation closure quantity, or source term configuration as input, and outputs node space source term sampling quantity, modal fluid update quantity, energy pool update quantity, or source term income / expenditure diagnostic quantity. The common constraint of the above internal source term coupling is that if a source term enters the node space nonlinear calculation, it uses the same angular nodes or weak integral sampling points, node weights, or mass matrices as the nonlinear fluid update quantity; if a source term is treated as a split source term, its output enters the pending modal update package or source term income / expenditure diagnostic quantity, and is accepted or rejected together with the modal diagnostic acceptance.

[0019] In this specification, the precision of preserved modes refers to the recoverability of modes in the diagonal direction of the preserved mode set or the preserved basis index set when the mode is back-projected after nodal transformation or weak integral sampling, or when the preserved basis is back-projected. That is, when the preserved mode is back-projected after nodal transformation or weak integral sampling, it can be restored to the original preserved mode under the condition of no filtering and no rejection by diagnostic acceptance. For real-valued physical fields, the back-projection also maintains or restores the conjugate relationship between positive and negative modes.

[0020] To achieve the above objectives, the angular spectral magnetohydrodynamic three-dimensional nonlinear coupled solution method is organized according to the same processing chain. This processing chain includes the following technical steps: 1. Reading or forming angular spectral modal states on the basis computational grid, wherein the angular spectral modal states include modal fluid states, modal magnetic flux states, and angular-preserved mode sets or preserved basis index sets; 2. Transforming the angular spectral modal states to the angular node space or weak integral sampling space according to preset dealiasing conditions, so that the sampling space has node states or weak integral states for three-dimensional original variable recovery, reconstruction, surface flux calculation, angular flux calculation, geometric source term sampling, and magnetic update driving quantity calculation; 3. Calculating nonlinear fluid update quantities and node space magnetic update driving quantities in the same sampling space, wherein the nonlinear... The fluid update quantity is formed by the three-dimensional state of the sampling space, rather than directly given by the linearized flux in the preserved modal space; Fourth, the nonlinear fluid update quantity and the magnetic update driving quantity in the node space are back-projected to the angular preserved mode set or the preserved basis index set to obtain the modal fluid update quantity and the modal magnetic update driving quantity; Fifth, the modal fluid update to be accepted is formed based on the modal fluid update quantity, and the modal magnetic flux update to be accepted is formed based on the modal magnetic update driving quantity; Sixth, diagnostic acceptance is performed based on modal magnetic divergence, conjugate mode error, aliasing energy, axis regularity or adoptability, and the modal fluid update and the modal magnetic flux update to be accepted are accepted after the acceptance is passed.

[0021] The above technical steps are used to illustrate the data relationship between angular spectrum mode state, dealiasing angular node space or weak integral sampling space, node space nonlinear calculation, mode back projection, mode topological magnetic update and mode diagnosis and acceptance, and do not set additional embodiment steps numbering that are repeated with the steps in the claims.

[0022] In one implementation, the angularly preserved mode set includes a zero-order mode and at least one pair of positive and negative angular modes, with node weights, mode-to-node transformation operators, and node-to-mode backprojection operators sharing the same conjugate mode pairing relationship. The basis computation mesh includes one or more of the following: radial-axial mesh, cylindrical coordinate section mesh, curvilinear coordinate section mesh, locally body-fitted section mesh, or locally circumferential section mesh. The basis functions include one or more of the following: complex Fourier basis, real-valued sine and cosine basis, Fourier leaf basis compatible with the device field periodic boundary conditions, Fourier-Bessel radial spectral basis, Fourier-Zenick section spectral basis, de Rham-compatible discrete form basis, or basis functions compatible with device periodicity and satisfying the same angularly periodic boundary conditions. This configuration enables the sampling space computation to generate nonlinear products, geometric source terms, and magnetic update driving forces while preserving the cost of finite modes.

[0023] In one implementation, node space computation employs finite volume, conserved finite difference, element residuals, or weakly formal residuals to form local conserved updates. The node space three-dimensional magnetohydrodynamic flux tensor includes radial flux, axial flux, and angular flux. The angular flux term can be represented by the modal flux derivative, weak derivative matrix, or circumferential flux difference satisfying the circumferential conservation relation in the angular node space. Fluid surface flux, angular flux terms, cylindrical or curvilinear geometric source terms, non-ideal electric field terms, and node space magnetic update driving quantities are formed under the same angular node or weak integral sampling point, the same node weight or mass matrix, and the same geometric metric. In this way, fluid update quantities and magnetic update driving quantities use the same sampling state, avoiding the need for separate discretization based on different angular sampling states.

[0024] In one implementation, the finite volume calculation of the nodal space may include restoring the original variables of the nodal space, limiting the slope of the nodal space or reconstructing higher-order parameters, constructing the surface states of the nodal space, solving for the surface fluxes of the nodal space, and forming the angular flux terms. The surface flux solution may employ Harten-Lax-van Leer family fluxes (HLL / HLLD / HLLE), local Lax-Friedrichs fluxes, or other approximate Riemannian solvers. In the non-finite volume implementation, the nonlinear fluid update quantities of the nodal space may be formed using conserved finite difference flux differences or element residuals. A common constraint of the above implementations is that the output nonlinear fluid update quantities of the nodal space are back-projected onto the same angularly preserved mode set together with the magnetic update driving quantities of the nodal space.

[0025] In one implementation, the node space sampling quantities may include neutral-plasma exchange terms, anomalous transport terms, high-energy particle deposition terms, neutral particle injection deposition terms, or radio-frequency heated deposition terms. These sampling quantities participate in node space nonlinear calculations and backprojections only as internal source terms, external source terms, or closure parameters, and must still use the same angular nodes or weakly integrated sampling points, node weights or mass matrices, and diagnostic acceptance rules as the fluid update quantities and node space magnetic update driving quantities. In this way, this sub-solution is compatible with mode processing of neutral fluids, anomalous transport, external heating, or high-energy particle deposition, but does not limit its state progression or engineering source term formation process.

[0026] In one embodiment of extended magnetofluidic term access, the node-space nonlinear calculation module 103 generates nonideal electric field terms corresponding to the Hall electric field, electron pressure gradient, resistivity correction, and electron inertia or anisotropic transport correction at the same dealiasing angle node or weak integral sampling point. These nonideal electric field terms are input into the modal backprojection module 104 as the node-space magnetic update driving quantity, the additive contribution of the node-space edge electric field circulation, or the modal magnetic update residual term, and are then used by the modal topological magnetic update module 105 to generate the magnetic flux update of the mode to be accepted. In this way, the extended magnetofluidic term and the nonlinear fluid update quantity share the sampling state and diagnostic acceptance boundary, preventing the nonideal magnetic update from bypassing the modal topology path.

[0027] In one internal source term coupling embodiment, the node space nonlinear calculation module 103 receives the state equation closure quantity, source term configuration, and angular node space state to form node space source term sampling quantities corresponding to neutral-plasma exchange, anomalous transport, high-energy particle deposition, neutral particle injection deposition, radio frequency heating deposition, radiation loss, or two-temperature energy exchange. These source term sampling quantities can be back-projected along with the nonlinear fluid update quantity to become modal fluid update quantities, or they can be entered into the modal diagnosis and acceptance module 106 as source term income and expenditure diagnostic quantities. Regardless of the entry method, the same source term is not simultaneously counted as a node space right-end term and a split source term within the same time sub-stage.

[0028] In one implementation, modal topological magnetic updates take the modal surface magnetic flux or the discrete state of the modal surface magnetic flux, or the state of the same modal discrete two-form degrees of freedom as the modal surface magnetic flux, which can be derived into magnetic fields, non-ideal electric field constitutive quantities, or magnetic energy inner products through modal Hodge metric mapping, as the update object. The modal magnetic flux update to be accepted is formed through modal edge quantities, modal surface magnetic flux increments, modal magnetic update residual terms, or modal magnetic update driving quantities that meet the modal magnetic constraint residual acceptance conditions. Modal topological magnetic updates can be driven by modal edge quantities, modal surface magnetic flux increments, de Rham-compatible discrete complexes, or modal magnetic update residuals. When implemented using modal constrained transport (CT), the modal edge quantities are modal edge electric field circulations. When implemented using modal de Rham, the magnetic update is formed by the external differential relations between discrete forms 1, 2, and 3. Nodal space boundary conditions, material interface responses, or control inputs are only used as nodal space boundary states, source term samples, or diagnostic inputs in nodal space computation, and must not bypass modal topological magnetic updates to directly cover modal surface flux. This approach avoids external boundaries or control inputs disrupting the consistent update path of modal flux states.

[0029] In one implementation, modal-to-node transformation operators and node-to-modal backprojection operators, or transformation operators and weak projection operators that preserve the basis to the weak integral sampling space, satisfy the requirement of preserving modal accuracy. When backprojecting nonlinear fluid updates and nodal spatial magnetic update driving quantities, node-to-modal backprojection or weak projection maintains or restores the conjugate relationship between positive and negative modes. If real-valued sine / cosine modes are used, backprojection can convert positive and negative mode pairs into real-valued paired coefficients; if complex modes are used, backprojection can check whether a negative mode is a conjugate of a positive mode. In this way, backprojection not only undertakes dimensionality reduction storage but also undertakes the consistency of real-valued physical fields and diagnostic acceptance boundaries.

[0030] In one implementation, diagnostic acceptance includes checks on modal divergence, conjugate mode error, aliasing energy, axis regularity, and admissibility. The aliasing energy can be obtained from the unretained modal energy before and after back projection, the filtered energy, or the residuals of higher-order nodes; the axis regularity can be determined based on the parity, radial order, or regular boundary conditions of different angular modes at the axis. When the conjugate error, modal divergence, aliasing energy, axis regularity, or admissibility exceeds a threshold, the diagnostic acceptance outputs a decision to shorten the time step, recalculate, or reject the current time sub-stage.

[0031] Implementations of angular spectral modes can be categorized into complex Fourier mode representation, real-valued sine / cosine mode pairing representation, mode representation compatible with device field period, Fourier-Bessel radial spectrum representation, Fourier-Zenick cross-sectional spectrum representation, de Rham-compatible discrete form representation, and spiral phase, rotating phase, or axis-normalized vector combination representation. All of these representations accept modal states or preserved basis index sets as input, output preserved mode updates, and satisfy dealiasing sampling space nonlinear calculation and modal back projection constraints. Each core representation is expanded segmentally in specific implementations.

[0032] Implementations of modal topological magnetic updates can be categorized into modal edge quantity driving, node space magnetic update driven quantity back projection, magnetic update residual driving, and Deram-compatible discrete complex driving. All of these implementations use modal flux states as the update object, output modal flux updates to be accepted, and undergo modal magnetic constraint diagnosis. Each core magnetic update method is elaborated in segments within the specific implementation.

[0033] Initialization, modal regularization, and nodal space boundary correction of angular spectral modal states are supporting components of the key dependent embodiments. Initialization may include axisymmetric equilibrium projection, three-dimensional initial field low-mode projection, linear eigenmode perturbation projection, experimental diagnostic perturbation projection, analytical perturbation, or random low-mode perturbation; modal regularization may include acceptability projection, conjugate mode recovery, axis regularization projection, mode-preserving precise filtering, spectral viscosity limiting, modal divergence cleanup, or modal magnetic constraint residual correction; nodal space boundary correction may include angular periodic boundary matching, field periodic boundary phase matching, axis regularization boundary projection, wall or material interface nodal space boundary state projection, free boundary sampling limiting, or modal boundary residual diagnosis. The above objects are processed as pending modal updates, nodal space boundary states, source term samples, or diagnostic inputs, and must not bypass modal topological magnetic updates to directly cover modal surface magnetic flux.

[0034] The present invention also provides a three-dimensional nonlinear coupling solution device for angular spectral magnetohydrodynamics. The device includes a modal state management module 101, an anti-aliasing node transformation module 102, a node space nonlinear calculation module 103, a modal back projection module 104, a modal topological magnetic update module 105, and a modal diagnosis and acceptance module 106.

[0035] In some embodiments, the modal state management module 101 is used to maintain the modal fluid state, modal flux state, and angular retention mode set; the dealiasing node transformation module 102 is used to transform the modal state to the dealiasing angular node space or weak integral sampling space; the node space nonlinear calculation module 103 is used to calculate the three-dimensional nonlinear fluid update quantity and the node space magnetic update driving quantity; the modal back projection module 104 is used to back project the node space calculation results to the retention modes; the modal topological magnetic update module 105 is used to form the magnetic flux update of the modal to be accepted; and the modal diagnosis and acceptance module 106 is used to accept or reject the modal update to be accepted based on the modal magnetic divergence, conjugate mode error, aliasing energy, axis regularity, or acceptability.

[0036] In one possible implementation, the present invention also provides an electronic device. The electronic device includes a processor and a memory, the memory storing processor-executable instructions; when the processor executes the instructions, it invokes the corresponding functions of the modal state management module 101, the dealiasing node transformation module 102, the node space nonlinear calculation module 103, the modal back projection module 104, the modal topological magnetic update module 105, and the modal diagnosis and acceptance module 106 to realize the aforementioned angular spectrum magnetohydrodynamic three-dimensional nonlinear coupling solution method.

[0037] In some embodiments, the present invention also provides a computer-readable storage medium. The computer-readable storage medium stores computer program instructions, which, when executed by a processor, implement the steps of angular spectral mode state acquisition, dealiasing node transformation, three-dimensional nonlinear calculation of node space, modal back projection, modal flux update, and diagnostic acceptance in the above-described method.

[0038] The present invention also provides a computer program product. The computer program product includes a computer program or instructions, which, when executed by a processor, cause an electronic device to perform the above-described method. The configuration forms of the computer program product include: 1. a local program package; 2. a simulation software plugin; 3. a container image, cloud task, or cluster job script.

[0039] Compared with related technologies, although two-dimensional axisymmetric models are computationally efficient, they are difficult to describe low-order non-axisymmetric perturbations. This invention expresses three-dimensional perturbations at a lower cost by preserving angular modes and calculating three-dimensional states in nodal space.

[0040] To address the issue of aliasing errors that can easily arise from direct modal space nonlinear truncation, this invention reduces nonlinear convolution aliasing by performing node space nonlinear flux calculation that satisfies the dealiasing sampling conditions.

[0041] In one possible implementation, the node space fluid flux and node space magnetic update driving forces are sampled using the same dealiasing angular node. This avoids inconsistencies that can arise from fluid flux and magnetic flux updates being based on different angular states.

[0042] The discrete representation of modal flux states includes: 1. Modal surface flux; 2. Discrete states with the same modal discrete two-form degrees of freedom as the modal surface flux, which can be derived into magnetic fields, non-ideal electric field constitutive quantities, or magnetic energy inner products via modal Hodge metric mapping; 3. Modal magnetic update residual states that satisfy the modal magnetic constraint residual acceptance conditions. All of the above discrete representations are used to form the modal flux update to be accepted through modal edge quantities, modal surface flux increments, or modal magnetic update residual terms, in order to maintain the modal discrete magnetic divergence constraint.

[0043] The beneficial effects of this invention include: it can express three-dimensional nonlinear magnetohydrodynamic flux containing radial, axial and angular fluxes at low-order angular modal costs; it can reduce nonlinear convolutional aliasing by dealiasing node space; it can maintain modal discrete magnetic divergence constraints by modal topological magnetic updates; and it can improve computational reliability by preserving modal accuracy, conjugate modes, axis regularity, geometric source terms and aliasing energy diagnosis. Attached Figure Description

[0044] Figure 1 A schematic diagram of the structure of the angular spectrum magnetohydrodynamic three-dimensional nonlinear coupling solution device according to an embodiment of the present invention is shown.

[0045] Figure 2 A flowchart illustrating the three-dimensional nonlinear coupling closed-loop process of an embodiment of the present invention is shown.

[0046] Figure 3 This diagram illustrates a discrete complex with preserved basis block de Rham constraint transport (CT) according to an embodiment of the present invention. Detailed Implementation

[0047] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. The described embodiments are only some embodiments of the present invention, and not all embodiments; other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are all within the protection scope of the present invention.

[0048] Figure 1 A three-dimensional nonlinear coupled solution device for angular spectrum magnetohydrodynamics is shown. The device includes a modal state management module 101, an anti-aliasing node transformation module 102, a node space nonlinear calculation module 103, a modal back projection module 104, a modal topological magnetic update module 105, and a modal diagnosis and acceptance module 106.

[0049] The modal state management module 101 maintains modal fluid conservations and modal surface magnetic fluxes on radial-axial finite volume grids, cross-sectional spectral bases, or de Rham-compatible discrete complexes. The modal state management module 101 maintains the preserved basis function coefficients of the magnetohydrodynamic field variables, rather than the propulsion states of particle positions or velocities. The set of preserved basis indices can be expressed as shown in Equation 1:

[0050] In Equation 1, 𝒜 represents the set of basis-preserving indices, m represents the angular, circumferential, or field periodic mode number, ℓ represents the radial, sectional, or weak-form basis function number, and q represents the physical component, discrete-form order, or vector basis function component. For a pure Fourier angular spectrum implementation, ℓ and q can be degenerated into single values ​​to obtain the angular-preserving mode set ℳ; for a Fourier-Bessel implementation, ℓ can represent the Bessel radial zero or radial order; for a Fourier-Zenick implementation, ℓ can represent the Zenick radial order; for a Dram-compatible implementation, q can represent 0-form, 1-form, 2-form, or 3-form degrees of freedom. For field variables near the axis, scalar components and axial components satisfy corresponding even / odd and amplitude order constraints according to the modal order, while radial and angular components can be subject to compatible regularity constraints through transverse combination quantities; different physical components must not be replaced by the same fixed even / odd template.

[0051] The dealiasing node transformation module 102 transforms the retained basis function coefficients to the angular node space, the cross-sectional quadrature node space, or the weak integral sampling space, as shown in Equation 2:

[0052] In Equation 2, xⱼ represents the j-th dealiasing angular node, cross-sectional quadrature node, or weak integral sampling point; the summation coefficients represent the fluid conservation coefficients on the α-th retained basis function and the modal flux coefficients on the β-th retained 2-form basis function, respectively; the corresponding basis functions represent the fluid quantity basis function and the surface magnetic flux or the basis function used in the corresponding discrete 2-form. The basis functions used by the dealiasing node transformation module 102 have their retained basis index set and modal coefficients stored by the modal state management module 101, and the modal back projection module 104 uses the same basis set to back-project the node space calculation results.

[0053] In one embodiment of the Fourier angular spectrum, the explicit basis functions are expanded only along the angular, circumferential, or field periodic directions, and the angular mode indices are used as the retained basis indices. This embodiment can be given in the following form, as shown in Equation 3:

[0054] In Equation 3, i represents the imaginary unit, and m represents the angular, circumferential, or field periodic mode number. In this embodiment, ℓ and q can degenerate into a single value. The node space nonlinear calculation module 103 forms a three-dimensional nonlinear quantity on the angular nodes, and the modal backprojection module 104 backprojects the node results to the angular preserved mode set.

[0055] In one embodiment of the Fourier-Bessel radial spectrum, the basis functions simultaneously include angular Fourier factors and Bessel radial factors determined by radial boundary conditions. This embodiment can be given in the following form, as shown in Equation 4:

[0056] In Equation 4, J m Let m represent the m-th order Bessel function. The radial spectral parameters are jointly indexed by the mode number m and the radial order ℓ, and determined by the radial boundary conditions or regularity conditions; ℓ represents the radial zero or radial order. In this embodiment, the weak integral sampling space receives the radial-angular basis function values, node weights, or mass matrix, and writes the back projection results back to the same set of preserved basis indices.

[0057] In one embodiment of the Fourier-Zenick cross-section spectrum, the basis functions include cross-sectional Zenick basis functions and Fourier factors in the circumferential or field periodic directions. This embodiment can be given in the following form, as shown in Equation 5:

[0058] In Equation 5, the Zernike cross-sectional basis functions are indexed by the radial order ℓ and the angular order m, where n represents the mode number in the circumferential or field periodic direction. In this embodiment, the radial, angular, and circumferential orders together constitute the retained basis index, and the cross-sectional quadrature nodes or weak integral sampling space outputs are still back-projected to the same retained basis set by the modal back-projection module 104.

[0059] In other embodiments, the basis functions may also be Fourier leaf bases compatible with the device field periodic boundary conditions, vector combination basis functions satisfying axis regularity constraints, or Deram-compatible discrete basis functions. The aforementioned basis functions or discrete basis functions are saved, nodeified, and back-projected by the modal state management module 101, the dealiasing node transformation module 102, and the modal back projection module 104, respectively, without changing the data boundary between the node space nonlinear calculation module 103 and the modal topological magnetic update module 105.

[0060] The number of angular nodes, radial / section quadrature nodes, or weak integral order satisfies the preset dealiasing sampling conditions to analytically preserve the modal nonlinear convolution terms, or satisfies the 3 / 2 sampling rule, the 2 / 3 truncation rule, the oversampling integration rule, or a rule that satisfies the same dealiasing error acceptance condition. For second-order nonlinear flux, the number of sampling nodes is selected to be the number of nodes that can analytically preserve the convolution results within the basis set; for higher-order nonlinear terms, the model configuration can increase the number of nodes or adopt filtering rules that satisfy the same dealiasing error acceptance condition. The energy on the unpreserved modes can be used as a diagnostic quantity for aliasing energy and input into the modal diagnosis and acceptance module 106. In this way, the nonlinear product is first formed in the node space and then projected back according to the basis set, reducing convolution aliasing caused by direct modal truncation.

[0061] The node-space nonlinear calculation module 103 performs original variable recovery, finite volume reconstruction, Riemann surface solving, weak form residual assembly, and node-space magnetic update driving quantity calculation at angular nodes, cross-sectional quadrature nodes, or weak integral sampling points to obtain the node-space nonlinear fluid right-hand term and the node-space magnetic update driving quantity. When constrained transport (CT) is used, the node-space magnetic update driving quantity is the node-space edge electric field circulation. The node-space calculation can be used for ideal magnetofluids, non-ideal magnetofluids, magnetofluids with source terms, or magnetofluids with free boundaries.

[0062] In one possible implementation, the nodal space flux includes radial and axial finite volume surface fluxes and modal flux derivatives, weak derivative matrices, or circumferential flux differences satisfying circumferential conservation relations in the angular direction. The right-hand side of the nodal space nonlinear fluid is calculated from the three-dimensional original variables, reconstructed states, and Riemann surface fluxes at angular nodes, cross-sectional quadrature nodes, or weak integral sampling points, and is not directly given by the linearized flux in the preserved modal space.

[0063] In this way, the angular spectrum representation performs full three-dimensional nonlinear finite volume flux, weak-form residual, or corresponding flux calculations, expressing the three-dimensional nonlinear effects in the sampling space and excluding linearized flux paths in the preserved mode space from this stage. The angular flux derivative can be formed by the angular derivative relationship of the product of the imaginary unit i and the mode number m acting on the m-th mode, or by the circumferential flux difference, weak derivative matrix, or quadrature matrix on the dealiased angular nodes; all of the above representations yield the fluid right-hand term within the preserved mode or preserved basis set after back projection.

[0064] In some embodiments, the cylindrical coordinate geometric source term, magnetic tension term, angular momentum term, and nodal space magnetic update driving quantity use the same dealiasing angular nodal sampling or weak integral sampling, the same nodal weights or mass matrix, and the same geometric metric; when implemented using constrained transport (CT), the nodal space magnetic update driving quantity is the nodal space upwind circulation. In this way, it is possible to avoid the fluid flux and magnetic update using different angular states.

[0065] In one embodiment of flux load construction, the nodal space magnetic renewal drive is formed by the flux load output by the nodal space Riemann surface solver. This embodiment receives the reconstructed state, surface flux, and geometric measure at the same angular node or weakly integrated sampling point, and outputs the projectible nodal space edge electric circulation, surface magnetic flux increment, or magnetic renewal drive.

[0066] In one embodiment of wave velocity weighting, the nodal space magnetic update driving force is formed by wave velocity weights, lateral magnetic field jumps, or upwind corrections. This embodiment shares wave velocity estimation, nodal weights, or mass matrices with nodal space nonlinear fluid computation, and the output is still back-projected to the same angularly preserved mode set by the modal back-projection module 104.

[0067] In one embodiment of magnetic update residual construction, the nodal space magnetic update driving quantity is formed by magnetic update residual terms, non-ideal electric field terms, or free boundary sampling terms. All of the above construction results are back-projected to form modal magnetic update driving quantities; when implemented using constrained transport (CT), the nodal space windward edge circulation is back-projected to form modal edge electric field circulation. Original variable recovery, Riemannian solution, optional non-ideal electric field, free boundary sampling, and source term sampling are all completed in the nodal space and then uniformly back-projected to the preserved modes.

[0068] Modal back projection module 104 projects the right-hand side terms of the nodal space nonlinear fluid and the circulation of the nodal space edge electric field back to the basis set. The back projection operator uses the same nodal weights, mass matrix, and conjugate mode pairing relations as the nodalization; for nonorthogonal or weakly formal bases, mass matrix projection can be used, as shown in Equation 6:

[0069] In Equation 6, the mass matrix with the superscript U represents the mass matrix of the fluid quantity basis functions, and the mass matrix with the superscript I represents the mass matrix of the form-1 or edge quantity basis functions; wⱼ represents the node or quadrature weight, the capped flux at the sampling point represents the nonlinear surface flux or right-hand side term, and e represents the edge electric field circulation or magnetic renewal driving quantity at that sampling point. The form-1 basis functions are used to form weak projections, and the back projection results are the modal fluid flux coefficients and modal edge quantity coefficients, respectively. When there are orthogonal Fourier bases and uniform node weights, Equation 6 degenerates into a discrete Fourier back projection; in Fourier-Bessel, Fourier-Zenick, or de Rham compatible discrete bases, Equation 6 represents a mass matrix projection or the corresponding weak projection. The modal-to-node transformation operator and the node-to-modal back projection operator are configured to satisfy the requirement of preserving modal accuracy; the same node weight or the same mass matrix is ​​used simultaneously for the back projection of nonlinear fluid renewal quantities and node spatial magnetic renewal driving quantities. For real-valued physical fields, the back projection results still maintain or restore the conjugate relationship between positive and negative modes; when the conjugate error exceeds the threshold, the modal diagnosis acceptance is marked as failed.

[0070] The modal topological magnetic update module 105 uses modal magnetic update driving quantities to form the surface magnetic flux of the modes to be accepted; when implemented using constrained transport (CT), the modal magnetic update driving quantities are modal edge electric field circulations. In a delamin-compatible implementation, the 0-form, 1-form, 2-form, and 3-form discrete spaces and the corresponding discrete exterior differential operators constitute the following discrete complex: Figure 3 The object numbers in the attached figures are used to explain the modal topological magnetic update path. Here, A1 represents modal edge quantities, A2 represents modal surface magnetic flux, A3 represents modal discrete magnetic divergence, and A4 represents modal constrained transport (CT) constraints. A1 to A4 are all... Figure 3 The discrete object or constraint object number in the module is not used as the device module number; the device module is still the modal topology magnetic update module 105 and the modal diagnosis and acceptance module 106.

[0071] Formula 7:

[0072] And it satisfies the complex identity, as shown in Equation 8:

[0073] In Equations 7 and 8, Λ h 0 Λ h ¹、Λ h ² and Λ h ³ represents the spaces of discrete 0 form, discrete 1 form, discrete 2 form, and discrete 3 form, respectively, and G represents these spaces. h C represents the discrete gradient or exterior differential operator from form 0 to form 1. hD represents the discrete curl or exterior differential operator from form 1 to form 2. h This represents the discrete divergence or exterior differential operator from form 2 to form 3. The complex can be formed by the point, edge, surface, and volume relationships of a finite volume mesh, or by assembling finite element exterior differential spaces, hybrid spline finite elements, polar coordinate splines, spectral element discrete forms, or weak form basis functions. Metrics, curvilinear coordinates, material coefficients, constitutive coefficients, magnetic field derivatives, and magnetic energy inner products are entered through mass matrix or discrete Hodge metric mapping without changing the topological complex identity in Equation 8.

[0074] In this discrete complex, each order of space can be spanned by the corresponding discrete basis functions, and the edge electric circulation, surface magnetic flux, and mass matrices of each order can be expressed as shown in Equation 9:

[0075] The discrete curl and discrete divergence are determined by the directed correlation coefficient or the corresponding weak derivative matrix, as shown in Equation 10:

[0076] In Equations 9 and 10, the edge electric circulation is defined according to the ℓth directed edge or a 1-degree-of-freedom form, and the surface magnetic flux is defined according to the f-th directed surface or a 2-degree-of-freedom form; the 1-, 2-, and 3-form basis functions are used to represent edge quantities, surface quantities, and volume quantities, respectively; the mass matrix with superscript r represents the r-th order discrete form of the mass matrix, ⋆ h Let C represent the discrete Hodge metric map, where the two directed correlation coefficients represent the correlation directions of edge-face and face-volume, respectively. For the angular spectral mode realization, C h and D h It can be represented as a modal block matrix containing angular derivative terms; for pure Fourier modes, the angular derivative terms can be given by the product of the imaginary unit i and the modal index m; for Fourier-Bessel, Fourier-Zenick, spectral element, or Drum-compatible weak formal bases, C h and D h Assembled by weak derivative matrices or incidence matrices, the quadrature matrix, mass matrix, and Hodge metric map are used for projection, deriving magnetic fields, forming constitutive quantities, and forming energy inner products, but the divergence-curl composition must still be kept zero.

[0077] When using constrained transport updates, the magnetic flux of the surface to be inspected is formed by the following formula, as shown in Equation 11:

[0078] From equation 8, we can obtain equation 12:

[0079] In Equations 11 and 12, the magnetic flux with a wavy line represents the magnetic flux of the modal surface to be accepted in the next time layer, and the edge electric field circulation with a wavy line represents the edge electric field circulation to be accepted or the corresponding magnetic update driving quantity obtained by nonlinear calculation and back projection of the node space. Δt represents the current time sub-stage step size. Therefore, when using Modal Constrained Transport (CT), if the initial modal magnetic flux satisfies the modal divergence-free condition, the Modal Constrained Transport (CT) update will not generate new modal discrete magnetic divergence. When using modal topological magnetic update that satisfies the modal magnetic constraint residual acceptance condition, the corresponding magnetic update operator should be able to map to the discrete complex described in Equations 7 to 12 or satisfy the same discrete divergence-curl composition zero relationship. In this way, modal fluid update and modal surface magnetic flux update can be accepted at the same diagnostic acceptance point, avoiding the direct overlay of modal surface magnetic flux by non-modal topological magnetic update paths.

[0080] The modal diagnostic acceptance module 106 checks modal discrete magnetic divergence, conjugate modal error, axis regularity, aliasing energy, and acceptability. For real-valued physics, positive and negative modes satisfy complex conjugate relations; node-to-modal backprojection maintains or restores the positive and negative modal conjugate relations when backprojecting nonlinear fluid update quantities and node space magnetic update driving quantities; the amplitude of the m-th mode near the axis satisfies the regularity condition compatible with the modal order. Axis regularity serves not only as a diagnostic quantity but also as an acceptability constraint before and after entering the node space for computation; modal states that do not satisfy the constraint can be projected to the regularized subspace, limited, or triggered to reject the current time sub-stage. Conjugate modal error can be obtained from the norm of the difference between positive and negative modes, axis regularity can be obtained from the amplitude order of adjacent elements on the axis or by comparing thresholds, and aliasing energy can be obtained from the unretained modal energy or filtered energy before and after backprojection. When any diagnostic quantity exceeds the threshold, the output decides to shorten the time step, recalculate, or reject the current time sub-stage.

[0081] If the diagnosis fails, the modal fluid update, modal edge quantity, modal edge electric field circulation, modal surface magnetic flux update, and modal diagnostic quantity to be accepted will not be written into the state of the next time sub-stage. The system will perform actions such as shortening the time step, recalculating the current time sub-stage, increasing the number of dealiasing nodes, or switching to a more robust flux form, depending on the failure type.

[0082] In one embodiment of complex Fourier modes, the modal state management module 101 stores complex positive and negative angular modal coefficients, and the positive and negative modes maintain the real-valued physical field through complex conjugate relations. This embodiment receives the set of angularly preserved modes and modal fluid / magnetic flux states, outputs the nodal states that can be evaluated in the angular nodal space, and the preserved modal coefficients are recovered by the modal back projection module 104.

[0083] In one real-valued sine / cosine modal embodiment, the modal state management module 101 stores the paired coefficients of the real-valued sine and cosine modes. This embodiment differs from the complex Fourier modal embodiment in that the coefficient storage format is different, but it still receives the same modal state input, outputs the same angular node space state, and forms a real-valued paired modal update after back projection.

[0084] In one field-periodic mode embodiment, the retained mode set is selected based on the corresponding Fourier leaf basis or field-periodic mode family according to the field-periodic boundary conditions of the device. This embodiment is used for magnetic confinement devices with repetitive field periods, where nodalization and back projection still satisfy the same angular periodic boundary conditions.

[0085] In one embodiment of the Fourier-Bessel radial spectrum, the radial basis functions are determined by the zeros of the Bessel functions, the radial order, or the boundary conditions, and together with the angular Fourier factors, form the retained basis index. This embodiment outputs radial-angular sampling states or weakly integrated sampling states, and after backprojection, forms the pending mode updates within the retained basis set.

[0086] In one embodiment of the Fourier-Zenick cross-section spectrum, the cross-section basis functions are jointly determined by the radial and angular Zenick orders and combined with the circumferential or field periodic Fourier factors. This embodiment is used to satisfy the axial regularity or weak integral sampling condition of the cross-section, and the output still enters the nodal space nonlinear calculation module 103 and the modal back projection module 104.

[0087] In a de Rham-compatible discrete form embodiment, the 0-form, 1-form, 2-form, and 3-form degrees of freedom correspond to scalar degrees of freedom, edge quantities, surface magnetic flux, and magnetic divergence diagnostic quantities, respectively. This embodiment can be implemented using finite element external differential space, hybrid spline finite element, polar coordinate spline, spectral element discrete form, or finite volume relation, and the divergence-curl composition is kept zero by the discrete external differential relation. The mesh geometry, material coefficients, constitutive relations, and energy inner product are entered into the corresponding weak form through Hodge metric mapping or mass matrix without changing the above topological chain.

[0088] In one embodiment combining helical phase, rotating phase, or axis-regular vectors, the basis index carries the phase tracking relation, rotating reference relation, or axis-regular constraint. This embodiment is applicable to tilted, rotating, or low-order non-axisymmetric perturbations, and the output still needs to pass through the dealiasing sampling space and modal backprojection link to enter the pending mode update.

[0089] In a 3 / 2 sampling embodiment, a preset dealiasing condition is used to parse the quadratic nonlinear convolution term by an angular node number higher than the number of modes to be retained. This embodiment outputs the node-space nonlinear flux and node-space magnetic update driving quantity for back projection, and writes the aliasing energy into the modal diagnostic acceptance module 106.

[0090] In a two-thirds truncation embodiment, a preset dealiasing condition reduces folding error by truncating higher-order modes or filtering out unacceptable convolutional terms. This embodiment differs from the two-thirds sampling embodiment in that unacceptable spectral components are filtered out first, but the same update package for the unacceptable modes is still output.

[0091] In one implementation of oversampling, exact convolution, dealiasing filtering, or adaptive node count, the model configuration increases the node count or adjusts the filtering rules based on the nonlinearity order, aliasing energy, or acceptability diagnostics. These rules represent other optional implementations of the dealiasing conditions, all without altering the same sampling space, the same backprojection link, or the same diagnostic acceptance boundary.

[0092] In one embodiment of finite volume nonlinear calculation, the node space nonlinear calculation module 103 uses the radial-axial finite volume surface flux and the derivative of the angular modal flux to jointly form the node space fluid update quantity. This embodiment receives the original three-dimensional variables and reconstructed state of the node space, and outputs the node space nonlinear fluid update quantity and the node space magnetic update driving quantity.

[0093] In one embodiment of angular node flux difference, the node spatial nonlinear calculation module 103 forms circumferential flux differences on the angular nodes to represent the angular flux terms. This embodiment shares the same angular nodes, node weights, and geometric metrics as the finite volume surface flux embodiment, and the output is still back-projected by the modal back-projection module 104.

[0094] In a conserved finite difference, element residual, or weak form residual embodiment, the node space nonlinear calculation module 103 generates node space nonlinear fluid update quantities through local conserved flux differences, element boundary fluxes, or weak residuals. In this embodiment, the fluid surface flux, angular flux term, geometric source term, nonideal electric field term, and node space magnetic update driving quantity are formed using the same dealiasing angular node sampling or weak integral sampling, the same node weights or mass matrix, and the same geometric metrics.

[0095] In one modal edge quantity driven embodiment, the modal topological magnetic update module 105 uses modal edge quantities to form the magnetic flux update of the mode to be accepted via discrete curl operators on the corresponding reserved basis; when constrained transport (CT) is used, the modal edge quantity is the modal edge electric field circulation.

[0096] In one embodiment of nodal spatial edge electric field circulation back projection, the nodal spatial edge electric field circulation first forms the nodal spatial magnetic flux increment, and then is back projected by the modal back projection module 104 as the modal surface magnetic flux increment. The output of this embodiment still updates the modal surface magnetic flux or the same discrete 2-degree-of-freedom form.

[0097] In a Deram-compatible discrete complex embodiment, the edge electric field circulation belongs to form 1 degree of freedom, the surface magnetic flux belongs to form 2 degree of freedom, the magnetic divergence diagnostic quantity belongs to form 3 degree of freedom, and the divergence-curl composition is kept to zero by the edge-surface correlation matrix, the surface-volume correlation matrix, or the corresponding weak derivative matrix.

[0098] In one embodiment of modal magnetic update residuals, the modal topological magnetic update module 105 uses modal magnetic update residual terms that meet the modal magnetic constraint residual acceptance conditions to form the modal flux update to be accepted. All the above magnetic update embodiments use the modal surface magnetic flux or the discrete state of the modal surface magnetic flux, or a state with the same modal discrete two-form degrees of freedom as the modal surface magnetic flux and capable of being mapped to derive magnetic fields, non-ideal electric field constitutive quantities, or magnetic energy inner products via modal Hodge metric mapping, as the update object, and all maintain the modal flux state from being directly covered by non-topological paths.

[0099] In one initialization embodiment, the angular spectrum modal state is obtained by projecting a two-dimensional axisymmetric equilibrium projector to obtain the zero-order mode and superimposing it with low-order perturbation modes, or by projecting a three-dimensional initial field, linear eigenmodes, experimental diagnostic perturbations, analytical tilt perturbations, rotational perturbations, helical perturbations, or random low-mode perturbations onto the angularly preserved mode set. The above initialization methods are subordinate support content, all of which restore or check the conjugate relationship between positive and negative modes and perform modal magnetic constraint residual checks on the modal flux state.

[0100] In one implementation of modal regularization, the modal diagnostic acceptance module 106 or the pre-admissibility processing performs modal admissibility projection, conjugate mode recovery, axis regularization projection, mode-preserving accurate filtering, spectral viscosity limiting, modal divergence cleaning, modal magnetic constraint projection based on Hodge metric mapping, or modal magnetic constraint correction that meets the modal magnetic constraint residual acceptance criteria. All of the above regularizations target the update of the mode to be accepted or the diagnostic quantity of the mode to be accepted, and are only accepted after the modal diagnostic acceptance is passed.

[0101] In one embodiment of nodal space boundary correction, the nodal space boundary state is formed by angular periodic boundary matching, field periodic boundary phase matching, axis canonical boundary projection, wall boundary state projection, material interface nodal space boundary state projection, free boundary sampling limiting, boundary leakage diagnosis, or modal boundary residual diagnosis. The above boundary corrections are optional subordinate implementations, and the results are used as nodal space boundary states, source term samples, or diagnostic inputs in nodal space nonlinear calculations. They must not bypass modal topological magnetic updates to directly cover modal surface magnetic flux.

[0102] In addition to the above embodiments, without changing the core constraint relationships of the present invention, angular basis functions, radial or cross-sectional spectrum basis, Delamin-compatible discrete form basis, dealiasing rules, low-mode initialization, mode regularization, nodal space boundary correction, nodal space nonlinear calculation method, back projection operator, and modal topological magnetic update method can be implemented in different combinations. The core constraint relationships include: the modal state can be transformed to the angular nodal space or weak integral sampling space; the sampling space performs nonlinear fluid update and magnetic update driving quantity calculation on the same dealiasing angular nodal sampling or weak integral sampling; the calculation results are entered into the angular retained mode set or retained basis index set through back projection that satisfies the accuracy of mode retention; the fluid update and magnetic flux update of the mode to be accepted are only accepted after the diagnostic acceptance is passed. All combinations that satisfy the above relationships are optional implementations of the present invention. In one application scenario, the method is used for the calculation of tilt mode, rotation mode, or low-order non-axisymmetric perturbation evolution of field anticonfiguration plasma. The device configuration provides a radial-axial mesh and an initial equilibrium profile. The angular spectrum branch retains the zero-order mode and one or more pairs of positive and negative low-order modes. Nonlinear compression, magnetic tension, and finite volume flux are calculated in the dealiased angular node space or weak integral sampling space, and the results are back-projected to the retained modes.

[0103] In various application scenarios, the method is used for the evolution calculation of local low-mode perturbations in tokamaks, low-mode responses of stellarator field periods, and non-axisymmetric end perturbations in magnetic mirrors or start-up devices. Walls, boundary conditions, or control inputs in the local computation segment, used as boundary states, source term samples, or diagnostic inputs in the sampling space, must not bypass the modal topological magnetic update to directly cover the modal surface magnetic flux; when implemented using CT, the modal surface magnetic flux must not be covered by bypassing the modal constrained transport (CT) path.

[0104] In some embodiments, the above-described angular spectral magnetohydrodynamic three-dimensional nonlinear coupling solution method can be executed by an electronic device. The electronic device may include a central processing unit, a graphics processing unit, an accelerator card, a field-programmable gate array, a cluster node, or a combination thereof, and stores modal states, node spatial states, back projection operators, modal flux states, and diagnostic thresholds in memory.

[0105] In one possible implementation, the computer program instructions stored in the computer-readable storage medium are executed by the processor to enable the processor to complete the acquisition of angular spectrum mode states, dealiasing node transformation, three-dimensional nonlinear calculation of node space, modal back projection, magnetic flux update of the mode to be accepted, and modal diagnostic acceptance.

[0106] The computer program product can take the form of: 1. an installable program; 2. a simulation software plug-in; or 3. a container image or cloud task configuration. When the computer program product is executed, it generates an acceptance mode update, aliasing energy diagnosis, conjugate mode error diagnosis, and a reserved mode state for the next time sub-stage.

Claims

1. A three-dimensional nonlinear coupled solution method for angular spectral magnetohydrodynamics, characterized in that, The process includes the following steps: S1. Obtaining the angular spectrum modal state on the base computational grid, the angular spectrum modal state including modal fluid state, modal magnetic flux state, and angular retained mode set; S2. Transforming the angular spectrum modal state to the angular node space or weak integral sampling space according to preset dealiasing conditions; S3. Calculating the nonlinear fluid update quantity in the angular node space or weak integral sampling space based on the three-dimensional original variables, reconstructed state, or surface flux, and calculating the node space magnetic update driving quantity for modal topological magnetic update. The nonlinear fluid update quantity and the node space magnetic update driving quantity are formed based on the same dealiasing angular node or weak integral sampling point, the same node weight or mass matrix, and the same geometric metric. The nonlinear fluid update quantity is calculated from the three-dimensional state of the angular node space or weak integral sampling space, rather than being directly given by the linearized flux in the retained modal space. S4. Using the same node-to-modal back projection operator, back project the nonlinear fluid update amount and the node spatial magnetic update driving amount to the angular preserved mode set to obtain the modal fluid update amount and modal magnetic update driving amount. S5. A modal fluid update to be accepted is formed based on the modal fluid update amount, and a modal magnetic flux update to be accepted is formed based on the modal magnetic update driving amount. The modal fluid update to be accepted and the modal magnetic flux update to be accepted form the same modal update package to be accepted. S6. Diagnostic acceptance is performed based on modal magnetic divergence, conjugate mode error, aliasing energy, axis regularity or adoptability. After acceptance, the modal fluid update to be accepted and the modal magnetic flux update to be accepted are jointly accepted. If acceptance fails, the current time sub-stage is jointly rejected or recalculated.

2. The method according to claim 1, characterized in that, The preset dealiasing conditions include one or more of the following: 3 / 2 sampling rule, 2 / 3 truncation rule, oversampling integration rule, precise convolution sampling rule, dealiasing filtering rule, or adaptive node number rule; the same dealiasing angular node sampling or weak integral sampling includes forming the fluid surface flux, angular flux term, cylindrical or curvilinear geometric source term, non-ideal electric field term, and node spatial magnetic update driving quantity under the same angular node, the same weak integral sampling point, the same node weight, the same mass matrix, and the same geometric metric.

3. The method according to claim 1, characterized in that, When the angular node space or weak integral sampling space calculation is implemented using finite volume, conserved finite difference, element residual, or weak form residual, the three-dimensional magnetohydrodynamic flux tensor of the sampling space includes radial flux, axial flux, and angular flux. The angular flux term is represented by the modal flux derivative, weak derivative matrix, or circumferential flux difference satisfying the circumferential conservation relationship in the angular node space. The right-hand side term of the nonlinear fluid in the node space is calculated from the three-dimensional original variables of the node space or weak integral sampling space, rather than being directly given by the linearized flux in the retained modal space.

4. The method according to claim 1, characterized in that, The modal flux update to be accepted is formed by modal topological magnetic update. The modal topological magnetic update takes the modal surface flux or the discrete state of the modal surface flux having the same modal discrete two-form degrees of freedom and capable of deriving magnetic fields, non-ideal electric field constitutive quantities, or magnetic energy inner products through modal Hodge metric mapping as the update object. It is formed by modal edge quantities, modal surface flux increments, modal magnetic update residual terms, or modal magnetic update driving quantities that meet the modal magnetic constraint residual acceptance conditions. The modal Hodge metric mapping is used to introduce mesh geometric metrics, material coefficients, constitutive relations, or energy inner products without changing the topological composite relationship of discrete curl and discrete divergence. The modal topological magnetic update includes the following implementations: when using modal constrained transport (CT), modal surface flux is formed from modal edge quantities through discrete curl operators on the corresponding preserved basis; when using modal delaminar (de... During the Rham update, the edge electric field circulation is treated as a discrete-form 1 degree of freedom, and the modal surface magnetic flux is treated as a discrete-form 2 degree of freedom. The magnetic flux update is formed by the edge-surface and surface-volume discrete relations that satisfy the composition of the divergence operator and the curl operator to zero; or an update is adopted that satisfies the composition of the discrete divergence operator and the discrete curl operator on the corresponding retained basis to zero and passes the modal magnetic constraint residual acceptance.

5. The method according to claim 1, characterized in that, The modal-to-node transformation operator and the node-to-modal backprojection operator or weak projection operator satisfy the requirement of preserving modal accuracy. The preservation of modal accuracy means that when the preserved mode is backprojected after node transformation or weak integral sampling, it recovers to the original preserved mode under the condition of no filtering and no diagnostic acceptance rejection. The node-to-modal backprojection or weak projection maintains or restores the conjugate relationship between positive and negative modes when backprojecting the nonlinear fluid update amount and the node spatial magnetic update driving amount.

6. The method according to claim 1, characterized in that, The diagnostic acceptance includes checks on modal magnetic divergence, conjugate mode error, axis regularity, aliasing energy, or adoptability; the aliasing energy is obtained from the unretained modal energy, filtered energy, or higher-order node residuals before and after back projection.

7. The method according to claim 1, characterized in that, The basis functions, initialization, regularization, or boundary corrections of the angular spectral modal states include at least one of the following:

1. Complex Fourier modes, real-valued sine / cosine modes, family of field periodic modes, spiral phase modes, rotating phase-tracking modes, Fourier-Bessel radial spectral basis, Fourier-Zernike spectral basis, delaminar spectral basis. Rham-compatible discrete-form basis or vector mode combination satisfying axis regularity; II. Two-dimensional axisymmetric equilibrium projection, three-dimensional initial field low-mode projection, linear eigenmode perturbation projection, experimental diagnostic perturbation projection, or random low-mode perturbation projection; III. Conjugate mode recovery, axis regularization projection, mode-preserving precise filtering, spectral viscosity limiting, modal adoptability projection, or modal magnetic confinement residual correction; IV. Angular periodic boundary matching, field periodic boundary phase matching, wall or material interface node spatial boundary state projection, free boundary sampling limiting, or modal boundary leakage diagnosis; The initialization, regularization, or boundary correction results, after passing modal diagnostic acceptance, are entered into the angular node space or weak integral sampling space calculation. The calculation or next time sub-stage state; when the internal source term includes neutral-plasma exchange term, anomalous transport term, high-energy particle deposition term or external heating deposition term, the internal source term is processed according to the same mode transformation, dealiasing, back projection and diagnostic acceptance rules, and does not change the processing chain of this method with nodal space nonlinear calculation, mode back projection and mode topological magnetic update as the core; when the nodal space magnetic update driving quantity includes extended magnetofluid term, Hall electric field term, electron pressure gradient term, electron inertia term, resistivity correction term or anisotropic transport correction term participate in the same back projection and diagnostic acceptance as nodal space magnetic update driving quantity, non-ideal electric field term or mode magnetic update residual term, and does not directly cover the modal surface magnetic flux.

8. A three-dimensional nonlinear coupled solution device for angular spectrum magnetohydrodynamics, characterized in that, The system includes a modal state management module 101, an anti-aliasing node transformation module 102, a node space nonlinear calculation module 103, a modal back projection module 104, a modal topological magnetic update module 105, and a modal diagnosis and acceptance module 106. The modal state management module 101 is used to execute S1 in claim 1; the anti-aliasing node transformation module 102 is used to execute S2; the node space nonlinear calculation module 103 is used to execute S3; the modal back projection module 104 is used to execute S4 and form the fluid update of the modal to be accepted in S5; the modal topological magnetic update module 105 is used to form the magnetic flux update of the modal to be accepted in S5; and the modal diagnosis and acceptance module 106 is used to execute S6.

9. An electronic device, characterized in that, It includes a processor and a memory for storing processor-executable instructions; the processor is configured to invoke the instructions stored in the memory to perform the method of any one of claims 1 to 7.

10. A computer-readable storage medium or computer program product, characterized in that, The computer-readable storage medium stores computer program instructions, or the computer program product includes a computer program or instructions; when the computer program instructions, computer program, or instructions are executed by a processor, they implement the method of any one of claims 1 to 7.

Images