Superconducting sub-cable multi-scale numerical calculation method and system based on asymptotic homogenization

Abstract

The invention relates to the technical field of numerical calculation, and discloses a superconductor cable multi-scale numerical calculation method and system based on asymptotic homogenization, and the method comprises the steps: building a microscopic representative unit considering core wire inclusion and a matrix, and determining a microscopic balance equation of the microscopic representative unit and a periodic boundary condition of the microscopic balance equation; solving the microscopic balance equation to obtain an equivalent elastic constant; establishing a macroscopic homogenization model based on the equivalent elastic constant to obtain macroscopic strain; and substituting the macroscopic strain and the characteristic displacement field function into a microcosmic stress expression to obtain microcosmic stress distribution of the microcosmic representative unit. A microcosmic-macroscopic-microcosmic numerical calculation model is established through an asymptotic homogenization method, so that prediction of overall mechanical behaviors and local contact force in sub-cables and inversion and evaluation of core wire microcosmic stress are realized at the same time.

Description

Technical Field

[0001] The present invention relates to the technical field of numerical calculation, and particularly to a multi-scale numerical calculation method and system for superconducting sub-cables based on asymptotic homogenization. Background Art

[0002] The cable-in-conduit conductor (CICC) is a key component of the superconducting magnet in ITER (International Thermonuclear Experimental Reactor). These CICCs are composed of copper and superconducting wires twisted step by step, which can transmit high currents while reducing AC losses. As a typical multi-level helical composite structure, the complex helical geometry generated by the gradual winding of CICC sub-cables at different structural levels poses a major challenge to mechanical analysis. Due to the limited ability of experimental methods to capture internal information of complex helical structures, it is very important to study the internal mechanical response of CICC sub-cables and reveal the mechanical interaction between the sub-cable and filament scales through other effective methods such as computational homogenization and theoretical modeling.

[0003] As a rigorously derived mathematical method, the asymptotic homogenization method (AHM) has been proposed to simplify periodic composite structures into simple materials with equivalent parameters. AHM is widely used in multi-scale mechanical analysis of various periodic engineering structures. There are usually two common methods to implement AHM in numerical models to determine the RVE equivalent parameters: the thermal stress method and the non-intrusive asymptotic homogenization (NIAH) method. The mechanical properties of multi-level composite helical structures affect each other within different structural levels. The local contact behavior between strands and the composite material characteristics at the filament level make it difficult for traditional methods to simultaneously predict the macroscopic and microscopic mechanical properties of sub-cables. Therefore, there is an urgent need to establish a numerical calculation model that can simultaneously calculate the overall mechanical response, local contact behavior, and microscopic stress distribution to characterize the mechanical behavior of sub-cables at different scales. Summary of the Invention

[0004] The purpose of the present invention is to provide a multi-scale numerical calculation method and system for superconducting sub-cables based on asymptotic homogenization, which establishes a microscopic-macroscopic-microscopic numerical calculation model through the asymptotic homogenization method to simultaneously achieve the prediction of the overall mechanical behavior and local contact forces within the sub-cable and the inversion and evaluation of the microscopic stresses of the filaments.

[0005] To achieve the above purpose, the following technical solutions are adopted:

[0006] In the first aspect, the present invention provides a multi-scale numerical calculation method for superconducting sub-cables based on asymptotic homogenization, and the method includes:

[0007] Establish a microscopic representative unit considering the filament inclusions and matrix, and determine the microscopic equilibrium equation and its periodic boundary conditions of the microscopic representative unit;

[0008] Solve the microscopic equilibrium equation to obtain the equivalent elastic constants;

[0009] Based on the equivalent elastic constants, establish a macroscopic homogenization model to obtain the macroscopic strain;

[0010] Substitute the macroscopic strain and the characteristic displacement field function into the microscopic stress expression to obtain the microscopic stress distribution of the microscopic representative unit.

[0011] Furthermore, the microscopic equilibrium equation of the microscopic representative unit is expressed as:

[0012]

[0013] where D ijkl and D ijmn are the first and second components of the elastic tensor, is the characteristic displacement field function, v i is the trial function, y j and y l are the first and second components of the microscopic scale coordinates, and Y is the periodic length of the microscopic representative unit region.

[0014] Furthermore, the microscopic equilibrium equation of the microscopic representative unit satisfies the following periodic boundary conditions:

[0015]

[0016] where n is a positive integer, Θ represents the internal region of the microscopic representative unit, represents the boundary of the microscopic representative unit, represents the characteristic displacement component at the microscopic scale, and onthevertices ofΘ represents the corner positions of the microscopic representative unit.

[0017] Furthermore, use the UEXPAN subroutine to solve the microscopic equilibrium equation to obtain the equivalent elastic constants, and the equivalent elastic constants are expressed as:

[0018]

[0019] where the superscript H represents homogenization, κ mn is the thermal conductivity, ΔT is the temperature change, represents the characteristic displacement component.

[0020] Furthermore, the thermal conductivity and the temperature change have the following relationship:

[0021] κ mn ΔT=diag(-1,-1,-1,-1,-1,-1) (4)

[0022] where diag represents a diagonal matrix.

[0023] Further, substituting the macroscopic strain and the characteristic displacement field function into the microscopic stress expression, the microscopic stress distribution of the microscopic representative unit is obtained, expressed as:

[0024]

[0025] where σ ij (y) represents the microscopic stress of the microscopic representative unit, represents the macroscopic strain.

[0026] Further, a macroscopic homogenization model is established based on the equivalent elastic constants to obtain the macroscopic strain, including:

[0027] Establish a macroscopic homogenization model, the macroscopic homogenization model includes helical superconducting wires wound around each other, the helical superconducting wires include an isotropic copper outer layer and a transversely isotropic uniform superconducting inner region, and the uniform superconducting inner region is provided with a core wire bundle and a bronze matrix;

[0028] Based on the equivalent elastic constants, establish an effective constitutive equation for the core wire bundle;

[0029] Based on the effective constitutive equation of the core wire bundle, establish the constitutive relationship of the macroscopic homogenization model and determine the macroscopic strain.

[0030] Further, the effective constitutive equation of the core wire bundle is expressed as:

[0031]

[0032] where [σ1, σ2, σ3, τ 12 , τ 13 , τ 23 and [ε1, ε2, ε3, ε 12 , ε 13 , ε 23 are the stress and strain components of the homogenized region respectively, represents the transverse tensile stiffness, represents the in-plane coupled tensile stiffness, represents the out-of-plane coupled tensile stiffness, represents the longitudinal tensile stiffness, represents the longitudinal shear stiffness, σ1 represents the longitudinal stress in the plane perpendicular to the helix tangent, σ2 represents the transverse stress in the plane perpendicular to the helix tangent, σ3 represents the stress along the helix tangent, τ 12 represents the shear stress in the plane parallel to the helix tangent, τ 13 represents the longitudinal shear stress in the plane perpendicular to the helix tangent plane, τ 23denotes the transverse shear stress in the plane perpendicular to the helix tangent plane, ε1 denotes the longitudinal strain in the plane perpendicular to the helix tangent, ε2 denotes the transverse strain in the plane perpendicular to the helix tangent, ε3 denotes the strain along the helix tangent, ε 12 denotes the shear strain in the plane parallel to the helix tangent, ε 13 denotes the longitudinal shear strain in the plane perpendicular to the helix tangent, ε 23 denotes the transverse shear strain in the plane perpendicular to the helix tangent.

[0033] Furthermore, the constitutive relation of the macroscopic homogenization model is expressed as:

[0034]

[0035] where E3, ν 23 , G 23 , G 12 , κ 12 respectively denote the axial Young's modulus, axial Poisson's ratio, axial shear modulus, transverse shear modulus, transverse bulk modulus of the material, and v 12 denotes the transverse Poisson's ratio.

[0036] In a second aspect, the present invention provides a superconducting cable multi-scale numerical calculation system based on asymptotic homogenization, and the system includes:

[0037] A microscopic representative unit establishment module, configured to establish a microscopic representative unit considering the core wire inclusion and the matrix, and determine the microscopic equilibrium equation and its periodic boundary conditions of the microscopic representative unit;

[0038] An equation solving module, configured to solve the microscopic equilibrium equation to obtain equivalent elastic constants;

[0039] A macroscopic strain acquisition module, configured to establish a macroscopic homogenization model based on the equivalent elastic constants to obtain macroscopic strain;

[0040] A microscopic stress distribution calculation module, configured to substitute the macroscopic strain and the characteristic displacement field function into the microscopic stress expression to obtain the microscopic stress distribution of the microscopic representative unit.

[0041] The beneficial effects of the present invention are:

[0042] 1. The present invention establishes a numerical homogenization model of the CICC cable considering the superconducting core wire distribution, realizing the micro-macro-micro multi-scale numerical analysis of the cable.

[0043] 2. The present invention can accurately predict the overall mechanical response, local contact behavior and microscopic stress distribution of the cable, and the calculation efficiency is one order of magnitude faster than the refined model.

[0044] 3. While retaining the local contact force and micro-stress information, the model constructed in the present invention provides a useful reference for realizing large-scale accurate calculation of CICC multi-stage sub-cables.

[0045] 4. The present invention can consider the mechanical behaviors of sub-cables under different types of external loads and establish a complete evaluation system for the mechanical properties of sub-cables including overall mechanical quantities and local contact forces. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 Shows the flow of a multi-scale numerical calculation method for superconducting sub-cables based on asymptotic homogenization according to an embodiment of the present invention Figure 1 .

[0047] Figure 2 Shows the flow of a multi-scale numerical calculation method for superconducting sub-cables based on asymptotic homogenization according to an embodiment of the present invention Figure 2 .

[0048] Figure 3 Shows a schematic diagram of meshing the representative volume element (RVE) at the microscale according to an embodiment of the present invention.

[0049] Figure 4 Shows a schematic diagram of the macroscopic homogenization model according to an embodiment of the present invention, where each strand contains a homogenized region and a copper outer layer.

[0050] Figure 5 Shows a structural diagram of a multi-scale numerical calculation system for superconducting sub-cables based on asymptotic homogenization according to an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0051] The following specific examples illustrate the embodiments of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that, without conflict, the following embodiments and the features in the embodiments can be combined with each other.

[0052] The following further describes in detail the specific embodiments of the present invention in conjunction with the drawings and embodiments.

[0053] An embodiment of the present invention provides a multi-scale numerical calculation method for superconducting sub-cables based on asymptotic homogenization, as Figure 1As shown, the method includes two parts: the microscale and the macroscale. Numerical modeling is performed on the two parts separately, and multiscale calculation is achieved by transferring the equivalent modulus and strain tensor. Generally speaking, when calculating, the method includes two core steps: microscale analysis and macroscale analysis. Based on microscale analysis and macroscale analysis, the microstress of a specified element can be determined. During microscale analysis, the unit strain tensor ε o(kl) is obtained according to the microscale modeling of the RVE, and then the characteristic strain tensor ε *(kl) is calculated, and the equivalent elastic tensor is calculated. During macroscale analysis, based on the equivalent elastic tensor , finite element modeling and progressive homogenization of the sub-cable are carried out to obtain the macro displacement field. Based on the macro displacement field, the macro strain tensor can be determined. Finally, by combining the equivalent elastic constant with the macro strain tensor, the microstress of the specified element can be obtained.

[0054] Specifically, as Figure 2 shown, the multiscale numerical calculation method for superconducting sub-cables based on asymptotic homogenization includes the following steps S100 - S400.

[0055] S100: Establish a microscopic representative unit considering the core wire inclusion and the matrix, and determine the microscopic equilibrium equation and its periodic boundary conditions of the microscopic representative unit.

[0056] Exemplarily, a microscopic representative unit (RVE) considering the core wire inclusion and the matrix is established. The size of the RVE is: W y = 6.6 mm, W x = 4 mm, the thickness is taken as 1 mm, and the radius of the core wire is 1.32 mm. As Figure 3 shown, it is a schematic diagram of meshing the microscopic representative unit (RVE). The length is W y , the width is W x , the radius of the core wire is R, and the number of meshes after division is 2.9 * 10 5 . Under periodic boundary conditions, the equivalent elastic tensor {ε} marco is the macro strain. The microscopic equilibrium equation of the RVE can be expressed as:

[0057]

[0058] where D ijkl and D ijmn are the first and second components of the elastic tensor, is the characteristic displacement field function, v i is the trial function, y j and y lare the first and second components of the microscopic coordinates, and Y is the period length of the microscopic representative unit area. The superscripts i, j, k, l, m, and n range from 1 to 3.

[0059] And the periodic boundary conditions are met:

[0060]

[0061] Where n is a positive integer, Θ represents the internal area of the microscopic representative unit, represents the boundary of the microscopic representative unit, represents the characteristic displacement component at the microscopic scale, and onthevertices ofΘ represents the corner position of the microscopic representative unit.

[0062] S200: Solve the microscopic equilibrium equations to obtain the equivalent elastic constants;

[0063] For example, the equivalent elastic constant can be obtained by solving the above equations in combination with the UEXPAN subroutine

[0064]

[0065] The superscript H represents homogenization, κ mn is the thermal conductivity, ΔT is the temperature change, represents the characteristic displacement component.

[0066] Thermal conductivity and temperature change have the following relationship:

[0067] κ mn ΔT=diag(-1,-1,-1,-1,-1,-1) (4)

[0068] Where diag represents a diagonal matrix.

[0069] S300: Establish a macroscopic homogenization model based on the equivalent elastic tensor to obtain macroscopic strain.

[0070] Specifically, the subcable consists of spiral superconducting strands wound around each other, each superconducting wire consisting of a superconducting material region (Nb3Sn core strands and bronze matrix) and a copper outer layer. For the homogenized model, the spiral consists of two material regions: an isotropic copper outer layer and a transversely isotropic homogeneous superconducting inner region. A representative volume element (RVE) is selected to determine the equivalent elastic tensor of the superconducting region, such as Figure 4 After obtaining the equivalent elastic constants according to formula (3), based on the Voigt symbol rule, the effective constitutive equation of the core bundle can be written as:

[0071]

[0072] Among them, [σ1, σ2, σ3, τ 12 , τ 13 , τ 23 and [ε1, ε2, ε3, ε 12 , ε 13 , ε 23 are the stress and strain components in the homogenized region respectively, represents the transverse tensile stiffness, represents the in-plane coupled tensile stiffness, represents the out-of-plane coupled tensile stiffness, represents the longitudinal tensile stiffness, represents the longitudinal shear stiffness. σ1 represents the longitudinal stress in the plane perpendicular to the helix tangent, σ2 represents the transverse stress in the plane perpendicular to the helix tangent, σ3 represents the stress along the helix tangent, τ 12 represents the shear stress in the plane parallel to the helix tangent, τ 13 represents the longitudinal shear stress in the plane perpendicular to the helix tangent, τ 23 represents the transverse shear stress in the plane perpendicular to the helix tangent, ε1 represents the longitudinal strain in the plane perpendicular to the helix tangent, ε2 represents the transverse strain in the plane perpendicular to the helix tangent, ε3 represents the strain along the helix tangent, ε 12 represents the shear strain in the plane parallel to the helix tangent, ε 13 represents the longitudinal shear strain in the plane perpendicular to the helix tangent, ε 23 represents the transverse shear strain in the plane perpendicular to the helix tangent.

[0073] Among them, the subscript "3" represents the tangent direction of the helical core wire, the plane "12" represents the plane located perpendicular to the helix tangent, and the subscripts "1, 2" represent two mutually perpendicular directions in this plane. Furthermore, the five engineering elastic constants {E3, ν 23 , G 23 , G 12 , κ 12} can be calculated by the following formula:

[0074]

[0075] Among them, E3, ν 23 , G 23 , G 12 , κ 12 represent the axial Young's modulus, axial Poisson's ratio, axial shear modulus, transverse shear modulus, transverse bulk modulus of the material respectively, and v 12 represents the transverse Poisson's ratio.

[0076] The constitutive relation of the macroscopic homogenization model can be established using Equation (7). After applying the corresponding boundary conditions to the macroscopic homogenization model and solving it, the macroscopic strain field distribution of the model can be obtained. Select the elements at specific positions, extract their macroscopic strain vectors, and the extracted macroscopic strain vectors are used in conjunction with subsequent steps to obtain the microscopic stress of the specified elements.

[0077] S400: Substitute the macroscopic strain and the characteristic displacement field function into the microscopic stress expression to obtain the microscopic stress distribution of the microscopic representative element.

[0078] In some embodiments, based on a macroscopic homogenization model can be established to obtain the macroscopic strain By substituting and into the microscopic stress expression, the microscopic stress distribution on the RVE can be obtained:

[0079]

[0080] where σ ij (y) represents the microscopic stress of the microscopic representative element, represents the macroscopic strain.

[0081] Exemplarily, after applying the corresponding boundary conditions to the macroscopic homogenization model and solving it, the macroscopic strain field distribution of the model can be obtained. Select the elements at specific positions, extract their macroscopic strain vectors, and by combining Equation (5), the microscopic stress of the specified elements can be obtained as follows:

[0082] {σ} el =[D] el ([ε 0(kl) el +[ε *(kl) el ){ε} marco

[0083] where [D] el (6×6) is the element stiffness matrix, [ε 0(kl) el (6×9) is the RVE strain matrix obtained from the nodal displacement fields corresponding to the application of 6 unit strain fields. [ε *(kl) el (6×9) is the characteristic strain matrix, {σ} el (9×1) and {ε} marco (9×1) are the matrix forms of the microscopic stress tensor components and the macroscopic strain tensor, respectively.

[0084] ​​​​It should be noted that in the above steps S100 - S400, steps S100 and S200 are at the microscale, step S300 is at the macroscale, and step S400 is at the microscale. Therefore, this method realizes a micro - macro - micro numerical solution process based on asymptotic homogenization, thereby enabling accurate prediction of the overall mechanical response, local contact behavior, and micro - stress distribution of the sub - cable.

[0085] An embodiment of the present invention also provides a multi - scale numerical calculation system for a superconducting sub - cable based on asymptotic homogenization. Please refer to Figure 5 , and this system includes:

[0086] A micro - representative unit establishment module 501, configured to establish a micro - representative unit considering the core wire inclusions and the matrix, and determine the micro - equilibrium equation and its periodic boundary conditions of the micro - representative unit;

[0087] An equation solving module 502, configured to solve the micro - equilibrium equation to obtain the equivalent elastic constants;

[0088] A macro - strain acquisition module 503, configured to establish a macro - homogenization model based on the equivalent elastic constants to obtain the macro - strain;

[0089] A micro - stress distribution calculation module 504, configured to substitute the macro - strain and the characteristic displacement field function into the micro - stress expression to obtain the micro - stress distribution of the micro - representative unit.

[0090] It should be noted that this multi - scale numerical calculation system for a superconducting sub - cable based on asymptotic homogenization belongs to the same technical concept as the method described above, and has the same technical principle and beneficial effects, so it will not be elaborated here.

[0091] The above embodiments are only used to illustrate the present invention, rather than to limit the present invention. Those of ordinary skill in the relevant technical field can also make various changes and modifications without departing from the spirit and scope of the present invention. Therefore, all equivalent technical solutions also belong to the scope of the present invention, and the patent protection scope of the present invention should be defined by the claims.

Claims

1. A multi-scale numerical calculation method for superconducting cables based on asymptotic homogenization, characterized in that The method includes: Establish a microscopic representative unit considering the core wire inclusion and the matrix, and determine the microscopic equilibrium equation of the microscopic representative unit and its periodic boundary conditions; Solve the microscopic equilibrium equation to obtain the equivalent elastic constants; Based on the equivalent elastic constants, establish a macroscopic homogenization model to obtain the macroscopic strain; Substitute the macroscopic strain and the characteristic displacement field function into the microscopic stress expression to obtain the microscopic stress distribution of the microscopic representative unit.

2. The method according to claim 1, wherein The microscopic equilibrium equation of the microscopic representative unit is expressed as: where D ijkl and D ijmn are the first and second components of the elastic tensor, is the characteristic displacement field function, v i is the trial function, y j and y l are the first and second components of the microscopic scale coordinates, and Y is the periodic length of the microscopic representative unit region.

3. The method according to claim 2, wherein The microscopic equilibrium equation of the microscopic representative unit satisfies the following periodic boundary conditions: where n is a positive integer, Θ represents the internal region of the microscopic representative unit, represents the boundary of the microscopic representative unit, represents the characteristic displacement component at the microscopic scale, and onthevertices of Θ represents the corner positions of the microscopic representative unit.

4. The method according to claim 3, wherein The UEXPAN subroutine is used to solve the microscopic balance equation to obtain the equivalent elastic constant, and the equivalent elastic constant is expressed as: where the superscript H represents homogenization, κ mn is the thermal conductivity, ΔT is the temperature change, represents the characteristic displacement component.

5. The method according to claim 4, wherein The thermal conductivity has the following relationship with the temperature change: κ mn ΔT = diag(-1, -1, -1, -1, -1, -1) (4) Where diag represents a diagonal matrix.

6. The method according to claim 5, wherein Substitute the macroscopic strain and the characteristic displacement field function into the microscopic stress expression to obtain the microscopic stress distribution of the microscopic representative unit, which is expressed as: where σ ij (y) represents the microscopic stress of the microscopic representative element, represents the macroscopic strain.

7. The method according to claim 6, wherein Based on the equivalent elastic constants, establish a macroscopic homogenization model to obtain the macroscopic strain, including: Establish a macroscopic homogenization model, the macroscopic homogenization model includes helical superconducting wires wound around each other, the helical superconducting wires include an isotropic copper outer layer and a transversely isotropic uniform superconducting inner region, and the uniform superconducting inner region is provided with a core wire bundle and a bronze matrix; Based on the equivalent elastic constants, establish an effective constitutive equation for the core wire bundle; Based on the effective constitutive equation of the core wire bundle, establish the constitutive relationship of the macroscopic homogenization model and determine the macroscopic strain.

8. The method according to claim 7, wherein The effective constitutive equation of the core wire bundle is expressed as: Among them, [σ1, σ2, σ3, τ 12 , τ 13 , τ 23 , and [ε1, ε2, ε3, ε 12 , ε 13 , ε 23 are the stress and strain components in the homogenized region respectively, represents the transverse tensile stiffness, represents the in-plane coupled tensile stiffness, represents the out-of-plane coupled tensile stiffness, represents the longitudinal tensile stiffness, represents the longitudinal shear stiffness. σ1 represents the longitudinal stress in the plane perpendicular to the helix tangent, σ2 represents the transverse stress in the plane perpendicular to the helix tangent, σ3 represents the stress along the helix tangent, τ 12 represents the shear stress in the plane parallel to the helix tangent, τ 13 represents the longitudinal shear stress in the plane perpendicular to the helix tangent, τ 23 represents the transverse shear stress in the plane perpendicular to the helix tangent, ε1 represents the longitudinal strain in the plane perpendicular to the helix tangent, ε2 represents the transverse strain in the plane perpendicular to the helix tangent, ε3 represents the strain along the helix tangent, ε 12 represents the shear strain in the plane parallel to the helix tangent, ε 13 represents the longitudinal shear strain in the plane perpendicular to the helix tangent, ε 23 represents the transverse shear strain in the plane perpendicular to the helix tangent.

9. The method according to claim 8, characterized in that, The constitutive relationship of the macroscopic homogenization model is expressed as: Among them, E3, ν 23 , G 23 , G 12 , κ 12 respectively represent the axial Young's modulus, axial Poisson's ratio, axial shear modulus, transverse shear modulus and transverse bulk modulus of the material, and v 12 represents the transverse Poisson's ratio.

10. A multi-scale numerical calculation system for superconducting sub-cables based on asymptotic homogenization, characterized in that, The system includes: A microscopic representative unit establishment module, configured to establish a microscopic representative unit considering the core wire inclusion and the matrix, and determine the microscopic equilibrium equation of the microscopic representative unit and its periodic boundary conditions; An equation solving module, configured to solve the microscopic equilibrium equation to obtain the equivalent elastic constants; A macroscopic strain acquisition module, configured to establish a macroscopic homogenization model based on the equivalent elastic constants to obtain the macroscopic strain; A microscopic stress distribution calculation module, configured to substitute the macroscopic strain and the characteristic displacement field function into the microscopic stress expression to obtain the microscopic stress distribution of the microscopic representative unit.

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