## 3 Theoretical Basis for Magnetic Coupled Field Simulation  
### Magnet-Averaged Finite Element Model  
The adoption of a gradient magnet with a small superconducting wire diameter and a higher number of turns and layers result in a significant increase in the number of meshes and node density when using a detailed finite element model. This can lead to reduced convergence in model calculations. Therefore, an average finite element model [17; 18; 19] is introduced to treat the magnet equivalently. This approach considers different windings of superconducting wire or windings of the same type of superconducting wire with different duty cycles as independent domains for the solution. It is worth noting that, after vacuum impregnation or mixing with epoxy resin, the mechanical properties of the coil will no longer exhibit isotropy, and it is necessary to analyze its anisotropic parameters. According to this theory, each coil winding comprises a continuous anisotropic material that is equivalent to the winding. Consequently, the equivalent physical properties of the material must be determined using the composite law. When employing the average finite element model for simulation calculations, the impact of the original material's discontinuity on the results is not considered.  
As depicted in Figure 6, the average finite element model of the magnet utilizes the parity layer winding method for its winding. The main feature of this coil winding method is that the odd-numbered layers in the winding have one more turn than the even-numbered layers, and the wires in adjacent layers alternate with each other in the central axis direction, forming a regular hexagonal structure with a center angle of 60\({}^{\circ}\). The smallest unit of the winding is a positive triangle, with superconducting wires constituting 90% and epoxy resin constituting 10%. Based on the above conclusion, the material of the superconducting coil is regarded as a composite material composed of epoxy resin and superconducting wire, which is equivalent to epoxy resin as the substrate, superconducting wire as fibers, with a fiber ratio of 0.9. The fibers are evenly distributed along the direction of the wire in the substrate.  
For a circular superconducting wire with a rectangular cross-section, the equivalent Young's modulus \(E_{2}\), equivalent shear modulus \(G_{12}\), and equivalent Poisson's ratio \(v_{12}\) are determined by the following empirical equations as follows [20]:  
\[\frac{E_{2}}{E_{m}}=\frac{1+2\eta V_{f}}{1-\eta V_{f}} \tag{1}\]  
\[\eta=\frac{\left(E_{f}/E_{m}\right)-1}{\left(E_{f}/E_{m}\right)+2} \tag{2}\]  
Figure 6: Finite element model of average magnet properties.  
\[\frac{G_{12}}{G_{m}}=\frac{1+\eta V_{f}}{1-\eta V_{f}} \tag{3}\]  
\[\eta=\frac{\left(G_{f}/G_{m}\right)-1}{\left(G_{f}/G_{m}\right)+1} \tag{4}\]  
\[v_{12}=v_{f}V_{f}+v_{m}V_{m} \tag{5}\]  
The remaining variables in the formula are as follows: [PERSON]'s modulus of epoxy resin--\(E_{m}\); [PERSON]'s modulus of superconductor--\(E_{f}\) shear modulus of epoxy resin--\(G_{m}\); shear modulus of superconductor--\(G_{f}\); [PERSON]'s ratio of epoxy resin--\(v_{m}\); [PERSON]'s ratio of superconductor--\(\eta\); volume fraction of epoxy resin--\(V_{m}\); volume fraction of superconductor--\(V_{f}\).  
The anisotropic physical property parameters of the equivalent material can be obtained, as shown in Table 6.