Distributed Inter-Strand Coupling Current Model for Finite Element Simulations of Rutherford Cables

TL;DR

This paper addresses the computational burden of simulating transient magnetic responses in Rutherford cables by introducing a DISCC-based homogenization that captures inter-strand coupling currents without modeling every strand. The approach couples a linear DISCC description with ROHM and ROHF strand models to reproduce hysteresis, eddy, IF, and ohmic losses, and offers two FE formulations ($h$-$$ and $h$-$$-$a$) for efficient implementation. The authors validate the linear DISCC model against a detailed reference, tune key scaling parameters, and demonstrate excellent agreement in field distributions and loss across frequency, field directions, and transport-current scenarios, with substantial speedups (often $>20 imes$). They further extend to nonlinear strand dynamics and verify the model in both uniform and nonuniform excitations, including a stack of cables via the $h$-$$-$a$ formulation, demonstrating robustness and practical applicability for magnet cross-section simulations. The work lays a foundation for fast, accurate electro-magneto-thermal analysis of complex superconducting magnets and points to future 3D extensions and temperature-dependent realizations.

Abstract

In this paper, we present the Distributed Inter-Strand Coupling Current (DISCC) model. It is a finite element (FE) model based on a homogenization approach enabling efficient and accurate simulation of the transient magnetic response of superconducting Rutherford cables without explicitly representing individual strands. The DISCC model reproduces the inter-strand coupling current dynamics via a novel mixed FE formulation, and can be combined with the Reduced Order Hysteretic Magnetization (ROHM) and Flux (ROHF) models applied at the strand level in order to reproduce the internal strand dynamics: hysteresis, eddy, and inter-filament coupling currents, as well as ohmic effects. We first analyze the performance of the DISCC model alone, as a linear problem. We then extend the analysis to include the internal strand dynamics that make the problem nonlinear. In all cases, the DISCC model offers a massive reduction of the computational time compared to conventional fully detailed FE models while still accounting for all types of loss, magnetization and inductance contributions. Rutherford cables homogenized with the DISCC model can be directly included in FE models of magnet cross-sections for efficient electro-magneto-thermal simulations of their transient response. We present two possible FE formulations for the implementation of the DISCC model, a first one based on the h-phi-formulation, and a second one based on the h-phi-a-formulation, which is well suited for an efficient treatment of the ferromagnetic regions in magnet cross-sections.

Figures (29)

• Figure 1: Sequence of models leading to a magnet cross-section with homogenized cables. (b-c-e-f) Illustrative solutions with external field and transport current excitation. Dashed arrows from (a) to (b) and from (d) to (e) represent the reduction of the 3D periodic structure of strands and cables to the 2D models with the CATI method as done in dular2024coupled and dular2024simulation, respectively. Solid orange arrows from (b) to (c) and from (e) to (f) represent homogenization steps, described in dular2024vector and this paper, respectively. Solid black arrows from (c) to (e) and from (f) to (g) illustrate the inclusion of a homogenized model into a larger-scale model.
• Figure 2: Structure of the paper. The linear models contain only the IS coupling current dynamics. The nonlinear models include the IS coupling current dynamics as well as the internal strand dynamics: hysteresis (hyst), eddy, IF, and ohmic (ohm) effects.
• Figure 3: Loss contributions in a Rutherford cable: adjacent IS coupling loss $p_{\text{IS,a}}$, crossing IS coupling loss $p_{\text{IS,c}}$, hysteresis loss $p_{\text{hyst}}$, eddy current loss $p_{\text{eddy}}$, IF coupling loss $p_{\text{IF}}$, and ohmic loss $p_{\text{ohm}}$. The line between the two layers represents a resistive strip.
• Figure 4: Reference cable model for $N_{\text{s}} = 10$ strands. Strands in (a) are represented without adjacent contacts for better visualization. View (b) is a cross-section of the cable, with FE mesh, along one of the green dash-dotted lines in (a). The electrical circuit in (c) defines connections between axial and transverse currents, with adjacent ($R_{\text{a}}$) and crossing ($R_{\text{c}}$) contact resistances, these connections are assumed to be concentrated in a cross-section of the cable along one of the orange dash-dotted lines in (a). Arrows show the convention for positive currents.
• Figure 5: Strand currents and IS coupling currents in the cable for a sinusoidal transverse magnetic field excitation of amplitude of $10$ mT in two directions: (top) along $\hat{\boldsymbol{e}}_{y}$ with frequency $10$ Hz, (bottom) along $\hat{\boldsymbol{e}}_{x}$ with frequency $1000$ Hz. Values are those at the time instant when the field is maximum. The results of the reference cable model (with $N_{\text{s}} = 36$) with the CATI method are shown. Two different scales are used in the upper plot for better visibility. The legend is the same for both plots. The strand currents and adjacent IS coupling currents are those of the top layer of strands in the cable, as illustrated in the cable schematics in the bottom right corner of the upper plot (the model describes IS coupling currents as flowing in a middle cross-section). Currents in the bottom layer can be deduced by symmetry, see also Fig. \ref{['disc_interpretation']}.