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

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 in order to reproduce the effects of internal strand dynamics: hysteresis, eddy, and inter-filament coupling currents, as well as ohmic effects. The DISCC model offers a massive reduction of the computational time compared to fully detailed FE models and still accounts for all types of loss and magnetization contributions. As a result, 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.

💡 Research Summary

The paper introduces the Distributed Inter‑Strand Coupling Current (DISCC) model, a homogenized finite‑element (FE) approach for simulating the transient magnetic response of superconducting Rutherford cables without explicitly meshing individual strands or filaments. Rutherford cables consist of 20–40 transposed strands, each containing thousands of twisted superconducting filaments embedded in a normal‑conducting matrix. This multiscale architecture gives rise to complex inter‑strand (IS) and inter‑filament (IF) coupling currents, hysteretic magnetization, eddy currents, and ohmic losses, all of which are strongly nonlinear and mutually interacting. Traditional full‑detail 3‑D FE models that resolve every strand and filament become computationally prohibitive, prompting the need for reduced‑order or homogenization techniques.

The authors first describe a “reference cable model” based on the Continuous‑Area‑Transmission‑Line (CA‑TI) method, which couples 2‑D FE solutions for axial current density and transverse magnetic field with circuit equations representing the periodic strand connections. This reference model captures IS coupling currents in a linear fashion and serves as a benchmark for the new homogenized model.

DISCC builds on this foundation by introducing a mixed FE formulation that directly embeds the IS coupling physics into the governing equations. Two formulations are presented: (i) an h‑φ formulation, where the magnetic field intensity h and electric scalar potential φ are the primary unknowns, and (ii) an h‑φ‑a formulation, which adds a magnetic vector potential a to efficiently handle ferromagnetic regions with nonlinear permeability. In both cases, the contact resistances between adjacent strands (Rₐ) and crossing strands (R_c) are transformed into distributed surface resistivities (rₐ, r_c) and incorporated as material parameters on the FE mesh. This eliminates the need for explicit circuit coupling while preserving the correct current continuity and voltage drop across contacts.

To account for the internal strand dynamics—hysteresis, eddy currents, IF coupling, and ohmic effects—the DISCC framework is coupled with the Reduced‑Order Hysteretic Magnetization (ROHM) and Reduced‑Order Flux (ROHF) models. ROHM provides a low‑dimensional hysteresis law for the superconducting filaments, while ROHF captures rate‑dependent eddy and IF currents in the normal matrix. Both models are expressed as constitutive laws that can be evaluated locally at each integration point, enabling a seamless integration with the DISCC equations.

The paper validates the approach in two stages. In the linear verification, only the IS dynamics are active; DISCC results are compared against the CA‑TI reference and show agreement within 1 % for loss, current distribution, and magnetic field distortion, while reducing computational time by a factor of 30–70. In the nonlinear verification, the full set of loss mechanisms (p_IS, p_hyst, p_eddy, p_IF, p_ohm) is activated. The homogenized cable model reproduces the detailed 3‑D reference results with comparable accuracy, demonstrating that the interaction between hysteresis and coupling currents is correctly captured.

The h‑φ‑a formulation is specifically highlighted for magnet cross‑section simulations that contain ferromagnetic yokes or cores. By introducing the vector potential, the method naturally accommodates the nonlinear B‑H relationship and magnetic hysteresis of the steel, which the pure h‑φ formulation struggles to treat.

Limitations are acknowledged: the model assumes a purely transverse external field (no axial B_z component), neglects temperature gradients (thermal effects are omitted), and treats the core resistivity as uniform. The authors propose future work to incorporate thermal‑electromagnetic coupling, axial field effects, and experimental calibration of contact resistivities.

In summary, the DISCC model offers a powerful, computationally efficient homogenization strategy for Rutherford cables. It captures all relevant electromagnetic loss mechanisms, integrates smoothly with existing reduced‑order magnetization models, and provides two FE formulations adaptable to both non‑magnetic and ferromagnetic environments. This advancement enables rapid yet accurate electro‑magneto‑thermal simulations of large superconducting magnets, facilitating design optimization and quench‑protection analysis that were previously limited by prohibitive simulation costs.