Homogenization Methods for Transient Simulations of Superconductors

To design advanced superconducting magnets and gain deeper insights into the physics of the superconducting phase transition, it is essential to simulate their transient electromagnetic behavior. However, this presents several challenges: (i) the governing equations are highly nonlinear due to the intrinsic properties of super-conducting materials, (ii) the geometry involves fine superconducting regions interwoven with comparatively large normal-conducting areas, leading to a multi-scale problem, and (iii) accurately capturing the rapid electromagnetic dynamics requires small time steps. Consequently, full-scale simulations of superconducting magnets using conventional modeling techniques remain computationally challenging. A well-established strategy in Finite Element (FE) simulations for handling multi-scale problems is homogenization. Classical homogenization techniques for non-superconducting materials rely on capturing material behavior within a small reference cell, which is then represented in a larger-scale model with a coarser geometric resolution, significantly improving computational efficiency. However, the strong nonlinearities and time-dependent behavior of superconductors prevent direct application of these conventional techniques to superconducting magnets.The proposed homogenization framework for superconducting magnets follows a two-stage approach. First, the macroscopic behavior of a composite strand, treated as the highest-resolution element, is abstracted into an effective cable representation. In the second stage, this homogenized cable model is incorporated into a full magnet-scale simulation. This two-stage homogenization must accurately reproduce the magnetization and inductive properties of the strands and cables to account for power losses without requiring full-field computations.This thesis presents a solution for homogenizing the macroscopic inductance effects of composite strands and cables by modeling their internal flux response to transport current variations. This is achieved through a novel reduced-order hysteretic flux model, which can be calibrated against full-scale reference simulations to approximate the associated voltage as well as power losses of a reference cell under arbitrary transport current transients.The second part of this work addresses the issue of parameter identification for the proposed homogenization model and introduces an algorithmic strategy. Both the homogenization approach and its fitting procedure are validated using a twisted multifilament Niobium-Titanium (Nb-Ti) strand as a case study. This represents the first stage of the homogenization process within a Low Temperature Superconductor (LTS) wound magnet, such as those used in the Large Hadron Collider (LHC). However, this homogenization approach is not purely limited to LTS. With further development, it has the potential to be successfully applied to multi-scale High Temperature Superconductor (HTS) magnet designs, contributing to the advancement of next-generation collider projects.