3-D thermal-electric finite element model of a Nb$_3$Sn coil during a quench

High field superconducting magnets for particle accelerators often exhibit premature quenches. Once a normal zone is generated within the conductor, the quench may propagate causing temperature and resistive voltage rise along the coil. The resulting thermal gradients can potentially cause new peak stresses that might exceed the tolerable limits, degrading the conductor. The computation of the strain state in the coils during quench then becomes of paramount importance for magnet design, and requires a complete three-dimensional (3-D) analysis of quench phenomena. The objective of this paper is to present the first multiphysics modeling activities towards a new full 3-D methodology for the analysis of magnet mechanics during quench. As a first step, a 3-D thermal-electric finite element model of a Nb$_3$Sn superconducting coil is developed and explained here. The model uses direct coupled-field elements to solve the system of thermal and electrical equations. A solving algorithm has also been implemented in order to investigate the physics behind quench transients. The output from this model, built in ANSYS APDL, can be easily coupled in a later stage to a mechanical model in order to estimate the strain state in the coil windings. A very good agreement has been observed between the numerical results and experimental tests performed in individual superconducting cables and real superconducting magnets.