Scalable modeling of 3D eddy current problem for magnetic fusion devices

Abstract

A novel three-dimensional time-domain eddy current solver, ERRAHI (Eddy cuRRent Analysis on Hierarchical Inductance), is presented. The solver is built upon a vector potential formulation derived from the electric field integral equation (EFIE), with degrees of freedom systematically identified using a spanning tree technique. Topological holes in the domain give rise to additional degrees of freedom associated with global cycles, and these global cycles are efficiently identified and optimized using a robust algorithm.

Designed for high-performance computing (HPC) environments, ERRAHI integrates the fast multipole method (FMM) with hierarchical matrix compression. In particular, matrix compression is performed via a randomized embedding scheme with FMM, which efficiently constructs low-rank blocks without assembling or storing the full matrix. This approach achieves empirical scaling of $O\left(N^{1.5}\right)$ and asymptotic scaling of $O\left(N^2\right)$ for total computation time. Leveraging FMM-based hierarchical compression and solving the resulting system with the generalized minimal residual method (GMRES), the code enables scalable analysis of complex tokamak CAD geometries.

To enhance locality and compressibility of the system matrix, global cycles are decomposed into local basis functions subject to additional constraints. This decomposition, combined with global cycle optimization, significantly improves the compressibility and structural coherence of the system matrix while maintaining accuracy. The solver has been validated against the TEAM7 benchmark, showing excellent agreement. Furthermore, large-scale simulations of the full KSTAR conductor model successfully reproduce the Rogowski coil and magnetic probe measurements from KSTAR vacuum experiments, demonstrating both the validity and applicability of the method in realistic tokamak scenarios.