Reduced Order Modeling for Real-Time Monitoring of Structural Displacements due to Electromagnetic Forces in Large Scale Tokamaks

TL;DR

This work addresses real-time estimation of VV deformations under electromagnetic loads during vertical displacement events in tokamaks by marrying Volume Integral Equation (VIE) electromagnetics with FEM-based structural analysis through Parametric Model Order Reduction (MOR). It constructs coil-specific EM-ROMs via Proper Orthogonal Decomposition (POD) and a single Structural-ROM that aggregates nonlinear EM loads into a reduced representation, enabling fast time-domain simulations. The coupling exploits precomputed maps to translate EM states into body forces, with optional DEIM to further reduce full-space projections. Validation on axisymmetric D-shaped and full 3D ITER-like VVs shows substantial compression using Hierarchical Matrices ($ abla$-matrix/H-Matrix) and real-time performance with high accuracy, supporting a viable Digital Twin for active monitoring and control of fusion devices.

Abstract

The real-time monitoring of the structural displacement of the Vacuum Vessel (VV) of thermonuclear fusion devices caused by electromagnetic (EM) loads is of great interest. In this paper, Model Order Reduction (MOR) is applied to the Integral Equation Methods (IEM) and the Finite Elements Method (FEM) to develop Electromagnetic and Structural Reduced Order Models (ROMs) compatible with real-time execution which allows for the real-time monitoring of strain and displacement in critical positions of Tokamaks machines. Low-rank compression techniques based on hierarchical matrices are applied to reduce the computational cost during the offline stage when the ROMs are constructed. Numerical results show the accuracy of the approach and demonstrate the compatibility with real-time execution in standard hardware.

Figures (6)

• Figure 1: Steps for construction of $\mathcal{H}$-matrix. (a) Reordering of DoFs in preprocessing. (b) Construction of block cluster tree of the product set $I\times J$ for the identification of low-rank (admissible) blocks.
• Figure 2: (a) Cross section of axisymmetric VV. Set of CS coils, and PF coils. The position of the plasma centroid at $t=0$ is marked with the red cross, while that at $t=0.63$ s with the red circle. The trajectory is also drawn. The equivalent current loops mimicking the plasma centroid movement are labeled with circular marks from 1 to 12. (b) The trend of plasma current $I_p(t)$ and that of equivalent current loops.
• Figure 3: (a) Example of D-shaped VV sector support module. (b) Fixed displacement area.
• Figure 4: Trend of total force over the VV during the VDE evaluated with the 2D axisymmetric model in COMSOL®, the FOM, and the ROM.
• Figure 5: (a) Norm of the displacement field at $t=0.63$ s evaluated with the 2D axisymmetric model in COMSOL®. (b) The magnitude of the displacement field at $t=0.63$ s evaluated with the ROM. Graphically the displacement is amplified by a factor $\alpha=300$.