Algorithms and optimizations for global non-linear hybrid fluid-kinetic finite element stellarator simulations

TL;DR

The paper tackles the challenge of simulating stellarator plasmas with hybrid fluid-kinetic models in a non-axisymmetric geometry. It develops a globally coupled particle-in-cell projection scheme inside the JOREK finite element framework, solving a single linear system that couples all toroidal harmonics, and accelerates matrix assembly with FFT while localizing particles via a 3D R-Tree. Quantitative convergence tests on realistic Wendelstein 7-X geometries demonstrate spectral convergence for the coupled scheme, in contrast to poor performance of uncoupled approaches. The resulting validated tool enables high-fidelity predictive analysis and optimization of next-generation stellarator designs, facilitating HPC-enabled exploration of kinetic effects on MHD dynamics.

Abstract

Predictive modeling of stellarator plasmas is crucial for advancing nuclear fusion energy, yet it faces unique computational difficulties. One of the main challenges is accurately simulating the dynamics of specific particle species that are not well captured by fluid models, which necessitates the use of hybrid fluid-kinetic models. The non-axisymmetric geometry of stellarators fundamentally couples the toroidal Fourier modes, in contrast to what happens in tokamaks, requiring different numerical and computational treatment. This work presents a novel, globally coupled projection scheme inside the JOREK finite element framework. The approach ensures a self-consistent and physically accurate transfer of kinetic markers to the fluid grid, effectively handling the complex 3D mesh by constructing and solving a unified linear system that encompasses all toroidal harmonics simultaneously. To manage the computational complexity of this coupling, the construction of the system's matrix is significantly accelerated using the Fast Fourier Transform (FFT). The efficient localization of millions of particles is made possible by implementing a 3D R-Tree spatial index, which supports this projection and ensures computational tractability at scale. On realistic Wendelstein 7-X stellarator geometries, the fidelity of the framework is rigorously shown. In sharp contrast to the uncoupled approaches' poor performance, quantitative convergence tests verify that the coupled scheme attains the theoretically anticipated spectral convergence. This study offers a crucial capability for the predictive analysis and optimization of next-generation stellarator designs by developing a validated, high-fidelity computational tool.

Figures (17)

• Figure 1: Feedback loop between electromagnetic fields and plasma in kinetic-fluid simulations. The Lorentz force acts on charged particles, influencing their motion and altering currents and charge densities, which in turn affect the electromagnetic fields Introduction_to_stellarators.
• Figure 2: A computer graphic representation of the two different types of reactors tokamak_vs_stellarator. On the left, the tokamak; on the right, the stellarator.
• Figure 3: A single 2D isoparametric Bézier element in an axisymmetric R-Z space. The geometry is parameterized by a mapping from the local s-t parent domain using Bernstein polynomial basis functions, which are also used to interpolate field variables like temperature.
• Figure 4: Comparison of particle trajectories between stellarator and tokamak models in tokamak configuration.
• Figure 5: Particle Tracing in a stellarator case using the stellarator model