Algorithms and optimizations for global non-linear hybrid fluid-kinetic finite element stellarator simulations
📝 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.
💡 Analysis
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.
📄 Content
Nowadays, nuclear fusion is one of the most important scientific and engineering problems of our time due to the quest for clean, sustainable energy. Fusion reactors promise an almost infinite energy source with major safety and environmental advantages over traditional fission and fossil fuel technologies by exploiting a similar process that powers stars [2]. The confinement of a high-temperature plasma in a magnetic field is the most promising method for accomplishing controlled fusion on Earth. However, high-fidelity computational simulations are required due to the enormous expense and complexity of building and running experimental reactors [40]. These simulations are essential resources for developing new approaches, deciphering experimental data, and investigating plasma phenomena that are challenging or impossible to measure directly. Fluid and kinetic models are the two main categories into which plasma simulation paradigms fall. Largescale instabilities and general plasma behavior are effectively captured by fluid models, which are governed by Magneto-HydroDynamics (MHD) equations and treat the plasma as a continuum [24]. Kinetic models, on the other hand, follow the paths of individual particles and offer a more accurate, albeit more computationally expensive, representation of the plasma [25]. The development of hybrid fluid-kinetic approaches has been prompted by the prohibitive computational cost of fully kinetic simulations for reactor-scale plasmas over MHDrelevant timescales. These models create a self-consistent framework by combining the accuracy of a kinetic treatment for certain, important particle populations with the computational efficiency of a fluid description for the bulk plasma [35]. Numerous specialized numerical codes have been created to handle the various physical regimes found in magnetically confined plasmas. Usually, each of these tools is tailored for a particular set of scales, equations, or configurations. In particular, for hybrid MHD-kinetic simulations many codes have been developed, for instance: JOREK [19], NIMROD [39], M3D-C1-K [13,28], MEGA [41] and XHMGC [43] are some of the leading codes in the field, designed to couple a fluid description of the bulk plasma with a kinetic treatment for specific particle species. Among them, specific interest has sparked in the simulation of stellarator reactors, leading to the development of specific extensions to account for the three-dimensionality of the plasma [7,33,47], underscoring the interest and need of the scientific community for this kind of reactor. The tokamak and the stellarator are the two main device concepts that have emerged in the field of magnetic confinement fusion. Both configurations are actively researched as promising concepts for a fusion reactor with different strengths and weaknesses. The tokamak’s intrinsic toroidal symmetry makes its physical model and numerical implementation easier, but the stellarator’s non-axisymmetric, three-dimensional magnetic field poses a much greater computational challenge. The need for sophisticated simulation tools that can faithfully capture the unique physics of stellarator designs is growing in importance, triggered in particular by the successful operation of the W7-X. As a result, although there are established hybrid simulation frameworks for tokamaks in codes like JOREK [19], there is still a vital and ongoing research need to extend them to stellarator geometries. The breakdown of toroidal axisymmetry is the main obstacle to applying the available hybrid models to stellarators. This symmetry permits a spectral decomposition of the mesh geometry in the toroidal direction, in which Fourier harmonics are decoupled, in tokamak simulations. The numerical problem is greatly simplified by this decoupling since each harmonic’s kinetic particle data can be independently projected onto the fluid grid. These harmonics are intrinsically coupled in a stellarator due to the three-dimensional nature of the geometry. The basis functions utilized for the finite element discretization are functions of all three spatial dimensions since the physical form of the poloidal cross-section varies with the toroidal angle. The uncoupled projection scheme would produce nonphysical results if applied to stellarators and would fail to accurately represent the physical plasma. By creating, putting into practice, and validating the algorithms necessary for a global non-linear hybrid fluidkinetic simulation capability within the JOREK stellarator model, this thesis directly tackles this challenge. The design of a novel particle-in-cell projection scheme that completely takes into account the coupling between toroidal harmonics that results from the non-axisymmetric mesh geometry is the main contribution. This approach allows for a precise and self-consistent transfer of information from the kinetic particles to the fluid grid by accurately formulating and solving the global linear system fo
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