Surrogate approximation of the Grad–Shafranov free boundary problem via stochastic collocation on sparse grids

• Published in: Journal of computational physics Vol. 448; p. 110699
• Main Authors: Elman, Howard C., Liang, Jiaxing, Sánchez-Vizuet, Tonatiuh
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.01.2022; Elsevier Science Ltd
• Subjects: Coils; Computational physics; Confinement; Elongation; Free boundaries; Free boundary Grad-Shafranov equation; Hydrostatic pressure; Magnetic permeability; Parameters; Partial differential equations; Physical properties; Plasma; Plasma equilibrium; Sparse grid; Stochastic collocation; Uncertainty; Uncertainty quantification; Plasma equilibrium; Uncertainty quantification; Stochastic collocation; Sparse grid; Free boundary Grad-Shafranov equation
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2021.110699

•Surrogate model of magnetic confinement equilibrium with stochastic parameters.•The surrogate function bypasses the solution of the free boundary problem.•Sparse grids in parameter space reduce the cost of tensor product grids.•Reduction of the time for Monte Carlo simulations by factors between 7 and 30. In magnetic confinement fusion devices, the equilibrium configuration of a plasma is determined by the balance between the hydrostatic pressure in the fluid and the magnetic forces generated by an array of external coils and the plasma itself. The location of the plasma is not known a priori and must be obtained as the solution to a free boundary problem. The partial differential equation that determines the behavior of the combined magnetic field depends on a set of physical parameters (location of the coils, intensity of the electric currents going through them, magnetic permeability, etc.) that are subject to uncertainty and variability. The confinement region is in turn a function of these stochastic parameters as well. In this work, we consider variations on the current intensities running through the external coils as the dominant source of uncertainty. This leads to a parameter space of dimension equal to the number of coils in the reactor. With the aid of a surrogate function built on a sparse grid in parameter space, a Monte Carlo strategy is used to explore the effect that stochasticity in the parameters has on important features of the plasma boundary such as the location of the x-point, the strike points, and shaping attributes such as triangularity and elongation. The use of the surrogate function reduces the time required for the Monte Carlo simulations by factors that range between 7 and over 30.