TY - JOUR

T1 - Surrogate approximation of the Grad–Shafranov free boundary problem via stochastic collocation on sparse grids

AU - Elman, Howard C.

AU - Liang, Jiaxing

AU - Sánchez-Vizuet, Tonatiuh

N1 - Funding Information: Howard Elman has been partially supported by the U.S. Department of Energy under grant DE-SC0018149 and by the National Science Foundation under grant DMS1819115 . Jiaxing Liang has been partially funded by the U.S. Department of Energy under grant DE-SC0018149 . Tonatiuh Sánchez-Vizuet has been partially funded by the U.S. Department of Energy under Grant DE-FG02-86ER53233 . Funding Information: All the free boundary computations were carried over using the code FEEQS.M [17,18]. The authors are deeply grateful to Holger Heumann, the INRIA CASTOR team, and all the development team of the CEDRES++ free boundary solver for sharing access to the code and for helping us get up to speed with its usage. The authors would also want to thank Antoine Cerfon (NYU-Courant) for his valuable insights. Howard Elman has been partially supported by the U.S. Department of Energy under grant DE-SC0018149 and by the National Science Foundation under grant DMS1819115. Jiaxing Liang has been partially funded by the U.S. Department of Energy under grant DE-SC0018149. Tonatiuh S?nchez-Vizuet has been partially funded by the U.S. Department of Energy under Grant DE-FG02-86ER53233. Publisher Copyright: © 2021 Elsevier Inc.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - 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.

AB - 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.

KW - Free boundary Grad-Shafranov equation

KW - Plasma equilibrium

KW - Sparse grid

KW - Stochastic collocation

KW - Uncertainty quantification

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U2 - https://doi.org/10.1016/j.jcp.2021.110699

DO - https://doi.org/10.1016/j.jcp.2021.110699

M3 - Article

SN - 0021-9991

VL - 448

JO - Journal of Computational Physics

JF - Journal of Computational Physics

M1 - 110699

ER -