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Surrogate Approximation of the Grad-Shafranov Free Boundary Problem via Stochastic Collocation on Sparse Grids
TLDR
In this article, 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.Abstract:
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.read more
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The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique
O. C. Zienkiewicz,J. Z. Zhu +1 more
TL;DR: In this article, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes, which has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems.
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High-Order Collocation Methods for Differential Equations with Random Inputs
Dongbin Xiu,Jan S. Hesthaven +1 more
TL;DR: A high-order stochastic collocation approach is proposed, which takes advantage of an assumption of smoothness of the solution in random space to achieve fast convergence and requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods.
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The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity
O. C. Zienkiewicz,J. Z. Zhu +1 more
TL;DR: In this paper, the authors derived a theorem showing the dependence of the effectivity index for the Zienkiewicz-Zhu error estimator on the convergence rate of the recovered solution.
Journal ArticleDOI
High dimensional polynomial interpolation on sparse grids
TL;DR: The error bounds show that the polynomial interpolation on a d-dimensional cube, where d is large, is universal, i.e., almost optimal for many different function spaces.
Journal ArticleDOI
Note on a Method for Calculating Corrected Sums of Squares and Products
TL;DR: In this paper, a method for calculating corrected sum of squares and products is presented. But this method is not suitable for counting the number of items in a set. And it is computationally difficult.