Multiconstrained variational problems in magnetohydrodynamics: Equilibrium and slow evolution

Turkington, Bruce; Lifschitz, Alexander; Eydeland, Alexander; Spruck, Joel

doi:10.1016/s0021-9991(83)71107-1

Cited by 10 publications

(3 citation statements)

References 17 publications

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“…These schemes generally fall into two categories. Eulerian or "direct" solvers use a prescribed mesh to calculate the unknown function [13,17,19,21,29], while Lagrangian or "inverse" solvers find the mapping of the plasma geometry in terms of magnetic coordinates [7,16,25,27,42]. The advantages and disadvantages of one formulation as compared to the other depend on the application of interest [43], on the plasma geometry, and on the type of inputs that plasma stability and transport codes require.…”

Section: Introductionmentioning

confidence: 99%

A fast, high-order solver for the Grad–Shafranov equation

Pataki

Cerfon

Freidberg

et al. 2013

Journal of Computational Physics

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We present a new fast solver to calculate fixed-boundary plasma equilibria in toroidally axisymmetric geometries. By combining conformal mapping with Fourier and integral equation methods on the unit disk, we show that high-order accuracy can be achieved for the solution of the equilibrium equation and its first and second derivatives. Smooth arbitrary plasma cross-sections as well as arbitrary pressure and poloidal current profiles are used as initial data for the solver. Equilibria with large Shafranov shifts can be computed without difficulty. Spectral convergence is demonstrated by comparing the numerical solution with a known exact analytic solution. A fusion-relevant example of an equilibrium with a pressure pedestal is also presented.

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Section: Introductionmentioning

confidence: 99%

A fast, high-order solver for the Grad–Shafranov equation

Pataki

Cerfon

Freidberg

et al. 2013

Journal of Computational Physics

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“…There are basically two different families of solution methods for axisymmetric plasma equilibrium problems. The first family are the so-called flux or Lagrangian coordinate methods, determining the localization of level lines that have equidistant flux-values (Lao et al 1985(Lao et al , 1981Ling & Jardin 1985;Turkington et al 1993;Gruber et al 1987;Degtyarev & Drozdov 1991;DeLucia et al 1980;Jardin et al 1986;DeLucia et al 1980;Degtyarev & Drozdov 1985) (see also (Jardin 2010, section 5.5)). A second family of methods uses standard finite difference methods on rectangular grids (Feneberg & Lackner 1973;Helton & Wang 1978;Johnson et al 1979;Lackner 1976) or finite element methods on triangular grids (Blum et al 1981).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quasi-static free-boundary equilibrium of toroidal plasma with CEDRES++: Computational methods and applications

et al. 2015

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We present a comprehensive survey of the various computational methods in CEDRES++ for finding equilibria of toroidal plasma. Our focus is on free-boundary plasma equilibria, where either poloidal field coil currents or the temporal evolution of voltages in poloidal field circuit systems are given data. Centered around a piecewise linear finite element representation of the poloidal flux map, our approach allows in large parts the use of established numerical schemes. The coupling of a finite element method and a boundary element method gives consistent numerical solutions for equilibrium problems in unbounded domains. We formulate a new Newton method for the discretized nonlinear problem to tackle the various non-linearities, including the free plasma boundary. The Newton method guarantees fast convergence and is the main building block for the inverse equilibrium problems that we can handle in CEDRES++ as well. The inverse problems aim at finding either poloidal field coil currents that ensure a desired shape and position of the plasma or at finding the evolution of the voltages in the poloidal field circuit systems that ensure a prescribed evolution of the plasma shape and position. We provide equilibrium simulations for the tokamaks ITER and WEST to illustrate the performance of CEDRES++ and its application areas.Here again, this is a straightforward and simple calculation for all mappings except one: the mapping J P,h that is related to the non-linear current profile in the plasma domain. The mapping J P,h is given by

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A gradient bound for the Grad-Kruskal-Kulsrud functional

Laurence

Stredulinsky

1996

Comm. Pure Appl. Math.

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We establish the existence of minimizers with a bounded gradient for a variational problem arising in the Grad‐Kruskal‐Kulsrud model for the equilibrium of a confined plasma. The variational problem involves derivatives of the nondecreasing rearrangement of minimizers. Results are derived for bounded convex domains in R 2. Our results answer in the negative (in the case of convex domains) a question raised by Grad concerning the possibility of singular behavior of the magnetic field at the point of maximum flux. The main approach is to use an approximating free boundary problem to handle the nonlinear nonlocal nature of the variational functional. Limiting minimizers of the variational problem are shown to have bounded gradient and to satisfy a weak equation that for one model problem takes the form $$\Delta\psi=-\psi^{*\,\prime \prime }(\mu_\psi(\psi)),$$ where ψ*, μψ are, respectively, the nondecreasing rearrangement and the distribution function of ψ. © 1996 John Wiley & Sons, Inc.

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Multiconstrained variational problems in magnetohydrodynamics: Equilibrium and slow evolution

Cited by 10 publications

References 17 publications

A fast, high-order solver for the Grad–Shafranov equation

A fast, high-order solver for the Grad–Shafranov equation

Quasi-static free-boundary equilibrium of toroidal plasma with CEDRES++: Computational methods and applications

A gradient bound for the Grad-Kruskal-Kulsrud functional

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