David V. Anderson

Methods for the efficient calculation of the magnetohydrodynamic (MHD) stability properties of magnetically confined fusion

A magnetohydrodynamic (MHD) model is applied to the problem of the stability of magnetically confined ther monuclear plasmas of interest in the pursuit of fusion power. Previous studies limited to two-dimensional con figurations are here generalized to three-dimensional toroidal plasmas. Using finite Fourier representations in the angle coordinates and finite hybrid elements in the radial direction, we solve the discretized Euler-Lagrange equations to determine the linear stability properties of the plasma.

Computation of E‐Layer and Plasma Equilibria in Astron

The equilibrium magnetic field for astron configurations is computed; a given vacuum magnetic field and the fields due to the E layer and plasma are included, assuming axial symmetry and Bθ≡0. The components Br(r, z) and Bz(r, z) are derived from the stream function ψ(r, z) = rAθ. The E‐layer is described by a distribution function that depends on the canonical angular momentum and the energy. Various models of the E layer have been used and are discussed. The function ψ(r, z) satisfies a nonlinear partial differential equation (Amperes law) that is solved by finite‐difference methods, using an iterative procedure. The object is to find equilibrium solutions with a reversed field in the central region that can contain a plasma. Examples of reversed field equilibria corresponding to various E layer distributions are presented. Simple models for the plasma are assumed, and equilibria of the coupled system of E layer plus plasma are computed.

Fully three‐dimensional ideal magnetohydrodynamic stability analysis of low‐n modes and Mercier modes in stellarators

The terpsichore three‐dimensional linear ideal magnetohydrodynamic (MHD) stability code [Theory of Fusion Plasmas, Proceedings of the Joint Varenna–Lausanne International Workshop, Chexbres, Switzerland, 1988 (Editrice Compositori, Bologna, Italy, 1989), p. 93; Controlled Fusion and Plasma Heating, Proceedings of the 17th European Conference, Amsterdam, The Netherlands (European Physical Society, Petit‐Lancy, Switzerland, 1990), Vol. 14B, Part II, p. 931; Theory of Fusion Plasmas, Proceedings of the Joint Varenna–Lausanne International Workshop, Valla Monastero, Varenna, Italy, 1990 (Editrice Compositori, Bologna, Italy, 1990), p. 655] has been extended to the full MHD equations. The new code is used to calculate the physical growth rates of nonlocal low‐n modes for l=2 torsatron configurations. A comprehensive investigation of the relation between the Mercier modes and the low‐n modes has been performed. The unstable localized low‐n modes are found to be correlated with the Mercier criterion. Finite grow...

Hybrid Ordered Particle Simulation (HOPS) code for plasma modelling on vector-serial, vector-parallel, and massively parallel computers

Abstract The Hybrid Ordered Particle Simulation (HOPS) code has been developed to provide a more efficient method for carrying out particle-in-cell (PIC) simulations of plasma phenomena. Conventional PIC methods store the particle attributes in tables that tend to have a random spatial order, which was appropriate for older serial-scalar computers. Problems associated with excessive accessing of memory, indirect indexing, and with many-to-one mappings in the deposition phase can make these codes inefficient on vector-serial, vector-parallel, and massively parallel machines. In the HOPS code we employ a sorting scheme to keep the particles ordered with respect to their spatial positions. By doing so, we have reduced memory accesses, recovered substantial direct indexing, and most importantly removed the many-to-one mapping problem. A low overhead sorting and reordering scheme is presented which allows HOPS to be most efficient on vector-serial machines and which scales linearly to various kinds of parallel computers. This paper focuses on the Cray C-90 vector-parallel computer but also discusses aspects of a massively parallel implementation.

Accurate calculations of field-reversed axisymmetric equilibria and their MHD stability properties☆

Abstract Finite elements with bi-cubic, B-spline basis functions are used to represent the flux functions of an axisymmetric, field-reversed plasma. Equilibria, in this representation, are obtained by the collocation method and a generalized ICCG algorithm. For the infinite, toroidal-mode number limit of the MHD energy principle, coupled Sturm-Liouville-like equations result on each flux line. A secant-bisection shooting method employing the gridless Gear integrator is used to find very accurate eigenfunctions, eigenvalues, and growth rates from the continuous-spline representation of the fields. Precise agreement with some exact analytic work is obtained.

Parallel computing and multitasking

Abstract Over the past decade we have witnessed an evolution of scientific computers in which more and more concurrent or parallel arithmetic operations are allowed. The segmented pipeline arithmetic functional units, direct vectorization, indirect vectorization, multiprocessing and finally multitasking represent stages of development of parallel computation. Algorithms for the solution of physics problems must be tailored, if possible, to the forms required for these various kinds of parallelism. We report on some experiences we have had building and running various parallelized physics codes with particular emphasis on the Cray-2. We show that the implementation of multitasking and the subsequent debugging effort are straightforward. These techniques are applicable to more methods, including implicit ones, than was originally predicted. We present arguments that favor the use of interactive timesharing operating systems, particularly for the multitasking situation.

Calculations of 3D mirror equilibria using a finite element vector potential representation

Abstract Several applications of plasma equilibrium models require a suitable continuous representation for the magnetic field. For example, numerical generation of single particle orbits in these fields requires that the continuum representation maintain ▿ · B = 0 or at least keep it very small. Performing the interpolation analysis in terms of the vector potential A guarantees ▿ · B = 0, whereas using the magnetic induction B does not. The code presented here uses A as the primary dependent variable. A finite element representation employing tricubic splines significantly reduces spatial truncation errors compared to conventional finite difference methods. The theoretical equilibrium model allows pressure functions of the form P ( B , ψ = P( B ) ω(ψ). A modified long-thin approximation is derived which includes field line curvature effects; it agrees well with the results obtained from the code. Some results pertinent to the MFTF- B experiment are presented.

ILUBCG2–11: Solution of 11-banded nonsymmetric linear equation systems by a preconditioned biconjugate gradient routine

Abstract The biconjugate gradient method (BCG) provides an attractive alternative to the usual conjugate gradient algorithms for the solution of sparse systems of linear equations with nonsymmetric and indefinite matrix operators. A preconditioned algorithm is given, whose form resembles the incomplete L-U conjugate gradient scheme (ILUCG2) previously presented. Although the BCG scheme requires the storage of two additional vectors, it converges in a significantly lesser number of iterations (often half), while the number of calculations per iteration remains essentially the same.

Three-Dimensional Ideal Magnetohydrodynamic Stability on Parallel Machines

On the path towards a thermonuclear fusion reactor there are several technological and physical uncertainties to be understood and solved. One of the most fundamental problems is the appearance of many sorts of instabilities which can either enhance the energy outflow or even destroy the magnetic confinement of the fusion plasma. The knowledge of such instabilities is a prerequisite to a good understanding of the behaviour of actual experiments, and to the design of new devices. Most of the effort is devoted to the study of axisymmetric toroidal configurations such as tokamaks or spheromaks and to helically twisted toroidal devices such as stellarators.

Ideal magnetohydrodynamic stability computations for three-dimensional magnetic fusion devices

Reference CRPP-ARTICLE-1991-020doi:10.1016/0045-7825(91)90067-GView record in Web of Science Record created on 2008-04-16, modified on 2017-05-12