Transport and dynamcis in toroidal fusion systems. Final report, 1992--1995 Page: 4 of 51
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SAIC-95/1323:APPAT-170
space to maximize the accuracy of the calculation. Thus placed, the points are
connected with line elements that form the edges of triangles. These triangles are
the Eulerian control volumes that form the basis for the finite representation of the
appropriate fluid equations. In the logical data structure that describes the mesh,
mesh points (and associated triangles) can easily be added or deleted dynamically
based on pre-defined accuracy criteria. The spatial representation can thus adapt to
evolving spatial structures without the mesh distortion problems associated with
Lagrangian formulations.
Techniques based on unstructured, adaptive meshes have come to maturity
in computational fluid dynamics (CFD), where quantitative predictions in real
geometry have become essential in the design of aircraft and gas turbine engines6.
These methods are generally based the solution of a Riemann problem at each
triangle interface (edge) to determine the fluxes of energy, mass, and momentum7.
The simplest extension of the hydrodynamic model that is appropriate for the
description of magnetic fusion plasmas is magnetohydrodynamics (MHD).
In this document we describe an extension of these spatial gridding
techniques to an MHD model suitable for the description of the dynamics of toroidal
fusion devices. Since the dominant MHD modes in these devices have relatively
long toroidal wavelength, the toroidal coordinate is approximated with finite
Fourier series. The unstructured, triangular mesh is used to describe the details of
the poloidal geometry. The hydrodynamic variables are treated in a manner
analogous to that used in CFD. These quantities (mass, energy, and momentum) are
volume based densities that satisfy scalar or vector conservation laws. The
electromagnetic variables (the magnetic flux density B and the electric current
density J) are area based densities that satisfy pseudo-vector conservation laws, and
have no counterpart in fluid dynamics. These variables are also constrained to
remain solenoidal. These quantities are represented on the triangular mesh in a
new manner that is an extension of that used on rectangular, structured meshes.
In this work we have chosen to solve the primitive (instead of reduced) MHD
equations in order to make the resulting codes and techniques more generally
applicable to problems beyond the narrow scope of tokamak plasmas. The temporal
stiffness problems inherent in this description of tokamak dynamics that motivate
the reduced MHD model are addressed here with the semi-implicit method of time
integration8. Finally, we remark that, while the present work deals strictly with the
MHD equations, other volume based fluid descriptions, such as diffusive transport,
could easily be adapted to these techniques and coupled with the description of the
electromagnetic field presented here.
This document is organized as follows. In Section 2 we discuss the properties
of structured and unstructured meshes, and the data structures useful for describing
them. Issues related to the triangulation of an arbitrary set of points in a plane are
also discussed. In Section 3 we derive a finite volume approximation to the
resistive MHD equations suitable for use on an unstructured, triangular mesh in
toroidal geometry. Boundary conditions are discussed here. The specific MHD
model, and its implementation on the unstructured mesh, is discussed in Section 4.
In Section 5 we discuss methods of time integration, and describe our
implementation of semi-implicit and fully implicit algorithms. Examples of the
2
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Schnack, D. D. Transport and dynamcis in toroidal fusion systems. Final report, 1992--1995, report, September 15, 1995; San Diego, California. (https://digital.library.unt.edu/ark:/67531/metadc667919/m1/4/: accessed March 28, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.