Page 95 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 79

sult of the FE approximation. In real systems, only interfaces of different media
where the surface density of the EM forces is physically meaningful can be re-
garded as surfaces of discontinuity. If we point an outer normal out of medium
1 to medium 2, we obtain the following equation for the forces acting on a unit
area of the media interface [11]:

f = B 2 n H − B n 1 H − 0.5( BH 2 − BH ) ⋅ n. fS=(B2nH −B1nH )−0.5(B H −
)
(
2
1
1
2
1
S
1
2
2
2
B H )⋅n.
For any surface that is not a surface of discontinuity, B = B and, hence, it 1 1
2
1
should be f = 0.
S
On the other hand, the calculation of a volume force acting on any medium
domain is known to be reduced to the computation of stresses experienced by
the domain surface. In the case of an EM field, the stresses occurred at that point
are described by components of the Maxwell stress tensor [11]:
B 2
i
BB k
T ik = µµ − 2 0 δ , Tik=BiBkµ µ−B 2µ µδik,
ik
2
µµ
0
0
0
where δ is the Kronecker delta, and i, k ∈ 1,2,3.
ik
From these expressions, it follows that the volume and surface forces acting
on any domain of a medium are determined by a field at the boundary of that
domain. Consequently, the numerical computations of both can be done using
the same algorithmic approach.
The forces and moments acting on structural components can be calcu-
lated by reducing the system of volume forces f to a system of Maxwell’s
stresses [11]:
∫
∫ V fdV = S TdS, ∫VfdV=∇STndS,
n
where T is the stress tensor determined on surface S enveloping the volume V.
The computational procedure is based on the FE approximation, and the inte-
gration is done using FE volumes and facets.
∫
The magnetic flux Φ = B ds that comes through a closed surface S re- Φ=∇SBds
S
flects the solution accuracy, as Φ always equals zero under the requirement
∇·B = 0. Time changes in a magnetic flux coming through a given surface
determine the induced ‘vortex’ electromotive force along the surface contour.
−5
−3
In practice, tolerable relative error for these computations is 10 –10 . The
desired computational accuracy is achieved using a numerical integration
across the facets of FEs. To this end, the Gauss–Legendre quadrature [21–23]
is usually employed. For a hexahedron with trilinear functions, integration
can be performed analytically. A similar approach is applicable to other types
of FEs.
Let any arbitrary convex quadrangle be taken with vertices determined by
radii vectors r , r , r r and specified vectors B , B , B and B at each vertex.
3, 4
2
1
1
4
3
2