Crank J. Free and Moving Boundary Problems
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392
Numerical solution
of
free-boundary problems
On
a smooth
part
of
CCN
U
CAN
the
exterior angle
(3
between the
tangents
is
7r
so V = -!.
Hansen (1980,1983) introduced local and analytic solutions
at
points
on the boundary
C where intervals with different boundary conditions
meet, in order to handle the singularities there.
At
such points and
at
points corresponding to edges, y
is
singular. Local expansions
of
the type
'"
</>
= L a,.p(n-!)('1T/I3) sin(n -!)7r8/(3,
(8.124)
n=1
are
commonly incorporated into the solution
at
an edge point
on
C for
example,
at
which
</>
= 0
on
one
side where 8 = 0 and
a</>/an
= 0
on
the
other
side where 8 =
(3;
a1>
a
2
,
•••
are
constants, and p
is
the distance from
the
edge (Motz 1946; Fox 1979).
The
integral equation
is
to
be
satisfied only
at
a finite number of
collocation points which in general
are
chosen to include all the singu-
larities.
If
Land
s+
denote two neighbouring singularities
on
C, and
S1>
S2,
•••
,SM
are the collocation points in
the
interval between, so that
S1
=
Land
SM
=
S+,
the function y in the closed interval L
to
S+
is
written
as
y(S) =
(s+
-s)"'+(s
-L)"'-f(s),
where the constants
a_
and
a+
are such that f is finite and non-zero.
The
function
f(s)
is
approximated by a function which is continuous between
s_
and
s+
and
is
a first degree polynomial in each sub-interval between
neighbouring nodal points
Sj
and
Sj+1>
for
2~j~M-2.
In
the intervals
s_
to
S2
and
SM-1
to
s+,
the approximating function is determined from the
first two terms of the appropriate local expansion
of
type (8.124).
If
all
the angles
(3
in a particular problem are rational fractions of
7r,
y(s) can
be
replaced in the integral of (8.123) by an analytical function of
0'
where
O'm
=
s.
When
s'
is
not
equal
to
Sj
or
Sj+1
the
integral
is
evaluated by
Simpson's rule.
For
s'
=
s,
B(s',
s) has a logarithmic singularity,
but
the
integral of the logarithmic term can
be
evaluated by a quadrature formula
which gives the integral
of
a function P(s)ln
lsi
exactly
if
P
is
a second
degree polynomial.
In
the neighbourhood of such a point
Sn,
therefore, C
is approximated by a parabola passing through
Sn-1>
Sn,
and
Sn+1'
The
anode surface extends to infinity
and
so for the infinite interval beyond
the
last collocation point f is derived from an asymptotic solution of
Laplace's equation. Finally, at an edge point P with an angle
(3
between
the
tangents associated with
CANU
CCN,
v changes discontinuously from
v = !((3/
7r)
-1
to v =
-!
where the boundary is smooth in
the
neighbour-
hood of P. Hansen (1983) avoided the numerical difficulty by omitting
such a point P from
the
selected collocation points. From
the
solution of
Integral equation methods
393
the system of linear algebraic equations obtained by Gaussian elimina-
tion, the values of
a<f>/aN
were evaluated at the collocation points and the
changes
(Ar,
Az) in
the
anode coordinates obtained from the equivalent
of (8.119). Hansen evaluated results for a geometry resembling an
experimental one available to him
and
made the comparisons mentioned
in §2.12.3.
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