A fortran program (RELAX3D) to solve the 3 dimensional poisson (Laplace) equation

program has found many problems
2)
Method Finite difference methods, dating back to Gauss 6) can be used to obtain for all static electric and fields where eddy currents can The
W
, W
, W
x
Ch 2 x
y
Ch 2 y
z
i =-Ch 2 z
and
For cylindrical
co-ordinates
with rotational
W
symmetry
¢0 = Wx(¢l+~2 ) + Wy(¢3+¢4)+ - - ~ (~4-~3)
- 0/C
where
C =-2 h2 x
+ -2 , h2 y
others4, 5) with the idea in mind that appropriate for it at some future date when computers Fortunately,
could run it in a practical manner.
our view into the crystal ball turned out to be accurate and the to solve both at TRIUMF and elsewhere.
is shown in Fig. I for the Poisson equation equation
W z (~ 5+~ 6) = - -I
in the finite difference
¢0 = Wx(¢ I+~ 2) +
W y (¢ 3+¢ 4) + : - -i
0/C
where
C : - - + - - 12 2 2 , h2 h2 h2 x y z 0 = 0 (x0,Y0,Z0)
the boundary having the dominant effect. Although restricting the boundaries to lie on the grid (mesh) points limits the resolution with which one can describe a particular geometry (given the maximum number of allowed grid points) this greatly simplifies the program structure and consequently its execution speed.
I < R < 2
The patterns called
of points used to form the finite difference "molecules" or "stencils".
equations
are commonly
"templates",
Y
Let the uniform grid spacing along the x, y , and z axes
by replacing
the old ~ by
*Current address Physics Department, Vancouver, B.C., Canada.
University
of British Columbia,
99
Cold
+ R * (¢0 - ~old ) where usually
W
= - -1
, W
x
Ch 2 x
y
i =-Ch 2 2 Y and axis of symmetry is along x axis and radius R is along y axis, W : while for R = 0 (on axis)
¢0 = Wx(¢i+¢2) + w
2 +4 C =--- , h2 h2 x y
Y
¢~W
p/c
=--
i Ch 2' x
W
=-y
4 Ch 2 y
x
p = p (x0,Y0) = p(z,R)
100
3.
Boundaries RELAX3D handles three types of boundaries. The first type is a DIRICHLET
¢ at each grid point (that is allowed equation.
to vary) on each iteration is The over-relaxation method
replaced using the finite - difference accelerates the convergence
be hx, hy, and h z respectively where in general be different. these may all
~Y
/
~
o hx
'1
0~/
/T hx ~._x
hy
Z
~3
The 3 dimensional resulting molecule
FIG,'J.
V~ = 0
the problem case, the iterations
and the contour plotting of any desired slices etc. in 1973 3) based on work by applications would be found
Basic parts of the program were developed
partial differential difference interest, equations
for the field, being replaced by a set of finite the volume of the potential ~ at
at discrete points in a mesh occupying connecting
of Dirichlet
and Neumann boundaries
regular 3 dimensional mesh. solved using a successive of commands, supplemented
The finite difference
equations
at these nodes are
A Fortran Program (RELAX3D) to Solve the 3 Dimensional Poisson (Laplace) Equation
H. Houtman*,
C.J. Kost
TRIUMF, University of British Columbia Vancouver, B.C., Canada
boundaries must be restricted to lie on the grid points their precise position is not very important to the fields a few mesh points away, only the existence of
user written subroutine BND (describing the user problem geometry) and consequently will not be allowed to change during the relaxation procedure. The second type is a NEUMANN boundary. All or some of the points on the The
numerical magnetic
solutions,
to any desired accuracy,
fields in uniform media and time-varying We will restrict ourselves equation
be neglected.
to problems given in TABLE I.
boundary, that is a boundary (usually metal) on which the potential ~ is known (and hence fixed). These boundary points are always specified as negative in the
results in a large number of equations
each point with those of adjacent points and is solved by an iterative procedure known as relaxation. potential The basis of the iteration procedure is that the
faces of the 3 dimensional box (cylinder) can be of this type (default).
charge density, permittivity (where applicable) and potential on adjacent points orthogonal to such a symmetry plane point (one inside the box and the other a "fictitious" one outside the box) are assumed to be equal and hence the normal gradients of charge density, permittivity and potential are zero for these symmetry boundary points. NEUMANN boundaries. The third type is a dielectric boundary between two media of different permittivity (currently restricted to 2D problems). Although dielectric Thus equipotential lines always cross perpendicular to