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{ TWOZ_DIRECT.PDE
This problem constructs two noncoplanar spheres inside a box by constructing
a single dividing surface to delimit both spheres.
The domain consists of three layers.
layer 1 is the space below the spheres
layer 2 contains the sphere bodies, and is of zero thickness outside the spheres
layer 3 is the space above the spheres
The sphere interiors are Void, and are thus excluded from analysis. You could
just as well fill them with material if you wanted to model the insides.
The bounding surfaces of layer 2 are specified as a slope perpendicular to the
centerline of the spheres and overridden by regional expressions within
the (X,Y) extent of each sphere.
Click "Controls>Domain Review" to watch the mesh construction process.
See TWOZ_PLANAR.PDE, TWOZ_EXPORT.PDE and TWOZ_IMPORT.PDE for other methods of
treating spheres with centers on differing Z coordinates.
}
title 'Two Spheres in 3D  direct surface matching'
coordinates
cartesian3
Variables
u
definitions
K = 1 { dielectric constant of box filler (vacuum?) }
box = 1 { bounding box size }
{ read sphere specs from file, to guarantee that they are the same as those in surfgen }
#include "sphere_spec.inc"
{ sphere shape functions }
sphere1_shape = SPHERE ((x1,y1,0),R1)
sphere2_shape = SPHERE ((x2,y2,0),R2)
{ construct an extrusion surface running through both sphere diameters
by building an embankment between the spheres }
Rc = sqrt((x2x1)^2+(y2y1)^2)R1R2
Rx = Rc*(x2x1)/Rc/4
Ry = Rc*(y2y1)/Rc/4
xm = (x1+x2)/2
ym = (y1+y2)/2
xa = xm  Rx
ya = ym  Ry
xb = xm + Rx
yb = ym + Ry
xc = xm + Ry
yc = ym  Rx
slope = PLANE((xa,ya,z1), (xb,yb,z2), (xc,yc,0))
zbottom = min(z2,max(z1,slope))
ztop = zbottom
equations
U: div(K*grad(u)) = 0
extrusion
surface "box_bottom" z=box
surface "sphere_bottoms" z = zbottom
surface "sphere_tops" z = ztop
surface "box_top" z=box
boundaries
surface "box_bottom" natural(u) = 0 {insulating boundaries top and bottom }
surface "box_top" natural(u) = 0
Region 1 { The bounding box }
start(box,box) line to (box,box) to (box,box) to (box,box) to close
limited region 2 { sphere 1 }
mesh_spacing = R1/5 { force a dense mesh on the sphere }
zbottom = Z1sphere1_shape { shape of surface 2 in sphere 1}
ztop = Z1+sphere1_shape { shape of surface 3 in sphere 1}
layer 2 void
surface 2 value(u)=V1 { specify sphere1 voltage on top and bottom }
surface 3 value(u)=V1
start (x1+R1,y1)
arc(center=x1,y1) angle=360
limited region 3 { sphere 2 }
mesh_spacing = R2/5 { force a dense mesh on the sphere }
zbottom = Z2sphere2_shape { shape of surface 2 in sphere 2}
ztop = Z2+sphere2_shape { shape of surface 3 in sphere 2}
layer 2 void
surface 2 value(u)=V2 { specify sphere2 voltage on top and bottom }
surface 3 value(u)=V2
start (x2+R2,y2)
arc(center=x2,y2) angle=360
plots
grid(x,y,z)
grid(x,z) on y=y1 paintregions as "Ycut through lower sphere"
contour(u) on y=y1 as "Solution on Ycut through lower sphere"
grid(x,z) on y=y2 paintregions as "Ycut through upper sphere"
contour(u) on y=y2 as "Solution on Ycut through upper sphere"
grid(x*sqrt(2),z) on xy=0 paintregions as "Diagonal cut through both spheres"
contour(u) on xy=0 as "Solution on Diagonal cut through both spheres"
end