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{ TWOZ_PLANAR.PDE
This problem constructs two spheres inside a box by constructing multiple planar
extrusion layers.
It presents an alternate method for comparison to that of TWOZ_EXPORT.PDE and TWOZ_IMPORT.PDE.
The domain consists of five layers.
layer 1 is the space below the lower sphere
layer 2 contains the lower sphere body, and is of zero thickness outside the sphere
layer 3 is the space between the spheres
layer 4 contains the upper sphere body, and is of zero thickness outside the sphere
layer 5 is the space above the upper sphere
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 layers 2 and 4 are specified as planes at the level of the sphere
center, overridden by regional expressions within the (X,Y) extent of each sphere.
Click "Controls>Domain Review" to watch the mesh construction process.
}
title 'Two Spheres in 3D  planar formulation'
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)
zbottom1 = z1
ztop1 = z1
zbottom2 = z2
ztop2 = z2
equations
U: div(K*grad(u)) = 0
extrusion
surface "box_bottom" z = box
surface "sphere1_bottom" z = zbottom1
surface "sphere1_top" z = ztop1
surface "sphere2_bottom" z = zbottom2
surface "sphere2_top" z = ztop2
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 }
zbottom1 = Z1sphere1_shape { shape of surface 2 in sphere 1}
ztop1 = 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 }
zbottom2 = Z2sphere2_shape { shape of surface 2 in sphere 2}
ztop2 = Z2+sphere2_shape { shape of surface 3 in sphere 2}
layer 4 void
surface 4 value(u)=V2 { specify sphere2 voltage on top and bottom }
surface 5 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