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{ TWO_SPHERES.PDE
This problem constructs two spheres inside a box. The centers of the spheres lie
on a single z-plane, which simplifies the domain construction.
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. They could just as
well be filled with material if one wanted to model the insides.
The bounding surfaces of layer 2 are specified as a default, over-ridden by regional
expressions within the (X,Y) extent of each sphere.
See TWOZ_PLANAR.PDE, TWOZ_DIRECT.PDE, TWOZ_EXPORT.PDE and TWOZ_IMPORT.PDE
for methods of treating spheres with centers on differing Z coordinates.
}
title 'Two Spheres in 3D'
coordinates
cartesian3
variables
u
definitions
K = 1 { dielectric constant of box filler (vacuum?) }
box = 1 { bounding box size }
R1 = 0.25 { sphere 1 radius, center and voltage}
x1 = -0.5
y1 = -0.5
V1 = -1
R2 = 0.4 { sphere 2 radius, center and voltage}
x2 = 0.2
y2 = 0.2
V2 = 2
{ sphere shape functions }
sphere1_shape = SPHERE ((x1,y1,0),R1)
sphere2_shape = SPHERE ((x2,y2,0),R2)
{ default position of layer 2 surfaces }
zbottom = 0
ztop = 0
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 = -sphere1_shape { shape of surface 2 in sphere 1}
ztop = 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 = -sphere2_shape { shape of surface 2 in sphere 2}
ztop = 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 as "Grid on Y-cut at sphere 1 center"
contour(u) on y=y1 as "Solution on Y-cut at sphere 1 center"
grid(x,z) on y=y2 as "Grid on Y-cut at sphere 2 center"
contour(u) on y=y2 as "Solution on Y-cut at sphere 2 center"
{ sqrt(2) is needed to plot true distance along diagonal: }
grid(x*sqrt(2),z) on x-y=0 as "Grid on x=y diagonal"
contour(u) on x-y=0 as "Solution on x=y diagonal"
end