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# 3d_spherespec

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# 3d_spherespec   { 3D_SPHERESPEC

This problem demonstrates the use of the SPHERE function for construction

of a spherical domain in 3D.  It is a modification of the example problem 3D_SPHERE.PDE.

}

title '3D Sphere'

coordinates

cartesian3

variables

u

definitions

K = 0.1                 { conductivity }

R0 = 1                 { radius }

H0 = 1                 { total heat input }

heat =3*H0/(4*pi*R0^3) { volume heat source }

zs = sphere((0,0,0),R0) { the top hemisphere }

equations

U: div(K*grad(u)) + heat   = 0

extrusion

surface z = -zs         { the bottom hemisphere }

surface z = zs         { the top hemisphere }

boundaries

surface 1 value(u) = 0 { fixed value on sphere surfaces }

surface 2 value(u) = 0

Region 1

start  (R0,0)

arc(center=0,0) angle=360

plots

grid(x,y,z)

grid(x,z) on y=0

contour(u) on x=0

contour(-4*pi*k*(x*dx(u)+y*dy(u)+z*dz(u))/sqrt(x^2+y^2+z^2)) on x=0 as "normal flux"

contour(-4*pi*k*(x*dx(u)+y*dy(u)+z*dz(u))/sqrt(x^2+y^2+z^2)) on y=0 as "normal flux"

contour(4*pi*normal(-k*grad(u))) on surface 1 as "4*pi*Normal Flux=1" { bottom surface }

contour(4*pi*normal(-k*grad(u))) on surface 2 as "4*pi*Normal Flux=1" { top surface }

surface(4*pi*normal(-k*grad(u))) on surface 1 as "4*pi*Normal Flux=1" { bottom surface }

surface(4*pi*normal(-k*grad(u))) on surface 2 as "4*pi*Normal Flux=1" { top surface }

summary

report(sintegral(normal(-k*grad(u)),1)) as "Bottom current :: 0.5 "

report(sintegral(normal(-k*grad(u)),2)) as "Top current :: 0.5 "

report(vintegral(heat)) as "Total heat :: 1"

report(sintegral(normal(-k*grad(u)))) as "Total Flux :: 1"

end