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{ 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*magnitude(k*grad(u))) on x=0

  contour(4*pi*magnitude(k*grad(u))) on y=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"

  vector(-grad(u)) on x=0

  vector(-grad(u)) on y=0

 

  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