3d_sphere

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3d_sphere

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{ 3D_SPHERE.PDE

 

 This problem considers the construction of a spherical domain in 3D.

 

 The heat equation is  Div(-K*grad(U)) = h, wth U the temperature and

 h the volume heat source.

 

 A sphere with uniform heat source h will generate a total amount of heat

   H = (4/3)*Pi*R^3*h, from which  h = 3*H/(4*Pi*R^3).

 

 The normal flux at the surface will be Fnormal = -K*grad(U) <dot> Normal,

 where Normal is the surface-normal unit vector.  On the sphere, the unit

 normal is [x/R,y/R,z/R].

 At the surface, the flux will be uniform, so the surface integral of flux is

   TOTAL = 4*pi*R^2*normal(-K*grad(U)) = H

 or  normal(-K*grad(u)) = H/(4*pi*R^2)  =  R*h/3.

 

 In the following, we set R=1 and H = 1, from which

   h = 3/(4*pi)

   normal(-k*grad(u)) = 1/(4*pi)

}  

 

title '3D Sphere'  

 

coordinates  

   cartesian3  

 

variables  

   u  

 

definitions  

   K = 0.1   { conductivity }  

   R0 = 1   { radius }  

   H0 = 1   { total heat }  

  { volume heat source }

   heat =3*H0/(4*pi*R0^3)                  

 

equations  

   U: div(K*grad(u)) + heat   = 0  

 

 

extrusion  

  surface z = -SPHERE ((0,0,0),R0)       { the bottom hemisphere }  

  surface z = SPHERE ((0,0,0),R0)       { 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