Sample Problems > Usage > 3D_domains > 3d

{ 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

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 }

3d_sphere

3d_sphere