ff3d-problem with elliptic cylinder

Boundary value problem

• Domain:

  G := {(x,y,z)|  1 < (x/rx)² + (y/ry)² < a²;  0 < z < L }

  where rx=1, ry=0.5, a=2, L=5. (This resembles a section of an elliptic hollow cylinder.)
• Differential equation:

  Δ u = 0   in G.

• Boundary condition:

  u(x,y,z) = (|x,y|-1)(a-|x,y|)(L-z)/L   at δG

  with

  |x,y|:=sqrt( (x/rx)² + (y/ry)²  ).

• One can employ the symmetry of the domain and the boundary condition to save computational costs. For the computations only the quarter

  Q:={(x,y,z)∈G| x>0; y>0}

  of G is used. At the additional boundaries at x=0 and y=0 natural boundary conditions are imposed (These are the default boundary conditions in ff3d therefore no additional statement is needed in the ff3d input file).

Graphical representation of the boundary condition

ff3d results -- the problem.

• The computed solution u(x,y,z) doesnot yield the boundary condition at z=0:
  
  
• The computed solution is not reasonable at the middle height of the cylinder section:
  
• The computed solution is not equal to zero at the top cap:
  
  

stderr output of ff3d:

Files

• The ff3d input: cable-ellipse-annulus.ff3d
  
• The corresponding povray-file cable-ellipse-annulus-num.pov
  
  If one adds two light sources, a camera, and some background color PoVRay produces the following picture:
  
• The call of ff3d:
  
• The gnuplot file for producing the pictures: cable-ellipse-annulus.gp
  

Comparision with getdp result

gmsh generated 3d-view at the results

Potential u at z=1.0 (getdp result)

Files

• Geometry description in gmsh
  
• Problem description in getdp