Border conditions wih thermo-elastic problem

Dear All,

I’m a beginner in using freefem and finite elements computation and I need help to better understand in particular border conditions.

I’m modeling a circular mirror in vacuum, held in his mount by the rear surface, with a laser hitting the center of the front surface.
I want to find temperature and deformation of the front and rear surfaces.

I converted 3D equations to be 2D axisymmetric.
thus, I have 2 coupled equations for temperature and stress tensor.
I “translated” the problem in a variationnal formulation to integrate border conditions which are :

  • for the temperature: laser heating on the front surface and radiation on all surfaces.
  • for the deformation (ur,uz) : the mirror is held on a small part of the border and motion free everywhere else.

for the temperature field, the variational formulation involved the flux of temperature on the surface related to incident laser density and radiation, thus it is quite natural to include them to the equation. it’s fine.

but for the deformation vector field, the variational formulation involves the normal (to the surface) stress tensor.
my questions are then :

  • is a free moving equivalent to this normal stress tensor to be zero ?

  • how to take into account the mirror held in his mount ?
    I defined a border of the mirror to take into account this constrain.
    then, I just need to use “on (Held_border, ur=0, uz=0)” ?
    or do I need to express in the same time the expression of the remaining normal stress tensor which is non zero on this part of the mirror ?
    or something else ?

thanks a lot for reading and for your help

best regards

ronicSummary.pdf (54.4 KB)

I added the equations.
I think the 1st variational equation is OK.
my questions are for the 2nd variational equation :

  • is the normal stress tensor zero if there is no external force on the mirror (if gravity is neglected) on border 4 ?
  • what to do with the normal stress tensor on border 3 ?
    do I need to keep it in some 1D integral if I add the condition “on (3,ur=0,uz=0)” ?

thanks again

problem solved.
the answers are :

  • yes, for a mirror without external force, the normal stress tensor is zero
  • one can use 2 different conditions on a particular border (normal stress tensor = 0 AND no deformation in a particular direction, uz=0 for example)