Dear Yves Renard and getfem users.
I am trying to add boundaries to MeshImData. However, in case of a fictitious domain method with boundaries specified by a levelset, I was not able to find a way to select boundary regions and include it in MeshIm as we do for conforming meshes. The problem is that MeshImData doesn't work with MeshIm produced with expressions other than 'all(a)'.
I tried these
1. Because I wanted to add boundary to MeshIm object, I first tried
mim = gf.MeshIm('levelset',mesh_ls,'inside(a)+boundary(a)', gf.Integ('IM_TRIANGLE(6)'))
but the function does not accept expressions like 'inside(a)+boundary(a)'.
2. In case I use the following script
mim = gf.MeshIm('levelset',mesh_ls,'inside(a)', gf.Integ('IM_TRIANGLE(6)'))
mim.set_integ(4)
This also failed at the line
mim_data = gf.MeshImData(mim, -1, [2, 2])
And even if I use 'all(a)' there seems to be no way to provide the boundary regions to Neumann term and to add_Dirichlet_condition_with_Nitsche_method function.
3. I also tried to separately generate MeshIm,mim = gf.MeshIm('levelset',mesh_ls,'inside(a)', gf.Integ('IM_TRIANGLE(6)'))
mim_bound = gf.MeshIm('levelset',mesh_ls,'boundary(a)', gf.Integ('IM_TRIANGLE(6)'))
but this also failed at mim_bound_data = gf.MeshImData(mim_bound, -1, [2, 2])
4. using cut_mesh() of MeshLevelSet
These meshes are composed of rectangular elements and triangle elements ( rectangular elements cut by a level set). And I was not able to define MeshIm that works with this composition.
Am I missing something here?
Respectfully yours.
Dear Heungson Lee,
There is a priori two known approaches for the used of Nitsche's methode for plasticity. The easiest is to consider some additional Gauss points on the boundary in order to have the plastic strain there as you mention, or to extrapolate it from the internal Gauss points as it is mentionned for instance in Seitz thesis https://mediatum.ub.tum.de/doc/1454145/1454145.pdf
For the first approach (additional Gauss points on the boundary) you can normally specify a mesh region where you include both the domain and the boundary where you want to prescribe the Dirichlet condition, which means that the region will contain both some elements and some faces of elements. Then you create the mim_data object with this mesh region.
Concerning the variants with nonzero theta parameter, this is indeed complicated for nonlinear material laws, since the constitutive law has to be derivated. This is much more complicated for plasticity where the constitutive law is only piecewise regular and thus cannot be derivated twice for the solve with Newton algorithm. So it seems that, unless the use of a regularization, the only variant of Nitsche's method that can be use for plasticity is the one with theta = 0.
Best regards,
Yves
----- Mail original -----
De: "Heungson Lee" <address@hidden>
À: "getfem-users" <address@hidden>
Envoyé: Lundi 18 Mai 2020 09:07:18
Objet: Imposing Dirichlet B.C. using Nitsche's method in fictitious domains for elastoplasticity.
Dear getfem++ users. I am Heungson Lee.
I am trying to impose Dirichlet B.C. using Nitsche's method in fictitious
domains for elastoplasticity. For linear elastic problems, it just works
fine. But for elastoplastic problems such as Prandtl Reuss plasticity, I
was not able to make it work.
I think the major problem is:
1. Stress tensor and its variation on the Dirichlet boundary.
-> Nitsche's method brick requires Neumann term on the boundary which uses
plastic strain. And the elastoplasticity brick requires plastic strain to
be stored in MeshImData. I was not able to find a way to compute plastic
strain on the boundary and store it in MeshImData for this purpose.
These two problems might be not that important right now.
2. Nitsche's method brick does not recognize nonlinear variables for small
strain elastoplasticity brick.
-> So I temporarily modified the getfem source code and python interface
related to the function 'add_Dirichlet_condition_with_Nitsche_method' to
receive an additional boolean parameter from users which tells if the
problem is linear or nonlinear problem. It seems that this walk-around
works.
3. Fails to calculate the derivative of the Neumann term when theta (one of
Nitsche's method parameter) is not zero.
-> It seems that Neumann term includes the Heaviside function which is
originated from the yield function to distinguish whether it's in elastic
or plastic region. Inside 'add_Dirichlet_condition_with_Nitsche_method'
function, derivative of the Neumann term is symbolically evaluated, but it
fails because of the Heaviside function.
Please refer to the attached file to see what I am trying to do.
If you have any idea how to make this work. please let me know. I would
really appreciate it.
Respectfully yours.
plate_with_fixed_hole_nitsche_PrandlReuss.py