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From: | Jean-François Barthélémy |
Subject: | Re: [Getfem-users] Elastic interface |
Date: | Tue, 30 Dec 2014 23:33:19 +0100 |
Dear Jean-François,
The "interpolate tranformation" tool is a generic tool but it can indeed be time consuming because for each Gauss point on the boundary, a search of the corresponding element on the other mesh is done. Although this search is optimized (rtree structures), it can be of course far more expensive than a standard assembly (however, in 2D it is done on a 1D interface so that it should be relatively fast).
However, in your case, the simpler way should be to do the job with a single mesh defining two different regions. You can define two different fields on the two regions by restricting the same "mesh_fem" to the dof of the corresponding region (with the partial_mesh_fem object, for instance, or with a filtered variable of the model object).
Then, you have to define two integration methods, one on each region to add the two different elasticity terms. Then, you can use the generic assembly without "interpolate transformation" to add your elasticity term on the interface (you have to define a mesh region by selecting the faces corresponding to your interface on only one side of the interface).
Another possibility is indeed to use a level-set and the mesh_fem_level_set object which describe a finite element space cut by the level-set. However, there is no tool for the moment to deal directly with the jump at the interface in the generic assembly (I agree that this would be interesting in many situations). So that, the easier way is again two define two different fields and use the level-set tools of getfem to define cut integrations methods (mesh_im_level_set) still using the generic assembly to add your elasticity term on the interface (using a integration method on the level-set, also provided by mesh_im_level_set). All is available on the python/matlab/scilab interface. You have some examples in the interface test programs:
interface/tests/python/demo_fictitious_domains.py
interface/tests/matlab/demo_fictitious_domains.m
interface/tests/matlab/demo_fictitious_domains_laplacian.m
interface/tests/matlab/demo_structural_optimization.m
Best regards,
Yves.
_______________________________________________
----- Original Message -----
From: "Jean-François Barthélémy" <address@hidden>
To: address@hidden
Sent: Sunday, December 28, 2014 12:15:47 PM
Subject: [Getfem-users] Elastic interface
Dear Getfem users,
I am trying to find an easy and efficient way to take into account an elastic linear interface between two different materials in Getfem (using the python interface and/or in C++). The problem is defined on a uniform elastic matrix containing an elastic inclusion (with different moduli from that of the matrix) such that the matrix/inclusion interface is ruled by a relationship of the form T=K.[u] where T is the stress vector acting on the interface, K is the interface stiffness and [u] is the displacement jump (simple bilateral contact). Some finite element codes address this problem by allowing to build "joint elements".
I have already succeeded in solving this problem but in a very inefficient way I think. I have indeed defined two differents meshes, one for the matrix and one for the inclusion. Although occupying the same geometrical domain, the interface boundaries are different from one mesh to the other (pairs of points at the same place but one in the matric mesh and the other one in the inclusion). I then defined two MeshFems, two variables (umat and uinc), two integration methods,... and the contribution of the interface to the global elastic stiffness "(umat-uinc).K.(Test_umat-Test_uinc)" by using the "interpolate transformation" to project fields expressed on one mesh to the other one (kind of mortar method on consistent boundaries). It works fine on a small mesh in 2D but is too time consuming on large meshes or in 3D.
Is there a better way to do it either by using one or two meshes in which the interface correspond to element edges or, even better, by resorting to a levelset to define the interface independently of the mesh and xfem ?
In the case of a levelset, how is it possible to build the discontinuous field of elastic moduli (the elements crossed by the levelset contain two different materials but how to correctly define all the degrees of freedom including those related to the Heaviside function) ? And finally how can I define the interface contribution to the stiffness ie integral_{interface} [ (umat-uinc).K.(Test_umat-Test_uinc) ] dS, possibly in a high-level generic assembly procedure in Python since I don't know how to access to discontinuity terms in the high-level language ?
I haven't really found solutions to my questions in the examples (either related to mortar, cracks, contacts...) but I may have missed something.
Thank you very much in advance for any help or piece of code.
Best regards,
Jean-François Barthélémy
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