CGAL users discussion list

[Mael Rouxel-Labbé <>]

Hello,

Assuming you cannot modify your FEM formulation to simply use a single 
mesh as Iordan was describing, do you wish to have the facets to be 
aligned on the side of the cube to identify them as boundary, or because 
it is necessary in your FEM formulation that they geometrically lie on 
the cube's side ? If it's not a geometrical constraint, extracting a 
canonical block of tetrahedral elements with well-identified boundary 
faces that reproduces periodically in all three directions (basically, 
what you see in the drawing in the manual) is very easy to do.

Best,
Mael

Does the FEM formulation require the boundary tets to have their 
boundary face fully aligned with a side of the cube? After all,

On 17/03/2021 01:19, Matthew Russell ( via 
cgal-discuss Mailing List) wrote:
> Dear Iordan,
>
> Thanks for this, and, as far as I'm concerned, no need to apologise.
> It's an interesting way to look at the the problem and it makes a lot
> of sense conceptually. However, in my experience, it is not how PBCs
> are usually implemented in finite element solvers. Do you know of any
> examples describing how to reformulate a periodic problem as a problem
> on a torus or it's numerical implementation? The problem I'm solving
> is Stokes flow in a periodic unit cell, with no-slip on the
> non-periodic part of the boundary (the part that's kind-of
> "inward-facing"; see `gmsh.jpg` in my first post).
>
> Thanks,
> Matt
>
> On Tue, 16 Mar 2021 at 23:30, Iordan Iordanov <> wrote:
> > Dear Matthew, Monique, all,
> >
> > Sorry to appear out of nowhere like this, but I would like to offer another 
> > perspective, perhaps.
> >
> > Matthew, you mention that in your FEM problem you want to define periodic 
> > boundary conditions.
> > The need for specifying periodic boundary conditions in FEM problems comes 
> > from the fact that
> > traditional meshes do not have a way to represent the periodic medium over 
> > which the equation
> > to be solved is defined. In other words, you need to discretize your domain 
> > so that the vertices and
> > faces coincide neatly, so that you know how to compute interactions and value 
> > propagations
> > periodically, from one side to the other.
> >
> > The thing is, with a periodic mesh, you don't need to specify any boundary 
> > conditions, because there
> > is no boundary. The CGAL periodic 3D mesh package represents a domain that "wraps 
> > around" in
> > 3 dimensions, as the name implies, so the only things you need to define are 
> > the equation itself and the
> > initial conditions. In this setting, boundary conditions actually do not make 
> > sense -- if you try to apply
> > boundary conditions, you would end up splitting a torus only to glue it back 
> > together.
> >
> > Why not just use the torus? :)
> >
> > I am guessing it will require some re-formulating of your problem, but you 
> > will get much better results
> > in the end, and your computation will be more efficient. The reason is that, 
> > when you define a traditional
> > mesh, you need to store the boundary elements (vertices/faces) twice. 
> > Moreover, you need to keep track
> > of which vertices coincide and when not to count interactions twice, so it is 
> > a bit messy. With the periodic
> > mesh, you really don't save anything extra, and all the elements are unique. 
> > Every element knows its
> > neighbors so computations are much more straightforward conceptually.
> >
> > I hope this has some meaning for you, and, once more, sorry for butting in 
> > like this.
> >
> > Best of luck and best regards,
> >    - Iordan
> >
> > On Wed, Mar 17, 2021 at 3:29 AM Matthew Russell <> wrote:
> > > Dear Monique,
> > >
> > > Thanks - I saw that example and have tried using a unit cube (defined
> > > implicitly) as the second function in the vector but I don't think it
> > > solves my problem. Is this what you had in mind? The resulting mesh is
> > > similar to the example attached to my original email, except that it
> > > can also include the region with the opposite sign (which I don't
> > > require), depending on the chosen sign vectors.
> > >
> > > Apologies if I described the problem poorly! I would like to create a
> > > collection of different meshes, one for each of a collection of
> > > implicitly-defined regions, so I don't think it's a multiple domains
> > > problem. On each mesh I will run a separate finite element simulation
> > > with periodic boundary conditions. We can ignore the multiple meshes
> > > here and consider just one. To make it easy to identify the periodic
> > > boundaries in the FEM code, my first thought was to try to create
> > > meshes where the cells at the boundaries lie on the facets of the
> > > bounding cube as I mentioned earlier. It seems this isn't possible
> > > with the periodic mesh package, but there may be a workaround using
> > > the regular 3D mesh package as Mael suggested. I was wondering if
> > > someone had any further hints on how to achieve this? It seems like a
> > > difficult problem to manually place vertices on the intersection
> > > between the bound cube's facets and the implicit region. I really
> > > appreciate all your help with this!
> > >
> > > Thanks,
> > > Matt
> > >
> > > On Tue, 16 Mar 2021 at 15:44, Monique Teillaud
> > >
> > >
> > >
> > > On Tue, 16 Mar 2021 at 15:44, Monique Teillaud
> > >
> > > <> wrote:
> > > > Dear Matthew,
> > > >
> > > > If I understand correctly (I am not sure) you would like to mesh multiple 
> > > > domains.
> > > > Does this example help you?
> > > > https://doc.cgal.org/latest/Periodic_3_mesh_3/index.html#title19
> > > >
> > > > Best,
> > > > --
> > > > Monique Teillaud
> > > > https://members.loria.fr/Monique.Teillaud/
> > > > INRIA Nancy - Grand Est, LORIA
> > > > Institut National de Recherche en Informatique et Automatique
> > > >
> > > > ----- Le 16 Mar 21, à 15:48, cgal-discuss  a écrit :
> > > >
> > > > > Dear Monique,
> > > > >
> > > > > I agree that adding points on the facets of the bounding cube in the
> > > > > right places would be a way to solve this. But this technique requires
> > > > > a description of the regions on each facet that should be
> > > > > triangulated. This is difficult because many of the surfaces I am
> > > > > dealing with are defined implicitly and do not cover the entirety of
> > > > > the cube's facets. Is there something in CGAL that would help do this?
> > > > >
> > > > > However, I suppose my actual issue is being able to accurately specify
> > > > > boundary conditions. I was trying to make this easy by forcing the
> > > > > boundaries of the mesh to align with bounding cube.
> > > > >
> > > > > With a triangulation generated by the periodic mesh package, is there
> > > > > a way to colour or otherwise somehow mark the cell faces on each
> > > > > boundary? I'm not sure this is a meaningful question given the
> > > > > irregular nature of the boundaries!
> > > > >
> > > > > Sorry for the basic questions - I'm new to CGAL and still finding my way 
> > > > > around!
> > > > >
> > > > > Many thanks,
> > > > > Matt
> > > > >
> > > > >
> > > > > On Tue, 16 Mar 2021 at 10:54, Monique Teillaud
> > > > > <> wrote:
> > > > > > Dear Matthew,
> > > > > >
> > > > > > In fact, if you add points on the facets of the cube and you ensure that the
> > > > > > facets are triangulated as mael suggests, then you are enforcing the
> > > > > > periodicity through these points. You are not using the functionality that is
> > > > > > offered by the periodic mesh package.
> > > > > > In other words, this should rather be done with the non-periodic meshing
> > > > > > package.
> > > > > >
> > > > > > (In fact I would be curious why you need the facets to be triangulated... Our
> > > > > > algorithm is precisely designed to avoid adding such artificial points...)
> > > > > >
> > > > > > Best regards,
> > > > > > --
> > > > > > Monique Teillaud
> > > > > > https://members.loria.fr/Monique.Teillaud/
> > > > > > INRIA Nancy - Grand Est, LORIA
> > > > > > Institut National de Recherche en Informatique et Automatique
> > > > > >
> > > > > > ----- Le 15 Mar 21, à 17:21, cgal-discuss <> a écrit :
> > > > > >
> > > > > > Hi Mael,
> > > > > >
> > > > > > Thanks for your reply.
> > > > > >
> > > > > > It's a shame the periodic mesher can't do this currently. Would you be able to
> > > > > > give some hints on how to implement your idea of using the standard 3D mesh
> > > > > > package? I'm slightly wary about manually moving vertices around - won't this
> > > > > > reduce the quality of the resulting mesh?
> > > > > >
> > > > > > Thanks,
> > > > > > Matt
> > > > > >
> > > > > > On Mon, 15 Mar 2021, 08:04 Mael Rouxel-Labbé,
> > > > > > <> wrote:
> > > > > > > Hello,
> > > > > > >
> > > > > > > This is not currently possible in a straightforward manner in the
> > > > > > > periodic mesh generator, as there is no 3D constrained Delaunay
> > > > > > > triangulations in CGAL yet.
> > > > > > >
> > > > > > > In a less straight forward manner, if you wish your elements to lie
> > > > > > > exactly on the boundary of the periodic domain, you might be able to do
> > > > > > > away with the periodic mesh generation algorithm: generate a 3D mesh of
> > > > > > > the implicit function clipped by the periodic domain, and then ensure
> > > > > > > that the vertices are identical on both sides.
> > > > > > >
> > > > > > > Best,
> > > > > > > Mael
> > > > > > >
> > > > > > > On 15/03/2021 02:05, Matthew Russell ( via
> > > > > > > cgal-discuss Mailing List) wrote:
> > > > > > > > Dear all,
> > > > > > > >
> > > > > > > > I would like to generate periodic 3D meshes of regions bounded by
> > > > > > > > implicit functions and so I was pleased to find examples such as [1]
> > > > > > > > that do almost exactly what I want. The problem that I'm having is
> > > > > > > > that cells in the meshes generated from this example are able to lie
> > > > > > > > across the bounding unit cube, but to use the meshes in finite element
> > > > > > > > simulations I need the faces of the cells to align with the boundaries
> > > > > > > > of the cube.
> > > > > > > >
> > > > > > > > For example, the CGAL example [1] (with a different implciit function)
> > > > > > > > generates meshes like [cgal.jpg], but I would like one like
> > > > > > > > [gmsh.jpg].
> > > > > > > >
> > > > > > > > I haven't been able to find a way to do this in CGAL. Is it possible?
> > > > > > > >
> > > > > > > > Many thanks,
> > > > > > > > Matt
> > > > > > > >
> > > > > > > > [1]
> > > > > > > > https://doc.cgal.org/latest/Periodic_3_mesh_3/Periodic_3_mesh_3_2mesh_implicit_shape_8cpp-example.html
> > > > > > > --
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> > > > > >
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> >
> > --
> >     Iordan Iordanov, PhD
> >     https://imiordanov.github.io/
> >
> >
> >
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