CGAL users discussion list

[Matthew Russell <>]

Hi Mael,

It was to be able to easily identify the boundaries in code, by
selecting boundary faces within an epsilon of x == 0 etc. If it's easy
to identify the faces on each boundary of a general periodic mesh
using CGAL and export that data along with a single instance of the
mesh, then that might also work.

I can imagine a naive algorithm that for each boundary face, it loops
over all boundary faces again looking for a counterpart separated by a
translation, labelling the faces accordingly. Is there a nicer way to
achieve this using CGAL functionality? I haven't come across any
examples so far.

Thanks,
Matt

On Wed, 17 Mar 2021 at 08:24, Mael Rouxel-Labbé
<> wrote:
>
> 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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> >>>>>>> https://sympa.inria.fr/sympa/info/cgal-discuss
> >>>>>>>
> >>>>>>>
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> >>>>>>
> >>>>>>
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> >>>>>>
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> >>>>> https://sympa.inria.fr/sympa/info/cgal-discuss
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> >>>
> >>>
> >>
> >> --
> >>     Iordan Iordanov, PhD
> >>     https://imiordanov.github.io/
> >>
> >>
> >>
> >> --
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> >>
>
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