[Dune] parametrized AMR and cell divergence

[Carsten Gräser]

Am 25.02.2015 um 11:03 schrieb Marco Cisternino:
> Thanks Carsten and Andreas, 
> I understand the situation but I cannot understand the cure Carsten proposed. 
> Could you be more specific, Carsten, please? 
Assume you have a grid with hanging vertices resulting from local
refinement and a _continuous_ P1-function f.

Then each hanging vertex p on level k is contained in
a face F of an element on level l<k without being a vertex
of F. Hence f(p) is given by linear interpolation of the values
f(p_(F,i)) at the vertices p_(F,i) of F, i.e. it's a linear
combination of those values.

If some of the p_(F,i) are hanging themselves, the values f(p_(F,i))
are again linear combinations of values at vertices of even coarser
element. In the end f(p) is a linear combination of values
f(q_i) at nonhanging vertices q_i.

If you restrict your deformations to the subspace of such
_continuous_ P1-functions instead of all P1-functions, you will no
longer have holes or overlapping element (at least if the deformation
is small enough).

This subspace is often called the conforming P1-space for
the nonconforming mesh. Please refer to the discretization
module of your choice for the implementation: I know that
it's implemented in dune-fufem and dune-pdelab, and I'm
pretty sure that it's also implemented in dune-fem.

Best,
Carsten

> Thanks again, 
> 
> Marco 
> 
> 
> 
> Da: "Carsten Gräser" <graeser at mi.fu-berlin.de> 
> 
> A: "Marco Cisternino" <marco.cisternino at optimad.it>, dune at dune-project.org 
> Inviato: Mercoledì, 25 febbraio 2015 0:28:11 
> Oggetto: Re: [Dune] parametrized AMR and cell divergence 
> 
> Hi Marco, 
> with hanging node you will have precisely the situation mentioned 
> by Andreas. This does not change in 3d. 
> 
> The problem is that GeometryGrid moves individual vertices, i.e, 
> the grid is deformed by a P1-function living on the host grid. 
> Unfortunately a P1-function with arbitrary values for each vertex 
> is not continuous in general. Hence you get holes or overlapping 
> elements as described by Andreas. 
> 
> However you can cure this by using conforming P1-functions 
> leading to continuous deformations. This can be achieved by 
> interpolating the values at hanging nodes by their non-hanging 
> neighbors. 
> 
> Best, 
> Carsten 
> 
> Am 24.02.2015 um 19:17 schrieb Andreas Dedner: 
>> All I'm saying is that one element is a pentagon and GeometryGrid does 
>> not change the type of the element. So that the situation will not 
>> be correctly resolved by GeometryGrid since it uses the element geometry 
>> types of the hostgrid. For example element.geometry().volume() would 
>> be wrong. You would need a grid that can handle this pentagon. 
>> So the underlying codim zero elements do not conform to the dune 
>> structure if you use GeometryGrid. 
>>
>> One could still get the intersection correctly I guess. But not the way 
>> it is implemented as you can see in 
>> dune/grid/geometrygrid/intersection.hh, As you can see it uses the 
>> geometry of 
>> the inside element and as discussed above that is not correct. 
>> Andreas 
>> FieldVector< ctype, dimensionworld > 
>> outerNormal ( const FieldVector< ctype, dimension-1 > &local ) const 
>> { 
>> const ReferenceElement< ctype, dimension > &refElement 
>> = ReferenceElements< ctype, dimension>::general( 
>> insideGeo_.type() ); 
>>
>> FieldVector< ctype, dimension > x( geometryInInside().global( 
>> local ) ); 
>> const typename ElementGeometryImpl::JacobianInverseTransposed 
>> &jit = insideGeo_.jacobianInverseTransposed( x ); 
>> const FieldVector< ctype, dimension > &refNormal = 
>> refElement.integrationOuterNormal( indexInInside() ); 
>> FieldVector< ctype, dimensionworld > normal; 
>> jit.mv( refNormal, normal ); 
>> return normal; 
>> } 
>>
>>
>>
>>
>>
>> On 24/02/15 17:17, Marco Cisternino wrote: 
>>> This is exactly what I mean, but I was talking about 3D. 
>>> Anyway, from what you're saying I cannot understand the use of 
>>> GeometryGrid. 
>>> If I can move nodes using a deformation function and I can refine 
>>> locally (in Dune style), the "moved" situation is quite common. 
>>> Moreover, in this case I'd sum over the 5 intersections of the right 
>>> element asking for normals and intersections length, what is the 
>>> relevance of being a cube for the element? Intersections are "cube". 
>>> I can agree if you speak about the volume of the right element. 
>>> Who's Z?? You meant Y? 
>>> Thanks, 
>>>
>>> Marco 
>>>
>>>
>>> -- 
>>> ----------------------------------------------- 
>>> Marco Cisternino, PhD 
>>> Software Developer 
>>> OPTIMAD Engineering s.r.l. 
>>> Via Giacinto Collegno 18 
>>> 10143 Torino - Italy 
>>> www.optimad.it 
>>> marco.cisternino at optimad.it 
>>> +39 011 19719782 
>>> ----------------------------------------------- 
>>>
>>> ------------------------------------------------------------------------ 
>>> *Da: *"Andreas Dedner" <a.s.dedner at warwick.ac.uk> 
>>> *A: *"Marco Cisternino" <marco.cisternino at optimad.it> 
>>> *Cc: *dune at dune-project.org 
>>> *Inviato: *Martedì, 24 febbraio 2015 18:00:30 
>>> *Oggetto: *Re: [Dune] parametrized AMR and cell divergence 
>>>
>>> So you have the following situation 
>>> Original Moved 
>>> X-----------A----------------X X--------A------------------X 
>>> | | | | 
>>> \ | 
>>> | | | | 
>>> \ | 
>>> |------------Y | |-----------Y | 
>>> | | | | 
>>> / | 
>>> | | | | 
>>> / | 
>>> X-----------B----------------X X-------B-------------------X 
>>> Perhaps somebody else can add his thoughts but the right element after 
>>> movement is not a cube any more so it still will have the straight edge 
>>> between AB and your grid elements would not be non overlapping. That 
>>> is not allowed in DUNE. 
>>>
>>> The right element does have 5 intersections and I guess two of those 
>>> could be between AZ and ZB... But the main problem is the non DUNE 
>>> style grid. 
>>>
>>> Andreas 
>>>
>>> On 24/02/15 16:35, Marco Cisternino wrote: 
>>>
>>> Absolutely non conforming. 
>>> I know I have a hanging node, but I cannot understand what you 
>>> say: why do you talk about coarse and fine intersections?? 
>>> If I move a hanging node I'm moving 4 intersections, isn't it? And 
>>> my coarse element has 9 intersections, right? 
>>> Through its intersections the coarse element knows about the 
>>> hanging node or not? 
>>> If not I cannot imagine the use of intersections. 
>>>
>>> Thanks again. 
>>> Marco 
>>>
>>>
>>> ------------------------------------------------------------------------ 
>>> *Da: *"Andreas Dedner" <a.s.dedner at warwick.ac.uk> 
>>> *A: *"Marco Cisternino" <marco.cisternino at optimad.it>, 
>>> dune at dune-project.org 
>>> *Inviato: *Martedì, 24 febbraio 2015 16:55:38 
>>> *Oggetto: *Re: [Dune] parametrized AMR and cell divergence 
>>>
>>> Hi. 
>>> Conforming or non conforming? If non conforming you have a hanging 
>>> node and if you move coarse and fine intersections will not match 
>>> (the 
>>> coarse grid element does not know about the hanging nor where it 
>>> is). Could that be your problem? 
>>> Best 
>>> Andreas 
>>>
>>>
>>> On 24/02/15 15:39, Marco Cisternino wrote: 
>>>
>>> Hi Andreas, thank you for your reply. 
>>> I'm using GeometryGrid and ALU together. I provide ALU with a 
>>> DiscreteDeformationFunction (my coarse element 
>>> parametrization). Any time a new point appears in the mesh 
>>> after a refinement I move that point using the 
>>> DiscreteDeformationFunction. 
>>> The problem appears for local refinement. On the other hand, 
>>> if I globally refine the mesh I have no problem about the 
>>> divergence of the cells: it is zero everywhere. 
>>> However, if I refine only a bunch of cells, then the coarse 
>>> ones, sharing intersection with the fine ones, have non-zero 
>>> divergence. 
>>> I'm doing all this using the following methods: 
>>> intersection centerOuterUnitNormal 
>>> and codimension 1 volume method (choosing the element face 
>>> corresponding to the intersection) 
>>> I hope it is clearer. 
>>> Thanks, 
>>>
>>> Marco 
>>>
>>>
>>>
>>> ------------------------------------------------------------------------ 
>>> *Da: *"Andreas Dedner" <a.s.dedner at warwick.ac.uk> 
>>> *A: *dune at dune-project.org 
>>> *Inviato: *Martedì, 24 febbraio 2015 12:14:33 
>>> *Oggetto: *Re: [Dune] parametrized AMR and cell divergence 
>>>
>>> I'm not quite sure I understand the question. Are you seeing 
>>> problems 
>>> when refining the grid using ALU's standard globalRefine 
>>> method or do the problems 
>>> only appear when you use your local parametrization and after 
>>> moving nodes around 
>>> (I'm not quite clear how you do that with ALU....). 
>>> Andreas 
>>>
>>>
>>> On 24/02/15 11:00, Marco Cisternino wrote: 
>>>
>>> Good morning, 
>>> I would like to ask duners about deformed AMR. 
>>> I'm working with body fitted mesh and ALUGrid. 
>>>
>>> In order to refine my mesh I compute a local 
>>> parametrization for every coarse element, then I refine 
>>> the mesh moving new nodes using my parametrization. 
>>> Therefore I compute the cell divergence summing over the 
>>> cell intersection 
>>> normal . flux * area 
>>> where intersection normals and areas are computed by Dune 
>>> methods centerUnitNormal and volume(this for a face), 
>>> while flux is (1,1,1) everywhere. 
>>> I expect 0 over the entire mesh and this is true on the 
>>> coarse grid, but for a refined grid this is not true for 
>>> coarse cells sharing intersections with finer cells. 
>>> Does anyone see something like this? And is there a cure? 
>>> Am I doing something wrong? 
>>>
>>> Thanks for any hint. 
>>> Bests, 
>>>
>>> Marco