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Question about the physical sizes of quadrants/octants at a specified level

Question about the physical sizes of quadrants/octants at a specified level

From:
Ethan Hereth
Date:
2015-03-10 @ 19:06
Hey all,

I hope I can make this clear; I want to know if it is possible to have a
p4est with quadrants at a specific level having physically different
dimensions?

In a non-isotropic or stretched forest, I can see how this could happen,
but if all of the trees in the forest are isotropic I do not think that any
quadrant at given level in the entire forest can have differing physical
dimensions. This implies, I think, that every (macro) tree in the forest
can have a maximum of six neighbor (macro) trees; or, in another way, no
(macro) tree can have more than one neighbor (macro) tree per face. Am I
correct?

What this should buy me is the ability to store one physical
length/area/volume per global level verses a length/area/volume per
quadrant.

Have I misunderstood anything by making this assumption? I hope my question
is clear.

Cheers!

Ethan Alan

Re: [p4est] Question about the physical sizes of quadrants/octants at a specified level

From:
Tobin Isaac
Date:
2015-03-10 @ 19:43
Hi Ethan,

On Tue, Mar 10, 2015 at 03:06:19PM -0400, Ethan Hereth wrote:
> Hey all,
> 
> I hope I can make this clear; I want to know if it is possible to have a
> p4est with quadrants at a specific level having physically different
> dimensions?
> 
> In a non-isotropic or stretched forest, I can see how this could happen,
> but if all of the trees in the forest are isotropic I do not think that any
> quadrant at given level in the entire forest can have differing physical
> dimensions.

The property that you need for this is for the mapping from the trees
reference domain [-1,1]^d into physical space to be an affine
mapping.  In that case, the volume each quadrant at a given level will
be 2^{-d*level} times the volume of the tree.

> This implies, I think, that every (macro) tree in the forest
> can have a maximum of six neighbor (macro) trees; or, in another way, no
> (macro) tree can have more than one neighbor (macro) tree per face. Am I
> correct?

The connectivity ("coarse mesh") of the forest is required to be a
conformal quadrilateral/hexahedral mesh, which guarantees at most one
neighbor per face.  No ouroboros meshes, sorry.

> 
> What this should buy me is the ability to store one physical
> length/area/volume per global level verses a length/area/volume per
> quadrant.

As I said above, if the mappings of the trees into space are affine,
then you can store one value per *tree*, and compute the lengths of
refined quadrants based only on their levels of refinement.

Cheers,
  Toby

> 
> Have I misunderstood anything by making this assumption? I hope my question
> is clear.
> 
> Cheers!
> 
> Ethan Alan

Re: [p4est] Question about the physical sizes of quadrants/octants at a specified level

From:
Ethan Hereth
Date:
2015-03-10 @ 20:29
Toby,

Thank you very much for your prompt response.

On Tue, Mar 10, 2015 at 3:43 PM, Tobin Isaac <tisaac@ices.utexas.edu> wrote:

>
> Hi Ethan,
>
> On Tue, Mar 10, 2015 at 03:06:19PM -0400, Ethan Hereth wrote:
> > Hey all,
> >
> > I hope I can make this clear; I want to know if it is possible to have a
> > p4est with quadrants at a specific level having physically different
> > dimensions?
> >
> > In a non-isotropic or stretched forest, I can see how this could happen,
> > but if all of the trees in the forest are isotropic I do not think that
> any
> > quadrant at given level in the entire forest can have differing physical
> > dimensions.
>
> The property that you need for this is for the mapping from the trees
> reference domain [-1,1]^d into physical space to be an affine
> mapping.  In that case, the volume each quadrant at a given level will
> be 2^{-d*level} times the volume of the tree.
>
> > This implies, I think, that every (macro) tree in the forest
> > can have a maximum of six neighbor (macro) trees; or, in another way, no
> > (macro) tree can have more than one neighbor (macro) tree per face. Am I
> > correct?
>
> The connectivity ("coarse mesh") of the forest is required to be a
> conformal quadrilateral/hexahedral mesh, which guarantees at most one
> neighbor per face.  No ouroboros meshes, sorry.
>

At the risk of sounding dense, I want to ask the following question: does
your last two statements imply that IF I have a conformal coarse/skeleton
mesh which is NOT of the ouroboros type, (had to look that one up!) the
mapping that p4est does from physical to computational space is affine?
These type of connectivities (conformal) are the only kind I need, so far!


> >
> > What this should buy me is the ability to store one physical
> > length/area/volume per global level verses a length/area/volume per
> > quadrant.
>
> As I said above, if the mappings of the trees into space are affine,
> then you can store one value per *tree*, and compute the lengths of
> refined quadrants based only on their levels of refinement.
>

A follow up question here: is there a case where a quadrant/octant at level
'l' in a particular tree has dimension different from another
quadrant/octant at the same level 'l' in another tree? If there is, I don't
see/understand it. This would imply that there could be trees of differing
dimension in the forest which doesn't make sense in an isotropic
connectivity. Surely I misunderstand you. I envision creating something
like

faceLenths[level] = {1.0, 0.25, 0.0125, ...}
faceAreas[level] = {1.0, 0.0625, 0.015625, ...}
etc.

I think I can store this at a global/forest level, not at a tree level. Am
I wrong?

Thanks again for your help, I must say you have always been very helpful
and for that I'm grateful.

Cheers,

Ethan Alan


> Cheers,
>   Toby
>
> >
> > Have I misunderstood anything by making this assumption? I hope my
> question
> > is clear.
> >
> > Cheers!
> >
> > Ethan Alan
>

Re: [p4est] Question about the physical sizes of quadrants/octants at a specified level

From:
Tobin Isaac
Date:
2015-03-11 @ 15:48
On Tue, Mar 10, 2015 at 04:29:27PM -0400, Ethan Hereth wrote:
> Toby,
> 
> Thank you very much for your prompt response.
> 
> On Tue, Mar 10, 2015 at 3:43 PM, Tobin Isaac <tisaac@ices.utexas.edu> wrote:
> 
> >
> > Hi Ethan,
> >
> > On Tue, Mar 10, 2015 at 03:06:19PM -0400, Ethan Hereth wrote:
> > > Hey all,
> > >
> > > I hope I can make this clear; I want to know if it is possible to have a
> > > p4est with quadrants at a specific level having physically different
> > > dimensions?
> > >
> > > In a non-isotropic or stretched forest, I can see how this could happen,
> > > but if all of the trees in the forest are isotropic I do not think that
> > any
> > > quadrant at given level in the entire forest can have differing physical
> > > dimensions.
> >
> > The property that you need for this is for the mapping from the trees
> > reference domain [-1,1]^d into physical space to be an affine
> > mapping.  In that case, the volume each quadrant at a given level will
> > be 2^{-d*level} times the volume of the tree.
> >
> > > This implies, I think, that every (macro) tree in the forest
> > > can have a maximum of six neighbor (macro) trees; or, in another way, no
> > > (macro) tree can have more than one neighbor (macro) tree per face. Am I
> > > correct?
> >
> > The connectivity ("coarse mesh") of the forest is required to be a
> > conformal quadrilateral/hexahedral mesh, which guarantees at most one
> > neighbor per face.  No ouroboros meshes, sorry.
> >
> 
> At the risk of sounding dense, I want to ask the following question: does
> your last two statements imply that IF I have a conformal coarse/skeleton
> mesh which is NOT of the ouroboros type, (had to look that one up!) the
> mapping that p4est does from physical to computational space is affine?
> These type of connectivities (conformal) are the only kind I need, so far!
> 

In general, no: if you simply specify an embedding by the location of
the 2^d coordinates, the mapping is bilinear / trilinear.  The mapping
will be affine only if the image of the tree is a parallelogram /
parallelepiped: otherwise, the Jacobian of the mapping changes.  There
may be some special cases where the Jacobian changes, but its
determinant does, but barring those the volumes of refined cells will
be different.

That being said, you can probably come up with some bounds on the
variation of cell volumes, as long as your mapping is a "nice" one.

> 
> > >
> > > What this should buy me is the ability to store one physical
> > > length/area/volume per global level verses a length/area/volume per
> > > quadrant.
> >
> > As I said above, if the mappings of the trees into space are affine,
> > then you can store one value per *tree*, and compute the lengths of
> > refined quadrants based only on their levels of refinement.
> >
> 
> A follow up question here: is there a case where a quadrant/octant at level
> 'l' in a particular tree has dimension different from another
> quadrant/octant at the same level 'l' in another tree? If there is, I don't
> see/understand it. This would imply that there could be trees of differing
> dimension in the forest which doesn't make sense in an isotropic
> connectivity. Surely I misunderstand you. I envision creating something
> like
> 
> faceLenths[level] = {1.0, 0.25, 0.0125, ...}
> faceAreas[level] = {1.0, 0.0625, 0.015625, ...}
> etc.

Imagine a degenerate quadrilateral, one that is almost a triangle:

```

  o-o
  |  \
  |   \
  |    \
  o-----o

```

If I map a quadtree onto this, the quadrants at the top will be
smaller than the quadrants at the bottom.

Cheers,
  Toby

> 
> I think I can store this at a global/forest level, not at a tree level. Am
> I wrong?
> 
> Thanks again for your help, I must say you have always been very helpful
> and for that I'm grateful.
> 
> Cheers,
> 
> Ethan Alan
> 
> 
> > Cheers,
> >   Toby
> >
> > >
> > > Have I misunderstood anything by making this assumption? I hope my
> > question
> > > is clear.
> > >
> > > Cheers!
> > >
> > > Ethan Alan
> >

Re: [p4est] Question about the physical sizes of quadrants/octants at a specified level

From:
Ethan Hereth
Date:
2015-03-12 @ 12:52
Morning Toby,

On Wed, Mar 11, 2015 at 11:48 AM, Tobin Isaac <tisaac@ices.utexas.edu>
wrote:

> On Tue, Mar 10, 2015 at 04:29:27PM -0400, Ethan Hereth wrote:
> > Toby,
> >
> > Thank you very much for your prompt response.
> >
> > On Tue, Mar 10, 2015 at 3:43 PM, Tobin Isaac <tisaac@ices.utexas.edu>
> wrote:
> >
> > >
> > > Hi Ethan,
> > >
> > > On Tue, Mar 10, 2015 at 03:06:19PM -0400, Ethan Hereth wrote:
> > > > Hey all,
> > > >
> > > > I hope I can make this clear; I want to know if it is possible to
> have a
> > > > p4est with quadrants at a specific level having physically different
> > > > dimensions?
> > > >
> > > > In a non-isotropic or stretched forest, I can see how this could
> happen,
> > > > but if all of the trees in the forest are isotropic I do not think
> that
> > > any
> > > > quadrant at given level in the entire forest can have differing
> physical
> > > > dimensions.
> > >
> > > The property that you need for this is for the mapping from the trees
> > > reference domain [-1,1]^d into physical space to be an affine
> > > mapping.  In that case, the volume each quadrant at a given level will
> > > be 2^{-d*level} times the volume of the tree.
> > >
> > > > This implies, I think, that every (macro) tree in the forest
> > > > can have a maximum of six neighbor (macro) trees; or, in another
> way, no
> > > > (macro) tree can have more than one neighbor (macro) tree per face.
> Am I
> > > > correct?
> > >
> > > The connectivity ("coarse mesh") of the forest is required to be a
> > > conformal quadrilateral/hexahedral mesh, which guarantees at most one
> > > neighbor per face.  No ouroboros meshes, sorry.
> > >
> >
> > At the risk of sounding dense, I want to ask the following question: does
> > your last two statements imply that IF I have a conformal coarse/skeleton
> > mesh which is NOT of the ouroboros type, (had to look that one up!) the
> > mapping that p4est does from physical to computational space is affine?
> > These type of connectivities (conformal) are the only kind I need, so
> far!
> >
>
> In general, no: if you simply specify an embedding by the location of
> the 2^d coordinates, the mapping is bilinear / trilinear.  The mapping
> will be affine only if the image of the tree is a parallelogram /
> parallelepiped: otherwise, the Jacobian of the mapping changes.  There
> may be some special cases where the Jacobian changes, but its
> determinant does, but barring those the volumes of refined cells will
> be different.
>
> That being said, you can probably come up with some bounds on the
> variation of cell volumes, as long as your mapping is a "nice" one.
>
>
Perfect, that's what I was looking for.


> >
> > > >
> > > > What this should buy me is the ability to store one physical
> > > > length/area/volume per global level verses a length/area/volume per
> > > > quadrant.
> > >
> > > As I said above, if the mappings of the trees into space are affine,
> > > then you can store one value per *tree*, and compute the lengths of
> > > refined quadrants based only on their levels of refinement.
> > >
> >
> > A follow up question here: is there a case where a quadrant/octant at
> level
> > 'l' in a particular tree has dimension different from another
> > quadrant/octant at the same level 'l' in another tree? If there is, I
> don't
> > see/understand it. This would imply that there could be trees of
> differing
> > dimension in the forest which doesn't make sense in an isotropic
> > connectivity. Surely I misunderstand you. I envision creating something
> > like
> >
> > faceLenths[level] = {1.0, 0.25, 0.0125, ...}
> > faceAreas[level] = {1.0, 0.0625, 0.015625, ...}
> > etc.
>
> Imagine a degenerate quadrilateral, one that is almost a triangle:
>
> ```
>
>   o-o
>   |  \
>   |   \
>   |    \
>   o-----o
>
> ```
>
> If I map a quadtree onto this, the quadrants at the top will be
> smaller than the quadrants at the bottom.
>

Right, I understand that completely; what I was trying to express by using
the term 'isotropic connectivity' was that degenerate
quadrilaterals/hexahedra like the one you sketched above don't exist in the
connectivity I'm using.

I think I'm good now!

Thanks again,

Ethan Alan


>
> Cheers,
>   Toby
>
> >
> > I think I can store this at a global/forest level, not at a tree level.
> Am
> > I wrong?
> >
> > Thanks again for your help, I must say you have always been very helpful
> > and for that I'm grateful.
> >
> > Cheers,
> >
> > Ethan Alan
> >
> >
> > > Cheers,
> > >   Toby
> > >
> > > >
> > > > Have I misunderstood anything by making this assumption? I hope my
> > > question
> > > > is clear.
> > > >
> > > > Cheers!
> > > >
> > > > Ethan Alan
> > >
>