[Dune-devel] Checking whether a given point is contained in a given element

[Martin Nolte]

Hi Oli, hi Andreas,

well, the restriction is only necessary for the local method on pyramid 
geometries (e.g., simplices, four-sided pyramid). Assuming that you will 
always use CachedMultiLinearGeometry, it will never be called for simplices, 
so you can safely remove the assertion. In the case of a 3d pyramid, however, 
the trouble will persist.

Please note, that the old generic geometry implementation behaved as described 
above with the assertion commented out. So this is not a new bug, but only a 
new assertion.

While the mapping might be globally well-defined (I did not check this), it 
becomes singular in the tip for a four-sided pyramid. This causes trouble with 
the Newton scheme used in local, because the derivative can no longer be 
computed. Maybe we need to resort to some bisection scheme in this case, but 
that might become extremely slow.

I am not sure, whether the problem can be resolved satisfactorily. Maybe we 
should revive the method checkInside on the geometry returning true if and 
only if a point in global coordinates lies within the image of the reference 
mapping (up to a tolerance). If the method returns true, it should be safe to 
call local. My feeling is that this method is simpler to implement that a 
local method that has to work for arbitrary points.

A last question: What is the exact semantics of local, if the point does not 
lie inside the geometric realization of an entity? For surface grids, you 
currently have the following (unintuitive) behavior: Assume that a point lies 
slightly "above" the actual geometric realization. Now, local will return the 
local coordinates of its orthogonal projection and checkInside will return 
true despite the fact that the point actually lies outside the geometric 
realization.

Best,

Martin



On 11/08/2012 02:29 PM, Oliver Sander wrote:
> Am 08.11.2012 13:20, schrieb Dedner, Andreas:
>> I think that the local method should not abort if the point is inside the
>> reference element
>> or not - and one of the reasons is that you need as Oliver said you need the
>> local method
>> to verify if a point is inside. Also I can think of applications where I
>> would like to
>> represent for example the barycenter of neighbors in the local coordinate
>> system
>> of an element - requiring me to use its local method...
>> I would thus suggest to remove that check (and think its mathematically fine
>> because these
>> transformations are global mappings in all the cases I can think of...)
> I certainly wouldn't object to that solution.  Strangely enough, though,
> after removing all assertions from MultiLinearGeometry, the code still
> crashed due to infs and nans appearing (only for simplices, now that I
> think about it).  Does the MLG code rely on the points being inside the
> reference element?
> best,
> Oliver
>
>>
>> Andreas
>> ________________________________________
>> From: dune-devel-bounces+a.s.dedner=warwick.ac.uk at dune-project.org
>> [dune-devel-bounces+a.s.dedner=warwick.ac.uk at dune-project.org] on behalf of
>> Oliver Sander [sander at igpm.rwth-aachen.de]
>> Sent: 08 November 2012 12:09
>> To: dune-devel at dune-project.org
>> Subject: [Dune-devel] Checking whether a given point is contained in a
>> given    element
>>
>> Dear Dune,
>> I have recently run into the following issue: I have a given point in world
>> space and an element.  I want to decide whether the element contains
>> the point.
>> So far, I have solved this by computing the coordinates of the point
>> wrt to the element (using Geometry::local(point)), and then calling
>> checkInside on the appropriate reference element.  That used to work
>> nicely.
>> Unfortunately, some grids have recently changed to MultiLinearGeometry
>> for their geometry implementations.  MLG has fairly tight preconditions:
>> in particular, its 'local' method aborts with an assertion failure if the
>> input argument is not contained in the element.
>> That does not seem like an unreasonable restriction to make.  After all,
>> strictly speaking, local element coordinates only exist on the element.
>> Unfortunately, now my old trick doesn't work anymore.  And I can't
>> think of another elegant why to decide whether a given point is contained
>> in a given element.
>>
>> Any ideas?
>>
>> cheers,
>> Oliver
>>
>>
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>>
>
>
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-- 
Dr. Martin Nolte <nolte at mathematik.uni-freiburg.de>

Universität Freiburg                                   phone: +49-761-203-5630
Abteilung für angewandte Mathematik                    fax:   +49-761-203-5632
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79104 Freiburg, Germany