[Dune] Handling checkinside for curvilinear elements

[Martin Nolte]

Hi Andreas, hi Aleksejs,

maybe the source of the confusion is an ongoing discussion on the behavior of 
local for points outside the set F( R ), where R denotes the reference element 
and ist the mapping described by the geometry.

If there is some x in R such that F( x ) = y, then the definition of local is 
clear and you can, e.g., apply Newton's method to find it. But what if there 
isn't? How far outside F( R ) does F have to be invertable? If we ignore 
surfaces for a second, this question is not an issue for (n-)linear mappings. 
However, for general nonlinear mappings, this might not be so easy. Consider the 
following example:

F(x_1,x_2) = (x_1^2 + x_2^2, x_2).

This map is perfectly invertable on the image of the refenrece triangle. But 
there is no inverse for points (y_1,y_2) with y_1 < y_2^2 (and they reside 
arbitrary near to the boundary of F( R )).

Best,

Martin

On 07/28/2014 06:10 PM, Andreas Dedner wrote:
> PS: for local I would have though you can use a Newton method.to find a root of
> F(y) = global(y)-x
> for some given global point x. That should be uniquely solvable otherwise your
> reference mapping is not
> one-on-one and that would be "!unfortunate".
> Andreas
>
> On 28/07/14 17:03, Andreas Dedner wrote:
>> Hi.
>> I think all your problems boil down to the last one: implementing the method
>> local.
>> If you have local then a point is a global point x is within an element if
>> local(x) is within the
>> reference element. That is what we use for the hoerarchicSearch class (a
>> utility class in
>> dune grid which should then be directly useable for your second point).
>> I don't think I understand your first problem since you are still using the
>> same reference element, right?
>> So a condition for a point being inside the reference element has nothing to
>> do with your global mapping,
>> be it straight faced or not.
>> Best
>> Andreas
>>
>> On 28/07/14 16:55, Aleksejs Fomins wrote:
>>> Dear Dune,
>>>
>>> I am currently implementing curvilinear tetrahedral grid for dune.
>>>
>>> The following functionality will be required:
>>>
>>> 1) dune/geometry/genericreferenceelements.hh has function isinside()
>>> which checks whether the point is inside of the tetrahedron. This
>>> functionality will not be valid for curvilinear tetrahedrons.
>>>
>>> Where should I implement the new functionality? Should I directly edit
>>> the genericreferenceelements.hh or should I define a new method inside
>>> my new version of multilineargeometry.hh, or sth else?
>>>
>>> Where is this functionality used?
>>>
>>> Also, if you have suggestions on how to mathematically check this,
>>> please advise me. I have a conjecture that the point is inside if it is
>>> on the correct side of each face, and it is on the correct side of the
>>> face if det(d, v1, v2) has the right sign, where d is the distance
>>> vector between this point and the closest point on the surface, and v1
>>> and v2 are the surface tangential unit vectors given by the parametric
>>> coordinates.
>>>
>>> 2) We personally would really need the functionality of finding the
>>> tetrahedron to which a given global coordinate belongs. Extending this
>>> to curvilinear is trivial - first find the element to which it would
>>> have belonged to if the tetrahedrons were straight-sided, then check
>>> whether it is actually inside this tetrahedron or one of its neighbors
>>> using the isinside() method.
>>>
>>> Is this (straight-sided) functionality already implemented in dune
>>> somewhere? Is it efficient? I mean, whether it loops over all elements
>>> and checks isinside(), or whether it uses something more sophisticated
>>> like octrees.
>>>
>>> 3) Another mathematics-related question. To map from local to global
>>> coordinates within an element I simply evaluate the linear combination
>>> of lagrange polynomials.
>>>
>>> How do I map from global to local?
>>>
>>> The best thing that comes to my mind is to write an iterative
>>> optimization algorithm minimizing the distance between given local
>>> coordinate and the expected one. It will have to be random like MCMC
>>> because the problem will have local minima. Please comment if you have
>>> good ideas :)
>>>
>>> Thank you,
>>> Aleksejs
>>>
>>>
>>>
>>> _______________________________________________
>>> Dune mailing list
>>> Dune at dune-project.org
>>> http://lists.dune-project.org/mailman/listinfo/dune
>>
>>
>>
>> _______________________________________________
>> Dune mailing list
>> Dune at dune-project.org
>> http://lists.dune-project.org/mailman/listinfo/dune
>
>
>
> _______________________________________________
> Dune mailing list
> Dune at dune-project.org
> http://lists.dune-project.org/mailman/listinfo/dune
>

-- 
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
Hermann-Herder-Straße 10
79104 Freiburg, Germany