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Hi.<br>
It might simply help to google Duffy transform(ation). That provides
the idea of how to<br>
go from a tensor product quadrature rule on a cube to a quadrature
on a simplex - but the<br>
resulting quadrature will be far from optimal w.r.t. the number of
points. <br>
Andreas<br>
<br>
<br>
<div class="moz-cite-prefix">On 06/03/15 10:21, Oliver Sander wrote:<br>
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<blockquote cite="mid:54F97FA2.4090605@igpm.rwth-aachen.de"
type="cite">
<pre wrap="">Hi Aleksejs,
</pre>
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<pre wrap="">So I have found the file
dune/geometry/quadraturerules/simplexquadrature.hh
In which it is stated that the highest quadrature order for a triangle is 12 and for a tetrahedron is 5. Is this correct?
</pre>
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<pre wrap="">
IIRC these are the highest available special-purpose simplex rules. You can get higher-order ones,
but those will be constructed by conical multiplication.
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<pre wrap="">What would you do if you need to integrate a higher order polynomial?
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<pre wrap="">
a) Simply request a higher-order formula from the quadrature rules cache.
const QuadratureRule<double,dim>& quad = QuadratureRules<double,dim>::rule(it->type(), your_order_here);
If this does not work, then
b) Add more rules to dune-geometry
The book by Stroud contain a lot of them, but I only have a print copy.
Look on the web for higher order quad rules. E.g., A library of such rules is at
<a class="moz-txt-link-freetext" href="http://www.cs.kuleuven.ac.be/~nines/research/ecf/ecf.html">http://www.cs.kuleuven.ac.be/~nines/research/ecf/ecf.html</a>
best,
Oliver
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<pre wrap="">
I would gladly read the book you have sent. Do you happen to have a digital version? I've found it on amazon for $148, but that is a bit too much I think, ignoring the fact that it is in US :)
Regards,
Aleksejs
On 06/03/15 09:55, Oliver Sander wrote:
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<pre wrap="">1) Does Dune possess only 1D quadrature rules, or 2D and 3D also?
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<pre wrap="">No. Dune specializes on partial differential equations in one-dimensional domains,
and therefore only 1d quadrature rules are needed.
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<pre wrap=""> I understand that a higher dimensional quadrature can be obtained from a tensor product of lower-dimensional quadratures, but this question is about quadratures specialised for reference element geometries.
2) If yes, for which reference elements, and to what order are these quadrature rules available.
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<pre wrap="">See dune-geometry/dune/geometry/quadraturerules
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3) If no, how does one usually integrate over, say, triangle, using a tensor product quadrature? Does one simply set the value of the function to 0 for all sample points outside the entity? How does that affect the accuracy of integration?
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</pre>
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<pre wrap=""><a class="moz-txt-link-freetext" href="https://openlibrary.org/books/OL4583400M/Approximate_calculation_of_multiple_integrals">https://openlibrary.org/books/OL4583400M/Approximate_calculation_of_multiple_integrals</a>
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</pre>
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<pre wrap="">4) Finally, what is known about using quadrature rules for integrating non-polynomial integrands? Can one estimate the quadrature order necessary to integrate, say, sqrt(x) to a given accuracy?
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</pre>
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<pre wrap=""><a class="moz-txt-link-freetext" href="https://openlibrary.org/books/OL4583400M/Approximate_calculation_of_multiple_integrals">https://openlibrary.org/books/OL4583400M/Approximate_calculation_of_multiple_integrals</a>
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<pre wrap="">Cheers,
Oliver
</pre>
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</pre>
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<pre wrap="">
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</pre>
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