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Re: Numerical integration over an arbitrary 2D domain

  • To: mathgroup at smc.vnet.net
  • Subject: [mg127151] Re: Numerical integration over an arbitrary 2D domain
  • From: alfonso.pagani at gmail.com
  • Date: Mon, 2 Jul 2012 22:17:02 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <jsrplf$d3r$1@smc.vnet.net>

Il giorno luned=EC 2 luglio 2012 11:29:19 UTC+2, Alexei Boulbitch ha scritt=
o:
> Hi all,
> I am a new user of Wolfram Mathematica. I need to integrate
> (numerically of course) generic functions above complex 2D domains.
> For example, i would like to find the moments of inertia of an airfoil
> cross-section. Is it possible?
> My idea was to import a triangular mesh (list of nodes and element
> connectivity) of the integration domain. Then integrate my functions
> using a trapezoidal integration. How could i do this?
> Thank you very much,
>
> Alfonso
>
>
> Hi, Alfonso,
>
> Mathematica can solve multiple integrals, and there are function NIntegra=
te=
> [ ] doing them numerically with a possibility  to use various numeric met=
ho=
> ds and precisions. Check Menu/Help/NIntegrate.
>
> You can, of course, also do a custom program, but I recommend to first tr=
y =
> this function. If you already have done so and faced difficulties, post d=
et=
> ails of your integrand and domain. Otherwise your question is too general=
 t=
> o help you.
>
> Have fun, Alexei
>
>
> Alexei BOULBITCH, Dr., habil.
> IEE S.A.
> ZAE Weiergewan,
> 11, rue Edmond Reuter,
> L-5326 Contern, LUXEMBOURG
>
> Office phone :  +352-2454-2566
> Office fax:       +352-2454-3566
> mobile phone:  +49 151 52 40 66 44
>
> e-mail: alexei.boulbitch at iee.lu<mailto:alexei.boulbitch at iee.lu>

Hi Alexei,
thanks for your prompt answer.
Before posting my question I already checked NIntegrate.
Moreover I read "ADVANCED NUMERICAL INTEGRATION IN MATHEMATICA" from Wolfram Mathematica Tutorial Collection.
However I have not found yet the solution to my problem.
My integrands are the product of very simple functions of x and y.

Subscript[F[x_,y_],s]:=x^l y^m;

Where l and m depend on s. For the sake of simplicity, let F_1 and F_2 be as follows:

Subscript[F[x_,y_],1]:=x;
Subscript[F[x_,y_],2]:=y^2;

For example, if I had a rectangular integration domain I would write as follows:

Subscript[E,1,2] = Integrate[(Subscript[F, 1][x, y] Subscript[F, 2][x, y]), {x, -b/2, b/2}, {y, -h/2, h/2}]

The problem is that I need to integrate F_1 * F_2 over an arbitrary complex domain (such as a potato section!).
As a further complication, my domain could also be not simply connected.
Therefore I need necessarily to import my domain in Mathematica.
That's why my idea was to import a numerical mesh of the domain, ei to define my domain as composed by elementary (triangular) elements.

I hope I was sufficiently clear. Thank you for your help,

Alfonso



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