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Numerical integration over an arbitrary 2D domain
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Alfonso Pagani
Jul 1, 2012, 2:15:31 AM7/1/12
to
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
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
Alexei Boulbitch
Jul 2, 2012, 5:29:19 AM7/2/12
to
Mathematica can solve multiple integrals, and there are function NIntegrate=
[ ] doing them numerically with a possibility to use various numeric metho=
ds and precisions. Check Menu/Help/NIntegrate.
You can, of course, also do a custom program, but I recommend to first try =
this function. If you already have done so and faced difficulties, post det=
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.b...@iee.lu<mailto:alexei.b...@iee.lu>
Fred Bartoli
Jul 2, 2012, 10:24:48 PM7/2/12
to
Alfonso Pagani a écrit :
Alfonso,
No need to mess with that.
Just use a weighting function and then integrate the weighted function
over a simple rectangular region that contains all your shape:
(* First define the integrating region *)
weightFunc[x_, y_] =
Piecewise[{{1, x^2 + y^2 <= 2 && x >= 0}, {1,
y >= 0 && y <= Sqrt[2] (x + 1)^2 && x >= -1 && x <= 0}, {1,
y <= 0 && y >= -Sqrt[2] (x + 1) && x >= -1 && x <= 0}}, 0];
(* Then plot it and check its shape *)
toLogic[v_] = Switch[v, 1, True, 0, False];
RegionPlot[toLogic@weightFunc[x, y], {x, -2, 2}, {y, -2, 2}]
(* Finally integrate the moment inertia about the (1/2,0) point *)
NIntegrate[((x - 1/2)^2 + y^2) weightFunc[x, y], {x, -2, 2}, {y, -2,
2}]
Integrate[((x - 1/2)^2 + y^2) weightFunc[x, y], {x, -2, 2}, {y, -2, 2}]
N[% - %%]
--
Thanks,
Fred.
No need to mess with that.
Just use a weighting function and then integrate the weighted function
over a simple rectangular region that contains all your shape:
(* First define the integrating region *)
weightFunc[x_, y_] =
Piecewise[{{1, x^2 + y^2 <= 2 && x >= 0}, {1,
y >= 0 && y <= Sqrt[2] (x + 1)^2 && x >= -1 && x <= 0}, {1,
y <= 0 && y >= -Sqrt[2] (x + 1) && x >= -1 && x <= 0}}, 0];
(* Then plot it and check its shape *)
toLogic[v_] = Switch[v, 1, True, 0, False];
RegionPlot[toLogic@weightFunc[x, y], {x, -2, 2}, {y, -2, 2}]
(* Finally integrate the moment inertia about the (1/2,0) point *)
NIntegrate[((x - 1/2)^2 + y^2) weightFunc[x, y], {x, -2, 2}, {y, -2,
2}]
Integrate[((x - 1/2)^2 + y^2) weightFunc[x, y], {x, -2, 2}, {y, -2, 2}]
N[% - %%]
--
Thanks,
Fred.
alfonso...@gmail.com
Jul 2, 2012, 10:16:09 PM7/2/12
to
Il giorno luned=EC 2 luglio 2012 11:29:19 UTC+2, Alexei Boulbitch ha scritt=
o:
te=
> [ ] doing them numerically with a possibility to use various numeric met=
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=
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
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=
> 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,
>
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=
> ds and precisions. Check Menu/Help/NIntegrate.
>
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.b...@iee.lu<mailto:alexei.b...@iee.lu>
Hi Alexei,
> 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.b...@iee.lu<mailto:alexei.b...@iee.lu>
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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