[Scilab-users] Int3D / Triple integration

[Stéphane Mottelet]

Hi Lester,

If I understand well, you are only interested by integrating on

[x1,x2] x [y1,y2] x [z1,z2]

and not a general volume, that's it ?

S.

Le 02/04/2021 à 11:53, arctica1963 a écrit :
> Hello all,
>
> An update on a solution. Following e-mail correspondence with a fellow
> Scilab user (Javier Domingo), he has worked out a general solution to
> vectorising the X,Y,Z arrays for the tetrahedrons required by int3d. So
> taking this part of the code and adding a call to int3d within a function we
> get a simpler route to doing triple integrals given a function (f) and lower
> and upper limits of integration defined by x1,x2,y1,y2,z1,z2:
>
> function [Integral, Error] = Integral_3d (f, x1, x2, y1, y2, z1, z2)
> // Divide prism (given by abscissa: x1, x2, ordinate: y1 to y2 and z1 to z2)
> into
> // 12 tetrahedra (not regular), starting from the center of the prism, cover
> all its volume;
> // providing the array IX (abscissa of the vertices of the triangles),
> // and the array IY (ordinate of the vertices of the triangles).
>     xc = (x1 + x2) / 2; yc = (y1 + y2) / 2; zc = (z1 + z2) / 2; // center of
> prism
> // coordinates of the prism tips (2 prisms on each face)
> // bottom -top- right -left- front -rear-
>     LX = [xc, xc, xc, xc, xc, xc, xc, xc, xc, xc, xc, xc;
>         x1, x1, x1, x1, x2, x2, x1, x1, x1, x1, x1, x1;
>         x2, x1, x2, x1, x2, x2, x1, x1, x2, x1, x2, x1;
>         x2, x2, x2, x2, x2, x2, x1, x1, x2, x2, x2, x2];
>     LY = [yc, yc, yc, yc, yc, yc, yc, yc, yc, yc, yc, yc;
>         y1, y1, y1, y1, y1, y2, y1, y2, y1, y1, y2, y2;
>         y1, y2, y1, y2, y2, y2, y2, y2, y1, y1, y2, y2;
>         y2, y2, y2, y2, y1, y1, y1, y1, y1, y1, y2, y2];
>     LZ = [zc, zc, zc, zc, zc, zc, zc, zc, zc, zc, zc, zc;
>         z1, z1, z2, z2, z1, z2, z1, z2, z1, z1, z1, z1;
>         z1, z1, z2, z2, z1, z1, z1, z1, z1, z2, z1, z2;
>         z1, z1, z2, z2, z2, z2, z2, z2, z2, z2, z2, z2];
>         
> [Integral, Error] = int3d (LX, LY, LZ, f, 1, [0,100000,1.d-5,1.d-7]);
>
> endfunction
>
> As a simple test one can define a function v=x^2 + y^2 + z^2 with limits of
> 0 to 1 - which simplifies to 1.0 as the sum of the iterated integrals.
>
> deff('v=f(xyz,numfun)','v=xyz(1)^2+xyz(2)^2+xyz(3)^2')
> x1=0;x2=1;y1=0;y2=1;z1=0;z2=1;
>
> --> [Integral, Error] = Integral_3d (f, x1, x2, y1, y2, z1, z2)
>   Integral  =
>
>     1.
>   Error  =
>
>     1.110D-14
>
> Thanks to Javier for his work on defining/clarifying the X,Y,Z arrays and
> logic for defining the equation in Scilab. As a suggestion it would seem
> reasonable to have this aspect either built into the function (int3d) or for
> a separate mesh3d function to build tetrahedrons in a format compatible with
> int3d.
>
> Always great to exchange ideas to formulate a solution to a problem.
>
> Code tested under Scilab version 6.1.0
>
> Lester
>
>
>
>
> --
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-- 
Stéphane Mottelet
Ingénieur de recherche
EA 4297 Transformations Intégrées de la Matière Renouvelable
Département Génie des Procédés Industriels
Sorbonne Universités - Université de Technologie de Compiègne
CS 60319, 60203 Compiègne cedex
Tel : +33(0)344234688
http://www.utc.fr/~mottelet