Rank: Member Groups: Registered
Joined: 02/05/2020(UTC) Posts: 22 Was thanked: 1 time(s) in 1 post(s)

Hi everyone, I'm looking for some tips to make applications about the integration of a function over a closed non rectangular domain (i.e Green's theorem) Edited by user 22 May 2020 15:54:46(UTC)
 Reason: Not specified 
La fisica è come il sesso: non c'è dubbio che praticandola si ottengano risultati pratici, ma non è per questo che la si fa. 



Rank: Advanced Member Groups: Registered
Joined: 04/07/2015(UTC) Posts: 4,585 Was thanked: 775 time(s) in 615 post(s)

In short: area of a closed contour goes by the rule of the polygon. Something in preparation ~ one day. Not so easy on some difficult f(x,y) that have no symbolic. Cheers ... Jean

1 user thanked Jean Giraud for this useful post.



Rank: Member Groups: Registered
Joined: 02/05/2020(UTC) Posts: 22 Was thanked: 1 time(s) in 1 post(s)

On a textbook I found some applications with a fictitious rettangular domain and a function (theta in the picture) defined from vertices coords, that accounts for the effective boundary. It certainly works, but maybe there is something more elegant, even for numerical applications ... Edited by user 22 May 2020 18:07:21(UTC)
 Reason: Not specified 
La fisica è come il sesso: non c'è dubbio che praticandola si ottengano risultati pratici, ma non è per questo che la si fa. 



Rank: Advanced Member Groups: Registered
Joined: 04/07/2015(UTC) Posts: 4,585 Was thanked: 775 time(s) in 615 post(s)

That was done multiple times, recently. Construct the polygons from the book data. Attach the Smath document ... like done. Cheers ... Jean




Rank: Advanced Member Groups: Registered
Joined: 15/04/2012(UTC) Posts: 1,576 Was thanked: 853 time(s) in 542 post(s)

Originally Posted by: SteelCat On a textbook I found some applications with a fictitious rettangular domain and a function (theta in the picture) defined from vertices coords, that accounts for the effective boundary. It certainly works, but maybe there is something more elegant, even for numerical applications ... You could use finite elements. Create a mesh e.g. of triangles and use some quadrature formula inside the triangles based on some interpolation formula between the vertices (and for second order approximation also midside node). This provides an approximation which gets better with finer mesh. So what you need is:  A triangulation method
 an generic element integrator and
 some integration loop over all elements.
The most simple integrator for linear triangles is the average of the nodal values times the area of the triangle, but you also could use some quadrature formula based on internal locations (like Gauss quadrature). If you have a closed form integral for a single triangle of arbitrary size, then you can create a triangulation just using the boundary points without further mesh generation. The result is exact in this case. 

2 users thanked mkraska for this useful post.

on 23/05/2020(UTC), on 23/05/2020(UTC)


Rank: Member Groups: Registered
Joined: 02/05/2020(UTC) Posts: 22 Was thanked: 1 time(s) in 1 post(s)

Originally Posted by: mkraska Originally Posted by: SteelCat On a textbook I found some applications with a fictitious rettangular domain and a function (theta in the picture) defined from vertices coords, that accounts for the effective boundary. It certainly works, but maybe there is something more elegant, even for numerical applications ... You could use finite elements. Create a mesh e.g. of triangles and use some quadrature formula inside the triangles based on some interpolation formula between the vertices (and for second order approximation also midside node). This provides an approximation which gets better with finer mesh. So what you need is:  A triangulation method
 an generic element integrator and
 some integration loop over all elements.
The most simple integrator for linear triangles is the average of the nodal values times the area of the triangle, but you also could use some quadrature formula based on internal locations (like Gauss quadrature). If you have a closed form integral for a single triangle of arbitrary size, then you can create a triangulation just using the boundary points without further mesh generation. The result is exact in this case. Many thanks for your point of view. 
La fisica è come il sesso: non c'è dubbio che praticandola si ottengano risultati pratici, ma non è per questo che la si fa. 



Rank: Advanced Member Groups: Registered
Joined: 04/07/2015(UTC) Posts: 4,585 Was thanked: 775 time(s) in 615 post(s)

Originally Posted by: SteelCat It certainly works, but maybe there is something more elegant, even for numerical applications ... More elegant than what ? Maths Polygon Area_Perimeter.sm (76kb) downloaded 10 time(s).1. Construct your 2 polygons. 2. From the big one: subtract area of small from big. 3. That tread goes no where from nothing.




Forum Jump
You cannot post new topics in this forum.
You cannot reply to topics in this forum.
You cannot delete your posts in this forum.
You cannot edit your posts in this forum.
You cannot create polls in this forum.
You cannot vote in polls in this forum.