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March 22, 2006, 17:49 
Numerical integration of 'T' across a 3D surface

#1 
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Hi folks,
Strictly speaking, this is not CFD but it's cloely related. I would be really grateful if anyone could help me out here: I have a threedimensional surface with approximately 50,000 nodes and a carpetlike distribution of temperature on it. I now need to compute the averaged value of temperature on this surface. Note that the nodes are nonuniformly spaced so that a simple statistical averaging is out of the question. For each node I have the x, y and zcoordinates and the value of the dependent variable T. There is no way that anyone could know the equation of the function T=f(x,y,z) so I need to perform a numerical integration (SIMPSON, TRAPEZOIDAL, etc.) in order to solve the triple integral . The question is: How? Do I start with the integration in x, then extend it to xy and then finally perform the same thing in xyz or do I better start with the whole thing by computing each node with respect to its surrounding nodes? The last point is really a question of what is easier to do unless there are mathpackages such as MATHCAD or MATLAB that have a buildin procedure implemented. Thanks a lot for your help. ACFDstudent 

March 23, 2006, 02:33 
Re: Numerical integration of 'T' across a 3D surfa

#2 
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The simplest possibility you can try is
T_avg=sum(i=1,n)t_ids_i/S . http://mathworld.wolfram.com/search/...al+integration+ points to the different kinds of integration you can try.But most of the integrations are done on a uniform mesh size in packages so it might be difficult to use matlab or mathcad. H 

March 23, 2006, 04:19 
Re: Numerical integration of 'T' across a 3D surfa

#3 
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One way of making the average would be to
1) make a triangulation of your surface 2) compute for each triangle the average temperature by taking the (nonweighted) average of the values in the 3 nodes 3) making the global average by weighting the values on each triangle by its surface 

March 23, 2006, 16:06 
Re: Numerical integration of 'T' across a 3D surfa

#4 
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Thanks Harish.
One question : What is T_ids and how do I compute this? Do you perhaps mean to subdivide the surface into a number of small faces (subareas 'S') where for each face the mean temperature t_ids is computed? I would really appreciate you coming back. Thanks. ACFDstudent 

March 23, 2006, 16:16 
Re: Numerical integration of 'T' across a 3D surfa

#5 
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Yves, thanks for your advise. I think I'll try this.
Should you or anyone else have any more thoughts on this, please post a note on this forum. Thanks guys. ACFDstudent 

March 23, 2006, 16:51 
Re: Numerical integration of 'T' across a 3D surfa

#6 
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What I meant was assuming all the small faces are square you can use the four corner points to calculate a mean average for that cell.There are many ways in doing this and is used extensively in Finite volumes.then use that approximation to for that small area as a constant and use it to average.
H 

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