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I have been trying to integrate over a surface of a 3D Mesh. The sample code looks like this

l0 = 0.1;
COD = 0.2;
X1 = 0.0;
Y1 = 0.0;
Z1 = 0.0;
X2 = 1.0;
Y2 = 8.0;
Z2 = 1.0;
Ymid = 0.5 (Y2 + Y1);
aDim = 0.5;
reg1=Rectangle[{X1,Y1},{X2,Y2}];
reg2=Triangle[{{X1,Ymid+COD},{X1,Ymid-COD},{aDim,Ymid}}];
reg3=RegionDifference[reg1,reg2];
reg4=RegionProduct[reg3,Line[{{Z1},{Z2}}]];
elVol =5(l0/2)^3/(6\[Sqrt]2);
MeshIn = ToElementMesh[reg4, MaxCellMeasure -> {"Volume" -> 0.0125}, 
"MeshOrder" -> 1, 
MeshRefinementFunction -> 
Function[{vertices, Vol}, 
Block[{x, y, z, tol}, {x, y, z} = Mean[vertices]; tol = 0.05; 
If[y < Ymid + tol && y > Ymid - tol && x > aDim - tol , 
 Vol > elVol, Vol > 5]]]]

I want to integrate a function

f[x_,y_,z_] = x y^2 + z^2;

over the surface x = 1. I can directly integrate using the command

NIntegrate[f[x,y,z],{y,0,8},{z,0,1}];

Is it possible to integrate over the finite element mesh using something like this

NIntegrate[f[x,y,z], {x, y, z} \[Element] MeshIn]

and I also want to choose x = 1. I need this because i want to integrate the finite element solution which is a global interpolation function over the surface of the mesh. I would appreciate any help in this regard.

Thanks

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  • $\begingroup$ what is the value for COD? $\endgroup$ Apr 24, 2018 at 18:26
  • $\begingroup$ @JoséAntonioDíazNavas my apologies i missed it. I have edited my question now. $\endgroup$
    – KVK318
    Apr 24, 2018 at 19:06
  • $\begingroup$ I get compile messages. I'd fix those since I assume the resulting mesh is not what you want otherwise. $\endgroup$
    – user21
    Apr 25, 2018 at 7:06
  • $\begingroup$ @user21 I am sorry for the sloppy posting. The mesh generation works now. $\endgroup$
    – KVK318
    Apr 25, 2018 at 19:29

1 Answer 1

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You could use:

NIntegrate[f[1, y, z], {x, y, z} \[Element] MeshIn]
172.063
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  • $\begingroup$ Thank you, this works correctly. $\endgroup$
    – KVK318
    Apr 25, 2018 at 19:33

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