Average Flux Across Edge

Hi. I need to compute the average flux across the edge in the diagram that is of length xa. Shouldn't the results from the function evaluateHeatFlux be weighted by the mesh segment lengths on that edge? For non-uniform meshes, as this is, the segment lengths are not all equal and thus the result given by evaluateHeatFlux should have less weight in a small segment versus a large segment (when averaging). Does Matlab have a built-in way of doing this?

results = solve(model);
[qx,qy] = evaluateHeatFlux(results);
Nodes_Xa = findNodes(model.Mesh,"region","Edge",Edge_Xa);
flux_Xa = qy(Nodes_Xa);
averageFlux = mean(flux_Xa); % SHOULDN'T THIS BE WEIGHTED BY THE ELEMENT EDGE LENGTH?

Thanks for any input.

Answer 1

[Torsten]

Should be

averageFlux = 1/(Nodes_Xa(end)-Nodes_Xa(1))*trapz(Nodes_Xa,flux_Xa)

[Paul Safier]

Actually, this would be the way. Need to use the coordinate values (xa) versus the node numbers (Nodes_Xa).

results = solve(model);
[qx,qy] = evaluateHeatFlux(results);
Nodes_Xa = findNodes(model.Mesh,"region","Edge",Edge_Xa); % Gives node numbers
flux_Xa = qy(Nodes_Xa);
xa = results.Mesh.Nodes(1,Nodes_Xa); % Gives coord values at nodes
averageFlux = 1/(xa(end)-xa(1))*trapz(xa,flux_Xa);

Thanks for the suggestion, @Torsten

[Paul Safier]

@Torsten what're your thoughts about needing to scale the results from evaluateHeatFlux in a plot like:

figure; plot(xa,flux_Xa,'ro')

Should the values in flux_Xa be scaled by the segment sizes of xa? Since heat flux is a "per unit area" quantity, it would seem that the area of each element (i.e. diff(xa)) would be important. Or, do you think that the built-in Matlab function evaluateHeatFlux handles the different areas of element edges on the domain boundary?

[Torsten]

Or, do you think that the built-in Matlab function evaluateHeatFlux handles the different areas of element edges on the domain boundary?

If you want to compute the total heat flux across the boundary in Watt, the edge lengths would be important because you take the integral over the edge. But using "evaluateHeatFlux" gives the local values in Watt/m^2 in the node points - thus the value is independent of the actual local edge lengths of your mesh.

[Paul Safier]

Thanks!

[Torsten]

I think "evaluateHeatRate" is what you are looking for:

https://uk.mathworks.com/help/pde/ug/pde.steadystatethermalresults.evaluateheatrate.html

It's the integrated heat flux normal to the boundary. To get the mean, you'll have to divide by the length of the edge, I guess.

[Paul Safier]

OK, I just tested that and it comes up as the exact same answer as the integral with trapz. Thanks!

[Ansel]

Thanks for the link, you saved my day.