Integrate pressure over area, from dataset points

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Francesco Brescia 2023-10-30

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https://ww2.mathworks.cn/matlabcentral/answers/2040626-integrate-pressure-over-area-from-dataset-points

评论： Francesco Brescia 2023-11-2

Hi, i want to compute the integral over the area of the pressure distribution p(x,y). I have vectors x and y ,say each one of size n by 1,containing the coordinates at which the pressure is known, and a vector of pressures, size n by 1, with the pressure at each coordinate. I thought i could use trapz 2 times to compute the integral first in one direction and then the other, but to do this i would need a Matrix of pressure, instead i have a vector of the same size of the coordinates x and y. I also know that the area I have is an anulus surface, as i can see from plotting using stem3(x,y,p). The coordinates are taken w.r.t the center of the anulus, in cartesian coordinates. Hope you can help me figure this out, thanks.

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Dyuman Joshi 2023-10-31

Why not just sum the discrete values of Pressure-force at each point?

Torsten 2023-10-31

编辑：Torsten 2023-10-31

If your suggestion was used, the density of the measurement points had to mirror the areas associated with them. But imagine data accumulating in a certain area while in another, there are almost no measurements. In this case, the weights for the datapoints in the integration couldn't be equally chosen.

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Answer 1

Torsten 2023-10-30

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https://ww2.mathworks.cn/matlabcentral/answers/2040626-integrate-pressure-over-area-from-dataset-points#answer_1343701

编辑：Torsten 2023-10-30

在 MATLAB Online 中打开

Maybe you want to get mean pressure over the face. In this case, divide "sol" by area = pi*(5^2-1^2) as done below.

x = load("x.mat")

x = struct with fields:

x1: [1323×1 double]

y = load("y.mat")

y = struct with fields:

y1: [1323×1 double]

p = load("p.mat")

p = struct with fields:

p: [1323×1 double]

plot(x.x1,y.y1,'o')

F = scatteredInterpolant(x.x1,y.y1,p.p);

fun = @(r,theta)F(r.*cos(theta),r.*sin(theta)).*r;

sol = integral2(fun,1,5,0,2*pi)

Warning: Reached the maximum number of function evaluations (10000). The result fails the global error test.

sol = 37.9349

mean_sol = sol/(pi*(5^2-1^2))

mean_sol = 0.5031

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Torsten 2023-10-31

编辑：Torsten 2023-10-31

在 MATLAB Online 中打开

I don't have your data - so I cannot comment on this.

If you integrate from 1 to 5, interpolation of your data within the hole is irrelevant.

At the points where you supply your data for pressure, the interpolated values should be equal to the original values. What "scatteredInterpolant" makes in between depends on the noisiness of your data.

Even for the random values above, the integration with "scatteredInterpolant" is near to the expected. So I'm quite surprised that you get unexpected results.

Since your data are gridded, applying trapz two times should also work:

x = load("x.mat");

x = x.x1;

y = load("y.mat");

y = y.y1;

p = load("p.mat");

p = p.p;

plot(x,y,'o')

rho = sqrt(x.^2+y.^2);

theta = atan2(y,x);

theta = wrapTo2Pi(theta);

plot(rho,theta,'o')

rho = reshape(rho,21,[]);

theta = reshape(theta,21,[]);

p = reshape(p,21,[]);

I1 = trapz(theta(1,:),trapz(rho(:,1),p.*rho,1)) % integrate in rho-direction first

I1 = 37.1605

I1 = I1/(pi*(5^2-1^2))

I1 = 0.4929

I2 = trapz(rho(:,1),trapz(theta(1,:),p,2).*rho(:,1)) % integrate in theta-direction first

I2 = 37.1605

I2 = I2/(pi*(5^2-1^2))

I2 = 0.4929

Torsten 2023-11-1

编辑：Torsten 2023-11-1

Study the example

Multiple Numerical Integrations

under

https://uk.mathworks.com/help/matlab/ref/trapz.html

to learn about the necessary format of your function data in order to use "trapz" for a 2d integration.

If you have problems to make your data compatible with the required format for "trapz", use the "scatteredInterpolant" approach - the setup is easy and the results won't differ significantly in my opinion.

Francesco Brescia 2023-11-2