Interpolating a Surface: Finding Approximations for Thermodynamic Properties

[Saladsamurai]

So I have this surface that I am trying to make approximations on. Now I have values for $f(x_1,y_1)$ and $f(x_2,y_2)$. I want to find the value of $f(x,y)$ where x and y lie between (x1, x2) and (y1, y2). I am willing to assume that f changes linearly in both x and y for this small interval. Would the appropriate interpolation function then be

$$f(x,y)=f(x_1,y_1) + \frac{f(x_2,y_2) - f(x_1,y_1)}{x_2 - x_1} *(x - x_1) + \frac{f(x_2,y_2) - f(x_1,y_1)}{y_2 - y_1} *(y - y_1)$$

?

I feel like it would be ... but I also feel like it could be

$$f(x,y)=<br /> f(x_1,y_1) + <br /> \sqrt{<br /> [\frac{f(x_2,y_2) - f(x_1,y_1)}{x_2 - x_1} *(x - x_1)]^2<br /> + <br /> [\frac{f(x_2,y_2) - f(x_1,y_1)}{y_2 - y_1} *(y - y_1)]^2<br /> }$$What do you think? I am really confusing myself here First or second one?

 

[LCKurtz]

Your second possibility isn't first degree in its variables, which is what I think you mean. Let's say f(x₁,y₁) = z₁ and f(x₂,y₂) = z₂

Then a general plane through the first point would be:

z = a(x-x₁) + b(y - y₁) + z₁.

Making the plane go through the second plane requires:

z₂ = a(x₂-x₁) + b(y₂ - y₁) + z₁

This does not completely determine a and b. The problem is that there are infinitely many planes, first degree in each variable, passing through those two points, so there isn't any reasonable way to pick a best one. If you think about the straight line joining your two points in 3D, any plane containing that line interpolates your two points. You need another condition.

 

[Saladsamurai]

Hi LCKurtz I think I am more confused now haha. I am thinking along the lines of something like this: I have some value f(x₁,y₁). Now couldn;t I say that for some local region "around" the point (x₁,y₁) I can approximate f(x,y) = f(x₁,y₁) + Δf where Δf is the "finite analog" to $df = (\partial{f} /\partial{x})*dx + (\partial{f}/\partial{y})*dy$.

Am I still missing your point? Or does that work? Note that this is a "real problem" that I am working on here. I have some tabulated data and I am making some assumptions about the nature of the unknown funciton.

Thanks.

 

[LCKurtz]

Your second possibility isn't first degree in its variables, which is what I think you mean. Let's say f(x₁,y₁) = z₁ and f(x₂,y₂) = z₂

Then a general plane through the first point would be:

z = a(x-x₁) + b(y - y₁) + z₁.

Making the plane go through the second plane requires:

z₂ = a(x₂-x₁) + b(y₂ - y₁) + z₁

This does not completely determine a and b. The problem is that there are infinitely many planes, first degree in each variable, passing through those two points, so there isn't any reasonable way to pick a best one. If you think about the straight line joining your two points in 3D, any plane containing that line interpolates your two points. You need another condition.

Hi LCKurtz I think I am more confused now haha. I am thinking along the lines of something like this: I have some value f(x₁,y₁). Now couldn;t I say that for some local region "around" the point (x₁,y₁) I can approximate f(x,y) = f(x₁,y₁) + Δf where Δf is the "finite analog" to $df = (\partial{f} /\partial{x})*dx + (\partial{f}/\partial{y})*dy$.

Am I still missing your point? Or does that work? Note that this is a "real problem" that I am working on here. I have some tabulated data and I am making some assumptions about the nature of the unknown funciton.

Thanks.

Yes, you could with more information. That is the same thing as my suggestion z = a(x-x₁) + b(y - y₁) + z₁ where your two partials are my a and b. The problem is that you don't have enough information to estimate the two partials f_x and f_y, which are a and b.

 

[Saladsamurai]

Yes, you could with more information. That is the same thing as my suggestion z = a(x-x₁) + b(y - y₁) + z₁ where your two partials are my a and b. The problem is that you don't have enough information to estimate the two partials f_x and f_y, which are a and b.

I am sorry, I still don't follow. I thought that I did estimate the two partials

$$(\partial{f} /\partial{x}) \approx \frac{f(x_2,y_2) - f(x_1,y_1)}{x_2 - x_1} \text{ and }(\partial{f} /\partial{y}) \approx\frac{f(x_2,y_2) - f(x_1,y_1)}{y_2 - y_1}$$

 

[LCKurtz]

But you wouldn't estimate the f_x by changing the y variable nor f_y by changing the x variable. You would hold the other variable constant.

Anyway, I'm guessing you haven't really posted what you want or need. Bivariate approximation is a big subject. Do you have a rectangular grid in the xy plane where you know the values? Or scattered values?

I would suggest you Google interpolation of surfaces and look around. You might find just what you need.

 

[Saladsamurai]

But you wouldn't estimate the f_x by changing the y variable nor f_y by changing the x variable. You would hold the other variable constant.

Anyway, I'm guessing you haven't really posted what you want or need. Bivariate approximation is a big subject. Do you have a rectangular grid in the xy plane where you know the values? Or scattered values?

I would suggest you Google interpolation of surfaces and look around. You might find just what you need.

Actually, I have been Googling around and got so confused that I came here

Let me post an example. I have a bunch of data tabulated as follows,

You can see that for some value of "P" in the header (for example P = 0.06 MPa) we have a bunch of Temperatures and corresponding values of properties. For example for
P = 0.06 MPa we can scroll down to T = 10 and see that the corresponding value of v is
v = 0.37861.

Similarly we can see that for P = 0.10 MPa and T = 30 we have v = 0.24216.

My question is what if I want to approximate a value of v for P = 0.08 MPa and T = 15 ?

I cannot seem to figure out what kind of bivariate interpolation this falls under, nor do I know what determines which kind of interpolation I should use. It seems like I have a regular rectangular grid here... I think. But not all of the tables like the ones above have the same temperature range. But working with the 2 above would be a good start.

 

[LCKurtz]

This is getting a bit out of my area of expertise. One method I have read a bit about is the quadratic Shepard's method. This interpolates 2d data with a smooth quadratic function. There are implementations in Fortran and I suppose other languages. For example, look at:

http://people.sc.fsu.edu/~jburkardt/f_src/qshep2d/qshep2d.html

I'm afraid beyond that, I can't help you much.

 

[Saladsamurai]

This is getting a bit out of my area of expertise. One method I have read a bit about is the quadratic Shepard's method. This interpolates 2d data with a smooth quadratic function. There are implementations in Fortran and I suppose other languages. For example, look at:

http://people.sc.fsu.edu/~jburkardt/f_src/qshep2d/qshep2d.html

I'm afraid beyond that, I can't help you much.

Ok. thanks for your help! I have never had to interpolate about 2 axes before. I have done plenty of regular linear interpolation and thought it was a simple jump to do 2D. I guess I thought wrong, eh?

 

[rootX]

You can see that for some value of "P" in the header (for example P = 0.06 MPa) we have a bunch of Temperatures and corresponding values of properties. For example for
P = 0.06 MPa we can scroll down to T = 10 and see that the corresponding value of v is
v = 0.37861.

Similarly we can see that for P = 0.10 MPa and T = 30 we have v = 0.24216.

My question is what if I want to approximate a value of v for P = 0.08 MPa and T = 15 ?

For these kind of thermodynamic problems, there can be different formulas for different input variable ranges e.g. PV = mRT is one simple formula. However, heat transfer has different formula for different ranges of the inputs.

Best possible thing I can think of at the moment is getting sufficient information to construct a plane. Two points A and B have infinite number of planes. If you get one or more points, you make a plane and use the points as estimation. This should work for smooth graphs without discontinuities.