Numerical differentiation of discrete data with noisy distribution need to be done with a suitable smoothing routine such as the so called "Savitzky-Golay smoothing filter", which is being used -as a built in function- by Matlab and many other numerical solver softwares.

However; one -very simple and straight forward- technique is by using OriginPro software, as follow:

Put you data as X, Y in an origin sheet, go to "Analysis" → "Mathematics" → "differentiate" → "Open Dialog" , tick the box of Savitzky-Golay smooth and select your desired smoothing polynomial order (i.e: 2 which is by default) and finally click OK.

...

And that is it, as clear as possible

Instead of using numerical differentiation, you can use autodifferentation. Autodifferentiation utilizes computational graphs (or operator overloading) along with the chain rule to get machine precision accurate derivatives since it doesn't have to take small epsilons. The Julia programming language is related to MATLAB and has very good packages for autodifferentiation, such as ForwardDiff.jl and ReverseDiff.jl

https://github.com/JuliaDiff/ForwardDiff.jl

https://github.com/JuliaDiff/ReverseDiff.jl

A deeper explanation of autodifferentiation can found in lots of places. Here's one common resource:

https://en.wikipedia.org/wiki/Automatic_differentiation

I assume a 2D set of data, for example on triangles. You can start using linear shape functions in each triangle in such a way that for the i-j-k nodes of the triangle, where you have the values fi,fj,fk you get

f(x,y)=Ni(x,y)*fi+Nj(x,y)*fj+Nk(x,y)*fk

Now, by definition of gradient

Grad f = i df/dx + j df/dy

you have the analytical form of the gradient in each triangle, also on the edge on the boundaries.

Quadratic shape functions can be built from the linear and you can get higher accuracy.

Thank you for all the responses. I've actually tried finding the gradient through the shape function before, but in my case, I have noises at the boundaries.

I will try the other suggestions...

you should show the results....noise (wiggles?) can be due to their presence in the set of discrete values not in the reconstruction method.

I presume that the plate that you model is supported and/or loaded on boundaries. The stress distribution is discretely disturbed at each node where you apply a constraint, or a nodal load. Using such variables in the shape functions will produce bad results. The modelling strategy is defined at the beginning of the analysis, based on the scope of this analysis. For the same, let say, simply supported plate with an out-of-plane load, a model made to calculate the deflection will be completely different with respect to a model build to study the hertzian contact on the boundary. The first one will be not used to extract the stresses near the boundary.

Thank you for your contributions. In one of my examples, I considered a cantilevered plate with shear load at the opposite edge, and it makes sense to me that the data is not so accurate at the clamped edge or where the load is applied. When I differentiate the stresses with the respect to the coordinates x and y (not with respect to the design variable), the error just gets worse.

I tried using filtering and it does not solve the problem. I am still looking into automatic differentiation. I don't understand how it works with discrete data yet. Also, it looks like adjoint method is a type of autodifferentiation. I am trying to understand how different it is from SGR (Spatial Gradient Reconstruction) which I have been using.

Hello Ochana Fernandes

Numerical derivatives obtained by using forward or backward differences are usually inacurate.

Thus the first derivative f'(a) ≈ ( f(a+h) - f(a))/h with an error proportional to h

A better approximation for the 1st derivative can be obtained by using

f'(a) = [(-3f(a) + 4f(a+h) -f(a+2h)] /(2h) with an error proportional to h2

Further improvements can be derived but they become cumbersome.

It is imperative that h is taken less than 1.

Mr. Anyaegbunam,

Thank you for your answer. I believe that SGR was more accurate for other examples.

You can approximate data, one of the known function classes and determine derivatives. You can use one of the difference schemes for example three or five points scheme. Sometimes, however, smoothing or filtration algorithms are necessary. Regards,