interpolation strain tensor field

I have a 3D+t strain tensor field (3D over time) delivered by FAM software (Abaqus). What I need is to interpolate the strain tensor on a fixed plane to gest a 2D tensor maps over time. Can I use the intrep3 function for that interpolating each component of a tensor? Is there any better solution for this problem. I have read that it is better to interpolate eigenvalues and eigenvectors but I did not found details about how to do this

Answer 1

[Nicola Bombace]

You can interpolate the strain components easily with the interp3 function, and the method linear.

Is it an elastic material? If so you could reconstruct the strain tensor by interpolating only the displacement and using the shape functions...

Which kind of elements are you using?

[BLP]

Thank you for quick answer. No, the material is not elastic, it is hyperelastic. The elemnts used in the FEM simulations are Eight-node brick element (C3D8).

[Nicola Bombace]

Hi. In this case, there is a stream of literature that considers the interpolation of second order tensors with eigenvalues and eigenvectors (for example here and here ). The functions you need to add to your workflow for such methodology is eig. To use it, you will have to construct the strain tensor in Matlab as a matrix. The interp3 function with linear option eig uses functions with the same order of continuity of your elements. Personally I have always used interpolation directly from the components, since is a very general approach. Hope this helps.

[Stephen23]

Beata Lesniak-Plewinska's "Answer" moved here:

Hi Nicola. Thank you for your answer. I have found the first paper, you suggested, but I did not find the second one. I am not very fluent in tensor calculus, and the second paper semms as a remedy for this? However, I will be grateful for explaining one thing to me. I am not sure what do you mean by constructing tensor as a matrix. Do you mean a 3D table?

[Stephen23]

Beata Lesniak-Plewinska's "Answer" moved here:

Sory Nicola, something must have stung me. What did you mean is that the tensor for a given node should be represented as here?

[Nicola Bombace]

Hi Beata,

the second paper is

"Kratz, Andrea, et al. "Visualization and analysis of second‐order tensors: Moving beyond the symmetric positive‐definite case." Computer Graphics Forum. Vol. 32. No. 1. Oxford, UK: Blackwell Publishing Ltd, 2013."

For the strain tensor what I meant is that Abaqus gives you the stress in Voigt notation while you want to transfer them in matrix notation as explained in this Wikipedia page. Let's suppose that your strains, at one moment in time, are saved as

[eps1, eps2, eps3 ,eps4, eps5, eps6]; %Voigt notation

E = [  eps1 eps6*0.5 eps5*0.5;
       eps6*0.5 eps2 eps4*0.5;
       eps5*0.5 eps4*0.5 eps3]; % Matrix notation

eigenVals = eig(E);  % Vector with eigenvalues

From here on you can continue with any algorithm that uses eigenvalues of the strain tensor.

[BLP]

Hi Nicola, Your help is invaluable. Thank you very much. I have the filling that you know very well both, Matlab and Abaqus. If yes, mayby you may suggest how to select nodes which strain tensors can be used to be interpolated on a constant plane? I thought about selecting those nodes which are within a certain distance from the plane and about selecting those nodes, which belong to one finite element cut by the plane. Unfortunately I get only numbers of the nodes and I do not know how to check which nodes belong to the same element.

[Nicola Bombace]

If I understand correctly you are using Abaqus Explicit. In this case, you should really use C3D8R, for possible volumetric locking of the elements. Using this element would give you three advantages:

1. Robust simulation, because of no locking.
2. Smaller Computational time, because of the reduced integration
3. You can get the strain directly from the integration point of the element (is an output of Abaqus), and this integration point is in the centre of the hexahedron.

Ps You are really welcome, I hope all this information helps :)

[BLP]

Thanks again. The problem is - its not me who is conducting the numerical simulatio. I am trying to use the results as a refernce for a physical model. Therefore, I do not know much about how it is done in Abaqus. But I will forward you suggestions to a colleague of mine.