Cauchy Green Strain eigenvalues

[ZAARAOUI999]

Hi Mr Andreas, I wish you were fine, so I'm just asking if you have any idea about the results of the code given below. The ranking of the eigenvalues isn't the same, or it has no meaning?

The code:

import numpy as np
from casadi import SX, Function
from matadi.math import eigvals

#
C = SX.sym('C',3,3)
#
_wc = eigvals(C)
wc = Function('wc',[C],[_wc])
#
F = np.eye(3)
F[0,0]= 0.25
F[0,2] = 0.5
F[2,1] = 0.3
#
C = F.T@F
#
λi_casadi = wc(C)
λi_numpy, _ = np.linalg.eig(C)
#
print('λi_casadi: ', λi_casadi)
print('λi_numpy: ', λi_numpy)

The results:

λi_casadi:  [1.48739, 0.0485864, 0.866525]
λi_numpy:  [0.0484878  1.48742393 0.86658827]

Thanks.

[adtzlr]

Hi, thanks, I'm fine! Hope you're fine too? It's because I added small numbers to the diagonal entries in the eig-function in matadi. If you use plain casadi, eigenvalues will be identical to numpy. This ensures a stable gradient in case of equal eigenvalues but introduces some (minimal) numeric errors.

Here's the code

matadi/src/matadi/math.py

Lines 158 to 161 in 6820ca5

	def eigvals(T, eps=1e-4):
	D = DM([[1, 0, 0], [0, -1, 0], [0, 0, 0]])
	return eig_symbolic(T + D * eps)

[ZAARAOUI999]

I tried with the following code, but the ranking is not the same,

import numpy as np
from casadi import SX, Function, eig_symbolic

#
C = SX.sym('C',3,3)
#
_wc = eig_symbolic(C)
wc = Function('wc',[C],[_wc])
#
F = np.eye(3)
F[0,0]= 0.25
F[0,2] = 0.5
F[2,1] = 0.3
#
C = F.T@F
#
λi_casadi = wc(C)
λi_numpy, _ = np.linalg.eig(C)
#
print('λi_casadi: ', λi_casadi)
print('λi_numpy: ', λi_numpy)

This can have an effect on some hyperelastic models that have a stretch-based form, taking for example a modified Yeoh model, where the strain energy is given by;
$$W = \sum_{n=1}^3 C_{n0}(\lambda_n-1)^n$$
So here the ranking of eigenvalues should have sense.

[adtzlr]

Oh, you mean the order, not the small differences of the results. Sorry for my wrong interpretation of your question. Even numpy has different results on that, try to use eig and eigh. Isotropic hyperelasticity has always identical treatments of the stretches. Otherwise it's not isotropic. For non-isotropic stuff you have to sort the results manually - not every algorithm does that automatically.

See e.g. the numpy docs of eig: ...The eigenvalues are not necessarily ordered...

[adtzlr]

Do you have a literature reference on your Yeoh equation? For me that formulation does not make sense, are you sure it is $\lambda_n$ and not $\lambda_\alpha$ with a second sum inside the brackets?

[ZAARAOUI999]

Thank you for your time Mr Andreas, I didn't know that before and I should read more about this valuable information Isotropic hyperelasticity has always identical treatments of the stretches..
All the best.

[adtzlr]

Never mind. To be fair, you haven't written about isotropy - I assumed that because you mentioned the Yeoh model formulation.

[ZAARAOUI999]

Do you have a literature reference on your Yeoh equation? For me that formulation does not make sense, are you sure it is λn and not λα with a second sum inside the brackets?

No it was just an example where the invariants are replaced by the stretches.