TensorPrincipalInvariants filter

ParaView Support
1

Hi

I am wondering if the expected behavior of the TensorPrincipalInvariants filter is to calculate the eigenvectors as sigma1(vector), sigma2(vector) and sigma3(vector).

If so, I am not able to match the results in my dataset.

These are the tensor definition (I assume in the order XX-YY-ZZ-XY-YZ-XZ)
{E993F715-2486-47BA-90C8-41DC89F7F77F}

And these is calculated sigma1 (vector):
{E7BEFB08-FE28-4F7F-9ABD-473AB459E6EF}

I could not match these values by manual calculation.

Would the plugin by any chance read the tensor in another order?

One more information: it seems that the sign of the sigma’s “Y” variable is inverted in this case… this means that the eigenvalues are ok, but the eigenvectors sometimes go to the wrong quadrant.

2

Here is how the data is recovered in the filter:

      tensor[0][0] = array->GetComponent(idx, 0); 
      tensor[1][1] = array->GetComponent(idx, 1); 
      tensor[2][2] = array->GetComponent(idx, 2); 
      tensor[0][1] = array->GetComponent(idx, 3); 
      tensor[1][0] = array->GetComponent(idx, 3); 
      tensor[1][2] = array->GetComponent(idx, 4); 
      tensor[2][1] = array->GetComponent(idx, 4); 
      tensor[0][2] = array->GetComponent(idx, 5); 
      tensor[2][0] = array->GetComponent(idx, 5);

Hence the order is:

XX YY ZZ XY YZ XZ

So looks like your expectation is correct, I do not know why you have different results.

3

If I understood everything correctly, these are the eigenvectors and eigenvalues.
So, this must be valid:

TENSOR * \sigma_1(vec) = \sigma_1(lambda) * \sigma_1 (vec)

I cannot get this equality withe the values calculated by the filter.
Look at the first eigenvector our of the ultra-simple python code below (the y component should be negative):

TENSOR:
[[ 1.36070e+04 -3.97784e+04 -2.82264e+01]
 [-3.97784e+04 -4.70215e+03  5.65523e+01]
 [-2.82264e+01  5.65523e+01  1.70698e+03]]
EVAL:
 [45270.72363685 -36365.81693694   1706.92330009]
EVEC:
 [[ 7.82391753e-01 -6.22784873e-01  1.46537903e-03] <== here
 [-6.22785208e-01 -7.82392812e-01 -2.71208641e-04]
 [-1.31540666e-03  7.00424981e-04  9.99998890e-01]]

Python:

import numpy as np

mxx = 13607
myy = -4702.15
mzz = 1706.98
mxy = -39778.4
myz = 56.5523
mxz = -28.2264
M = np.array( [[ mxx, mxy, mxz ], [ mxy, myy, myz ] , [ mxz, myz, mzz ]] )
print(f"TENSOR:\n{M}")
eval, evec = np.linalg.eig(M)
print( f"EVAL:\n {eval}" );
print ( f"EVEC:\n {evec}" )
4

I’m afraid investigation into the filter is needed to figure this out.