Alessandro Benedetto - 3 months ago 37

Python Question

I have some doubts on the numpy.dot product.

I define a matrix 6x6 like:

`C=np.zeros((6,6))`

C[0,0], C[1,1], C[2,2] = 129.5, 129.5, 129.5

C[3,3], C[4,4], C[5,5] = 25, 25, 25

C[0,1], C[0,2] = 82, 82

C[1,0], C[1,2] = 82, 82

C[2,0], C[2,1] = 82, 82

Then I recast it in a 4-rank tensor by using a multidimensional array

`def long2short(m, n):`

"""

Given two indices m and n of the stiffness tensor the function

return i the index of the Voigt matrix

i = long2short(m,n)

"""

if m == n:

i = m

elif (m == 1 and n == 2) or (m == 2 and n == 1):

i = 3

elif (m == 0 and n == 2) or (m == 2 and n == 0):

i = 4

elif (m == 0 and n == 1) or (m == 1 and n == 0):

i = 5

return i

c=np.zeros((3,3,3,3))

for m in range(3):

for n in range(3):

for o in range(3):

for p in range(3):

i = long2short(m, n)

j = long2short(o, p)

c[m, n, o, p] = C[i, j]

And then I would like to change the coordinate reference system of the tensor by using the rotation matrix that I define like:

`Q=np.array([[sqrt(2.0/3), 0, 1.0/sqrt(3)], [-1.0/sqrt(6), 1.0/sqrt(2), 1.0/sqrt(3)], [-1.0/sqrt(6), -1.0/sqrt(2), 1.0/sqrt(3)]])`

Qt = Q.transpose()

The matrix is orthogonal (althought the numerical precision is not perfect):

`In [157]: np.dot(Q, Qt)`

Out[157]:

array([[ 1.00000000e+00, 4.28259858e-17, 4.28259858e-17],

[ 4.28259858e-17, 1.00000000e+00, 2.24240114e-16],

[ 4.28259858e-17, 2.24240114e-16, 1.00000000e+00]])

But why then if I perform:

`In [158]: a=np.dot(Q,Qt)`

In [159]: c_mat=np.dot(a, c)

In [160]: a1 = np.dot(Qt, c)

In [161]: c_mat1=np.dot(Q, a1)

I get the expected value for c_mat (=c) but not for c_mat1? Is there some subtility to use dot on multidimensional arrays?

Answer

The issue is that `np.dot(a,b)`

for multidimensional arrays makes the dot product of the last dimension of `a`

with the second-to-last dimension of `b`

:

```
np.dot(a,b) == np.tensordot(a, b, axes=([-1],[2]))
```

As you see, it does not work as a matrix multiplication for multidimensional arrays. Using `np.tensordot()`

allows you to control in which `axes`

from each input you want to perform the dot product. For example, to get the same result in `c_mat1`

you can do:

```
c_mat1 = np.tensordot(Q, a1, axes=([-1],[0]))
```

Which is forcing a matrix multiplication-like behavior.