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test_tensors.py
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#!/usr/bin/env python
# -*- coding: utf-8 -*-
import numpy as np
import pprint
import mechkit
basic = mechkit.tensors.Basic()
con = mechkit.notation.Converter()
##########################################
# Helpers
def random_stiffness():
"""Create random tensor of fourth order with inner symmetry,
i.e. minor and major symmetry
"""
tensor = np.random.rand(3, 3, 3, 3)
t_mandel = con.to_mandel6(tensor)
return con.to_tensor(0.5 * (t_mandel + t_mandel.transpose()))
def define_K_G_in_stiffness(stiffness, K, G):
"""Replace isotropic part by known isotropic stiffness"""
s = con.to_mandel6(stiffness)
m = mechkit.material.Isotropic(K=K, G=G)
for r, row in enumerate(m.stiffness_mandel6):
for c, col in enumerate(row):
if not np.allclose(col, 0.0):
s[r, c] = col
return con.to_tensor(s)
##########################################
# Tests
def test_I2():
it = np.array(
[
[1.0, 0.0, 0.0],
[0.0, 1.0, 0.0],
[0.0, 0.0, 1.0],
]
)
assert np.allclose(basic.I2, it)
def test_I4():
it = np.array(
[
[
1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
1.0,
],
]
)
assert np.allclose(con.to_mandel9(basic.I4), it)
def test_I4s():
it = np.array(
[
[
1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
]
)
assert np.allclose(con.to_mandel9(basic.I4s), it)
def test_I4a():
it = np.array(
[
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
1.0,
],
]
)
assert np.allclose(con.to_mandel9(basic.I4a), it)
def test_biorthogonal_P1P2():
a = np.einsum("ijkl, klmn ->ijmn", basic.P1, basic.P2)
assert np.allclose(a, np.zeros((a.shape)))
def test_biorthogonal_P2P1():
a = np.einsum("ijkl, klmn ->ijmn", basic.P2, basic.P1)
assert np.allclose(a, np.zeros((a.shape)))
def test_idempotenz_P1():
a = np.einsum("ijkl, klmn ->ijmn", basic.P1, basic.P1)
assert np.allclose(a, basic.P1)
def test_idempotenz_P2():
a = np.einsum("ijkl, klmn ->ijmn", basic.P2, basic.P2)
assert np.allclose(a, basic.P2)
def test_norm_P1():
assert np.allclose(np.linalg.norm(basic.P1), 1.0)
def test_norm_P2():
assert np.allclose(np.linalg.norm(basic.P2), np.sqrt(5))
def test_effect_P1():
K = 100
G = 61
rs = random_stiffness()
s = define_K_G_in_stiffness(rs, K, G)
K_result = np.einsum("ijkl, ijkl ->", s, basic.P1) / 3.0
assert np.allclose(K, K_result)
def test_effect_P2():
"""Norm of"""
K = 145
G = 43
rs = random_stiffness()
s = define_K_G_in_stiffness(rs, K, G)
G_result = np.einsum("ijkl, ijkl ->", s, basic.P2) / 2.0 / 5.0
assert np.allclose(G, G_result)
def test_I6s():
# Note:
# Does A have to be symmetric for formula of JA being valid?
# Or is the implementation of I6s wrong?
A = np.einsum("ijkl, kl-> ij", basic.I4s, np.random.rand(3, 3))
I2 = basic.I2
JA = (
1.0
/ 4.0
* (
np.einsum("im, jn ->ijmn", A, I2)
+ np.einsum("in, jm ->ijmn", A, I2)
+ np.einsum("jn, im ->ijmn", A, I2)
+ np.einsum("jm, in ->ijmn", A, I2)
)
)
I6sA = np.einsum("ijklmn, mn->ijkl", basic.I6s, A)
pprint.pprint(con.to_mandel9(JA))
pprint.pprint(con.to_mandel9(I6sA))
assert np.allclose(JA, I6sA)
if __name__ == "__main__":
rs = random_stiffness()
s = define_K_G_in_stiffness(stiffness=rs)