DAMASK_EICMD/python/tests/test_mechanics.py

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import pytest
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import numpy as np
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from damask import mechanics
def Cauchy(P,F):
sigma = 1.0/np.linalg.det(F) * np.dot(P,F.T)
return mechanics.symmetric(sigma)
def deviatoric_part(T):
return T - np.eye(3)*spherical_part(T)
def eigenvalues(T_sym):
return np.linalg.eigvalsh(symmetric(T_sym))
def eigenvectors(T_sym,RHS=False):
(u,v) = np.linalg.eigh(symmetric(T_sym))
if RHS:
if np.linalg.det(v) < 0.0: v[:,2] *= -1.0
return v
def left_stretch(T):
return polar_decomposition(T,'V')[0]
def maximum_shear(T_sym):
w = eigenvalues(T_sym)
return (w[0] - w[2])*0.5
def Mises_strain(epsilon):
return Mises(epsilon,2.0/3.0)
def Mises_stress(sigma):
return Mises(sigma,3.0/2.0)
def PK2(P,F):
S = np.dot(_np.linalg.inv(F),P)
return symmetric(S)
def right_stretch(T):
return polar_decomposition(T,'U')[0]
def rotational_part(T):
return polar_decomposition(T,'R')[0]
def spherical_part(T,tensor=False):
sph = np.trace(T)/3.0
return sph if not tensor else np.eye(3)*sph
def strain_tensor(F,t,m):
F_ = F.reshape(1,3,3)
if t == 'V':
B = np.matmul(F_,transpose(F_))
w,n = np.linalg.eigh(B)
elif t == 'U':
C = np.matmul(transpose(F_),F_)
w,n = np.linalg.eigh(C)
if m > 0.0:
eps = 1.0/(2.0*abs(m)) * (+ np.matmul(n,_np.einsum('ij,ikj->ijk',w**m,n))
- np.einsum('ijk->ijk',np.eye(3)))
elif m < 0.0:
eps = 1.0/(2.0*abs(m)) * (- np.matmul(n,_np.einsum('ij,ikj->ijk',w**m,n))
+ np.einsum('ijk->ijk',np.eye(3)))
else:
eps = np.matmul(n,np.einsum('ij,ikj->ijk',0.5*np.log(w),n))
return eps.reshape(3,3)
def symmetric(T):
return (T+transpose(T))*0.5
def transpose(T):
return T.T
def polar_decomposition(T,requested):
u, s, vh = np.linalg.svd(T)
R = np.dot(u,vh)
output = []
if 'R' in requested:
output.append(R)
if 'V' in requested:
output.append(np.dot(T,R.T))
if 'U' in requested:
output.append(np.dot(R.T,T))
return tuple(output)
def Mises(T_sym,s):
d = deviatoric_part(T_sym)
return np.sqrt(s*(np.sum(d**2.0)))
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class TestMechanics:
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n = 1000
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c = np.random.randint(n)
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@pytest.mark.parametrize('vectorized,single',[(mechanics.deviatoric_part, deviatoric_part),
(mechanics.spherical_part, spherical_part)
])
def test_vectorize_1_arg_(self,vectorized,single):
print("done")
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test_data_flat = np.random.rand(self.n,3,3)
test_data = np.reshape(test_data_flat,(self.n//10,10,3,3))
for i,v in enumerate(np.reshape(vectorized(test_data),vectorized(test_data_flat).shape)):
assert np.allclose(single(test_data_flat[i]),v)
@pytest.mark.parametrize('vectorized,single',[
(mechanics.deviatoric_part, deviatoric_part),
(mechanics.eigenvalues , eigenvalues ),
(mechanics.eigenvectors , eigenvectors ),
(mechanics.left_stretch , left_stretch ),
(mechanics.maximum_shear , maximum_shear ),
(mechanics.Mises_strain , Mises_strain ),
(mechanics.Mises_stress , Mises_stress ),
(mechanics.right_stretch , right_stretch ),
(mechanics.rotational_part, rotational_part),
(mechanics.spherical_part , spherical_part ),
(mechanics.symmetric , symmetric ),
(mechanics.transpose , transpose ),
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])
def test_vectorize_1_arg(self,vectorized,single):
epsilon = np.random.rand(self.n,3,3)
epsilon_vec = np.reshape(epsilon,(self.n//10,10,3,3))
for i,v in enumerate(np.reshape(vectorized(epsilon_vec),vectorized(epsilon).shape)):
assert np.allclose(single(epsilon[i]),v)
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@pytest.mark.parametrize('vectorized,single',[
(mechanics.Cauchy,Cauchy),
(mechanics.PK2 ,PK2 )
])
def test_vectorize_2_arg(self,vectorized,single):
P = np.random.rand(self.n,3,3)
F = np.random.rand(self.n,3,3)
P_vec = np.random.rand(self.n//10,10,3,3)
F_vec = np.random.rand(self.n//10,10,3,3)
for i,v in enumerate(np.reshape(vectorized(P_vec,F_vec),vectorized(P,F).shape)):
assert np.allclose(single(P[i],F[i]),v)
@pytest.mark.parametrize('vectorized,single',[(mechanics.strain_tensor,strain_tensor)])
def test_vectorize_strain_tensor(self,vectroized,single):
F = np.random.rand(self.n,3,3)
F_vec = np.random.rand(self.n//10,10,3,3)
t = ['V','U'][np.random.randint(0,2)]
m = np.random.random()*10.0 -5.0
for i,v in enumerate(np.reshape(vectorized(F_vec,t,m),vectorized(F,t,m).shape)):
assert np.allcloase(single(F[i],t,m),v)
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@pytest.mark.parametrize('function',[mechanics.Cauchy,
mechanics.PK2,
])
def test_stress_measures(self,function):
"""Ensure that all stress measures are equivalent for no deformation."""
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P = np.random.rand(self.n,3,3)
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assert np.allclose(function(P,np.broadcast_to(np.eye(3),(self.n,3,3))),mechanics.symmetric(P))
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def test_deviatoric_part(self):
I_n = np.broadcast_to(np.eye(3),(self.n,3,3))
r = np.logical_not(I_n)*np.random.rand(self.n,3,3)
assert np.allclose(mechanics.deviatoric_part(I_n+r),r)
def test_polar_decomposition(self):
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"""F = RU = VR."""
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F = np.broadcast_to(np.eye(3),[self.n,3,3])*np.random.rand(self.n,3,3)
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R = mechanics.rotational_part(F)
V = mechanics.left_stretch(F)
U = mechanics.right_stretch(F)
assert np.allclose(np.matmul(R,U),
np.matmul(V,R))
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def test_strain_tensor_no_rotation(self):
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"""Ensure that left and right stretch give same results for no rotation."""
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F = np.broadcast_to(np.eye(3),[self.n,3,3])*np.random.rand(self.n,3,3)
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m = np.random.random()*20.0-10.0
assert np.allclose(mechanics.strain_tensor(F,'U',m),
mechanics.strain_tensor(F,'V',m))
def test_strain_tensor_rotation_equivalence(self):
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"""Ensure that left and right strain differ only by a rotation."""
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F = np.broadcast_to(np.eye(3),[self.n,3,3]) + (np.random.rand(self.n,3,3)*0.5 - 0.25)
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m = np.random.random()*5.0-2.5
assert np.allclose(np.linalg.det(mechanics.strain_tensor(F,'U',m)),
np.linalg.det(mechanics.strain_tensor(F,'V',m)))
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def test_strain_tensor_rotation(self):
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"""Ensure that pure rotation results in no strain."""
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F = mechanics.rotational_part(np.random.rand(self.n,3,3))
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t = ['V','U'][np.random.randint(0,2)]
m = np.random.random()*2.0 - 1.0
assert np.allclose(mechanics.strain_tensor(F,t,m),
0.0)
def test_rotation_determinant(self):
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"""
Ensure that the determinant of the rotational part is +- 1.
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Should be +1, but random F might contain a reflection.
"""
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x = np.random.rand(self.n,3,3)
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assert np.allclose(np.abs(np.linalg.det(mechanics.rotational_part(x))),
1.0)
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def test_spherical_deviatoric_part(self):
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"""Ensure that full tensor is sum of spherical and deviatoric part."""
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x = np.random.rand(self.n,3,3)
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sph = mechanics.spherical_part(x,True)
assert np.allclose(sph + mechanics.deviatoric_part(x),
x)
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def test_deviatoric_Mises(self):
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"""Ensure that Mises equivalent stress depends only on deviatoric part."""
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x = np.random.rand(self.n,3,3)
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full = mechanics.Mises_stress(x)
dev = mechanics.Mises_stress(mechanics.deviatoric_part(x))
assert np.allclose(full,
dev)
def test_spherical_mapping(self):
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"""Ensure that mapping to tensor is correct."""
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x = np.random.rand(self.n,3,3)
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tensor = mechanics.spherical_part(x,True)
scalar = mechanics.spherical_part(x)
assert np.allclose(np.linalg.det(tensor),
scalar**3.0)
def test_spherical_Mises(self):
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"""Ensure that Mises equivalent strrain of spherical strain is 0."""
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x = np.random.rand(self.n,3,3)
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sph = mechanics.spherical_part(x,True)
assert np.allclose(mechanics.Mises_strain(sph),
0.0)
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def test_symmetric(self):
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"""Ensure that a symmetric tensor is half of the sum of a tensor and its transpose."""
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x = np.random.rand(self.n,3,3)
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assert np.allclose(mechanics.symmetric(x)*2.0,
mechanics.transpose(x)+x)
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def test_transpose(self):
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"""Ensure that a symmetric tensor equals its transpose."""
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x = mechanics.symmetric(np.random.rand(self.n,3,3))
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assert np.allclose(mechanics.transpose(x),
x)
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def test_Mises(self):
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"""Ensure that equivalent stress is 3/2 of equivalent strain."""
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x = np.random.rand(self.n,3,3)
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assert np.allclose(mechanics.Mises_stress(x)/mechanics.Mises_strain(x),
1.5)
def test_eigenvalues(self):
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"""Ensure that the characteristic polynomial can be solved."""
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A = mechanics.symmetric(np.random.rand(self.n,3,3))
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lambd = mechanics.eigenvalues(A)
s = np.random.randint(self.n)
for i in range(3):
assert np.allclose(np.linalg.det(A[s]-lambd[s,i]*np.eye(3)),.0)
def test_eigenvalues_and_vectors(self):
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"""Ensure that eigenvalues and -vectors are the solution to the characteristic polynomial."""
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A = mechanics.symmetric(np.random.rand(self.n,3,3))
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lambd = mechanics.eigenvalues(A)
x = mechanics.eigenvectors(A)
s = np.random.randint(self.n)
for i in range(3):
assert np.allclose(np.dot(A[s]-lambd[s,i]*np.eye(3),x[s,:,i]),.0)
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def test_eigenvectors_RHS(self):
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"""Ensure that RHS coordinate system does only change sign of determinant."""
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A = mechanics.symmetric(np.random.rand(self.n,3,3))
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LRHS = np.linalg.det(mechanics.eigenvectors(A,RHS=False))
RHS = np.linalg.det(mechanics.eigenvectors(A,RHS=True))
assert np.allclose(np.abs(LRHS),RHS)
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def test_spherical_no_shear(self):
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"""Ensure that sherical stress has max shear of 0.0."""
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A = mechanics.spherical_part(mechanics.symmetric(np.random.rand(self.n,3,3)),True)
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assert np.allclose(mechanics.maximum_shear(A),0.0)