eigenvalues is more specific name than principal components

python/damask/mechanics.py

@@ -37,7 +37,7 @@ def PK2(F,P):
         S = np.dot(np.linalg.inv(F),P)
     else:
         S = np.einsum('ijk,ikl->ijl',np.linalg.inv(F),P)
-    return S
+    return symmetric(S)
 
 
 def strain_tensor(F,t,m):
@@ -168,27 +168,41 @@ def maximum_shear(x):
       Symmetric tensor of which the maximum shear is computed.
 
     """
-    w = np.linalg.eigvalsh(symmetric(x))                                                            # eigenvalues in ascending order
-    return (w[2] - w[0])*0.5 if np.shape(x) == (3,3) else \
-           (w[:,2] - w[:,0])*0.5
+    w = eigenvalues(x)
+    return (w[0] - w[2])*0.5 if np.shape(x) == (3,3) else \
+           (w[:,0] - w[:,2])*0.5
 
 
-def principal_components(x):
+def eigenvalues(x):
     """
-    Return the principal components of a symmetric tensor.
+    Return the eigenvalues, i.e. principal components, of a symmetric tensor.
 
-    The principal components (eigenvalues) are sorted in descending order, each repeated according to
+    The eigenvalues are sorted in ascending order, each repeated according to
     its multiplicity.
 
     Parameters
     ----------
     x : numpy.array of shape (:,3,3) or (3,3)
-      Symmetric tensor of which the principal compontents are computed.
+      Symmetric tensor of which the eigenvalues are computed.
 
     """
-    w = np.linalg.eigvalsh(symmetric(x))                                                            # eigenvalues in ascending order
-    return w[::-1] if np.shape(x) == (3,3) else \
-           w[:,::-1]
+    return np.linalg.eigvalsh(symmetric(x))
+
+
+def eigenvectors(x):
+    """
+    Return eigenvectors of a symmetric tensor.
+
+    The eigenvalues are sorted in ascending order of their associated eigenvalues.
+
+    Parameters
+    ----------
+    x : numpy.array of shape (:,3,3) or (3,3)
+      Symmetric tensor of which the eigenvectors are computed.
+
+    """
+    (u,v) = np.linalg.eigh(symmetric(x))
+    return v
 
 
 def transpose(x):

python/tests/test_mechanics.py

@@ -58,10 +58,23 @@ class TestMechanics:
                             mechanics.maximum_shear(x[self.c]))
 
 
-    def test_vectorize_principal_components(self):
+    def test_vectorize_eigenvalues(self):
          x = np.random.random((self.n,3,3))
-         assert np.allclose(mechanics.principal_components(x)[self.c],
-                            mechanics.principal_components(x[self.c]))
+         assert np.allclose(mechanics.eigenvalues(x)[self.c],
+                            mechanics.eigenvalues(x[self.c]))
+
+
+    def test_vectorize_eigenvectors(self):
+         x = np.random.random((self.n,3,3))
+         assert np.allclose(mechanics.eigenvectors(x)[self.c],
+                            mechanics.eigenvectors(x[self.c]))
+
+
+    def test_vectorize_PK2(self):
+         F = np.random.random((self.n,3,3))
+         P = np.random.random((self.n,3,3))
+         assert np.allclose(mechanics.PK2(F,P)[self.c],
+                            mechanics.PK2(F[self.c],P[self.c]))
 
 
     def test_vectorize_transpose(self):
@@ -104,6 +117,13 @@ class TestMechanics:
                             np.matmul(V,R))
 
 
+    def test_PK2(self):
+         """Ensure 2. Piola-Kirchhoff stress is symmetrized 1. Piola-Kirchhoff stress for no deformation."""
+         P = np.random.random((self.n,3,3))
+         assert np.allclose(mechanics.PK2(np.broadcast_to(np.eye(3),(self.n,3,3)),P),
+                            mechanics.symmetric(P))
+
+
     def test_strain_tensor_no_rotation(self):
          """Ensure that left and right stretch give same results for no rotation."""
          F = np.broadcast_to(np.eye(3),[self.n,3,3])*np.random.random((self.n,3,3))
@@ -186,3 +206,22 @@ class TestMechanics:
          x = np.random.random((self.n,3,3))
          assert np.allclose(mechanics.Mises_stress(x)/mechanics.Mises_strain(x),
                             1.5)
+
+
+    def test_eigenvalues(self):
+         """Ensure that the characteristic polynomial can be solved."""
+         A = mechanics.symmetric(np.random.random((self.n,3,3)))
+         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):
+         """Ensure that eigenvalues and -vectors are the solution to the characteristic polynomial."""
+         A = mechanics.symmetric(np.random.random((self.n,3,3)))
+         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)