def test_intersection(self):
h1 = Halfspace.from_hyperplane([[1, 0, 0], [0, 1, 0]], [1, 1, 1],
[0.9, 0.9, 0.9], True)
h2 = Halfspace.from_hyperplane([[0, 1, 0], [0, 0, 1]], [1, 1, 1],
[0.9, 0.9, 0.9], True)
h3 = Halfspace.from_hyperplane([[0, 0, 1], [1, 0, 0]], [1, 1, 1],
[0.9, 0.9, 0.9], True)
h4 = Halfspace.from_hyperplane([[-1, 0, 1], [0, -1, 1]], [1, 1, 0],
[0.9, 0.9, 0.9], True)
hi = HalfspaceIntersection([h1, h2, h3, h4], [0.9, 0.9, 0.9])
self.assertTrue(np.allclose(np.sum(hi.vertices, axis=0), [3, 3, 3]))
h5 = Halfspace.from_hyperplane([[1, 0, 0], [0, 1, 0]], [1, 2, 2],
[0.9, 0.9, 0.9], True)
hi = HalfspaceIntersection([h5, h2, h3, h4], [0.9, 0.9, 0.9])
self.assertTrue(np.allclose(np.sum(hi.vertices, axis=0), [2, 2, 6]))
for h, vs in zip(hi.halfspaces, hi.facets_by_halfspace):
for v in vs:
self.assertAlmostEqual(
np.dot(h.normal, hi.vertices[v]) + h.offset, 0)
for v, hss in zip(hi.vertices, hi.facets_by_vertex):
for i in hss:
hs = hi.halfspaces[i]
self.assertAlmostEqual(np.dot(hs.normal, v) + hs.offset, 0)
def test_infinite_vertex(self):
h1 = Halfspace.from_hyperplane([[1,0,0], [0,1,0]], [1,1,1], [0.9, 0.9, 0.9], True)
h2 = Halfspace.from_hyperplane([[0,1,0], [0,0,1]], [1,1,1], [0.9, 0.9, 0.9], True)
h3 = Halfspace.from_hyperplane([[0,0,1], [1,0,0]], [1,1,1], [0.9, 0.9, 0.9], True)
h4 = Halfspace.from_hyperplane([[1,0,0], [0,1,0]], [2,2,2], [0.9, 0.9, 0.9], True)
hi = HalfspaceIntersection([h1, h2, h3, h4], [0.9, 0.9, 0.9])
self.assertTrue(np.allclose(np.sum(hi.vertices, axis = 0), [np.inf,np.inf,np.inf]))
def test_halfspace(self):
h1 = Halfspace.from_hyperplane([[0, 1, 0], [1, 0, 0]], [1, 1, -100],
[2, 2, 2], True)
self.assertTrue(all(h1.normal == [0, 0, -1]))
self.assertEqual(h1.offset, -100)
h2 = Halfspace.from_hyperplane([[0, 1, 0], [1, 0, 0]], [1, 1, -100],
[2, 2, 2], False)
self.assertEqual(h2.offset, 100)
def test_non_unit_normals(self):
h1 = Halfspace([3, 0], -3)
h2 = Halfspace([0, 2], 2)
h3 = Halfspace([-2, -1], 0)
hi = HalfspaceIntersection([h1, h2, h3], [0.9, -1.1])
self.assertTrue(
np.any(np.all(hi.vertices == np.array([1, -2]), axis=1)))
self.assertTrue(
np.any(np.all(hi.vertices == np.array([1, -1]), axis=1)))
def test_infinite_vertex(self):
h1 = Halfspace.from_hyperplane([[1, 0, 0], [0, 1, 0]], [1, 1, 1],
[0.9, 0.9, 0.9], True)
h2 = Halfspace.from_hyperplane([[0, 1, 0], [0, 0, 1]], [1, 1, 1],
[0.9, 0.9, 0.9], True)
h3 = Halfspace.from_hyperplane([[0, 0, 1], [1, 0, 0]], [1, 1, 1],
[0.9, 0.9, 0.9], True)
h4 = Halfspace.from_hyperplane([[1, 0, 0], [0, 1, 0]], [2, 2, 2],
[0.9, 0.9, 0.9], True)
hi = HalfspaceIntersection([h1, h2, h3, h4], [0.9, 0.9, 0.9])
self.assertTrue(
np.allclose(np.sum(hi.vertices, axis=0), [np.inf, np.inf, np.inf]))
def constraint(self, p, normal, offset):
"Return halfspace that constrains this point to lie inside the half-plane."
# (s*a + t*b + x) * nx + (-t*a + s*b + y) * ny <= offset
# (a * nx + b * ny) * s + (b * nx - a * ny) * t + nx * x + ny * y <= offset
return Halfspace([
np.dot(p, normal), p[1] * normal[0] - p[0] * normal[1], normal[0],
normal[1]
], offset)
def test_intersection(self):
h1 = Halfspace.from_hyperplane([[1,0,0], [0,1,0]], [1,1,1], [0.9, 0.9, 0.9], True)
h2 = Halfspace.from_hyperplane([[0,1,0], [0,0,1]], [1,1,1], [0.9, 0.9, 0.9], True)
h3 = Halfspace.from_hyperplane([[0,0,1], [1,0,0]], [1,1,1], [0.9, 0.9, 0.9], True)
h4 = Halfspace.from_hyperplane([[-1,0,1], [0,-1,1]], [1,1,0], [0.9, 0.9, 0.9], True)
hi = HalfspaceIntersection([h1, h2, h3, h4], [0.9, 0.9, 0.9])
self.assertTrue(np.allclose(np.sum(hi.vertices, axis = 0), [3,3,3]))
h5 = Halfspace.from_hyperplane([[1,0,0], [0,1,0]], [1,2,2], [0.9, 0.9, 0.9], True)
hi = HalfspaceIntersection([h5, h2, h3, h4], [0.9, 0.9, 0.9])
self.assertTrue(np.allclose(np.sum(hi.vertices, axis = 0), [2,2,6]))
for h, vs in zip(hi.halfspaces, hi.facets_by_halfspace):
for v in vs:
self.assertAlmostEqual(np.dot(h.normal, hi.vertices[v]) + h.offset, 0)
for v, hss in zip(hi.vertices, hi.facets_by_vertex):
for i in hss:
hs = hi.halfspaces[i]
self.assertAlmostEqual(np.dot(hs.normal, v) + hs.offset, 0)
def test_intersection2d(self):
h1 = Halfspace([1, 0], -1)
h2 = Halfspace([0, 1], 1)
h3 = Halfspace([-2, -1], 0)
h_redundant = Halfspace([1, 0], -2)
hi = HalfspaceIntersection([h1, h2, h_redundant, h3], [0.9, -1.1])
for h, vs in zip(hi.halfspaces, hi.facets_by_halfspace):
for v in vs:
self.assertAlmostEqual(
np.dot(h.normal, hi.vertices[v]) + h.offset, 0)
for v, hss in zip(hi.vertices, hi.facets_by_vertex):
for i in hss:
hs = hi.halfspaces[i]
self.assertAlmostEqual(np.dot(hs.normal, v) + hs.offset, 0)
#redundant halfspace should have no vertices
self.assertEqual(len(hi.facets_by_halfspace[2]), 0)
self.assertTrue(
np.any(np.all(hi.vertices == np.array([1, -2]), axis=1)))
self.assertTrue(
np.any(np.all(hi.vertices == np.array([1, -1]), axis=1)))
def vertices(self):
"""
Calls qhull for determining the vertices of the polytope (it computes the vertices only when called).
"""
if self._vertices is None:
if self.empty:
self._vertices = []
return self._vertices
halfspaces = []
for i in range(0, self.n_facets):
halfspace = Halfspace(self.A[i,:].tolist(), (-self.b[i,0]).tolist())
halfspaces.append(halfspace)
polyhedron_qhull = HalfspaceIntersection(halfspaces, self.center.flatten().tolist())
self._vertices = polyhedron_qhull.vertices
return self._vertices
def visualize_3d_polytope(p, name, visualizer):
p = reorder_coordinates_visualizer(p)
halfspaces = []
# change of coordinates because qhull is stupid...
b_qhull = p.rhs_min - p.lhs_min.dot(p.center)
for i in range(p.lhs_min.shape[0]):
halfspace = Halfspace(p.lhs_min[i, :].tolist(),
(-b_qhull[i, 0]).tolist())
halfspaces.append(halfspace)
p_qhull = HalfspaceIntersection(
halfspaces,
np.zeros(p.center.shape).flatten().tolist())
vertices = p_qhull.vertices + np.hstack(
[p.center] * len(p_qhull.vertices)).T
mesh = Mesh(vertices.tolist(), p_qhull.facets_by_halfspace)
visualizer[name].setgeometry(mesh)
return visualizer
def vertices(self):
"""
Calls qhull for determining the vertices of the polytope (it computes the vertices only when called).
"""
if self._vertices is None:
if self.n_variables == 1:
self._vertices = [np.array([[self.rhs_min[i,0]/self.lhs_min[i,0]]]) for i in [0,1]]
return self._vertices
if self.empty:
self._vertices = []
return self._vertices
halfspaces = []
# change of coordinates because qhull is stupid...
b_qhull = self.rhs_min - self.lhs_min.dot(self.center)
for i in range(self.lhs_min.shape[0]):
halfspace = Halfspace(self.lhs_min[i,:].tolist(), (-b_qhull[i,0]).tolist())
halfspaces.append(halfspace)
polyhedron_qhull = HalfspaceIntersection(halfspaces, np.zeros(self.center.shape).flatten().tolist())
self._vertices = polyhedron_qhull.vertices
self._vertices += np.repeat(self.center.T, self._vertices.shape[0], axis=0)
self._vertices = [np.reshape(vertex, (vertex.shape[0],1)) for vertex in self._vertices]
return self._vertices
def get_chempot_range_map(self, limits=[[-2,16], [-4,4]]):
"""
Returns a chemical potential range map for each stable entry.
Args:
elements: Sequence of elements to be considered as independent
variables. E.g., if you want to show the stability ranges of
all Li-Co-O phases wrt to uLi and uO, you will supply
[Element("Li"), Element("O")]
Returns:
Returns a dict of the form {entry: [simplices]}. The list of
simplices are the sides of the N-1 dim polytope bounding the
allowable chemical potential range of each entry.
"""
tol = PourbaixAnalyzer.numerical_tol
all_chempots = []
facets = self._pd.facets
entries = self._pd.qhull_entries
for facet in facets:
chempots = self.get_facet_chempots(facet)
chempots["H+"] /= -0.0591
chempots["V"] = -chempots["V"]
chempots["1"] = chempots["1"]
all_chempots.append([chempots[el] for el in self._keys])
basis_vecs = []
on_plane_points = []
# Create basis vectors
for row in self._pd._qhull_data:
on_plane_points.append([0, 0, row[2]])
this_basis_vecs = []
norm_vec = [-0.0591 * row[0], -1 * row[1], 1]
if abs(norm_vec[0]) > tol:
this_basis_vecs.append([-norm_vec[2]/norm_vec[0], 0, 1])
if abs(norm_vec[1]) > tol:
this_basis_vecs.append([0, -norm_vec[2]/norm_vec[1], 1])
if len(this_basis_vecs) == 0:
basis_vecs.append([[1, 0, 0], [0, 1, 0]])
elif len(this_basis_vecs) == 1:
if abs(this_basis_vecs[0][0]) < tol:
this_basis_vecs.append([1, 0, 0])
else:
this_basis_vecs.append([0, 1, 0])
basis_vecs.append(this_basis_vecs)
else:
basis_vecs.append(this_basis_vecs)
# Find point in half-space in which optimization is desired
ph_max_contrib = -1 * max([abs(0.0591 * row[0])
for row in self._pd._qhull_data]) * limits[0][1]
V_max_contrib = -1 * max([abs(row[1]) for row in self._pd._qhull_data]) * limits[1][1]
g_max = (-1 * max([abs(pt[2]) for pt in on_plane_points])
+ ph_max_contrib + V_max_contrib) - 10
point_in_region = [7, 0, g_max]
# Append border hyperplanes along limits
for i in xrange(len(limits)):
for j in xrange(len(limits[i])):
basis_vec_1 = [0.0] * 3
basis_vec_2 = [0.0] * 3
point = [0.0] * 3
basis_vec_1[2] = 1.0
basis_vec_2[2] = 0.0
for axis in xrange(len(limits)):
if axis is not i:
basis_vec_1[axis] = 0.0
basis_vec_2[axis] = 1.0
basis_vecs.append([basis_vec_1, basis_vec_2])
point[i] = limits[i][j]
on_plane_points.append(point)
# Hyperplane enclosing the very bottom
basis_vecs.append([[1, 0, 0], [0, 1, 0]])
on_plane_points.append([0, 0, 2 * g_max])
hyperplane_list = [Halfspace.from_hyperplane(basis_vecs[i], on_plane_points[i], point_in_region)
for i in xrange(len(basis_vecs))]
hs_int = HalfspaceIntersection(hyperplane_list, point_in_region)
int_points = hs_int.vertices
pourbaix_domains = {}
self.pourbaix_domain_vertices = {}
for i in xrange(len(self._pd._qhull_data)):
vertices = [[int_points[vert][0], int_points[vert][1]] for vert in
hs_int.facets_by_halfspace[i]]
if len(vertices) < 1:
continue
pourbaix_domains[self._pd._qhull_entries[i]] = ConvexHull(vertices).simplices
# Need to order vertices for highcharts area plot
cx = sum([vert[0] for vert in vertices]) / len(vertices)
cy = sum([vert[1] for vert in vertices]) / len(vertices)
point_comp = lambda x, y: x[0]*y[1] - x[1]*y[0]
vert_center = [[v[0] - cx, v[1] - cy] for v in vertices]
vert_center.sort(key=cmp_to_key(point_comp))
self.pourbaix_domain_vertices[self._pd._qhull_entries[i]] =\
[[v[0] + cx, v[1] + cy] for v in vert_center]
self.pourbaix_domains = pourbaix_domains
return pourbaix_domains
def get_chempot_range_map(self, limits=[[-2,16], [-4,4]]):
"""
Returns a chemical potential range map for each stable entry.
Args:
elements: Sequence of elements to be considered as independent
variables. E.g., if you want to show the stability ranges of
all Li-Co-O phases wrt to uLi and uO, you will supply
[Element("Li"), Element("O")]
Returns:
Returns a dict of the form {entry: [simplices]}. The list of
simplices are the sides of the N-1 dim polytope bounding the
allowable chemical potential range of each entry.
"""
tol = PourbaixAnalyzer.numerical_tol
all_chempots = []
facets = self._pd.facets
for facet in facets:
chempots = self.get_facet_chempots(facet)
chempots["H+"] /= -0.0591
chempots["V"] = -chempots["V"]
chempots["1"] = chempots["1"]
all_chempots.append([chempots[el] for el in self._keys])
basis_vecs = []
on_plane_points = []
# Create basis vectors
for entry in self._pd.stable_entries:
ie = self._pd.qhull_entries.index(entry)
row = self._pd._qhull_data[ie]
on_plane_points.append([0, 0, row[2]])
this_basis_vecs = []
norm_vec = [-0.0591 * row[0], -1 * row[1], 1]
if abs(norm_vec[0]) > tol:
this_basis_vecs.append([-norm_vec[2]/norm_vec[0], 0, 1])
if abs(norm_vec[1]) > tol:
this_basis_vecs.append([0, -norm_vec[2]/norm_vec[1], 1])
if len(this_basis_vecs) == 0:
basis_vecs.append([[1, 0, 0], [0, 1, 0]])
elif len(this_basis_vecs) == 1:
if abs(this_basis_vecs[0][0]) < tol:
this_basis_vecs.append([1, 0, 0])
else:
this_basis_vecs.append([0, 1, 0])
basis_vecs.append(this_basis_vecs)
else:
basis_vecs.append(this_basis_vecs)
# Find point in half-space in which optimization is desired
ph_max_contrib = -1 * max([abs(0.0591 * row[0])
for row in self._pd._qhull_data]) * limits[0][1]
V_max_contrib = -1 * max([abs(row[1]) for row in self._pd._qhull_data]) * limits[1][1]
g_max = (-1 * max([abs(pt[2]) for pt in on_plane_points])
+ ph_max_contrib + V_max_contrib) - 10
point_in_region = [7, 0, g_max]
# Append border hyperplanes along limits
for i in range(len(limits)):
for j in range(len(limits[i])):
basis_vec_1 = [0.0] * 3
basis_vec_2 = [0.0] * 3
point = [0.0] * 3
basis_vec_1[2] = 1.0
basis_vec_2[2] = 0.0
for axis in range(len(limits)):
if axis is not i:
basis_vec_1[axis] = 0.0
basis_vec_2[axis] = 1.0
basis_vecs.append([basis_vec_1, basis_vec_2])
point[i] = limits[i][j]
on_plane_points.append(point)
# Hyperplane enclosing the very bottom
basis_vecs.append([[1, 0, 0], [0, 1, 0]])
on_plane_points.append([0, 0, 2 * g_max])
hyperplane_list = [Halfspace.from_hyperplane(basis_vecs[i], on_plane_points[i], point_in_region)
for i in range(len(basis_vecs))]
hs_int = HalfspaceIntersection(hyperplane_list, point_in_region)
int_points = hs_int.vertices
pourbaix_domains = {}
self.pourbaix_domain_vertices = {}
for i in range(len(self._pd.stable_entries)):
vertices = [[int_points[vert][0], int_points[vert][1]] for vert in
hs_int.facets_by_halfspace[i]]
if len(vertices) < 1:
continue
pourbaix_domains[self._pd.stable_entries[i]] = ConvexHull(vertices).simplices
# Need to order vertices for highcharts area plot
cx = sum([vert[0] for vert in vertices]) / len(vertices)
cy = sum([vert[1] for vert in vertices]) / len(vertices)
point_comp = lambda x, y: x[0]*y[1] - x[1]*y[0]
vert_center = [[v[0] - cx, v[1] - cy] for v in vertices]
vert_center.sort(key=cmp_to_key(point_comp))
self.pourbaix_domain_vertices[self._pd.stable_entries[i]] =\
[[v[0] + cx, v[1] + cy] for v in vert_center]
self.pourbaix_domains = pourbaix_domains
return pourbaix_domains
def test_halfspace(self):
h1 = Halfspace.from_hyperplane([[0,1,0], [1,0,0]], [1,1,-100], [2,2,2], True)
self.assertTrue(all(h1.normal == [0,0,-1]))
self.assertEqual(h1.offset, -100)
h2 = Halfspace.from_hyperplane([[0,1,0], [1,0,0]], [1,1,-100], [2,2,2], False)
self.assertEqual(h2.offset, 100)
