def is_in_element(position, element, allNodes):
"""
This function tests the inclusion of a point in an element. It calculates
the form functions. If one if negative, it means that the point is outside.
NOTE: We substract 1 from element.nodes since node ids start from 1, like
element ids.
example of allNodes array:
allNodes = np.array(list(zip(flow.nodes_X[0],flow.nodes_Y[0])))
"""
nodes_coords = allNodes[element.nodes - 1]
p1, p2, p3 = nodes_coords[[0, 1, 2]]
N1 = psi1(position, p1, p2, p3)
N2 = psi2(position, p1, p2, p3)
N3 = psi3(position, p1, p2, p3)
# It suffices that one interpolation function takes a negative value to
# deduce that the point is not inside the element.
if (N1 >= 0) and (N2 >= 0) and (N3 >= 0):
return True
else:
return False
def find_element_partrack(position, current_element, elements, allNodes):
"""
This function implements the particle tracer algorithm of
Lohner and Ambrosiano (1990)
"""
nodes_coords = allNodes[current_element.nodes - 1]
p1, p2, p3 = nodes_coords[[0, 1, 2]]
N1 = psi1(position, p1, p2, p3)
N2 = psi2(position, p1, p2, p3)
N3 = psi3(position, p1, p2, p3)
if (N1 >= 0) and (N2 >= 0) and (N3 >= 0):
# In this case, the particle is still inside 'current_element'.
return current_element
else:
# e.g.
# If N3 has the most negative value, then index_smallest = 2.
# Thus point_id_smallest (>= 0) is the farthest node from 'position'.
index_smallest = np.argmin([N1, N2, N3])
point_id_smallest = current_element.nodes[index_smallest]
next_element = find_opposite_neighbor(point_id_smallest,
current_element, elements)
return find_element_partrack(position, next_element, elements,
allNodes)
def Psi4_y(*args):
L1 = psi1(*args)
L2 = psi2(*args)
L1_y = psi1_y(*args)
L2_y = psi2_y(*args)
return 4 * (L1_y * L2 + L1 * L2_y)
def Psi4_x(*args):
L1 = psi1(*args)
L2 = psi2(*args)
L1_x = psi1_x(*args)
L2_x = psi2_x(*args)
return 4 * (L1_x * L2 + L1 * L2_x)
def Psi6_y(*args):
L1 = psi1(*args)
L3 = psi3(*args)
L1_y = psi1_y(*args)
L3_y = psi3_y(*args)
return 4 * (L3_y * L1 + L3 * L1_y)
def Psi6_x(*args):
L1 = psi1(*args)
L3 = psi3(*args)
L1_x = psi1_x(*args)
L3_x = psi3_x(*args)
return 4 * (L3_x * L1 + L3 * L1_x)
def Psi4(*args):
L1 = psi1(*args)
L2 = psi2(*args)
return 4 * L1 * L2
def Psi1_y(*args):
L1 = psi1(*args)
L1_y = psi1_y(*args)
return (4 * L1 - 1) * L1_y
def Psi1_x(*args):
L1 = psi1(*args)
L1_x = psi1_x(*args)
return (4 * L1 - 1) * L1_x
def Psi1(*args):
L1 = psi1(*args)
return L1 * (2 * L1 - 1)
def Psi6(*args):
L1 = psi1(*args)
L3 = psi3(*args)
return 4 * L3 * L1