def integrate_uAnlytical_minus_uFEM_squared(self, u_analytical,
local_u_FEM):
integral = 0
coords = self.coords
degree = self.degree
x1, t1 = coords[0]
x3, t3 = coords[2]
num_gps = math.ceil(degree + 1)
xg, wg = leggauss(num_gps)
for j in range(len(xg)):
gp_eta = xg[j]
gp_wt_eta = wg[j]
for i in range(len(xg)):
gp_xi = xg[i]
gp_wt_xi = wg[i]
N = hsf.shape_functions_2d(gp_xi, gp_eta, degree)
dNodal = hsf.shape_functions_derivatives_2d(gp_xi, gp_eta, 1)
nodal_coordinates = self.coords
jacobian = np.dot(dNodal, nodal_coordinates)
det_jacobian = np.linalg.det(jacobian)
x = x1 + 0.5 * (gp_xi + 1) * (x3 - x1)
t = t1 + 0.5 * (gp_eta + 1) * (t3 - t1)
u = u_analytical(x, t)
u_FEM = np.dot(N, local_u_FEM)
integral += (u -
u_FEM)**2 * gp_wt_xi * gp_wt_eta * det_jacobian
return integral
def get_solution_point_from_solution_vector(self, x, y, u):
coords = self.coords
degree = self.degree
x1, t1 = coords[0]
x3, t3 = coords[2]
xi = 2 / (x3 - x1) * (x - x1) - 1
eta = 2 / (t3 - t1) * (y - t1) - 1
N = hsf.shape_functions_2d(xi, eta, degree)
return np.dot(N, u)
def k_matrix(self):
"""
compute the k matrix of the element
Returns
-------
k_e : ndarray
stiffness matrix of the element
"""
degree = self.degree
num_dofs = (degree + 1)**2
k_e = np.zeros((num_dofs, num_dofs))
A = np.zeros((num_dofs, num_dofs))
B = np.zeros((num_dofs, num_dofs))
num_gps = math.ceil(degree + 1)
xg, wg = leggauss(num_gps)
for j in range(len(xg)):
gp_eta = xg[j]
gp_wt_eta = wg[j]
for i in range(len(xg)):
gp_xi = xg[i]
gp_wt_xi = wg[i]
N = hsf.shape_functions_2d(gp_xi, gp_eta, degree)
dN = hsf.shape_functions_derivatives_2d(gp_xi, gp_eta, degree)
dNodal = hsf.shape_functions_derivatives_2d(gp_xi, gp_eta, 1)
nodal_coordinates = self.coords
jacobian = np.dot(dNodal, nodal_coordinates)
inv_jacobian = np.linalg.inv(jacobian)
det_jacobian = np.linalg.det(jacobian)
dN_global = np.dot(inv_jacobian, dN)
dNdx = dN_global[0]
dNdt = dN_global[1]
A = np.outer(
N, dNdt) * self._c * gp_wt_xi * gp_wt_eta * det_jacobian
B = np.outer(
dNdx, dNdx) * self._k * det_jacobian * gp_wt_xi * gp_wt_eta
k_e += (A + B)
return k_e
def assemble_force_vector_uniform_mesh(self, force_global_function):
"""
assemble the force vector of the model.\n
This is an optimized version that uses the mesh's uniformity
Returns
-------
f : ndarray
the assembled force vector of the model
"""
if self.total_num_dofs is None:
return
num_dofs = self.total_num_dofs
f = np.zeros(num_dofs)
element_1_coords = self.get_element(1).coords
x1, t1 = element_1_coords[0]
x3, t3 = element_1_coords[2]
area = (x3 - x1) * (t3 - t1)
det_j = area / 4
shape_functions_info = dict()
num_gps = math.ceil(self.degree + 1)
xg, wg = leggauss(num_gps)
for j in range(len(xg)):
gp_eta = xg[j]
gp_wt_eta = wg[j]
for i in range(len(xg)):
gp_xi = xg[i]
gp_wt_xi = wg[i]
N = hsf.shape_functions_2d(gp_xi, gp_eta, self.degree)
shape_functions_info[(gp_xi, gp_eta, gp_wt_xi * gp_wt_eta)] = N
for element in self.elements:
fe = element.f_vector(force_global_function, shape_functions_info,
det_j)
element_dofs = element.dofs
for i in range(len(element_dofs)):
dof = element_dofs[i]
f[dof] += fe[i]
return f
def get_solution_to_plot(self, global_u_vector):
coords = self.coords
degree = self.degree
x1, t1 = coords[0]
x3, t3 = coords[2]
local_u = global_u_vector[self.dofs]
x = list()
t = list()
u = list()
for eta in np.linspace(-1, 1, degree + 1):
for xi in np.linspace(-1, 1, degree + 1):
N = hsf.shape_functions_2d(xi, eta, degree)
u.append(np.dot(N, local_u))
i_x = x1 + 0.5 * (xi + 1) * (x3 - x1)
i_t = t1 + 0.5 * (eta + 1) * (t3 - t1)
x.append(i_x)
t.append(i_t)
return np.array(x), np.array(t), np.array(u)
def f_vector(self,
force_global_function,
shape_functions_info=None,
det_j=None):
"""
compute the f vector of the element
Parameters
----------
force_global_function : callabe
the function of force applied on the model
shape_functions_info : dict
the gauss points local coordiates and the array of the shape functions
Returns
-------
f_e : ndarray
force vector of the element
"""
degree = self.degree
num_dofs = (degree + 1)**2
f_e = np.zeros(num_dofs)
coords = self.coords
x1, t1 = coords[0]
x3, t3 = coords[2]
if shape_functions_info and det_j:
for key, value in shape_functions_info.items():
xi, eta, weight = key
N = value
x = x1 + 0.5 * (xi + 1) * (x3 - x1)
t = t1 + 0.5 * (eta + 1) * (t3 - t1)
f = force_global_function(x, t)
f_e += f * N * det_j * weight
else:
num_gps = math.ceil(degree + 1)
xg, wg = leggauss(num_gps)
for j in range(len(xg)):
gp_eta = xg[j]
gp_wt_eta = wg[j]
for i in range(len(xg)):
gp_xi = xg[i]
gp_wt_xi = wg[i]
N = hsf.shape_functions_2d(gp_xi, gp_eta, degree)
dNodal = hsf.shape_functions_derivatives_2d(
gp_xi, gp_eta, 1)
j = np.dot(dNodal, coords)
det_j = np.linalg.det(j)
x = x1 + 0.5 * (gp_xi + 1) * (x3 - x1)
t = t1 + 0.5 * (gp_eta + 1) * (t3 - t1)
f = force_global_function(x, t)
f_e += f * N * det_j * gp_wt_xi * gp_wt_eta
return f_e