def _calculate_f_e(info, tables , e):
f_e = np.zeros(shape=(2,1))
e_tab = tables.edge_to_global
f1 = e_tab[e, 3] #numbers of functions in T^(r) -> linear_trial_funcs[f1]
f2 = e_tab[e, 4]
i1 = e_tab[e, 0]
i2 = e_tab[e, 1]
nr = e_tab[e, 2]
x1 = tables.nodes_to_coordinates[i1, 0]
y1 = tables.nodes_to_coordinates[i1, 1]
x2 = tables.nodes_to_coordinates[i2, 0]
y2 = tables.nodes_to_coordinates[i2, 1]
length = math.sqrt((x2-x1)**2+(y2-y1)**2)
integ2 = lambda xi: dn_funcs[nr](_ref_e_to_global(x1, x2, y1, y2, xi))*linear_trial_funcs_1D[1](xi) * length
integ1 = lambda xi: dn_funcs[nr](_ref_e_to_global(x1, x2, y1, y2, xi))*linear_trial_funcs_1D[0](xi) * length
f_e[0] = numeric.gauss_1D_6(integ1)
f_e[1] = numeric.gauss_1D_6(integ2)
return f_e
def _calculate_f_e(info, tables, e):
f_e = np.zeros(shape=(2, 1))
e_tab = tables.edge_to_global
f1 = e_tab[e, 3] #numbers of functions in T^(r) -> linear_trial_funcs[f1]
f2 = e_tab[e, 4]
i1 = e_tab[e, 0]
i2 = e_tab[e, 1]
nr = e_tab[e, 2]
x1 = tables.nodes_to_coordinates[i1, 0]
y1 = tables.nodes_to_coordinates[i1, 1]
x2 = tables.nodes_to_coordinates[i2, 0]
y2 = tables.nodes_to_coordinates[i2, 1]
length = math.sqrt((x2 - x1)**2 + (y2 - y1)**2)
integ2 = lambda xi: dn_funcs[nr](_ref_e_to_global(x1, x2, y1, y2, xi)
) * linear_trial_funcs_1D[1](xi) * length
integ1 = lambda xi: dn_funcs[nr](_ref_e_to_global(x1, x2, y1, y2, xi)
) * linear_trial_funcs_1D[0](xi) * length
f_e[0] = numeric.gauss_1D_6(integ1)
f_e[1] = numeric.gauss_1D_6(integ2)
return f_e
def assemble_equations(local_to_global, nodes_to_coordinates, borders, eps,
rho, neumann):
"""
Assembles the final system of linear equations to be solved.
Outputs a Matrix K_h and the right hand side vector f_h
"""
N_nodes = nodes_to_coordinates.shape[0]
N_fe = local_to_global.shape[0]
K_h = np.zeros(shape=(N_nodes, N_nodes))
f_h = np.zeros(shape=(N_nodes, 1))
for r in range(0, N_fe):
#K_r, f_r for each element
i1 = local_to_global[r, 0]
i2 = local_to_global[r, 1]
i3 = local_to_global[r, 2]
x1 = nodes_to_coordinates[i1, 0]
x2 = nodes_to_coordinates[i2, 0]
x3 = nodes_to_coordinates[i3, 0]
y1 = nodes_to_coordinates[i1, 1]
y2 = nodes_to_coordinates[i2, 1]
y3 = nodes_to_coordinates[i3, 1]
K_r = _calculate_K_r(x1, x2, x3, y1, y2, y3, eps)
f_r = _calculate_f_r(x1, x2, x3, y1, y2, y3, rho)
for alpha in range(0, 3):
i = local_to_global[r, alpha]
f_h[i, 0] = f_h[i, 0] + f_r[alpha, 0]
for beta in range(0, 3):
j = local_to_global[r, beta]
K_h[i, j] = K_h[i, j] + K_r[alpha, beta]
#neumann boundarys
#neumann = {"e2g": edge_to_global, "e2n": edge_to_number, "e2l": edge_to_local, "funcs":func_dict}
edge_to_global = neumann["e2g"]
edge_to_number = neumann["e2n"]
edge_to_local = neumann["e2l"]
func_dict = neumann["funcs"]
Ne = edge_to_local.shape[1]
for e in range(0, edge_to_global.shape[0]):
#only if e in neumann boundary TODO
#print(func_dict.keys())
if edge_to_number[e] in func_dict.keys():
e_number = edge_to_number[e]
#create fe2
fe2 = np.zeros(shape=(Ne, 1))
xe_1 = nodes_to_coordinates[edge_to_global[e, 0], 0]
xe_Ne = nodes_to_coordinates[edge_to_global[e, Ne - 1], 0]
ye_1 = nodes_to_coordinates[edge_to_global[e, 0], 1]
ye_Ne = nodes_to_coordinates[edge_to_global[e, Ne - 1], 1]
root = math.sqrt((xe_Ne - xe_1)**2 + (ye_Ne - ye_1)**2)
#x = (xe_Ne - xe_1) + xi1 + xe_1
#y = (ye_Ne - ye_1) + xi2 + ye_1
x_e = lambda xi: (xe_Ne - xe_1) + xi + xe_1
y_e = lambda xi: (ye_Ne - ye_1) + xi + ye_1
#phi_lin_tri
#int func(global_to_ref(xi)) phi_(p,e) root ds
l1 = int(edge_to_local[e, 0])
l2 = int(edge_to_local[e, 1])
#do this with for loop
fe2[0, 0] = numeric.gauss_1D_6(lambda xi: func_dict[e_number](x_e(
xi), y_e(xi)) * phi_lin_tri[l1](x_e(xi), y_e(xi)) * root)
fe2[1, 0] = numeric.gauss_1D_6(lambda xi: func_dict[e_number](x_e(
xi), y_e(xi)) * phi_lin_tri[l2](x_e(xi), y_e(xi)) * root)
for p in range(0, Ne):
i = edge_to_global[e, p]
f_h[i, 0] = f_h[i, 0] + fe2[p, 0]
#dirichlet boundarys
#borders2 = [b[0] for b in borders]
#borders3 = list(itertools.chain(*borders2))
values = np.zeros(shape=(N_nodes, 1))
is_set = np.zeros(shape=(N_nodes, 1))
for nodes, value in borders:
for n in nodes:
is_set[n] = 1
values[n] = value
b_tmp = values * is_set
for i in range(0, N_nodes):
for j in range(0, N_nodes):
if is_set[i] == 0:
f_h[i] = f_h[i] - K_h[i, j] * b_tmp[j]
for i in range(0, N_nodes):
for j in range(0, N_nodes):
if is_set[i] == 1 or is_set[j] == 1:
if i == j:
K_h[i, j] = 1
else:
K_h[i, j] = 0
if is_set[i] == 1:
f_h[i] = values[i]
return K_h, f_h
def assemble_equations(local_to_global, nodes_to_coordinates, borders, eps, rho, neumann):
"""
Assembles the final system of linear equations to be solved.
Outputs a Matrix K_h and the right hand side vector f_h
"""
N_nodes = nodes_to_coordinates.shape[0]
N_fe = local_to_global.shape[0]
K_h = np.zeros(shape=(N_nodes, N_nodes))
f_h = np.zeros(shape=(N_nodes,1))
for r in range(0, N_fe):
#K_r, f_r for each element
i1 = local_to_global[r, 0]
i2 = local_to_global[r, 1]
i3 = local_to_global[r, 2]
x1 = nodes_to_coordinates[i1, 0]
x2 = nodes_to_coordinates[i2, 0]
x3 = nodes_to_coordinates[i3, 0]
y1 = nodes_to_coordinates[i1, 1]
y2 = nodes_to_coordinates[i2, 1]
y3 = nodes_to_coordinates[i3, 1]
K_r = _calculate_K_r(x1, x2, x3, y1, y2, y3, eps)
f_r = _calculate_f_r(x1, x2, x3, y1, y2, y3, rho)
for alpha in range(0,3):
i = local_to_global[r,alpha]
f_h[i,0] = f_h[i,0] + f_r[alpha,0]
for beta in range(0,3):
j = local_to_global[r,beta]
K_h[i,j] = K_h[i,j] + K_r[alpha,beta]
#neumann boundarys
#neumann = {"e2g": edge_to_global, "e2n": edge_to_number, "e2l": edge_to_local, "funcs":func_dict}
edge_to_global = neumann["e2g"]
edge_to_number = neumann["e2n"]
edge_to_local = neumann["e2l"]
func_dict = neumann["funcs"]
Ne = edge_to_local.shape[1]
for e in range(0, edge_to_global.shape[0]):
#only if e in neumann boundary TODO
#print(func_dict.keys())
if edge_to_number[e] in func_dict.keys():
e_number = edge_to_number[e]
#create fe2
fe2 = np.zeros(shape=(Ne,1))
xe_1 = nodes_to_coordinates[edge_to_global[e, 0], 0]
xe_Ne = nodes_to_coordinates[edge_to_global[e, Ne-1], 0]
ye_1 = nodes_to_coordinates[edge_to_global[e, 0], 1]
ye_Ne = nodes_to_coordinates[edge_to_global[e, Ne-1], 1]
root = math.sqrt( (xe_Ne-xe_1)**2 + (ye_Ne-ye_1)**2 )
#x = (xe_Ne - xe_1) + xi1 + xe_1
#y = (ye_Ne - ye_1) + xi2 + ye_1
x_e = lambda xi: (xe_Ne - xe_1) + xi + xe_1
y_e = lambda xi: (ye_Ne - ye_1) + xi + ye_1
#phi_lin_tri
#int func(global_to_ref(xi)) phi_(p,e) root ds
l1 = int(edge_to_local[e,0])
l2 = int(edge_to_local[e,1])
#do this with for loop
fe2[0,0] = numeric.gauss_1D_6( lambda xi: func_dict[e_number](x_e(xi), y_e(xi)) * phi_lin_tri[l1](x_e(xi), y_e(xi)) * root )
fe2[1,0] = numeric.gauss_1D_6( lambda xi: func_dict[e_number](x_e(xi), y_e(xi)) * phi_lin_tri[l2](x_e(xi), y_e(xi)) * root )
for p in range(0, Ne):
i = edge_to_global[e,p]
f_h[i,0] = f_h[i,0] + fe2[p,0]
#dirichlet boundarys
#borders2 = [b[0] for b in borders]
#borders3 = list(itertools.chain(*borders2))
values = np.zeros(shape=(N_nodes,1))
is_set = np.zeros(shape=(N_nodes,1))
for nodes, value in borders:
for n in nodes:
is_set[n] = 1
values[n] = value
b_tmp = values * is_set
for i in range(0, N_nodes):
for j in range(0, N_nodes):
if is_set[i] == 0:
f_h[i] = f_h[i] - K_h[i,j] * b_tmp[j]
for i in range(0,N_nodes):
for j in range(0,N_nodes):
if is_set[i] == 1 or is_set[j] == 1:
if i == j:
K_h[i,j] = 1
else:
K_h[i,j] = 0
if is_set[i] == 1:
f_h[i] = values[i]
return K_h, f_h