stiffness = assembly_quads_stress_strain(nodes=nodes,
elements=elements,
thickness=1.0,
elasticity_matrix=d)
print("Evaluating force...")
def force_func(node):
if abs(node[0]) < 0.0000001:
return array([0.0, -q / m])
else:
return array([0.0, 0.0])
force = nodal_force(nodes=nodes, freedom=2, force_function=force_func)
print("Evaluating boundary conditions...")
for i in range(len(nodes)):
if abs(l - nodes[i, 0]) < 0.0000001:
assembly_initial_value(stiffness, force, 2 * i, 0.0)
assembly_initial_value(stiffness, force, 2 * i + 1, 0.0)
print("Solving a system of linear equations")
x = spsolve(stiffness, force)
u = x[0::2]
v = x[1::2]
print(min(u), " <= u <= ", max(u))
print(min(v), " <= Y <= ", max(v))
draw_vtk(nodes, elements, u, title="u", show_labels=True)
draw_vtk(nodes, elements, v, title="v", show_labels=True)
d = plane_strain_isotropic(e, nu)
print(d)
(nodes, elements) = rectangular_triangles(x_count=n, y_count=m, x_origin=0.0, y_origin=-c, width=l, height=2.0 * c)
stiffness = assembly_triangles_stress_strain(nodes, elements, d)
dimension = stiffness.shape[0]
force = zeros(dimension)
for i in range(len(nodes)):
if abs(nodes[i, 0]) < 0.0000001:
force[2 * i + 1] = -q / m
for i in range(len(nodes)):
if abs(l - nodes[i, 0]) < 0.0000001:
for j in range(dimension):
if stiffness[2 * i, j] != 0.0:
stiffness[2 * i, j] = 0.0
if stiffness[j, 2 * i] != 0.0:
stiffness[j, 2 * i] = 0.0
if stiffness[2 * i + 1, j] != 0.0:
stiffness[2 * i + 1, j] = 0.0
if stiffness[j, 2 * i + 1] != 0.0:
stiffness[j, 2 * i + 1] = 0.0
stiffness[2 * i, 2 * i] = 1.0
force[2 * i] = 0.0
stiffness[2 * i + 1, 2 * i + 1] = 1.0
force[2 * i + 1] = 0.0
stiffness = stiffness.tocsr()
x = spsolve(stiffness, force)
print(min(x[0::2]), " <= X <= ", max(x[0::2]))
print(min(x[1::2]), " <= Y <= ", max(x[1::2]))
draw_vtk(nodes, elements, x[0::2], title="u", show_labels=True)
draw_vtk(nodes, elements, x[1::2], title="v", show_labels=True)
],
[
0.0, shape_dy[0], 0.0, shape_dy[1], 0.0,
shape_dy[2], 0.0, shape_dy[3]
],
[
shape_dy[0], shape_dx[0], shape_dy[1], shape_dx[1],
shape_dy[2], shape_dx[2], shape_dy[3], shape_dx[3]
]])
sigma = d.dot(b).dot(displacement)
sigma_x[element[i]] += sigma[0]
sigma_y[element[i]] += sigma[1]
tau_xy[element[i]] += sigma[2]
adjacent[element[i]] += 1.0
sigma_x /= adjacent
sigma_y /= adjacent
tau_xy /= adjacent
print(min(u), " <= u <= ", max(u))
print(min(v), " <= v <= ", max(v))
print(min(sigma_x), " <= sigma x <= ", max(sigma_x))
print(min(sigma_y), " <= sigma y <= ", max(sigma_y))
print(min(tau_xy), " <= tau xy <= ", max(tau_xy))
print("Analytical sigma: " + str(e * alpha / (1 - nu)))
draw_vtk(nodes, elements, u, title="u", show_labels=True)
draw_vtk(nodes, elements, v, title="v", show_labels=True)
draw_vtk(nodes, elements, sigma_x, title="sigma x", show_labels=True)
draw_vtk(nodes, elements, sigma_y, title="sigma y", show_labels=True)
draw_vtk(nodes, elements, tau_xy, title="tau xy", show_labels=True)
elements=elements,
thickness=1.0,
freedom=3,
force_function=force_func,
gauss_order=3)
print("Evaluating boundary conditions...")
for i in range(len(nodes)):
if (abs(nodes[i, 0] - 0) <
0.0000001) or (abs(nodes[i, 0] - a) < 0.0000001) or (
abs(nodes[i, 1] - 0) < 0.0000001) or (abs(nodes[i, 1] - a)
< 0.0000001):
assembly_initial_value(stiffness, force, 3 * i, 0.0)
assembly_initial_value(stiffness, force, 3 * i + 1, 0.0)
assembly_initial_value(stiffness, force, 3 * i + 2, 0.0)
print("Solving a system of linear equations")
x = spsolve(stiffness, force)
w = x[0::3]
theta_x = x[1::3]
theta_y = x[2::3]
print(min(w), " <= w <= ", max(w))
print(min(theta_x), " <= theta x <= ", max(theta_x))
print(min(theta_y), " <= theta y <= ", max(theta_y))
print("Analytical bending: ",
0.00126 * q * a**4.0 * 12.0 * (1.0 - nu**2.0) / (e * h**3.0))
draw_vtk(nodes, elements, w, title="w", show_labels=True)
draw_vtk(nodes, elements, theta_x, title="theta x", show_labels=True)
draw_vtk(nodes, elements, theta_y, title="theta y", show_labels=True)
n = 51 # nodes in the first direction of computational domain
m = 11 # nodes in the second direction of computational domain
d = plane_strain_isotropic(e, nu)
print(d)
(nodes, elements) = rectangular_quads(x_count=n, y_count=m, x_origin=-l, y_origin=-c, width=2.0 * l, height=2.0 * c)
stiffness = assembly_quads_stress_strain(nodes, elements, d)
dimension = stiffness.shape[0]
force = zeros(dimension)
for i in range(len(nodes)):
if abs(c - nodes[i, 1]) < 0.0000001:
force[2 * i + 1] = -(q * 2.0 * l / (n - 1))
if (abs(l - nodes[i, 0]) < 0.0000001) or (abs(-l - nodes[i, 0]) < 0.0000001):
force[2 * i + 1] = -(q * 2.0 * l / (n - 1)) / 2.0
for i in range(len(nodes)):
if abs(-c - nodes[i, 1]) < 0.0000001 and (abs(l - nodes[i, 0]) < 0.0000001 or abs(-l - nodes[i, 0]) < 0.0000001):
for j in range(dimension):
if stiffness[2 * i + 1, j] != 0.0:
stiffness[2 * i + 1, j] = 0.0
if stiffness[j, 2 * i + 1] != 0.0:
stiffness[j, 2 * i + 1] = 0.0
stiffness[2 * i + 1, 2 * i + 1] = 1.0
force[2 * i + 1] = 0.0
stiffness = stiffness.tocsr()
x, info = cg(stiffness, force, tol=1e-8)
print(min(x[0::2]), " <= X <= ", max(x[0::2]))
print(min(x[1::2]), " <= Y <= ", max(x[1::2]))
draw_vtk(nodes, elements, x[0::2], title="u")
draw_vtk(nodes, elements, x[1::2], title="v")
sigma_x /= adjacent
sigma_y /= adjacent
tau_xy /= adjacent
mises /= adjacent
print(min(u), " <= u <= ", max(u))
print(min(v), " <= v <= ", max(v))
print(min(sigma_x), " <= sigma x <= ", max(sigma_x))
print(min(sigma_y), " <= sigma y <= ", max(sigma_y))
print(min(tau_xy), " <= tau xy <= ", max(tau_xy))
# draw_vtk(nodes, elements, u, title="u", show_labels=True)
# draw_vtk(nodes, elements, v, title="v", show_labels=True)
draw_vtk(nodes,
elements,
sigma_x,
title="sigma x",
show_labels=False,
use_gray=True,
contours_count=0,
colors_count=5)
draw_vtk(nodes,
elements,
sigma_y,
title="sigma y",
show_labels=False,
use_gray=True,
contours_count=0,
colors_count=5)
draw_vtk(nodes,
elements,
tau_xy,
title="tau xy",
for i in range(element_dimension):
ii = element[int(i / freedom)] * freedom + i % freedom
for j in range(i, element_dimension):
jj = element[int(j / freedom)] * freedom + j % freedom
geometric[ii, jj] += kg[i, j]
if ii != jj:
geometric[jj, ii] = geometric[ii, jj]
print("Assembly is done")
active = range(dimension)
for i in range(len(nodes)):
if (abs(nodes[i, 0] + a/2.0) < 0.0000001) or (abs(nodes[i, 0] - a/2.0) < 0.0000001) or (abs(nodes[i, 1] + a/2.0) < 0.0000001) or (abs(nodes[i, 1] - a/2.0) < 0.0000001):
active.remove(freedom * i)
print ("Boundary conditions are processed")
geometric = lil_matrix(geometric.tocsr()[:, active])
geometric = lil_matrix(geometric.tocsc()[active, :])
geometric = geometric.tocsr()
stiffness = lil_matrix(stiffness.tocsr()[:, active])
stiffness = lil_matrix(stiffness.tocsc()[active, :])
stiffness = stiffness.tocsr()
print ("Matrices are converted")
vals, vecs = eigsh(A=stiffness, M=geometric, sigma=0.0, which='LM')
print(vals)
for i in range(len(vals)):
x = zeros(dimension)
x[active] = vecs[:, i]
w = array(x[0::freedom])
draw_vtk(nodes=hstack((nodes, w.reshape(len(w), 1) / 50.0)), elements=elements, values=w, title="w, T = " + str(vals[i]),
show_labels=False, show_axes=True, use_gray=True, contours_count=0, colors_count=5)
# active.remove(freedom * i + 1)
# active.remove(freedom * i + 2)
# active.remove(freedom * i + 3)
active.remove(freedom * i + 4)
if (abs(nodes[i, 1] - 0) < 0.0000001) or (abs(nodes[i, 1] - b) <
0.0000001):
# active.remove(freedom * i)
active.remove(freedom * i + 1)
# active.remove(freedom * i + 2)
active.remove(freedom * i + 3)
# active.remove(freedom * i + 4)
if (abs(nodes[i, 0] - 0) <
0.0000001) or (abs(nodes[i, 0] - a) < 0.0000001) or (
abs(nodes[i, 1] - 0) < 0.0000001) or (abs(nodes[i, 1] - b)
< 0.0000001):
active.remove(freedom * i + 2)
geometric = lil_matrix(geometric.tocsr()[:, active])
geometric = lil_matrix(geometric.tocsc()[active, :])
geometric = geometric.tocsr()
stiffness = lil_matrix(stiffness.tocsr()[:, active])
stiffness = lil_matrix(stiffness.tocsc()[active, :])
stiffness = stiffness.tocsr()
vals, vecs = eigsh(A=stiffness, M=geometric, sigma=0.0, which='LM')
print(vals)
for i in range(6):
x = zeros(dimension)
x[active] = vecs[:, i]
w = x[2::freedom]
draw_vtk(nodes, elements, w, title="w", show_labels=True)
geometric[jj, ii] = geometric[ii, jj]
print("Assembly is done")
active = range(dimension)
for i in range(len(nodes)):
if (abs(nodes[i, 0] - 0) <
0.0000001) or (abs(nodes[i, 0] - a) < 0.0000001) or (
abs(nodes[i, 1] - 0) <
0.0000001) or (abs(nodes[i, 1] - a / factor) < 0.0000001):
active.remove(freedom * i)
print("Boundary conditions are processed")
# print (active)
geometric = lil_matrix(geometric.tocsr()[:, active])
geometric = lil_matrix(geometric.tocsc()[active, :])
geometric = geometric.tocsr()
stiffness = lil_matrix(stiffness.tocsr()[:, active])
stiffness = lil_matrix(stiffness.tocsc()[active, :])
stiffness = stiffness.tocsr()
print("Matrices are converted")
vals, vecs = eigsh(A=stiffness, M=geometric, sigma=0.0, which='LM')
print(vals)
x = zeros(dimension)
x[active] = vecs[:, 0]
w = array(x[0::freedom])
draw_vtk(nodes=hstack((nodes, w.reshape(len(w), 1) / 100.0)),
elements=elements,
values=w,
title="w",
show_labels=True,
show_axes=True)