def test_nat_freq_plate(plot=False, mode=0):
nx = 15
ny = 15
a = 0.3
b = 0.5
# Material Lastrobe Lescalloy
E = 203.e9 # Pa
nu = 0.33
rho = 7.83e3 # kg/m3
h = 0.01 # m
xtmp = np.linspace(0, a, nx)
ytmp = np.linspace(0, b, ny)
dx = xtmp[1] - xtmp[0]
dy = ytmp[1] - ytmp[0]
xmesh, ymesh = np.meshgrid(xtmp, ytmp)
ncoords = np.vstack((xmesh.T.flatten(), ymesh.T.flatten())).T
x = ncoords[:, 0]
y = ncoords[:, 1]
inner = np.logical_not(
isclose(x, 0) | isclose(x, a) | isclose(y, 0) | isclose(y, b))
np.random.seed(20)
rdm = (-1 + 2 * np.random.rand(x[inner].shape[0]))
np.random.seed(20)
rdm = (-1 + 2 * np.random.rand(y[inner].shape[0]))
x[inner] += dx * rdm * 0.45
y[inner] += dy * rdm * 0.45
nids = 1 + np.arange(ncoords.shape[0])
nid_pos = dict(zip(nids, np.arange(len(nids))))
nids_mesh = nids.reshape(nx, ny)
n1s = nids_mesh[:-1, :-1].flatten()
n2s = nids_mesh[1:, :-1].flatten()
n3s = nids_mesh[1:, 1:].flatten()
n4s = nids_mesh[:-1, 1:].flatten()
plate = read_isotropic(thickness=h, E=E, nu=nu, calc_scf=True)
K = np.zeros((DOF * nx * ny, DOF * nx * ny))
M = np.zeros((DOF * nx * ny, DOF * nx * ny))
quads = []
for n1, n2, n3, n4 in zip(n1s, n2s, n3s, n4s):
pos1 = nid_pos[n1]
pos2 = nid_pos[n2]
pos3 = nid_pos[n3]
pos4 = nid_pos[n4]
r1 = ncoords[pos1]
r2 = ncoords[pos2]
r3 = ncoords[pos3]
r4 = ncoords[pos4]
normal = np.cross(r2 - r1, r3 - r2)
assert normal > 0 # guaranteeing that all elements have CCW positive normal
quad = Quad4R()
quad.rho = rho
quad.n1 = n1
quad.n2 = n2
quad.n3 = n3
quad.n4 = n4
quad.scf13 = plate.scf_k13
quad.scf23 = plate.scf_k23
quad.h = h
quad.ABDE = plate.ABDE
update_K(quad, nid_pos, ncoords, K)
update_M(quad, nid_pos, ncoords, M)
quads.append(quad)
print('elements created')
# applying boundary conditions
# simply supported
bk = np.zeros(K.shape[0], dtype=bool) #array to store known DOFs
check = isclose(x, 0.) | isclose(x, a) | isclose(y, 0) | isclose(y, b)
bk[2::DOF] = check
#eliminating all u,v displacements
bk[0::DOF] = True
bk[1::DOF] = True
bu = ~bk # same as np.logical_not, defining unknown DOFs
# sub-matrices corresponding to unknown DOFs
Kuu = K[bu, :][:, bu]
Muu = M[bu, :][:, bu]
eigvals, U = eigh(a=Kuu, b=Muu)
omegan = eigvals**0.5
# vector u containing displacements for all DOFs
u = np.zeros(K.shape[0], dtype=float)
u[bu] = U[:, mode]
# theoretical reference
m = 1
n = 1
D = 2 * h**3 * E / (3 * (1 - nu**2))
wmn = (m**2 / a**2 + n**2 / b**2) * np.sqrt(D * np.pi**4 /
(2 * rho * h)) / 2
print('Theoretical omega123', wmn)
print('Numerical omega123', omegan[0:10])
if plot:
import matplotlib
matplotlib.use('TkAgg')
import matplotlib.pyplot as plt
plt.clf()
for n1, n2, n3, n4 in zip(n1s, n2s, n3s, n4s):
pos1 = nid_pos[n1]
pos2 = nid_pos[n2]
pos3 = nid_pos[n3]
pos4 = nid_pos[n4]
r1 = ncoords[pos1]
r2 = ncoords[pos2]
r3 = ncoords[pos3]
r4 = ncoords[pos4]
plt.plot([r1[0], r2[0]], [r1[1], r2[1]], 'k-')
plt.plot([r2[0], r3[0]], [r2[1], r3[1]], 'k-')
plt.plot([r3[0], r4[0]], [r3[1], r4[1]], 'k-')
plt.plot([r4[0], r1[0]], [r4[1], r1[1]], 'k-')
plt.contourf(xmesh, ymesh, u[2::DOF].reshape(xmesh.shape))
plt.show()
assert np.isclose(wmn, omegan[0], rtol=0.05)
if plot:
import matplotlib
matplotlib.use('TkAgg')
import matplotlib.pyplot as plt
plt.clf()
ax = plt.gca()
ax.set_aspect('equal')
for s in ax.spines.values():
s.set_visible(False)
ax.set_xticks([])
ax.set_yticks([])
plt.triplot(ncoords[:, 0], ncoords[:, 1], d.simplices, lw=0.5)
plt.plot(ncoords[:, 0], ncoords[:, 1], 'o', ms=2)
plt.show()
plate = read_isotropic(thickness=h, E=E, nu=nu, calc_scf=True)
print('scf', plate.scf_k13, plate.scf_k23)
K = np.zeros((DOF * nx * ny, DOF * nx * ny))
M = np.zeros((DOF * nx * ny, DOF * nx * ny))
trias = []
for s in d.simplices:
n1, n2, n3 = nids[s]
n1, n2, n3 = n2, n3, n1
pos1 = nid_pos[n1]
pos2 = nid_pos[n2]
pos3 = nid_pos[n3]
r1 = ncoords[pos1]
r2 = ncoords[pos2]
r3 = ncoords[pos3]
normal = np.cross(r2 - r1, r3 - r2)
def test_static_plate_quad_point_load(plot=False):
nx = 7
ny = 7
# geometry
a = 3
b = 7
h = 0.005 # m
# material
E = 200e9
nu = 0.3
plate = read_isotropic(thickness=h, E=E, nu=nu, calc_scf=True)
xtmp = np.linspace(0, a, nx)
ytmp = np.linspace(0, b, ny)
xmesh, ymesh = np.meshgrid(xtmp, ytmp)
ncoords = np.vstack((xmesh.T.flatten(), ymesh.T.flatten())).T
x = ncoords[:, 0]
y = ncoords[:, 1]
nids = 1 + np.arange(ncoords.shape[0])
nid_pos = dict(zip(nids, np.arange(len(nids))))
nids_mesh = nids.reshape(nx, ny)
n1s = nids_mesh[:-1, :-1].flatten()
n2s = nids_mesh[1:, :-1].flatten()
n3s = nids_mesh[1:, 1:].flatten()
n4s = nids_mesh[:-1, 1:].flatten()
# creating global stiffness matrix
K = np.zeros((DOF * nx * ny, DOF * nx * ny))
# creating elements and populating global stiffness
quads = []
for n1, n2, n3, n4 in zip(n1s, n2s, n3s, n4s):
pos1 = nid_pos[n1]
pos2 = nid_pos[n2]
pos3 = nid_pos[n3]
pos4 = nid_pos[n4]
r1 = ncoords[pos1]
r2 = ncoords[pos2]
r3 = ncoords[pos3]
normal = np.cross(r2 - r1, r3 - r2)
assert normal > 0 # guaranteeing that all elements have CCW positive normal
quad = Quad4R()
quad.n1 = n1
quad.n2 = n2
quad.n3 = n3
quad.n4 = n4
quad.scf13 = plate.scf_k13
quad.scf23 = plate.scf_k23
quad.h = h
quad.ABDE = plate.ABDE
update_K(quad, nid_pos, ncoords, K)
quads.append(quad)
print('elements created')
# applying boundary conditions
# simply supported
bk = np.zeros(K.shape[0], dtype=bool) #array to store known DOFs
check = np.isclose(x, 0.) | np.isclose(x, a) | np.isclose(
y, 0) | np.isclose(y, b)
bk[2::DOF] = check
# eliminating all u,v displacements
bk[0::DOF] = True
bk[1::DOF] = True
bu = ~bk # same as np.logical_not, defining unknown DOFs
# external force vector for point load at center
f = np.zeros(K.shape[0])
fmid = 1.
# force at center node
check = np.isclose(x, a / 2) & np.isclose(y, b / 2)
f[2::DOF][check] = fmid
assert f.sum() == fmid
# sub-matrices corresponding to unknown DOFs
Kuu = K[bu, :][:, bu]
fu = f[bu]
# solving static problem
uu = solve(Kuu, fu)
# vector u containing displacements for all DOFs
u = np.zeros(K.shape[0])
u[bu] = uu
w = u[2::DOF].reshape(nx, ny).T
# obtained with bfsplate2d element, nx=ny=29
wmax_ref = 6.594931610258557e-05
assert np.isclose(wmax_ref, w.max(), rtol=0.02)
if plot:
import matplotlib
matplotlib.use('TkAgg')
import matplotlib.pyplot as plt
plt.gca().set_aspect('equal')
levels = np.linspace(w.min(), w.max(), 300)
plt.contourf(xmesh, ymesh, w, levels=levels)
plt.colorbar()
plt.show()