def test_gauss_tri():
# Zero order
pts, wts = gauss.gauss_tri(order=1)
fun = lambda x, y: 1
inte = 0.0
for cont, pt in enumerate(pts):
x, y = pt
inte += 0.5 * wts[cont] * fun(x, y)
assert np.isclose(inte, 0.5)
# First order
pts, wts = gauss.gauss_tri(order=1)
a, b = np.random.uniform(0, 1, 2)
fun = lambda x, y: a*x + b*y
inte = 0.0
for cont, pt in enumerate(pts):
x, y = pt
inte += 0.5 * wts[cont] * fun(x, y)
assert np.isclose(inte, (a + b)/6)
# Second order
pts, wts = gauss.gauss_tri(order=2)
a, b, c, d, e, f = np.random.uniform(0, 1, 6)
fun = lambda x, y: a*x**2 + b*y**2 + c*x*y + d*x + e*y + f
inte = 0.0
for cont, pt in enumerate(pts):
x, y = pt
inte += 0.5 * wts[cont] * fun(x, y)
assert np.isclose(inte, (2*a + 2*b + c + 4*d + 4*e + 12*f)/24)
# Third order
pts, wts = gauss.gauss_tri(order=3)
a, b, c, d, e, f = np.random.uniform(0, 1, 6)
fun = lambda x, y: a*x**2 + b*y**2 + c*x*y + d*x + e*y + f
inte = 0.0
for cont, pt in enumerate(pts):
x, y = pt
inte += 0.5 * wts[cont] * fun(x, y)
assert np.isclose(inte, (2*a + 2*b + c + 4*d + 4*e + 12*f)/24)
# Seventh order
pts, wts = gauss.gauss_tri(order=7)
a, b, c, d, e, f, g, h = np.random.uniform(0, 1, 8)
fun = lambda x, y: a*x**7 + b*x**6*y + c*x**5*y**2 + d*x**4*y**3\
+ e*x**3*y**4 + f*x**2*y**5 + g*x*y**6 + h*y**7
inte = 0.0
for cont, pt in enumerate(pts):
x, y = pt
inte += 0.5 * wts[cont] * fun(x, y)
assert np.isclose(inte, (105*a + 15*b + 5*c + 3*d + 3*e + 5*f + 15*g
+ 105*h)/7560)
def elast_tri3(coord, params):
"""Triangular element with 3 nodes
Parameters
----------
coord : ndarray
Coordinates for the nodes of the element (3, 2).
params : tuple
Material parameters in the following order:
young : float
Young modulus (>0).
poisson : float
Poisson coefficient (-1, 0.5).
dens : float, optional
Density (>0).
Returns
-------
stiff_mat : ndarray
Local stiffness matrix for the element (6, 6).
mass_mat : ndarray
Local mass matrix for the element (6, 6).
Examples
--------
>>> coord = np.array([
... [0, 0],
... [1, 0],
... [0, 1]])
>>> params = [8/3, 1/3]
>>> stiff, mass = uel3ntrian(coord, params)
>>> stiff_ex = 1/2 * np.array([
... [4, 2, -3, -1, -1, -1],
... [2, 4, -1, -1, -1, -3],
... [-3, -1, 3, 0, 0, 1],
... [-1, -1, 0, 1, 1, 0],
... [-1, -1, 0, 1, 1, 0],
... [-1, -3, 1, 0, 0, 3]])
>>> np.allclose(stiff, stiff_ex)
True
"""
stiff_mat = np.zeros([6, 6])
mass_mat = np.zeros([6, 6])
C = fem.umat(params[:2])
if len(params) == 2:
dens = 1
else:
dens = params[-1]
gpts, gwts = gau.gauss_tri(order=2)
for cont in range(gpts.shape[0]):
r, s = gpts[cont, :]
H, B, det = fem.elast_diff_2d(r, s, coord, fem.shape_tri3)
factor = det * gwts[cont]
stiff_mat += 0.5 * factor * (B.T @ C @ B)
mass_mat += 0.5 * dens * factor * (H.T @ H)
return stiff_mat, mass_mat
def uel6ntrian(coord, params):
"""Triangular element with 6 nodes
Parameters
----------
coord : ndarray
Coordinates for the nodes of the element (6, 2).
params : tuple
Material parameters in the following order:
young : float
Young modulus (>0).
poisson : float
Poisson coefficient (-1, 0.5).
dens : float, optional
Density (>0).
Returns
-------
stiff_mat : ndarray
Local stiffness matrix for the element (12, 12).
mass_mat : ndarray
Local mass matrix for the element (12, 12).
Examples
--------
>>> coord = np.array([
... [0, 0],
... [1, 0],
... [0, 1],
... [0.5, 0],
... [0.5, 0.5],
... [0, 0.5]])
>>> params = [8/3, 1/3]
>>> stiff, mass = uel6ntrian(coord, params)
>>> stiff_ex = 1/6 * np.array([
... [12, 6, 3, 1, 1, 1, -12, -4, 0, 0, -4, -4],
... [6, 12, 1, 1, 1, 3, -4, -4, 0, 0, -4, -12],
... [3, 1, 9, 0, 0, -1, -12, -4, 0, 4, 0, 0],
... [1, 1, 0, 3, -1, 0, -4, -4, 4, 0, 0, 0],
... [1, 1, 0, -1, 3, 0, 0, 0, 0, 4, -4, -4],
... [1, 3, -1, 0, 0, 9, 0, 0, 4, 0, -4, -12],
... [-12, -4, -12, -4, 0, 0, 32, 8, -8, -8, 0, 8],
... [-4, -4, -4, -4, 0, 0, 8, 32, -8, -24, 8, 0],
... [0, 0, 0, 4, 0, 4, -8, -8, 32, 8, -24, -8],
... [0, 0, 4, 0, 4, 0, -8, -24, 8, 32, -8, -8],
... [-4, -4, 0, 0, -4, -4, 0, 8, -24, -8, 32, 8],
... [-4, -12, 0, 0, -4, -12, 8, 0, -8, -8, 8, 32]])
>>> np.allclose(stiff, stiff_ex)
True
"""
stiff_mat = np.zeros([12, 12])
mass_mat = np.zeros([12, 12])
C = fem.umat(params[:2])
if len(params) == 2:
dens = 1
else:
dens = params[-1]
gpts, gwts = gau.gauss_tri(order=3)
for i in range(gpts.shape[0]):
ri = gpts[i, 0]
si = gpts[i, 1]
ddet, B = fem.stdm6NT(ri, si, coord)
N = fem.sha6(ri, si)
factor = gwts[i] * ddet
stiff_mat += 0.5 * factor * (B.T @ C @ B)
mass_mat += dens * factor * (N.T @ N)
return stiff_mat, mass_mat