def find_physical_coordinates(self, N):
self.N = N
self.Nq = N + 1
if self.get_ndim() == 3:
self.Nfp = (N + 1) * (N + 1)
self.Np = (N + 1) * (N + 1) * (N + 1)
else:
self.Nfp = N + 1
self.Np = (N + 1) * (N + 1)
z_1, jnk = gauss_lobatto(1)
z_N, jnk = gauss_lobatto(N)
J = lagrange(z_N, z_1)
self.xe = []
self.ye = []
self.ze = []
if self.ndim == 3:
for e in range(self.get_num_elems()):
x = self.x[e, :]
y = self.y[e, :]
z = self.z[e, :]
xx = np.array([x[0], x[1], x[3], x[2], x[4], x[5], x[7], x[6]])
yy = np.array([y[0], y[1], y[3], y[2], y[4], y[5], y[7], y[6]])
zz = np.array([z[0], z[1], z[3], z[2], z[4], z[5], z[7], z[6]])
xe = kron(J, J, J, xx)
ye = kron(J, J, J, yy)
ze = kron(J, J, J, zz)
self.xe.append(xe)
self.ye.append(ye)
self.ze.append(ze)
else:
for e in range(self.get_num_elems()):
x = self.x[e, :]
y = self.y[e, :]
z = self.z[e, :]
xx = np.array([x[0], x[1], x[3], x[2]])
yy = np.array([y[0], y[1], y[3], y[2]])
zz = np.arraz([z[0], z[1], z[3], z[2]])
xe = kron(J, J, J, xx)
ye = kron(J, J, J, yy)
ze = kron(J, J, J, zz)
self.xe.append(xe)
self.ye.append(ye)
self.ze.append(ze)
self.xe = np.array(self.xe)
self.ye = np.array(self.ye)
self.ze = np.array(self.ze)
def reference_mass_matrix_2d(p):
z, w = gauss_lobatto(p)
n = p + 1
B = np.zeros((n * n, ), dtype=np.float64)
for k in range(n):
for j in range(n):
B[j * n + k] = w[j] * w[k]
return B
def box(x, y, z, M):
N = 1
n = N + 1
nn = n * n
m = M + 1
mm = m * m
zn, wn = gauss_lobatto(N)
zm, wm = gauss_lobatto(M)
J = lagrange(zm, zn)
x = x.reshape((nn, n))
y = y.reshape((nn, n))
z = z.reshape((nn, n))
Jx = np.dot(x, J.T)
Jy = np.dot(y, J.T)
Jz = np.dot(z, J.T)
x = Jx.reshape((n, n, m))
y = Jy.reshape((n, n, m))
z = Jz.reshape((n, n, m))
Jx = np.zeros((n, m, m))
Jy = np.zeros((n, m, m))
Jz = np.zeros((n, m, m))
for i in range(n):
Jx[i, :, :] = np.dot(J, x[i, :, :])
Jy[i, :, :] = np.dot(J, y[i, :, :])
Jz[i, :, :] = np.dot(J, z[i, :, :])
x = Jx.reshape((n, mm))
y = Jy.reshape((n, mm))
z = Jz.reshape((n, mm))
Jx = np.dot(J, x)
Jy = np.dot(J, y)
Jz = np.dot(J, z)
x = Jx.reshape((m, m, m))
y = Jy.reshape((m, m, m))
z = Jz.reshape((m, m, m))
return x, y, z
def box_2d(x, y, M):
N = 1
n = N + 1
m = M + 1
zn, wn = gauss_lobatto(N)
zm, wm = gauss_lobatto(M)
J = lagrange(zm, zn)
x = x.reshape((n, n))
y = y.reshape((n, n))
Jx = np.dot(x, J.T)
Jy = np.dot(y, J.T)
x = Jx.reshape((n, m))
y = Jy.reshape((n, m))
Jx = np.dot(J, x)
Jy = np.dot(J, y)
x = Jx.reshape((m, m))
y = Jy.reshape((m, m))
return x, y
def reference_derivative_matrix(p):
z, w = gauss_lobatto(p)
D = lagrange_derivative_matrix(z)
return D
def trapezoid(M):
N = 1
zn, wn = gauss_lobatto(N)
zm, wm = gauss_lobatto(M)
J = lagrange(zm, zn)
x = np.array([
-0.5,
0.5,
-1.0 / sqrt(2.0),
1.0 / sqrt(2.0),
-0.5,
0.5,
-1.0 / sqrt(2.0),
1.0 / sqrt(2.0),
])
y = np.array([
0.5,
0.5,
1.0 / sqrt(2.0),
1.0 / sqrt(2.0),
0.5,
0.5,
1.0 / sqrt(2.0),
1.0 / sqrt(2.0),
])
z = np.array([0.0, 0.0, 0.0, 0.0, 0.1, 0.1, 0.1, 0.1])
n = N + 1
nn = n * n
m = M + 1
mm = m * m
x = x.reshape((nn, n))
y = y.reshape((nn, n))
z = z.reshape((nn, n))
Jx = np.dot(x, J.T)
Jy = np.dot(y, J.T)
Jz = np.dot(z, J.T)
Jx = Jx.reshape((n, n, m))
Jy = Jy.reshape((n, n, m))
Jz = Jz.reshape((n, n, m))
JJx = np.zeros((n, m, m))
JJy = np.zeros((n, m, m))
JJz = np.zeros((n, m, m))
for i in range(n):
JJx[i, :, :] = np.dot(J, Jx[i, :, :])
JJy[i, :, :] = np.dot(J, Jy[i, :, :])
JJz[i, :, :] = np.dot(J, Jz[i, :, :])
X = JJx
Y = JJy
Z = JJz
for j in range(n):
for i in range(m):
Y[j, i, :] = Y[j, i, :] + (Y[j, i, :] - Y[j, 0, :]) * (
X[j, i, 0] * X[j, i, 0] - np.multiply(X[j, i, :], X[j, i, :]))
Jx = JJx.reshape((n, mm))
Jy = JJy.reshape((n, mm))
Jz = JJz.reshape((n, mm))
X = np.dot(J, Jx)
Y = np.dot(J, Jy)
Z = np.dot(J, Jz)
X = X.reshape((m, m, m))
Y = Y.reshape((m, m, m))
Z = Z.reshape((m, m, m))
return X, Y, Z
def reference_mass_matrix_1d(p):
z, w = gauss_lobatto(p)
return np.diag(w)