def project(u, v):
"""Projects vector u onto vector v."""
u = matrix_to_vector(Matrix(u), C)
v = matrix_to_vector(Matrix(v), C)
m_v = v.magnitude()
scal = u.dot(v) / m_v
pprint(scal)
return scal * (v / m_v)
def scipy_one_cell(points, nodes):
degree = len(nodes) - 1
x, y, z = sp.symbols("x y z")
a = [sp.Symbol("a{0:d}".format(i)) for i in range(degree + 1)]
# Find polynomial for variation in z-direction
poly = 0
for deg in range(degree + 1):
poly += a[deg] * y**deg
eqs = []
for node in nodes:
eqs.append(poly.subs(y, points[node][-2]) - points[node][-1])
coeffs = sp.solve(eqs, a)
transform = poly
for i in range(len(a)):
transform = transform.subs(a[i], coeffs[a[i]])
# Compute integral
C = CoordSys3D("C")
para = sp.Matrix([x, y, transform])
vec = matrix_to_vector(para, C)
cross = (vec.diff(x) ^ vec.diff(y)).magnitude()
expr = (transform + x * y) * cross
approx = sp.lambdify((x, y), expr)
ref = scipy.integrate.dblquad(approx, 0, 1, lambda x: 0,
lambda x: 1 - x)[0]
return ref
def test_SphericallySymmetricVolCondMEG_04(self):
'''dipole time series'''
# define symbols
N = sv.CoordSys3D('')
Q_x, Q_y, Q_z = sp.symbols('Q_x Q_y Q_z', real=True)
R_x, R_y, R_z = sp.symbols('R_x R_y R_z', real=True)
r_x, r_y, r_z = sp.symbols('r_x r_y r_z', real=True)
Q = sv.matrix_to_vector(sp.Matrix([Q_x, Q_y, Q_z]), N) # dipole moment
R = sv.matrix_to_vector(sp.Matrix([R_x, R_y, R_z]), N) # dipole loc.
r = sv.matrix_to_vector(sp.Matrix([r_x, r_y, r_z]), N) # meas. loc.
# eq. 25 in Sarvas et al. 1987:
a = r - R
a_ = sp.sqrt(a.dot(a))
r_ = sp.sqrt(r.dot(r))
F = a_ * (r_ * a_ + r_**2 - R.dot(r))
nabla_F = (a_**2 / r_ + a.dot(r) / a_ + 2 * a_ + 2 * r_
) * r - (a_ + 2 * r_ + a.dot(r) / a_) * R
H = (F * Q.cross(R) - Q.cross(R).dot(r) * nabla_F) / (4 * sp.pi * F**2)
# check some values
p = np.c_[np.eye(3), -np.eye(3)]
r_p = np.array([0.1, -0.2, 0.9])
r_s = np.array([-0.3, 0.1, 1.3])
H_gt = np.zeros((1, 3, 6))
for i, p_ in enumerate(p.T):
H_gt[0, :, i] = np.array(H.evalf(subs={
Q_x: p_[0],
Q_y: p_[1],
Q_z: p_[2],
R_x: r_p[0],
R_y: r_p[1],
R_z: r_p[2],
r_x: r_s[0],
r_y: r_s[1],
r_z: r_s[2],
}).to_matrix(N).tolist()).flatten()
meg = lfpykit.eegmegcalc.SphericallySymmetricVolCondMEG(
r=np.array([r_s]))
M = meg.get_transformation_matrix(r_p)
np.testing.assert_almost_equal(M @ p, H_gt)
def sympy_scipy(points, nodes, L, H):
"""Approximated integration of z + x*y over a surface where the z-coordinate
is only dependent of the y-component of the box. x in [0,L], y in
[0,H]
Input: points: All points of defining the geometry nodes: Points on
one of the outer boundaries varying in the y-direction
"""
degree = len(nodes) - 1
x, y, z = sp.symbols("x y z")
a = [sp.Symbol("a{0:d}".format(i)) for i in range(degree + 1)]
# Find polynomial for variation in z-direction
poly = 0
for deg in range(degree + 1):
poly += a[deg] * y**deg
eqs = []
for node in nodes:
eqs.append(poly.subs(y, points[node][-2]) - points[node][-1])
coeffs = sp.solve(eqs, a)
transform = poly
for i in range(len(a)):
transform = transform.subs(a[i], coeffs[a[i]])
# Compute integral
C = CoordSys3D("C")
para = sp.Matrix([x, y, transform])
vec = matrix_to_vector(para, C)
cross = (vec.diff(x) ^ vec.diff(y)).magnitude()
expr = (transform + x * y) * cross
approx = sp.lambdify((x, y), expr)
ref = scipy.integrate.nquad(approx, [[0, L], [0, H]])[0]
# Slow and only works for simple integrals
# integral = sp.integrate(expr, (y, 0, H))
# integral = sp.integrate(integral, (x, 0, L))
# ex = integral.evalf()
return ref
def unit(v):
v = matrix_to_vector(Matrix(v), C)
return v.normalize()
def magnitude(v):
v = matrix_to_vector(Matrix(v), C)
return v.magnitude()
def cross(u, v):
u = matrix_to_vector(Matrix(u), C)
v = matrix_to_vector(Matrix(v), C)
return u.cross(v)
def test_SphericallySymmetricVolCondMEG_03(self):
'''compare with (slow) sympy predictions'''
# define symbols
N = sv.CoordSys3D('')
Q_x, Q_y, Q_z = sp.symbols('Q_x Q_y Q_z', real=True)
R_x, R_y, R_z = sp.symbols('R_x R_y R_z', real=True)
r_x, r_y, r_z = sp.symbols('r_x r_y r_z', real=True)
Q = sv.matrix_to_vector(sp.Matrix([Q_x, Q_y, Q_z]), N) # dipole moment
R = sv.matrix_to_vector(sp.Matrix([R_x, R_y, R_z]), N) # dipole loc.
r = sv.matrix_to_vector(sp.Matrix([r_x, r_y, r_z]), N) # meas. loc.
# eq. 25 in Sarvas et al. 1987:
a = r - R
a_ = sp.sqrt(a.dot(a))
r_ = sp.sqrt(r.dot(r))
F = a_ * (r_ * a_ + r_**2 - R.dot(r))
nabla_F = (a_**2 / r_ + a.dot(r) / a_ + 2 * a_ + 2 * r_
) * r - (a_ + 2 * r_ + a.dot(r) / a_) * R
H = (F * Q.cross(R) - Q.cross(R).dot(r) * nabla_F) / (4 * sp.pi * F**2)
# compare sympy pred with our implementation w. different dipole
# moments in different measurement and dipole locations
for p_ in np.expand_dims(np.c_[np.eye(3), [0.5, -1.7, 0.74]].T, 2):
for r_p in np.array([[1, 0, 0],
[0, 1, 0],
[0, 0, 1],
[0.5, 0.65, 0.9]]):
for r_s in np.array([[2, 0, 0],
[0, 2, 0],
[0, 0, 2],
[0.95, -1.2, 0.75]]):
F_gt = float(F.subs({
R_x: r_p[0],
R_y: r_p[1],
R_z: r_p[2],
r_x: r_s[0],
r_y: r_s[1],
r_z: r_s[2],
}))
nabla_F_gt = np.array(nabla_F.evalf(subs={
R_x: r_p[0],
R_y: r_p[1],
R_z: r_p[2],
r_x: r_s[0],
r_y: r_s[1],
r_z: r_s[2],
}).to_matrix(N).tolist()).flatten()
H_gt = np.expand_dims(H.evalf(subs={
Q_x: p_[0, 0],
Q_y: p_[1, 0],
Q_z: p_[2, 0],
R_x: r_p[0],
R_y: r_p[1],
R_z: r_p[2],
r_x: r_s[0],
r_y: r_s[1],
r_z: r_s[2],
}).to_matrix(N).tolist(), 0)
meg = lfpykit.eegmegcalc.SphericallySymmetricVolCondMEG(
r=np.array([r_s]))
np.testing.assert_almost_equal(
F_gt,
meg._compute_F(r_p,
r_s,
np.linalg.norm(r_s - r_p),
np.linalg.norm(r_s)))
np.testing.assert_almost_equal(
nabla_F_gt,
meg._compute_grad_F(r_p,
r_s,
r_s - r_p,
np.linalg.norm(r_s - r_p),
np.linalg.norm(r_s)))
np.testing.assert_almost_equal(
H_gt,
meg.calculate_H(p_, r_p)
)
def convert(vec):
C = CoordSys3D('')
return matrix_to_vector(parse_matrix(vec), C)