def test_hessian():
from pyiga.approx import interpolate
# 2D test
kvs = 2 * (make_knots(3, 0.0, 1.0, 4),)
grid = 2 * (np.linspace(0, 1, 7),)
u = BSplineFunc(kvs, interpolate(kvs, lambda x,y: x**2 + 4*x*y + 3*y**2))
hess = u.grid_hessian(grid)
assert np.allclose(hess, [2.0, 4.0, 6.0]) # (xx, xy, yy)
# 3D test
kvs = 3 * (make_knots(3, 0.0, 1.0, 4),)
grid = 3 * (np.linspace(0, 1, 5),)
u = BSplineFunc(kvs, interpolate(kvs, lambda x,y,z: x**2 + 3*x*z + 2*y*z))
hess = u.grid_hessian(grid)
assert np.allclose(hess, [2.0, 0.0, 3.0, 0.0, 2.0, 0.0]) # (xx, xy, xz, yy, yz, zz)
# 2D vector test
kvs = 2 * (make_knots(3, 0.0, 1.0, 4),)
grid = 2 * (np.linspace(0, 1, 7),)
u = BSplineFunc(kvs, interpolate(kvs, lambda x,y: (x**2 + 4*x*y, 3*y**2)))
hess = u.grid_hessian(grid)
assert np.allclose(hess, [[2.0, 4.0, 0.0], [0.0, 0.0, 6.0]]) # (xx, xy, yy)
# 3D vector test
kvs = 3 * (make_knots(3, 0.0, 1.0, 4),)
grid = 3 * (np.linspace(0, 1, 5),)
u = BSplineFunc(kvs, interpolate(kvs, lambda x,y,z: (x**2, 3*x*z, 2*y*z)))
hess = u.grid_hessian(grid)
assert np.allclose(hess, # (xx, xy, xz, yy, yz, zz)
[[2.0, 0.0, 0.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 3.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 0.0, 2.0, 0.0]])
def test_assemble_asym():
kv1 = bspline.make_knots(4, 0.0, 1.0, 10)
kv2 = bspline.make_knots(1, 0.0, 1.0, 20)
K_12 = bsp_mixed_deriv_biform_1d_asym(kv1, kv2, 1, 0, quadgrid=kv2.mesh)
assert(K_12.shape[0] == kv2.numdofs)
assert(K_12.shape[1] == kv1.numdofs)
u = interpolate(kv1, lambda x: x**4)
v = interpolate(kv2, lambda x: 1.0)
itg = K_12.dot(u).dot(v)
assert abs(itg - 1.0) < 1e-10 # int(4*x^3, 0, 1) = 1
def test_interpolation():
kv = make_knots(3, 0.0, 1.0, 10)
# create random spline
coeffs = np.random.rand(kv.numdofs)
def f(x): return ev(kv, coeffs, x)
# interpolate it at Gréville points and check that result is the same
result = interpolate(kv, f)
assert np.allclose(coeffs, result)
##
# test for p=0
kv = make_knots(0, 0.0, 1.0, 10)
coeffs = np.random.rand(kv.numdofs)
def f(x): return ev(kv, coeffs, x)
# interpolate it at Gréville points and check that result is the same
result = interpolate(kv, f)
assert np.allclose(coeffs, result)
def test_animate_field():
kvs = 2 * (bspline.make_knots(2, 0.0, 1.0, 5),)
fields = [ bspline.BSplineFunc(kvs,
approx.interpolate(kvs, lambda x,y: np.sin(t+x) * np.exp(y)))
for t in range(3) ]
anim = animate_field(fields, geo=geometry.bspline_quarter_annulus(), res=10)
anim.to_jshtml()
def test_divdiv_geo_2d():
kv = bspline.make_knots(3, 0.0, 1.0, 15)
geo = geometry.bspline_quarter_annulus()
A = divdiv((kv,kv), geo, layout='packed')
# construct divergence-free function
from pyiga.approx import interpolate
u = interpolate((kv,kv), lambda x,y: (x,-y), geo=geo).ravel()
assert abs(A.dot(u)).max() < 1e-12
def test_trf_gradient():
geo = bspline_quarter_annulus()
u_coeffs = approx.interpolate(geo.kvs, lambda x, y: x - y, geo=geo)
u = BSplineFunc(geo.kvs, u_coeffs)
u_grad = u.transformed_jacobian(geo)
grd = 2 * (np.linspace(0, 1, 10), )
grads = u_grad.grid_eval(grd)
assert np.allclose(grads[:, :, 0], 1) and np.allclose(grads[:, :, 1], -1)
def test_mixed_deriv_biform():
kv = bspline.make_knots(4, 0.0, 1.0, 20)
DxxD0 = bsp_mixed_deriv_biform_1d(kv, 2, 0)
DxxDx = bsp_mixed_deriv_biform_1d(kv, 2, 1)
u = interpolate(kv, lambda x: x)
# second derivative of linear function x -> x is 0
assert abs(DxxD0.dot(u)).max() < 1e-10
assert abs(DxxDx.dot(u)).max() < 1e-10
def test_plot_field():
def f(x, y): return np.sin(x) * np.exp(y)
geo = geometry.quarter_annulus()
plot_field(f, physical=True, geo=geo, res=10)
#
kvs = 2 * (bspline.make_knots(2, 0.0, 1.0, 5),)
u = bspline.BSplineFunc(kvs, approx.interpolate(kvs, f))
plot_field(u, res=10)
plot_field(u, geo=geo, res=10)
def test_mass_asym():
kv1 = bspline.make_knots(4, 0.0, 1.0, 10)
kv2 = bspline.make_knots(1, 0.0, 1.0, 20)
M_12 = bsp_mass_1d_asym(kv1, kv2, quadgrid=kv2.mesh)
assert(M_12.shape[0] == kv2.numdofs)
assert(M_12.shape[1] == kv1.numdofs)
u = interpolate(kv1, lambda x: x**4)
itg = M_12.dot(u).dot(np.ones(kv2.numdofs))
assert abs(itg - 1.0/5) < 1e-10 # int(x^4, 0, 1) = 1/5
def test_solution_1d():
# solve -u''(x) = 1, u(0) = 0, u(1) = 1
kv = bspline.make_knots(2, 0.0, 1.0, 10)
A = stiffness(kv)
f = inner_products(kv, lambda x: 1.0)
LS = RestrictedLinearSystem(A, f, [(0,kv.numdofs-1), (0.0,1.0)])
u = LS.complete(np.linalg.solve(LS.A.A, LS.b))
u_ex = interpolate(kv, lambda x: 0.5*x*(3-x))
assert np.linalg.norm(u - u_ex) < 1e-12
def test_divdiv_geo_2d():
kv = bspline.make_knots(3, 0.0, 1.0, 15)
geo = geometry.bspline_quarter_annulus()
A = divdiv((kv,kv), geo, layout='packed', format='bsr')
# construct divergence-free function
u = interpolate((kv,kv), lambda x,y: (x,-y), geo=geo)
assert abs(A.dot(u.ravel())).max() < 1e-12
# test blocked layout
A = divdiv((kv,kv), geo, layout='blocked')
u_blocked = np.moveaxis(u, -1, 0) # move last axis to the front
assert abs(A.dot(u_blocked.ravel())).max() < 1e-12
def test_deriv():
# create linear spline
kv = make_knots(4, 0.0, 1.0, 25)
coeffs = interpolate(kv, lambda x: 1.0 + 2.5*x)
# check that derivative is 2.5
x = np.linspace(0.0, 1.0, 100)
drv = deriv(kv, coeffs, 1, x)
assert np.linalg.norm(drv - 2.5) < 1e-10
# create random spline
coeffs = np.random.rand(kv.numdofs)
# compare derivatives by two methods
derivs1 = deriv(kv, coeffs, 1, x)
derivs2 = deriv(kv, coeffs, 2, x)
allders = collocation_derivs(kv, x, derivs=2)
assert np.linalg.norm(derivs1 - allders[1].dot(coeffs), np.inf) < 1e-10
assert np.linalg.norm(derivs2 - allders[2].dot(coeffs), np.inf) < 1e-10
def test_assemble_nonsym_vec():
kvs = 2 * (bspline.make_knots(2, 0.0, 1.0, 5),)
geo = geometry.quarter_annulus()
problem = 'inner(as_matrix([[2,1],[0,0]]).dot(u), v) * dx'
A = assemble(problem, kvs, geo=geo, bfuns=[('u',2), ('v',2)], layout='packed', format='bsr')
u = interpolate(kvs, lambda x, y: (x*y, -2*x*y), geo=geo)
assert np.allclose(A @ u.ravel(), 0)
# test the blockwise assembling using multi_blocks
asm = instantiate_assembler(problem, kvs, args={'geo': geo}, bfuns=[('u',2), ('v',2)])
blocks = np.array(asm.multi_blocks([(0,0), (0,1), (2,1)]))
AA = A.A # bsr does not support slicing
assert np.array_equal(blocks[0], AA[0:2, 0:2])
assert np.array_equal(blocks[1], AA[0:2, 2:4])
assert np.array_equal(blocks[2], AA[4:6, 2:4])
# test blocked layout
A = assemble(problem, kvs, geo=geo, bfuns=[('u',2), ('v',2)], layout='blocked')
u_blocked = np.moveaxis(u, -1, 0) # move last axis to the front
assert np.allclose(A @ u_blocked.ravel(), 0)