def creationSpaces(degree, grid):
"""
Create the schoenberg spaces of degree @degree (and @degree-1)
Parameters
----------
@degree: int
degree of the first schoenberg space
@grid : list of float or numpy.ndarray
breakpoints of the domain
Returns
-------
V : Class SplineSpace
1D finite element space of schoenberg space of degree @degree
W: Class SplineSpace
1D finite element space of schoenberg space of degree (@degree - 1)
"""
V = SplineSpace(degree, grid=grid)
W = SplineSpace(degree - 1, grid=grid)
V.init_fem(quad_order=degree + 1)
W.init_fem(quad_order=degree + 1)
return V, W
def test_pdes_2d_2():
print('============ test_pdes_2d_2 =============')
# ... abstract model
V = H1Space('V', ldim=2)
v = TestFunction(V, name='v')
u = TestFunction(V, name='u')
c = Constant('c', real=True, label='mass stabilization')
a = BilinearForm((v, u), dot(grad(v), grad(u)) + c * v * u)
# ...
# ... discretization
# Input data: degree, number of elements
p1 = 1
p2 = 1
ne1 = 4
ne2 = 4
# Create uniform grid
grid_1 = linspace(0., 1., num=ne1 + 1)
grid_2 = linspace(0., 1., num=ne2 + 1)
# Create 1D finite element spaces and precompute quadrature data
V1 = SplineSpace(p1, grid=grid_1)
V1.init_fem()
V2 = SplineSpace(p2, grid=grid_2)
V2.init_fem()
# Create 2D tensor product finite element space
V = TensorFemSpace(V1, V2)
# ...
# ...
discretize_symbol(a, [V, V])
# print(mass.symbol.__doc__)
# ...
# ...
n1 = 21
n2 = 21
t1 = linspace(-pi, pi, n1)
t2 = linspace(-pi, pi, n2)
x1 = linspace(0., 1., n1)
x2 = linspace(0., 1., n2)
xs = [x1, x2]
ts = [t1, t2]
e = a.symbol(*xs, *ts, 0.25)
print('> c = 0.25 :: ', e.min(), e.max())
e = a.symbol(*xs, *ts, 0.6)
print('> c = 0.6 :: ', e.min(), e.max())
def creation_spaces_2d(p1, p2, ne1, ne2): """ Creation of the different spaces used in our solver: Require: -------- @p1 and @p2 are degrees of direction 1 and 2 @ne1 and @ne2 are number of elements in direction 1 and 2 Returns: -------- @V TensorFem which represents H^1 space @W TensorFem which represents L^2 space @Wdiv1 and @Wdiv2 represent (H¹, div) space """ grid_1 = np.linspace( 0., 1., num=ne1+1 ) grid_2 = np.linspace( 0., 1., num=ne2+1 ) # Create 1D finite element spaces and precompute quadrature data V1 = SplineSpace( p1, grid=grid_1); V1.init_fem(quad_order=p1+1) W1 = SplineSpace(p1-1, grid=grid_1); W1.init_fem(quad_order=p1+1) V2 = SplineSpace( p2, grid=grid_2 ); V2.init_fem(quad_order=p2+1) W2 = SplineSpace(p2-1, grid=grid_2); W2.init_fem(quad_order=p2+1) # Create 2D tensor product finite element space V = TensorFemSpace(V1, V2 ) W = TensorFemSpace(W1, W2) Wdiv1 = TensorFemSpace(V1, W2) Wdiv2 = TensorFemSpace(W1, V2) return V, Wdiv1, Wdiv2, W
def test_pdes_1d_1():
print('============ test_pdes_1d_1 =============')
# ... abstract model
V = H1Space('V', ldim=1)
v = TestFunction(V, name='v')
u = TestFunction(V, name='u')
a = BilinearForm((v,u), dot(grad(v), grad(u)))
# ...
# ... discretization
# Input data: degree, number of elements
p1 = 1
ne1 = 4
# Create uniform grid
grid_1 = linspace( 0., 1., num=ne1+1 )
# Create 1D finite element spaces and precompute quadrature data
V = SplineSpace( p1, grid=grid_1 ); V.init_fem()
# ...
# ...
discretize_symbol( a, [V,V] )
# print(mass.symbol.__doc__)
# ...
# ...
n1 = 21
t1 = linspace(-pi, pi, n1)
x1 = linspace(0.,1., n1)
xs = [x1]
ts = [t1]
e = a.symbol(*xs, *ts)
print('> ', e.min(), e.max())
def test_pdes_3d_1():
print('============ test_pdes_3d_1 =============')
# ... abstract model
V = H1Space('V', ldim=3)
v = TestFunction(V, name='v')
u = TestFunction(V, name='u')
a = BilinearForm((v, u), dot(grad(v), grad(u)))
# ...
# ... discretization
# Input data: degree, number of elements
p1 = 1
p2 = 1
p3 = 1
ne1 = 4
ne2 = 4
ne3 = 4
# Create uniform grid
grid_1 = linspace(0., 1., num=ne1 + 1)
grid_2 = linspace(0., 1., num=ne2 + 1)
grid_3 = linspace(0., 1., num=ne3 + 1)
# Create 1D finite element spaces and precompute quadrature data
V1 = SplineSpace(p1, grid=grid_1)
V1.init_fem()
V2 = SplineSpace(p2, grid=grid_2)
V2.init_fem()
V3 = SplineSpace(p3, grid=grid_3)
V3.init_fem()
# Create 2D tensor product finite element space
V = TensorFemSpace(V1, V2, V3)
# ...
# ...
discretize_symbol(a, [V, V])
# print(mass.symbol.__doc__)
# ...
# ...
n1 = 21
n2 = 21
n3 = 21
t1 = linspace(-pi, pi, n1)
t2 = linspace(-pi, pi, n2)
t3 = linspace(-pi, pi, n3)
x1 = linspace(0., 1., n1)
x2 = linspace(0., 1., n2)
x3 = linspace(0., 1., n3)
xs = [x1, x2, x3]
ts = [t1, t2, t3]
e = a.symbol(*xs, *ts)
print('> ', e.min(), e.max())
analytic = lambda x: np.sin(2 * np.pi * x)
# define dirichlet boundary conditions
dirichletleft = 0.
dirichletright = 0.
# SplineSpace
p = 3
ne = 60
grid = np.linspace(0., 1., ne + 1)
# Main part
print("Programm running ... ")
V = SplineSpace(p, grid=grid)
V.init_fem()
# building stiffness matrix in a sparse matrix
stiff = StiffnessMatrix(V)
stiff = stiff.todense()
rhs = assemblyRhs(V, sec)
# homogenous dirichlet boundary conditions
stiff = stiff[1:-1, 1:-1]
rhs = rhs[1:-1]
u = np.linalg.solve(stiff, rhs)
u = list(u)
u.insert(0, dirichletleft)
u.append(dirichletright)