def project(self, interp, facets=None):
"""Perform :math:`L^2` projection onto the basis on the boundary.
The values of the interior DOFs remain zero.
Parameters
----------
interp
An object of type :class:`~skfem.element.DiscreteField` which is a
function (to be projected) evaluated at global quadrature points at
the boundary of the domain. If a function is given, then
:class:`~skfem.element.DiscreteField` is created by passing
an array of global quadrature point locations to the function.
facets
Optionally perform the projection on a subset of facets. The
values of the remaining DOFs are zero.
"""
from skfem.utils import solve, condense
M, f = self._projection(interp)
if facets is not None:
return solve(*condense(M, f, I=self.get_dofs(facets=facets)))
return solve(*condense(M, f, I=self.get_dofs(facets=self.find)))
def project(self, interp, elements=None):
"""Perform :math:`L^2` projection onto the basis.
See :ref:`l2proj` for more information.
Parameters
----------
interp
An object of type :class:`~skfem.element.DiscreteField` which is a
function (to be projected) evaluated at global quadrature points.
If a function is given, then :class:`~skfem.element.DiscreteField`
is created by passing an array of global quadrature point locations
to the function.
elements
Optionally perform the projection on a subset of elements. The
values of the remaining DOFs are zero.
"""
from skfem.utils import solve, condense
M, f = self._projection(interp)
if elements is not None:
return solve(*condense(M, f, I=self.get_dofs(elements=elements)))
elif self.tind is not None:
return solve(*condense(M, f, I=self.get_dofs(elements=self.tind)))
return solve(M, f)
def calculate_axial_from_stress(self, R, J):
"""
Calculate the axial force by integrating sigma_zz over the area
Parameters:
R final residual
J final jacobian
"""
#pylint: disable=no-value-for-parameter, invalid-unary-operand-type
if self.state_np1.ndim == 1:
dx = self.state_np1.basis.interpolate(
self.state_np1.mesh.p[0]
)[0][0] * self.state_np1.basis.dx * 2.0 * np.pi
else:
dx = self.state_np1.basis.dx
fake_force = self._internal_force(self.state_np1.tangent[:, :, 2, 2])
de = utils.solve(*utils.condense(J, fake_force, D=self.edofs),
solver=utils.solver_direct_scipy())
fake_strain = self.calculate_strain(de)
integrand1 = np.einsum('iijk',
self.state_np1.tangent[2, 2] * fake_strain)
integrand2 = self.state_np1.tangent[2, 2, 2, 2]
self.state_np1.force = np.sum(self.state_np1.stress[2, 2] * dx)
self.state_np1.stiffness = np.sum(
(-integrand1 + integrand2) * dx) / self.state_np1.h
def linear_solve(self, J, R):
"""
Do the linear solve for a particular jacobian and residual
Parameters:
J Jacobian sparse matrix
R residual vector
"""
#pylint: disable=no-value-for-parameter
return utils.solve(*utils.condense(J, R, D=self.edofs),
solver=utils.solver_direct_scipy())
def test_simple_cg_solver():
m = MeshTri().refined(3)
basis = CellBasis(m, ElementTriP1())
A0 = laplace.coo_data(basis)
A1 = laplace.assemble(basis)
f = unit_load.assemble(basis)
D = m.boundary_nodes()
x1 = solve(*condense(A1, f, D=D))
f[D] = 0
x0 = A0.solve(f, D=D)
assert_almost_equal(x0, x1)
# pass 1A through the conductors
current = 1
J_inner_conductor = asm(unit_load, inner_conductor_basis)
# scale inner conductor current density to have an integral
# equal to current over domain
J_inner_conductor *= current / np.dot(J_inner_conductor, load_integral)
J_outer_conductor = asm(unit_load, outer_conductor_basis)
# scale outer conductor current density to have an integral
# equal to -current over domain
J_outer_conductor *= -current / np.dot(J_outer_conductor, load_integral)
A = solve(
*condense(K_mag, J_inner_conductor +
J_outer_conductor, D=dofs['boundary']))
# magnetic field energy from FEM
E_mag = 0.5 * np.dot(A, K_mag * A)
# energy stored in inductor: E = 0.5*L*I^2
# thus, L = 2*E/I^2
L = 2 * E_mag / (current**2)
print(f'L={L} H/m')
# assemble the parts of the stiffness matrix for each material separately
K_elec_inner_insulator = asm(laplace, inner_insulator_basis) * eps0 * eps_ptfe
K_elec_outer_insulator = asm(laplace, outer_insulator_basis) * eps0 * eps_fep
# use dummy value for permittivity, uniform U in conductor
K_elec_inner_conductor = asm(laplace, inner_conductor_basis) * eps0
# use dummy value for permittivity, uniform U in conductor
# pass 1A through the conductors
current = 1
J_inner_conductor = asm(unit_load, inner_conductor_basis)
# scale inner conductor current density to have an integral
# equal to current over domain
J_inner_conductor *= current / np.dot(J_inner_conductor, load_integral)
J_outer_conductor = asm(unit_load, outer_conductor_basis)
# scale outer conductor current density to have an integral
# equal to -current over domain
J_outer_conductor *= -current / np.dot(J_outer_conductor, load_integral)
A = solve(
*condense(K_mag, J_inner_conductor +
J_outer_conductor, D=dofs['boundary']))
# magnetic field energy from FEM
E_mag = 0.5 * np.dot(A, K_mag * A)
# energy stored in inductor: E = 0.5*L*I^2
# thus, L = 2*E/I^2
L = 2 * E_mag / (current**2)
print(f'L={L} H/m')
# assemble the parts of the stiffness matrix for each material separately
K_elec_inner_insulator = asm(laplace, inner_insulator_basis) * eps0 * eps_ptfe
K_elec_outer_insulator = asm(laplace, outer_insulator_basis) * eps0 * eps_fep
# use dummy value for permittivity, uniform U in conductor
K_elec_inner_conductor = asm(laplace, inner_conductor_basis) * eps0
# use dummy value for permittivity, uniform U in conductor
# pass 1A through the conductors
current = 1
J_inner_conductor = asm(unit_load, inner_conductor_basis)
# scale inner conductor current density to have an integral
# equal to current over domain
J_inner_conductor *= current / np.dot(J_inner_conductor, load_integral)
J_outer_conductor = asm(unit_load, outer_conductor_basis)
# scale outer conductor current density to have an integral
# equal to -current over domain
J_outer_conductor *= -current / np.dot(J_outer_conductor, load_integral)
A = solve(
*condense(K_mag, J_inner_conductor +
J_outer_conductor, D=dofs['boundary']))
# magnetic field energy from FEM
E_mag = 0.5 * np.dot(A, K_mag * A)
# energy stored in inductor: E = 0.5*L*I^2
# thus, L = 2*E/I^2
L = 2 * E_mag / (current**2)
print(f'L={L} H/m')
# assemble the parts of the stiffness matrix for each material separately
K_elec_inner_insulator = asm(laplace, inner_insulator_basis) * eps0 * eps_ptfe
K_elec_outer_insulator = asm(laplace, outer_insulator_basis) * eps0 * eps_fep
# use dummy value for permittivity, uniform U in conductor
K_elec_inner_conductor = asm(laplace, inner_conductor_basis) * eps0
# use dummy value for permittivity, uniform U in conductor
