fem.projectors#

Functions#

`get_dual_dofs`(Vh, f, domain_h[, ...])	return the dual dofs tilde_sigma_i(f) = < Lambda_i, f >_{L2} i = 1, .
`knot_insertion_projection_operator`(domain, ...)	Compute the projection operator based on the knot insertion technique.
`knots_to_insert`(coarse_grid, fine_grid[, tol])	Compute the point difference between the fine grid and coarse grid.

Classes#

Inheritance diagram of psydac.fem.projectors

`DirichletProjector`(fem_space, *[, bcs, ...])	A LinearOperator that applies homogeneous (unless manually given different bcs) Dirichlet boundary conditions.
`MultipatchDirichletProjector`(fem_space, *[, ...])	A LinearOperator (for multipatch domains) that applies homogeneous (unless manually given different bcs) Dirichlet boundary conditions.

Details#

knots_to_insert(coarse_grid, fine_grid, tol=1e-14)[source]#: Compute the point difference between the fine grid and coarse grid.

knot_insertion_projection_operator(domain, codomain)[source]#

Compute the projection operator based on the knot insertion technique.

Return a linear operator which projects an element of the domain to an element of the codomain. Domain and codomain are scalar spline spaces over a cuboid, built as the tensor product of 1D spline spaces. In particular, domain and codomain have the same multi-degree (p1, p2, …).

This function returns a LinearOperator K working at the level of the spline coefficients, which are represented by StencilVector objects.

Thanks to the tensor-product structure of the spline spaces, the projection operator is the Kronecker product of 1D projection operators K[i] operating between 1D spaces. Each 1D operators is represented by a dense matrix:

K = K[0] x K[1] x …

For each dimension i the 1D grids defined by the breakpoints of the two spaces are assumed to be identical, or one nested into the other. Let nd[i] and nc[i] be the number of cells along dimension i for domain and codomain, respectively. We then have three different cases:

nd[i] == nc[i]: The two 1D grids are assumed identical, and K[i] is the identity matrix.
nd[i] < nc[i]: The 1D grid of the domain is assumed nested into the 1D grid of the codomain, hence the 1D spline space of the domain is a subspace of the 1D spline space of the codomain. In this case we build K[i] using the knot insertion algorithm.
nd[i] > nc[i]: The 1D grid of the codomain is assumed nested into the 1D grid of the domain, hence the 1D spline space of the codomain is a subspace of the 1D spline space of the domain. In this case we build K[i] as the transpose of the matrix obtained using the knot insertion algorithm from the codomain to the domain.

Parameters:

domainTensorFemSpace: Domain of the projection operator.
codomainTensorFemSpace: Codomain of the projection operator.

Returns:

KroneckerDenseMatrix: Matrix representation of the projection operator. This is a LinearOperator acting on the spline coefficients.

get_dual_dofs(Vh, f, domain_h, backend_language='python', return_format='stencil_array')[source]#

return the dual dofs tilde_sigma_i(f) = < Lambda_i, f >_{L2} i = 1, .. dim(Vh)) of a given function f, as a stencil array or numpy array

Parameters:

VhFemSpace: The discrete space for the dual dofs
f<sympy.Expr>: The function used for evaluation
domain_h: The discrete domain corresponding to Vh
backend_language: <str>: The backend used to accelerate the code
return_format: <str>: The format of the dofs, can be ‘stencil_array’ or ‘numpy_array’

Returns:

tilde_f: <Vector|ndarray>: The dual dofs

class DirichletProjector(fem_space, *, bcs=None, space_kind=None)[source]#

Bases: LinearOperator

A LinearOperator that applies homogeneous (unless manually given different bcs) Dirichlet boundary conditions.

Parameters:

fem_spacepsydac.fem.basic.FemSpace: fem_space.coeff_space is domain and codomain of this LO. fem_space.kind.name determines the BCs that get applied.
bcsIterable | None: Iterable of sympde.topology.Boundary objects. Allows the user to apply different kinds of BCs. Must not be passed if a space_kind argument is passed.
space_kindstr | SpaceType | None: Necessary only if fem_space.kind.name is undefined and no bcs are passed. Must not be passed if a bcs argument is passed.

Notes

See examples/vector_potential_3d.py for a use case of such an operator.

property domain#: The domain of the linear operator - an element of Vectorspace

property codomain#: The codomain of the linear operator - an element of Vectorspace

property dtype#: The data type of the coefficients of the linear operator, upon convertion to matrix.

tosparse()[source]#: Convert to a sparse matrix in any of the formats supported by scipy.sparse.

toarray()[source]#: Convert to Numpy 2D array.

dot(v, out=None)[source]#

Apply the LinearOperator self to the Vector v.

The result is written to the Vector out, if provided.

Parameters:

vVector: The vector to which the linear operator (self) is applied. It must belong to the domain of self.
outVector: The vector in which the result of the operation is stored. It must belong to the codomain of self. If out is None, a new vector is created and returned.

Returns:

Vector: The result of the operation. If out is None, a new vector is returned. Otherwise, the result is stored in out and out is returned.

transpose(conjugate=False)[source]#

Transpose the LinearOperator .

If conjugate is True, return the Hermitian transpose.

property bcs#

class MultipatchDirichletProjector(fem_space, *, bcs=None, space_kind=None)[source]#

Bases: DirichletProjector

A LinearOperator (for multipatch domains) that applies homogeneous (unless manually given different bcs) Dirichlet boundary conditions.

Parameters:

fem_spacepsydac.fem.basic.FemSpace: fem_space.coeff_space is domain and codomain of this LO. fem_space.kind.name determines the BCs that get applied.
bcsIterable | None: Iterable of sympde.topology.Boundary objects. Allows the user to apply different kinds of BCs. Must not be passed if a space_kind argument is passed.
space_kindstr | SpaceType | None: Necessary only if fem_space.kind.name is undefined and no bcs are passed. Must not be passed if a bcs argument is passed.

Notes

See examples/vector_potential_3d.py for a use case of such an operator.

dot(v, out=None)[source]#

Apply the LinearOperator self to the Vector v.

The result is written to the Vector out, if provided.

Parameters:

vVector: The vector to which the linear operator (self) is applied. It must belong to the domain of self.
outVector: The vector in which the result of the operation is stored. It must belong to the codomain of self. If out is None, a new vector is created and returned.

Returns:

Vector: The result of the operation. If out is None, a new vector is returned. Otherwise, the result is stored in out and out is returned.