fem.splines#
Classes#
|
a 1D Splines Finite Element space |
Details#
- class SplineSpace(degree, knots=None, grid=None, multiplicity=None, parent_multiplicity=None, periodic=False, dirichlet=(False, False), basis='B', pads=None)[source]#
Bases:
FemSpacea 1D Splines Finite Element space
- Parameters:
- degreeint
Polynomial degree.
- knotsarray_like
Coordinates of knots (clamped or extended by periodicity).
- grid: array_like
Coordinates of the grid. Used to construct the knots sequence, if not given.
- multiplicity: int
Multiplicity of the knots in the knot sequence.
- parent_multiplicity: int
Multiplicity of the parent knot sequence, if the space is reduced space.
- periodicbool
True if domain is periodic, False otherwise. Default: False
- dirichlettuple, list
True if using homogeneous dirichlet boundary conditions, False otherwise. Must be specified for each bound Default: (False, False)
- basisstr
Set to “B” for B-splines (have partition of unity) Set to “M” for M-splines (have unit integrals)
- property histopolation_grid#
Coordinates of the N+1 points x[i] that define the N 1D edges (x[i], x[i+1]) for histopolation, where N is equal to the number of basis functions (i.e. the cardinality of the space).
In the non-periodic case x is simply the array of extended Greville points. In the periodic case we “unroll” the 1D edges to ensure that they correspond to positive, well-defined intervals with x[i] < x[i+1].
- init_interpolation(dtype=<class 'float'>)[source]#
Compute the 1D collocation matrix and factorize it, in preparation for the calculation of a spline interpolant given the values at the Greville points.
- init_histopolation(dtype=<class 'float'>)[source]#
Compute the 1D histopolation matrix and factorize it, in preparation for the calculation of a spline interpolant given the integrals within the cells defined by the extended Greville points.
- property ldim#
Parametric dimension.
- property periodic#
True if domain is periodic, False otherwise.
- property pads#
Padding for potential parallel assembly.
- property mapping#
Assume identity mapping for now.
- property vector_space#
Returns the topological associated vector space.
- property is_product#
Boolean flag that describes whether the space is a product space. If True, an element of this space can be decomposed into separate fields.
- property symbolic_space#
Symbolic space.
- eval_field(field, *eta, weights=None)[source]#
Evaluate field at location(s) eta.
- Parameters:
- fieldFemField
Field object (element of FemSpace) to be evaluated.
- etalist of float or numpy.ndarray
Evaluation point(s) in logical domain.
- weightsStencilVector, optional
Weights of the basis functions, such that weights.space == field.coeffs.space.
- Returns:
- valuefloat or numpy.ndarray
Field value(s) at location(s) eta.
- eval_field_gradient(field, *eta, weights=None)[source]#
Evaluate field gradient at location(s) eta.
- Parameters:
- fieldFemField
Field object (element of FemSpace) to be evaluated.
- etalist of float or numpy.ndarray
Evaluation point(s) in logical domain.
- weightsStencilVector, optional
Weights of the basis functions, such that weights.space == field.coeffs.space.
- Returns:
- valuefloat or numpy.ndarray
Value(s) of field gradient at location(s) eta.
- property is_scalar#
Only scalar field is implemented for now.
- property basis#
- property interpolation_grid#
- property nbasis#
Number of basis functions, i.e. cardinality of spline space.
- property degree#
Spline degree.
- property ncells#
Number of cells in domain.
- property dirichlet#
True if using homogeneous dirichlet boundary conditions, False otherwise.
- property knots#
Knot sequence.
- property multiplicity#
- property parent_multiplicity#
- property breaks#
List of breakpoints.
- property domain#
Domain boundaries [a,b].
- property greville#
Coordinates of all Greville points. Used for interpolation.
- property ext_greville#
Greville coordinates of ‘extended’ space with degree p+1. Used for histopolation.
- property scaling_array#
If self.basis==’M’, return array used to rescale B-splines to M-splines If self.basis==’B’, return None.
The length of the scaling array is (len(knots)-degree-1).
- compute_interpolant(values, field)[source]#
Compute field (i.e. update its spline coefficients) such that it interpolates a certain function \(f(x)\) at the Greville points.
- Parameters:
- valuesarray_like (nbasis,)
Function values \(f(x_i)\) at the ‘nbasis’ Greville points \(x_i\), to be interpolated.
- fieldFemField
Input/output argument: spline that has to interpolate the given values.
- compute_histopolant(values, field)[source]#
Compute field (i.e. update its spline coefficients) such that its integrals between the extended Greville points match the given values.
- Parameters:
- valuesarray_like (nbasis,)
Integral values between the ‘nbasis’ extended Greville cells \([x_i, x_{i+1}]\), to be matched by the spline.
- fieldFemField
Input/output argument: spline that has to match the given integral values.