boundary

boundary#

This submodule contains types that describe strong boundary condtitions. For now, only steady boundary conditions are used, though there are already types for the unsteady ones.

Boundary Condition Types#

The main class that would be in use at the moment would be BoundaryCondition2DSteady. It is derived from BoundaryCondition2D, which is also the parent for BoundaryCondition2DUnsteady, but these are not (yet) supported, so they are not exported by the module.

class mfv2d.boundary.BoundaryCondition2DSteady(form: KFormUnknown, indices: array_like, func: Function2D)[source]#

Boundary condition for a 2D problem with no time dependence.

These boundary conditions specifiy values of differential forms on the the given indices directly. These are enforced “strongly” by adding a Lagrange multiplier to the system.

Parameters:

  • form (KFormUnknown) – Form for which the value is to be prescribed.

  • indices (array_like) – One dimensional array of edges on which this is prescribed. Note that 0-based indexing is used, since orientation does not matter.

  • func ((array, array) -> array_like) – Function that can be evaluated to obtain values of differential forms at those points. For 1-froms it should return an array_like with an extra last dimension, which contains the two vector components.

form: KFormUnknown#
func: Function2D#
indices: ndarray[tuple[Any, ...], dtype[uint64]]#
class mfv2d.boundary.BoundaryCondition2DUnsteady(form: KFormUnknown, indices: npt.ArrayLike, func: Callable[[npt.ArrayLike, npt.ArrayLike], npt.NDArray[np.float64]])[source]#

Unsteady boundary condition for a 2D problem.

form: KFormUnknown#
func: Function2D#
indices: ndarray[tuple[Any, ...], dtype[uint64]]#
class mfv2d.boundary.BoundaryCondition2D(form: KFormUnknown, indices: array_like)[source]#

Base class for 2D boundary conditions.

form: KFormUnknown#
indices: ndarray[tuple[Any, ...], dtype[uint64]]#

Boundary Condition Construction#

There are also functions used to construct boundary condition constraints and evalue contributions of weak boundary conditions.

mfv2d.boundary._element_weak_boundary_condition(mesh: Mesh, element_idx: int, side: ElementSide, unknown_orders: UnknownOrderings, unknown_index: int, leaf_order_mapping: Mapping[int, int], dof_offsets: npt.NDArray[np.uint32], weak_terms: Sequence[tuple[float, KBoundaryProjection]], basis_cache: FemCache) → tuple[ElementConstraint, ...][source]#

Compute boundary conditions contributions the side of the element.

Parameters:

  • mesh (Mesh) – Mesh in which the element is in.

  • element_idx (int) – Index of the element to compute the contributions on.

  • side (ElementSide) – Side on which the boundary conditions should be computed.

  • unknown_orders (UnknownOrderings) – Ordering of the unknown degrees of freedom.

  • dof_offsets (FixedElementArray of uint32) – Offsets of the degrees of freedom in the element.

  • weak_terms (Sequence of (float, KBoundaryProjection)) – Pairs of boundary projection terms and their scaling coefficients.

  • basis_cache (FemCache) – Cache from which to get basis from.

Returns:

Tuple with ElementConstraint objects that contain the indices of equations where to apply in the dofs member and contributions of boundary terms in the coeffs member.

Return type:

tuple of ElementConstraint

mfv2d.boundary._element_strong_boundary_condition(mesh: Mesh, element_idx: int, side: ElementSide, unknown_orders: UnknownOrderings, unknown_index: int, leaf_order_mapping: Mapping[int, int], dof_offsets: npt.NDArray[np.uint32], strong_bc: BoundaryCondition2DSteady, basis_cache: FemCache, skip_first: bool, skip_last: bool) → tuple[ElementConstraint, ...][source]#

Compute strong boundary condition constraints on the side of the element.

Parameters:

  • mesh (Mesh) – Mesh in which the element is in.

  • element_idx (int) – Index of the element to compute the degrees of freedom on.

  • side (ElementSide) – Side on which the boundary conditions should be computed.

  • unknown_orders (UnknownOrderings) – Ordering of the unknown degrees of freedom.

  • dof_offsets (FixedElementArray of uint32) – Offsets of the degrees of freedom in the element.

  • strong_bc (BoundaryCondition2DStrong) – Boundary condition to compute.

  • basis_cache (FemCache) – Cache from which to get basis from.

  • skip_first (bool) – Should the first node on the edge not be constrained for a 0-form.

  • skip_last (bool) – Should the last node on the edge not be constrained for a 0-form.

Returns:

Tuple with ElementConstraint objects that contain the degrees of freedom to constrain dofs member and their respective values in the coeffs member.

Return type:

tuple of ElementConstraint

mfv2d.boundary.mesh_boundary_conditions(evaluatable_terms: Sequence[KSum], unknown_order: UnknownOrderings, mesh: Mesh, leaf_order_mapping: Mapping[int, int], dof_offsets: npt.NDArray[np.uint32], strong_bcs: Sequence[Sequence[BoundaryCondition2DSteady]], basis_cache: FemCache) → tuple[tuple[ElementConstraint, ...], tuple[ElementConstraint, ...]][source]#

Compute boundary condition contributions and constraints.

Parameters:

  • evaluatable_terms (Sequence of KSum) – Right sides of equations that contain boundary projections to be evaluated. Must be ordered according to weights.

  • unknown_order (UnknownOrderings) – Orders of unknown forms.

  • mesh (Mesh) – Mesh in which the element is in.

  • dof_offsets (FixedElementArray[np.uint32]) – Offsets of DoFs within each element.

  • strong_bcs (Sequence of Sequence of BoundaryCondition2DSteady) – Boundary conditions grouped per weight functions and correctly ordered to match the order of weight functions in the system.

  • caches (FemCache) – Cache to use for basis functions.

Returns:

  • strong (tuple of ElementConstraint) – Strong boundary conditions in a specific notation. Each of these means that for element given by ElementConstraint.i_e, all dofs with indices ElementConstraint.dofs should be constrained to value ElementConstraint.coeffs.

  • weak (tuple of ElementConstraint) – Weak boundary conditions in a specific notation. Each of these means that for element given by ElementConstraint.i_e, all equations with indices ElementConstraint.dofs should have the value ElementConstraint.coeffs added to them.

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