Boundary Conditions

Dirichlet(u::Symbol, ∂Ω::Set, f::Function, components=nothing)

Create a Dirichlet boundary condition on u on the ∂Ω part of the boundary. f is a function of the form f(x) or f(x, t) where x is the spatial coordinate and t is the current time, and returns the prescribed value. components specify the components of u that are prescribed by this condition. By default all components of u are prescribed.

For example, here we create a Dirichlet condition for the :u field, on the faceset called ∂Ω and the value given by the sin function:


# Obtain the faceset from the grid
∂Ω = getfaceset(grid, "boundary-1")

# Prescribe scalar field :s on ∂Ω to sin(t)
dbc = Dirichlet(:s, ∂Ω, (x, t) -> sin(t))

# Prescribe all components of vector field :v on ∂Ω to 0
dbc = Dirichlet(:v, ∂Ω, x -> 0 * x)

# Prescribe component 2 and 3 of vector field :v on ∂Ω to [sin(t), cos(t)]
dbc = Dirichlet(:v, ∂Ω, (x, t) -> [sin(t), cos(t)], [2, 3])

Dirichlet boundary conditions are added to a ConstraintHandler which applies the condition via apply! and/or apply_zero!.

PeriodicDirichlet(u::Symbol, face_mapping, components=nothing)
PeriodicDirichlet(u::Symbol, face_mapping, R::AbstractMatrix, components=nothing)
PeriodicDirichlet(u::Symbol, face_mapping, f::Function, components=nothing)

Create a periodic Dirichlet boundary condition for the field u on the face-pairs given in face_mapping. The mapping can be computed with collect_periodic_faces. The constraint ensures that degrees-of-freedom on the mirror face are constrained to the corresponding degrees-of-freedom on the image face. components specify the components of u that are prescribed by this condition. By default all components of u are prescribed.

If the mapping is not aligned with the coordinate axis (e.g. rotated) a rotation matrix R should be passed to the constructor. This matrix rotates dofs on the mirror face to the image face. Note that this is only applicable for vector-valued problems.

To construct an inhomogeneous periodic constraint it is possible to pass a function f. Note that this is currently only supported when the periodicity is aligned with the coordinate axes.

See the manual section on Periodic boundary conditions for more information.

collect_periodic_faces(grid::Grid, mset, iset, transform::Union{Function,Nothing}=nothing)

Match all mirror faces in mset with a corresponding image face in iset. Return a dictionary which maps each mirror face to a image face. The result can then be passed to PeriodicDirichlet.

mset and iset can be given as a String (an existing face set in the grid) or as a Set{FaceIndex} directly.

By default this function looks for a matching face in the directions of the coordinate system. For other types of periodicities the transform function can be used. The transform function is applied on the coordinates of the image face, and is expected to transform the coordinates to the matching locations in the mirror set.

See also: collect_periodic_faces!, PeriodicDirichlet.

collect_periodic_faces(grid::Grid, all_faces::Union{Set{FaceIndex},String,Nothing}=nothing)

Split all faces in all_faces into image and mirror sets. For each matching pair, the face located further along the vector (1, 1, 1) becomes the image face.

If no set is given, all faces on the outer boundary of the grid (i.e. all faces that do not have a neighbor) is used.

See also: collect_periodic_faces!, PeriodicDirichlet.

add!(dh::MixedDofHandler, fh::FieldHandler)

Add all fields of the FieldHandler fh to dh.

add!(dh::AbstractDofHandler, name::Symbol, dim::Int[, ip::Interpolation])

Add a dim-dimensional Field called name which is approximated by ip to dh.

The field is added to all cells of the underlying grid. In case no interpolation ip is given, the default interpolation of the grid's celltype is used. If the grid uses several celltypes, add!(dh::MixedDofHandler, fh::FieldHandler) must be used instead.

add!(ch::ConstraintHandler, dbc::Dirichlet)

Add a Dirichlet boundary condition to the ConstraintHandler.

add!(ch::ConstraintHandler, ac::AffineConstraint)

Add the AffineConstraint to the ConstraintHandler.


Closes dh and creates degrees of freedom for each cell.

If there are several fields, the dofs are added in the following order: For a MixedDofHandler, go through each FieldHandler in the order they were added. For each field in the FieldHandler or in the DofHandler (again, in the order the fields were added), create dofs for the cell. This means that dofs on a particular cell, the dofs will be numbered according to the fields; first dofs for field 1, then field 2, etc.


Close and finalize the ConstraintHandler.

update!(ch::ConstraintHandler, time::Real=0.0)

Update time-dependent inhomogeneities for the new time. This calls f(x) or f(x, t) when applicable, where f is the function(s) corresponding to the constraints in the handler, to compute the inhomogeneities.

Note that this is called implicitly in close!(::ConstraintHandler).

apply!(K::SparseMatrixCSC, rhs::AbstractVector, ch::ConstraintHandler)

Adjust the matrix K and right hand side rhs to account for the Dirichlet boundary conditions specified in ch such that K \ rhs gives the expected solution.


apply!(K, rhs, ch) essentially calculates

rhs[free_dofs] = rhs[free_dofs] - K[free_dofs, constrained_dofs] * a[constrained]

where a[constrained] are the inhomogeneities. Consequently, the sign of rhs matters (in contrast to for apply_zero!).

apply!(v::AbstractVector, ch::ConstraintHandler)

Apply Dirichlet boundary conditions and affine constraints, specified in ch, to the solution vector v.


K, f = assemble_system(...) # Assemble system
apply!(K, f, ch)            # Adjust K and f to account for boundary conditions
u = K \ f                   # Solve the system, u should be "approximately correct"
apply!(u, ch)               # Explicitly make sure bcs are correct

The last operation is not strictly necessary since the boundary conditions should already be fulfilled after apply!(K, f, ch). However, solvers of linear systems are not exact, and thus apply!(u, ch) can be used to make sure the boundary conditions are fulfilled exactly.

apply_zero!(K::SparseMatrixCSC, rhs::AbstractVector, ch::ConstraintHandler)

Adjust the matrix K and the right hand side rhs to account for prescribed Dirichlet boundary conditions and affine constraints such that du = K \ rhs gives the expected result (e.g. du zero for all prescribed degrees of freedom).

apply_zero!(v::AbstractVector, ch::ConstraintHandler)

Zero-out values in v corresponding to prescribed degrees of freedom and update values prescribed by affine constraints, such that if a fulfills the constraints, a ± v also will.

These methods are typically used in e.g. a Newton solver where the increment, du, should be prescribed to zero even for non-homogeneouos boundary conditions.

See also: apply!.


u = un + Δu                 # Current guess
K, g = assemble_system(...) # Assemble residual and tangent for current guess
apply_zero!(K, g, ch)       # Adjust tangent and residual to take prescribed values into account
ΔΔu = K \ g                # Compute the (negative) increment, prescribed values are "approximately" zero
apply_zero!(ΔΔu, ch)        # Make sure values are exactly zero
Δu .-= ΔΔu                  # Update current guess

The last call to apply_zero! is only strictly necessary for affine constraints. However, even if the Dirichlet boundary conditions should be fulfilled after apply!(K, g, ch), solvers of linear systems are not exact. apply!(ΔΔu, ch) can be used to make sure the values for the prescribed degrees of freedom are fulfilled exactly.

    local_matrix::AbstractMatrix, local_vector::AbstractVector,
    global_dofs::AbstractVector, ch::ConstraintHandler;
    apply_zero::Bool = false

Similar to apply! but perform condensation of constrained degrees-of-freedom locally in local_matrix and local_vector before they are to be assembled into the global system.

When the keyword argument apply_zero is true all inhomogeneities are set to 0 (cf. apply! vs apply_zero!).

This method can only be used if all constraints are "local", i.e. no constraint couples with dofs outside of the element dofs (global_dofs) since condensation of such constraints requires writing to entries in the global matrix/vector. For such a case, apply_assemble! can be used instead.

Note that this method is destructive since it, by definition, modifies local_matrix and local_vector.

    assembler::AbstractSparseAssembler, ch::ConstraintHandler,
    local_matrix::AbstractMatrix, local_vector::AbstractVector;
    apply_zero::Bool = false

Assemble local_matrix and local_vector into the global system in assembler by first doing constraint condensation using apply_local!.

This is similar to using apply_local! followed by assemble! with the advantage that non-local constraints can be handled, since this method can write to entries of the global matrix and vector outside of the indices in global_dofs.

When the keyword argument apply_zero is true all inhomogeneities are set to 0 (cf. apply! vs apply_zero!).

Note that this method is destructive since it modifies local_matrix and local_vector.

get_rhs_data(ch::ConstraintHandler, A::SparseMatrixCSC) -> RHSData

Returns the needed RHSData for apply_rhs!.

This must be used when the same stiffness matrix is reused for multiple steps, for example when timestepping, with different non-homogeneouos Dirichlet boundary conditions.

apply_rhs!(data::RHSData, f::AbstractVector, ch::ConstraintHandler, applyzero::Bool=false)

Applies the boundary condition to the right-hand-side vector without modifying the stiffness matrix.

See also: get_rhs_data.


Stores the constrained columns and mean of the diagonal of stiffness matrix A.


Initial conditions

    a::AbstractVector, dh::AbstractDofHandler, fieldname::Symbol, 
    f::Function, cellset=1:getncells(dh.grid))

Apply a solution f(x) by modifying the values in the degree of freedom vector a pertaining to the field fieldname for all cells in cellset. The function f(x) are given the spatial coordinate of the degree of freedom. For scalar fields, f(x)::Number, and for vector fields with dimension dim, f(x)::Vec{dim}.

This function can be used to apply initial conditions for time dependent problems.


This function only works for standard nodal finite element interpolations when the function value at the (algebraic) node is equal to the corresponding degree of freedom value. This holds for e.g. Lagrange and Serendipity interpolations, including sub- and superparametric elements.