BoundaryCondition

A BoundaryCondition is similar to a Kernel. It is formed out of the weak form of a PDE (see the explanation in Kernel) and computes a residual and a Jacobian.

However, there are two types of BoundaryCondition objects: IntegratedBC and NodalBC. In general, an IntegratedBC should be used to represent boundary integrals within a weak form. The NodalBC is mainly used for "Dirichlet" or "Type-1" or "Essential" boundary conditions.

NodalBC

As mentioned above a NodalBC represents a "Dirichlet" boundary condition. That is, a condition of the type \(u = value\) on the boundary nodes. In MOOSE that value can either be a constant, or even a nonlinear function of the solution variables at that node. One restriction on NodalBC objects is that they do not have access to gradient information (because gradients are typically discontinuous at the nodes).

Creating a NodalBC

To create a new NodalBC you need to create a sub-type of NodalBC like so:

" Boundary condition expressing `u = value` at a set of nodes "
type DirichletBC <: NodalBC
    " The variable the BC will be applied to "
    u::Variable

    " The boundary ID to apply this BC to "
    bids::Array{Int64}

    " The value on the boundary "
    value::Float64
end

Here we are showing the actual code from DirichletBC.jl for implementing a \(u=v\) type BC. A BoundaryCondition must provide u: the Variable the BC is being applied to and bids::Array{Int64}: the boundary IDs in the mesh where the BC will be applied.

After the type has been created then a computeQpResidual() function needs to be created similar to Kernels. However, to form the residual for a NodalBC you simply take the \(u=v\) equation and move everything to the left hand side: \(u-v=0\). What is on the left hand side is the "residual" and can be coded up like so:

function computeResidual(bc::DirichletBC)
    u = bc.u

    return u.nodal_value - bc.value
end

Similar to Kernels a computeQpJacobian() statement can be provided. It is only optional if you are using Automatic Differentiation. Since the value of any Lagrange shape function at a node is always 1... \(\frac{\partial u}{\partial u_j} = 1\) at the node. Therefore:

function computeJacobian(bc::DirichletBC, v::Variable{Float64})
    u = bc.u

    if u.id == v.id
        return 1.
    end

    return 0.
end

Just as in Kernels you should always return 0 in the case where MOOSE is looking for a derivative with respect to a variable this BoundaryCondition is not operating on.