Boundary Conditions

Boundary Conditions#

SymPDE & Psydac allows you to use both strong and weak boundary conditions. We start first by explaining how to identify a boundary in NCube domains, such as Line, Square and a Cube.

Identifying a boundary in NCube domains#

Since we’re dealing with NCube domains, the best way to identify a boundary is to use the couple (axis, extremity). The axis is usually defined as perpendicular to the boundary, while the extremity gives the lower or upper bound of the interval associated to the axis.

Let’s see it through the following examples;

For a Line, the following figure gives the value to access a boundary.

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For a Square, we have,

png

For a Cube, it is similar and would not be helpful to visualize it here.

Essential boundary conditions#

Essential boundary conditions are usually treated in two ways, either in the strong or weak form. The latter can be achieved through the use of Nitsche’s method. You can check the associated examples in the sequel.

For the essential boundary conditions, usually, one needs one of the following expressions;

\[\begin{split} \begin{align} u &= 0, \quad \text{(homogeneous case)}, \\ u &= f, \quad \text{(inhomogeneous case)}, \\ \mathbf{v} &= 0, \quad \text{(homogeneous case)}, \\ \mathbf{v} &= \mathbf{g}, \quad \text{(homogeneous case)}, \end{align} \end{split}\]

where \(u\) and \(f\) are scalar functions while \(\mathbf{v}\) and \(\mathbf{g}\) denote vector functions.

Let Gamma denotes the boundary \(\Gamma\). The normal derivative is an available operator in SymPDE, it is provided as Dn operator. Alternatively, you can also define it manually,

    nn = NormalVector('nn')
    dn = lambda a: dot(grad(a), nn)

we have the following equivalences,

Mathematical Expression

SymPDE Expression

Example

\(u = 0\) on \(\Gamma\)

EssentialBC(u, 0, Gamma)

\(u = f\) on \(\Gamma\)

EssentialBC(u, f, Gamma)

\(\partial_n u = 0\) on \(\Gamma\)

EssentialBC(dn(u), 0, Gamma)

\(\partial_n u = f\) on \(\Gamma\)

EssentialBC(dn(u), f, Gamma)

\(\mathbf{v} = 0\) on \(\Gamma\)

EssentialBC(v, 0, Gamma)

\(\mathbf{v} = \mathbf{g}\) on \(\Gamma\)

EssentialBC(v, g, Gamma)

\(\mathbf{v} \cdot \mathbf{n} = 0\) on \(\Gamma\)

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\(\mathbf{v} \times \mathbf{n} = 0\) on \(\Gamma\)

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\(\mathbf{v} \cdot \mathbf{n} = f,\mathbf{g}\) on \(\Gamma\)

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\(\mathbf{v} \times \mathbf{n} = f,\mathbf{g}\) on \(\Gamma\)

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