Cut Meshes#

CutFEM computations are assembled on the background mesh. The physical domain is represented by selectors and runtime quadrature, not by replacing the mesh used for the finite element space. Nevertheless, it is often useful to build an explicit mesh of the selected geometry for visualization, export, and diagnostics. This is the role of cutfemx.create_cut_mesh(...).

Geometry mesh versus computation mesh

Let \({\mathcal T}_h\) be the background mesh and let

\[ \Omega_h=\{x:\phi_h(x)<0\}. \]

The discrete finite element space \(V_h\) is defined on \({\mathcal T}_h\). A cut mesh is a derived geometric object representing pieces such as \(K\cap\Omega_h\) or \(K\cap\Gamma_h\). It is not the mesh on which the unknown was solved.

For a cut cell \(K\), CutCells reconstructs the selected region by simpler subcells,

\[ K\cap\Omega_h \approx \bigcup_i K_i^{\mathrm{cut}}. \]

This reconstruction is intended for output and inspection. The variational problem itself is still represented by the background function space and the runtime quadrature rules.

Creating A Cut Mesh#

Start from existing cut data and a selector:

cut_data = cutfemx.cut(phi)
cut_mesh = cutfemx.create_cut_mesh(cut_data, "phi<0", mode="full")

The selector determines which geometric part is reconstructed. The returned cutfemx.CutMesh stores both the visualization mesh and its relation to the original background mesh:

  • cut_mesh.mesh: DOLFINx mesh for visualization or output

  • cut_mesh.parent_index: parent background cell for each output cell

  • cut_mesh.is_cut_cell: marker for reconstructed cut output cells

The parent map is important because finite element functions are still known on the background mesh. It tells CutFEMx which background cell should be used when evaluating a function on each cell of the cut mesh.

Full And Cut-Only Modes#

Use mode="full" when the object of interest is a full physical region. The output contains ordinary background cells that lie completely in the selected region together with reconstructed subcells in cut cells:

omega_mesh = cutfemx.create_cut_mesh(cut_data, "phi<0", mode="full")

This is the natural choice for plotting a solution on \(\Omega_h\). It gives a visible domain rather than only the interface reconstruction.

Use mode="cut_only" when the object of interest is the cut geometry itself, for example an embedded boundary or a local interface:

gamma_mesh = cutfemx.create_cut_mesh(cut_data, "phi=0", mode="cut_only")

If mode is omitted, CutFEMx chooses a mode from the selector, but explicit mode selection is clearer in tutorials and diagnostics.

Interpreting mode

The two modes correspond to different mathematical objects:

\[ \texttt{full}: \Omega_h, \qquad \texttt{cut\_only}: \Gamma_h\ \text{or another local intersection}. \]

Use full for a domain visualization and cut_only for an interface visualization. Mixing these two interpretations is a common source of confusing plots.

Transfer Functions#

A background function can be evaluated on the cut mesh by using the parent-cell map. This produces a new DOLFINx function whose function space lives on the cut visualization mesh:

phi_cut = cutfemx.fem.cut_function(phi, cut_mesh)

For a solved background function:

u_cut = cutfemx.fem.cut_function(uh, cut_mesh)

Mathematically, this is the restriction of the background finite element function to the reconstructed geometry:

\[ u_h^{\mathrm{cut}}(x)=u_h(x), \qquad x\in\Omega_h^{\mathrm{cut}}. \]

The operation is not a projection or a new solve. It is evaluation of the same finite element field on a different geometric representation.

What To Check#

Cut meshes are particularly useful for checking whether the geometric part of a problem is correct before interpreting PDE results:

  • Does "phi<0" reconstruct the expected physical domain?

  • Does "phi=0" lie on the intended embedded boundary?

  • Are the cut cells concentrated where the level set crosses the mesh?

  • Does the transferred level-set function vanish on the interface mesh?

  • Are there empty cut meshes caused by an incorrect selector or sign convention?

The transferred function can be passed to dolfinx.plot.vtk_mesh(...) or written with DOLFINx IO facilities when the selected mesh is non-empty.