Manifolds

Linking parts of the mesh to manifolds is an interesting way to factor the real geometry of the mesh during refinements. This mesh treatment can lead to reduced geometric approximation error.

subsection manifolds

  # Sets the total number of specified manifolds
  set number = 0

  subsection manifold 0

      # Boundary mesh id specified in the .gmsh file
      set id                 = 0

      # Type of manifold. Choices are <spherical|cylindrical|iges|step|none>
      set type               = none

      # Center of the sphere or a point along the cylinder's axis (x, y, z)
      set point coordinates  = 0, 0, 0  #for 2D, set point coordinates = 0, 0

      # Direction vector used only for the cylinder (x, y, z)
      set direction vector   = 0, 1, 0

  end

end

First the number of manifolds is specified by the set number command. Then a subsection for each of the manifolds is created starting with the manifold 0. The boundary id is in this case set to 0 corresponding to the mesh boundary to which the manifold will be applied. Then the type of the manifold is specified.

Warning

The manifold id will always be the same as the prescribed boundary id. Similar to Boundary Conditions - CFD, The parameter id can also take a list of boundary ids. Hence, set number refers to the number of manifold subsections, and not the total number of manifolds in the mesh.

• Lethe supports four types of manifolds:

  • spherical manifold: The former can be used to describe any sphere, circle, hypesphere or hyperdisc in two or three dimensions and requires the center of the geometry to set the manifold.

  • cylindrical manifold: Used to describe cylinders in three dimensions. It uses the coordinates of a point and a direction vector located on the axis of the cylinder.

    Caution

    Cylindrical manifolds are not supported in 2D.

  • iges or step: manifold corresponding to a CAD geometry stored in an IGES or a STEP file, respectively. The last two lines of the manifold 0 subsection are replaced by the following command set cad file = file_name.iges (or file_name.step) where the path to the cad file is specified, and optionally by set cad scale factor. The boundary nodes are projected onto the CAD surfaces along the normal of the mesh.

    Caution

    OpenCASCADE reads the CAD geometry in millimeters. It is then multiplied by set cad scale factor (default 1e-3), which converts it to meters. The CAD and the mesh must be in the same units: for a mesh in meters (e.g., built in millimeters and read with set scale = 0.001 in the mesh subsection) keep the default, and for a mesh in millimeters use set cad scale factor = 1. CAD manifolds are only supported in 3D.

    Caution

    The projection direction is estimated from the CAD normals at the end points of each edge. On edges that lie on a sharp feature of the CAD (e.g., the 90 degree edge between a pipe wall and a flat step), this normal is ambiguous and points can be projected onto the wrong surface. Give the surfaces on each side of a sharp feature different boundary ids, each with its own CAD file containing only its smooth surfaces (flat faces need no manifold).

Note

For more information about manifolds and the reasons behind them, we invite you to read the documentation page of deal.II: Manifold description for triangulations.

Understanding Manifolds

We can clearly see that the geometry is more similar to an octagon than to a circle. This is a problem because the cells composing the mesh are not aware that they should morph to shape a cylinder. After every refinement, an extra node is added between the existing nodes in order to generate new cells. The octagonal shape is therefore conserved as it is illustrated below:

A really coarse mesh of a cylinder viewed from the top looks like this:

Manifolds are added to the geometry in order to solve this issue. Using a point and a direction vector along the axis of the cylinder, the cells of a part of the mesh are restricted to respect a certain geometry while refining the mesh. On the image below, the geometry on the left is obtained by adding a cylindrical manifold to the mesh. The geometry on the right is obtained if no manifold are specified.