Curved outer boundary

The purpose of this tutorial is to demonstrate how to create an unstructured mesh on a domain with a curved outer boundary. This outer boundary curve is defined by parametric equations and contains fine features as well as smooth regions.

The outer boundary, background grid and mesh are visualized for quality inspection. The tutorial also shows how to adjust the background and add a local refinement region in order to better resolve a portion of the curved boundary.

Synopsis

This tutorial demonstrates how to:

• Define a curved outer boundary using parametric equations.
• Add and adjust the background grid.
• Visualize an interactive mesh project.
• Add manual refinement to a local region of the domain.

Initialization

From a Julia REPL we load the HOHQMesh package as well as GLMakie, a backend of Makie.jl, to visualize the boundary curve, mesh, etc. from the interactive tool.

julia> using GLMakie, HOHQMesh

Now we are ready to interactively generate unstructured quadrilateral meshes!

We create a new project with the name "TheBlob" and assign "out" to be the folder where any output files from the mesh generation process will be saved. By default, the output files created by HOHQMesh will carry the same name as the project. For example, the resulting mesh file from this tutorial will be named TheBlob.mesh. If the folder out does not exist, it will be created automatically in the current file path.

blob_project = newProject("TheBlob", "out")

Add the outer boundary

The outer boundary curve for the domain of interest in this tutorial is given by the parametric equations

\[ \begin{aligned} x(t) &= 4\cos(2 \pi t) - \frac{3}{5}\cos^3(8 \pi t),\\[0.2cm] y(t) &= 4\sin(2 \pi t) - \frac{1}{2}\sin^2(11 \pi t),\\[0.2cm] z(t) &= 0 \end{aligned} \qquad t\in[0,1]\]

Parametric equations in HOHQMesh can be any legitimate equation and use intrinsic functions available in Fortran, e.g., $\sin$, $\cos$, exp. The constant pi is available for use. Exponentiation is done with ^. All number literals are interpreted as floating point numbers.

The following commands create a new curve for the parametric equations above

xEqn = "x(t) = 4 * cos(2 * pi * t) - 0.6 * cos(8 * pi * t)^3"
yEqn = "y(t) = 4 * sin(2 * pi * t) - 0.5 * sin(11* pi * t)^2"
zEqn = "z(t) = 0.0"
blob = newParametricEquationCurve("Blob", xEqn, yEqn, zEqn)

The name of this curve is assigned to be "Blob". This name is also the label that HOHQMesh will give to this boundary curve in the resulting mesh file.

Now that we have created the boundary curve it must be added as an outer boundary in the blob_project.

addCurveToOuterBoundary!(blob_project, blob)

Add a background grid

HOHQMesh requires a background grid for the mesh generation process. This background grid sets the base resolution of the desired mesh. HOHQMesh will automatically subdivide from this background grid near sharp features of any curved boundaries.

For a domain bounded by an outer boundary curve, this background grid is set by indicating the desired element size in the $x$ and $y$ directions. To start, we set the background grid for blob_project to have elements with side length two in each direction

addBackgroundGrid!(blob_project, [2.0, 2.0, 0.0])

We next visualize the outer boundary curve and background grid with the following

plotProject!(blob_project, MODEL+GRID)

Here, we take the sum of the keywords MODEL and GRID in order to simultaneously visualize the curves and background grid. The resulting plot is given below. The chain of outer boundary curves is called "Outer" and it contains a single curve "Blob" labeled in the figure by O.1.

From the visualization we see that the background grid is likely too coarse to produce a "good" quadrilateral mesh for this domain. We reset the background grid size to have elements with size one half in each direction

setBackgroundGridSize!(blob_project, 0.5, 0.5)

Note, that after we execute the command above the visualization updates automatically with the outer boundary curve and the new background grid.

The new background grid that gives a finer initial resolution looks suitable to continue to the mesh generation.

Initial mesh and user adjustments

We next generate the mesh from the information contained in the blob_project. This will output the following files to the out folder:

• TheBlob.control: A HOHQMesh control file for the current project.
• TheBlob.tec: A TecPlot formatted file to visualize the mesh with other software, e.g., ParaView.
• TheBlob.mesh: A mesh file with format ISM-V2 (the default format).

To do this we execute the command

generate_mesh(blob_project)

 *******************
 2D Mesh Statistics:
 *******************
    Total time             =    8.4328000000000000E-002
    Number of nodes        =          479
    Number of Edges        =          895
    Number of Elements     =          417
    Number of Subdivisions =            5

 Mesh Quality:
         Measure         Minimum         Maximum         Average  Acceptable Low Acceptable High       Reference
     Signed Area      0.00025286      0.36208840      0.11936333      0.00000000    999.99900000      1.00000000
    Aspect Ratio      1.00002882      2.58443761      1.26335838      1.00000000    999.99900000      1.00000000
       Condition      1.00000000      3.12867591      1.18585825      1.00000000      4.00000000      1.00000000
      Edge Ratio      1.00006176      4.77438730      1.51314025      1.00000000      4.00000000      1.00000000
        Jacobian      0.00011275      0.28170345      0.10292806      0.00000000    999.99900000      1.00000000
   Minimum Angle     29.38249259     89.99827733     73.08294187     40.00000000     90.00000000     90.00000000
   Maximum Angle     90.00132795    156.98701187    109.36585212     90.00000000    135.00000000     90.00000000
       Area Sign      1.00000000      1.00000000      1.00000000      1.00000000      1.00000000      1.00000000

The call to generate_mesh also prints mesh quality statistics to the screen and updates the visualization. The background grid is removed from the visualization when the mesh is generated.

Inspecting the mesh we see that the automatic subdivision in HOHQMesh does well to capture the fine features of the curved outer boundary. Although, we see that the mesh near the point $(-4, 0)$ is still quite coarse. To remedy this we manually add a RefinementCenter near this region of the domain to force HOHQMesh to increase the resolution in this area. We create and add this refinement region to the current project with

center = newRefinementCenter("region", "smooth", [-4.0, -0.5, 0.0], 0.4, 1.0)
addRefinementRegion!(blob_project, center)

Above we create a circular refinement region centered at the point $(-4, -0.5)$ with a desired resolution size $0.4$ and a radius of $1.0$. Upon adding this refinement region to blob_project, the visualization will update to indicate the location and size of the manual refinement region.

Final mesh

With the refinement region added to the project we can regenerate the mesh. Note, this will create and save new output files TheBlob.control, TheBlob.tec, TheBlob.mesh and update the figure.

generate_mesh(blob_project)

 *******************
 2D Mesh Statistics:
 *******************
    Total time             =    9.2778999999999986E-002
    Number of nodes        =          503
    Number of Edges        =          940
    Number of Elements     =          438
    Number of Subdivisions =            5

 Mesh Quality:
         Measure         Minimum         Maximum         Average  Acceptable Low Acceptable High       Reference
     Signed Area      0.00025286      0.36208840      0.11412688      0.00000000    999.99900000      1.00000000
    Aspect Ratio      1.00002883      2.47056840      1.26416662      1.00000000    999.99900000      1.00000000
       Condition      1.00000000      3.12867592      1.18429212      1.00000000      4.00000000      1.00000000
      Edge Ratio      1.00006176      4.77438730      1.51407242      1.00000000      4.00000000      1.00000000
        Jacobian      0.00011275      0.28170345      0.09814903      0.00000000    999.99900000      1.00000000
   Minimum Angle     29.38249259     89.99827897     72.80049710     40.00000000     90.00000000     90.00000000
   Maximum Angle     90.00132784    156.98701188    109.63478104     90.00000000    135.00000000     90.00000000
       Area Sign      1.00000000      1.00000000      1.00000000      1.00000000      1.00000000      1.00000000

Note, the circular region indicating the refinement center is removed from the plot when the mesh is generated.

Now we decide that we are satisfied with the mesh quality and resolution of the outer boundary curve.

Summary

In this tutorial we demonstrated how to:

• Define a curved outer boundary using parametric equations.
• Add and adjust the background grid.
• Visualize an interactive mesh project.
• Add manual refinement to a local region of the domain.

For completeness, we include a script with all the commands to generate the mesh displayed in the final image. Note, we do not include the plotting in this script.

# Interactive mesh with a curved outer boundary
#
# Create an outer boundary from a set of parametric equations.
# Add manual refinement in a small region around the point (-4, -0.5).
#
# Keywords: outer boundary, parametric equations, refinement center
using HOHQMesh

# Instantiate the project
blob_project = newProject("TheBlob", "out")

# Create and add the outer boundary curve
xEqn = "x(t) = 4 * cos(2 * pi * t) - 0.6 * cos(8 * pi * t)^3"
yEqn = "y(t) = 4 * sin(2 * pi * t) - 0.5 * sin(11* pi * t)^2"
zEqn = "z(t) = 0.0"
blob = newParametricEquationCurve("Blob", xEqn, yEqn, zEqn)
addCurveToOuterBoundary!(blob_project, blob)

# Add the background grid
addBackgroundGrid!(blob_project, [0.5, 0.5, 0.0])

# Create and add the refinement region
center = newRefinementCenter("region", "smooth", [-4.0, -0.5, 0.0], 0.4, 1.0)
addRefinementRegion!(blob_project, center)

# Generate the mesh
generate_mesh(blob_project)