Tutorial 10

Mesh size fields

In addition to specifying target mesh sizes at the points of the geometry (see Tutorial 1) you can use general mesh size "Fields".

import gmsh

gmsh.initialize(append!(["gmsh"], ARGS))

gmsh.model.add("t10")

Let's create a simple rectangular geometry:

lc = .15
gmsh.model.geo.addPoint(0.0, 0.0, 0, lc, 1)
gmsh.model.geo.addPoint(1, 0.0, 0, lc, 2)
gmsh.model.geo.addPoint(1, 1, 0, lc, 3)
gmsh.model.geo.addPoint(0, 1, 0, lc, 4)
gmsh.model.geo.addPoint(0.2, .5, 0, lc, 5)

gmsh.model.geo.addLine(1, 2, 1)
gmsh.model.geo.addLine(2, 3, 2)
gmsh.model.geo.addLine(3, 4, 3)
gmsh.model.geo.addLine(4, 1, 4)

gmsh.model.geo.addCurveLoop([1, 2, 3, 4], 5)
gmsh.model.geo.addPlaneSurface([5], 6)

gmsh.model.geo.synchronize()

Say we would like to obtain mesh elements with size lc/30 near curve 2 and point 5, and size lc elsewhere. To achieve this, we can use two fields: "Distance", and "Threshold". We first define a Distance field (Field[1]) on points 5 and on curve 2. This field returns the distance to point 5 and to (100 equidistant points on) curve 2.

gmsh.model.mesh.field.add("Distance", 1)
gmsh.model.mesh.field.setNumbers(1, "NodesList", [5])
gmsh.model.mesh.field.setNumber(1, "NNodesByEdge", 100)
gmsh.model.mesh.field.setNumbers(1, "EdgesList", [2])

We then define a Threshold field, which uses the return value of the `Distance' field 1 in order to define a simple change in element size depending on the computed distances

gmsh.model.mesh.field.add("Threshold", 2)
gmsh.model.mesh.field.setNumber(2, "IField", 1)
gmsh.model.mesh.field.setNumber(2, "LcMin", lc / 30)
gmsh.model.mesh.field.setNumber(2, "LcMax", lc)
gmsh.model.mesh.field.setNumber(2, "DistMin", 0.15)
gmsh.model.mesh.field.setNumber(2, "DistMax", 0.5)

Say we want to modulate the mesh element sizes using a mathematical function of the spatial coordinates. We can do this with the MathEval field:

gmsh.model.mesh.field.add("MathEval", 3)
gmsh.model.mesh.field.setString(3, "F",
                                "Cos(4*3.14*x) * Sin(4*3.14*y) / 10 + 0.101")

We could also combine MathEval with values coming from other fields. For example, let's define a Distance field around point 1

gmsh.model.mesh.field.add("Distance", 4)
gmsh.model.mesh.field.setNumbers(4, "NodesList", [1])

We can then create a MathEval field with a function that depends on the return value of the Distance field 4, i.e., depending on the distance to point 1 (here using a cubic law, with minimum element size = lc / 100)

gmsh.model.mesh.field.add("MathEval", 5)
gmsh.model.mesh.field.setString(5, "F", "F4^3 + " * string(lc / 100))

We could also use a Box field to impose a step change in element sizes inside a box

gmsh.model.mesh.field.add("Box", 6)
gmsh.model.mesh.field.setNumber(6, "VIn", lc / 15)
gmsh.model.mesh.field.setNumber(6, "VOut", lc)
gmsh.model.mesh.field.setNumber(6, "XMin", 0.3)
gmsh.model.mesh.field.setNumber(6, "XMax", 0.6)
gmsh.model.mesh.field.setNumber(6, "YMin", 0.3)
gmsh.model.mesh.field.setNumber(6, "YMax", 0.6)

Many other types of fields are available: see the reference manual for a complete list. You can also create fields directly in the graphical user interface by selecting Define->Size fields in the Mesh module.

Finally, let's use the minimum of all the fields as the background mesh field:

gmsh.model.mesh.field.add("Min", 7)
gmsh.model.mesh.field.setNumbers(7, "FieldsList", [2, 3, 5, 6])

gmsh.model.mesh.field.setAsBackgroundMesh(7)

The API also allows to set a global mesh size callback, which is called each time the mesh size is queried

function meshSizeCallback(dim, tag, x, y, z)
    return 0.02 * x + 0.01
end
gmsh.model.mesh.setSizeCallback(meshSizeCallback)

To determine the size of mesh elements, Gmsh locally computes the minimum of

the size of the model bounding box;
if Mesh.MeshSizeFromPoints is set, the mesh size specified at geometrical points;
if Mesh.MeshSizeFromCurvature is set, the mesh size based on the curvature and Mesh.MinimumElementsPerTwoPi;
the background mesh field;
any per-entity mesh size constraint;
the mesh size returned by the mesh size callback, if any.

This value is then constrained in the interval [Mesh.MeshSizeMin, Mesh.MeshSizeMax] and multiplied by Mesh.MeshSizeFactor. In addition, boundary mesh sizes (on curves or surfaces) are interpolated inside the enclosed entity (surface or volume, respectively) if the option Mesh.MeshSizeExtendFromBoundary is set (which is the case by default).

When the element size is fully specified by a background mesh (as it is in this example), it is thus often desirable to set

gmsh.option.setNumber("Mesh.MeshSizeExtendFromBoundary", 0)
gmsh.option.setNumber("Mesh.MeshSizeFromPoints", 0)
gmsh.option.setNumber("Mesh.MeshSizeFromCurvature", 0)

This will prevent over-refinement due to small mesh sizes on the boundary.

gmsh.model.mesh.generate(2)
gmsh.write("t10.msh")

Info    : Meshing 1D...
Info    : [  0%] Meshing curve 1 (Line)
Info    : [ 30%] Meshing curve 2 (Line)
Info    : [ 50%] Meshing curve 3 (Line)
Info    : [ 80%] Meshing curve 4 (Line)
Info    : Done meshing 1D (Wall 0.0534752s, CPU 0.053474s)
Info    : Meshing 2D...
Info    : Meshing surface 6 (Plane, Frontal-Delaunay)
Info    : Done meshing 2D (Wall 0.806367s, CPU 0.806319s)
Info    : 22761 nodes 45523 elements
Info    : Writing 't10.msh'...
Info    : Done writing 't10.msh'

gmsh.finalize()

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