Create a line element mesh:

Create a 1D ElementMesh with region holes from an ImplicitRegion:

Create an ElementMeshRegionProduct from a 1D ElementMesh :

Create a triangle element mesh:

Create an ElementMeshRegionProduct from a 2D ElementMesh and a 1D ElementMesh:

Create a quad element mesh:

Create a quad element mesh:

Visualize the node numbers in red and the element numbers in blue:

Create a mixed-element mesh:

Create a mixed-element mesh with quadratic elements:

A linear tetrahedron element mesh with markers:

Visualizing the index of the coordinates at their respective positions:

Create the mesh:

Visualize the mesh with the elements' markers:

Create an element mesh consisting of hexahedra:

Convert an Image to a mesh:

Convert an Image to a mesh without region holes:

Region marker are useful to specify multiple materials in a region and are explained further in the section Markers in the Element Mesh Generation.

Create a mixed-element mesh with region markers:

Boundary markers are useful to specify NeumannValue on a region boundary and are explained further in the section Markers in the Element Mesh Generation.

Create a rectangle with boundary markers:

Point markers are useful to specify DirichletCondition on a region boundary and are explained further in the section Markers in the Element Mesh Generation.

Create a rectangle with point markers:

Boundary markers are useful to specify NeumannValue on a region boundary and are explained further in the section Markers in the Element Mesh Generation.

Compute integer markers for edges depending on the coordinates of the edge elements:

Inspect the boundary elements:

If "PointElementMarkers" are also computed, those are accessible as the second argument in the function given to "BoundaryMarkerFunction".

Markers can be specified for quad mesh elements:

The "Continuation" method uses a curve continuation method that can in many cases resolve corners, cusps, and sharp changes quite well:

The "RegionPlot" method is based on improving output from RegionPlot and can sometimes be faster:

More information on the "BoundaryMeshGenerator" option can be found on the ToBoundaryMesh reference page.

The "OpenCascade" boundary mesh generator is well suited for 3D BooleanRegion expressions consisting of Graphics3D primitives.

Specify a region:

Use "OpenCascade" as a boundary mesh generator:

Generate colors for the boundary markers:

Visualize the element mesh with highlighted boundary markers:

Replace the internal mesh generator with a custom mesh generator:

The numerical region given to the custom mesh generators will contain a "BoundaryMesh". If the interface to the external mesh generator does not need a boundary mesh representation because the mesh generator interface works with a symbolic representation the generation of the boundary mesh can be avoided by setting "BoundaryMeshGenerator"->None.

Approximation of an ImplicitRegion is generally not exact. The "ImproveBoundaryPosition" option helps to improve the position of boundary nodes:

The sum of the squares of the boundary nodes is not necessarily all at value 1:

Automatic improvement is the default in 2D, and the results are better:

When the "ImproveBoundaryPosition" option is set to False, quadratic elements will not have their middle nodes on the boundary of a curved part of the region.

Create an element mesh:

Specify a circle of additional points:

Create a mesh with the include points:

Points added with "IncludePoints" will be addressable with DirichletCondition. Additionally, if the predicate in DirichletCondition is True, then the boundary condition will take effect at the include points.

Adding include points to a quadrilateral mesh:

With "MaxBoundaryCellMeasure"->m, a boundary cell size is chosen:

Specify a boundary cell size and a cell size:

With MaxCellMeasure->m, a cell size is chosen and a boundary cell size is set to be appropriate for good-quality cells in the embedding dimension:

Find the maximum mesh element area:

Set a smaller max cell measure:

Find the maximum mesh element area:

The max cell measure targets linear elements. In the case of a higher-order mesh for a curved boundary, the actual cell measure may be higher.

A specific length can be specified using MaxCellMeasure->{"Length"->len}:

If a specific length and area are specified using MaxCellMeasure->{"Length"->len,"Area"->area}, the more stringent requirement is satisfied:

Find the maximum mesh element area:

Set a maximum cell measure on the boundary:

Create an element mesh:

Mesh the same region and split the elements into five blocks:

For two-dimensional meshes based on TriangleElement, a minimum triangle angle can be specified via the "MeshElementConstraint" option:

With the default:

The minimum angle specified must be smaller then 33 degrees. Theoretically, the algorithm is guaranteed to terminate for minimum angles smaller than 22.7 degrees, but in practice it often succeeds for minimum angles up to 33 degrees.

For three-dimensional TetrahedronElement meshes, a minimum ratio of the tetrahedral radius to edge length can be specified via the "MeshElementConstraint" option:

With the default setting:

The radius-edge ratio must be larger than . Theoretically, the algorithm is guaranteed to terminate for a radius-edge ratio of 2, but in practice it often succeeds for radius-edge ratios down to .

Mesh a region with quad mesh elements:

Mesh the same region with triangle elements:

Quad elements are currently only generated for rectangular regions.

Create a second-order element mesh for a region:

Inspect the mesh order:

Integrate over the mesh and assess the accuracy of the integration:

Mesh the same region with a first-order mesh:

Inspect the mesh order:

Integrate over the mesh and assess the accuracy of the integration:

More information about the "MeshOrder" option can be found in this section of the Element Mesh Generation tutorial and this section of the Element Mesh Generation tutorial.

The quality measure is computed so that for regular mesh elements , and for degenerate mesh elements :

Triangulate with more regular triangles:

Create a function that returns True if a specific 2D triangle element needs to be refined:

Create a full mesh of the boundary mesh with the refinement function:

Visualize the resulting mesh:

Compare to a mesh generated without refinement:

Create a compiled function that returns True if a specific 2D triangle element needs to be refined:

Create a mesh of a disk with the refinement function:

Visualize the resulting mesh:

Create a gradual refinement toward the center:

Create a gradual refinement toward the boundary:

Create a refinement in a subregion:

Visualize the mesh with the refinement contour:

Use an image as a refinement function:

Create a distance transform of that image:

Visualize the distance transform of the image:

Create an interpolation function from the image distance transform:

Create a compiled refinement function calling the interpolation function:

Create a triangle mesh with a given refinement:

Visualize the mesh:

Create a function that returns True if a specific 2D triangle element needs to be refined:

Create a compiled function that returns True or False :

Create a full mesh of the boundary mesh with the compiled refinement function:

Visualize the resulting mesh:

Create a compiled function that returns True if a specific 3D tetrahedron element needs to be refined:

Create a boundary mesh:

Create a full mesh of the boundary mesh with the refinement function:

Visualize the resulting mesh:

In 1D use DiscretizeRegion to do the refinement:

Create an ElementMesh from a region:

Visualize the incidences of the mesh elements:

Create a mesh from the same region with node reordering:

The pattern of the incidences is much more organized:

A function for the node reordering can be provided:

"NodeReordering" is also known as matrix bandwidth reduction.

Point markers are useful to specify DirichletCondition on a region boundary and are explained further in the section Markers in the Element Mesh Generation.

Compute integer markers for edges depending on the coordinates of the edge nodes:

Create an element mesh with the point markers:

Inspect the boundary elements:

The point marker values of the mid-side nodes for second-order meshes are derived from the boundary element markers. Since none are given here, the values are arbitrary:

If "BoundaryMarkerFunction" is also computed, middle nodes will have values derived from the boundary element markers:

Point markers are useful to specify DirichletCondition on a region boundary and are explained further in the section Markers in the Element Mesh Generation.

The way point markers are set up in a mesh can be done in different ways. One possible option is to derive point markers from boundary markers.

Mesh the region and deduce the point markers from the boundary markers:

Browse through the surfaces:

Select from which surfaces the point markers should be constructed:

Visualize the points with the selected point marker:

Find the intersection such that only points connected to all surfaces are considered. Boundary element marker ID 7 is the large cylinder body surface:

Visualize the points with the selected point marker:

Create a boundary mesh with an internal region:

Create a full mesh from the boundary mesh:

Create a full mesh from the boundary mesh, with the interior specified as a region hole:

Create an element mesh with a hole in the middle:

Mesh the region, including the hole:

Mesh the region without the hole by explicitly giving a coordinate in the region that is a hole:

Create and visualize a region with holes:

Generate a full mesh from the boundary mesh:

Specify a region hole and a region marker:

Specify a region hole in 3D:

Visualize the boundary mesh and a point inside the interior box:

Create the full element mesh with a hole inside:

Visualize the full element mesh:

Create a boundary mesh:

The boundary element containing {1/2,3/4} should use the marker 1, and the region containing {1/2,1/4} should use the marker 2:

Additionally, a refinement parameter for the region with marker 1 is given to the markers:

Specify a region in 3D:

Create the full element mesh with subregions inside that follow different refinements:

Visualize the full element mesh:

For 2D triangle meshes, the insertion of Steiner points can be controlled. In this case, the boundary respects the boundary size and does not further subdivide:

In this case, the second-order mid-side nodes are still present:

Using a first-order mesh avoids mid-side nodes: