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MeshRefinementFunction

is an option for DiscretizeRegion and related functions that specifies a function to indicate whether mesh cells should be refined or not.

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MeshRefinementFunction

MeshRefinementFunction

is an option for DiscretizeRegion and related functions that specifies a function to indicate whether mesh cells should be refined or not.

Details

  • With MeshRefinementFunction->f, the function f[vlist,m] is applied to each simplex created, where vlist is a list of the vertices and m is the measure. If f[vlist,m] returns True, the simplex will be refined.

Examples

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Basic Examples  (2)

Discretize a Disk so that the triangles near the center are smaller:

Wolfram Language code: DiscretizeRegion[Disk[], MeshRefinementFunction -> Function[{vertices, area}, area > 0.0005(1 + 10Norm[Mean[vertices]])]]

Discretize a Rectangle so that the triangles in the first quadrant are smaller:

Wolfram Language code: DiscretizeRegion[Rectangle[{-1, -1}, {1, 1}], MeshRefinementFunction -> Function[{vertices, area}, Block[{x, y}, {x, y} = Mean[vertices];If[x > 0 && y > 0, area > 0.001, area > 0.01]]]]

Scope  (4)

MeshRefinementFunction specifies a function to determine if further refinement is needed:

Wolfram Language code: f = Function[{vertices, area}, area > 0.0005(21 - 20Norm[Mean[vertices]])];

Further refinement is performed on triangles for which the function returns True:

Wolfram Language code: DiscretizeRegion[Disk[], MeshRefinementFunction -> f]

Using If makes it possible to refine triangles satisfying an implicit equation:

Wolfram Language code: f = Function[{vertices, area}, Block[{x, y}, {x, y} = Mean[vertices];If[x ^ 4 + y ^ 4 ≤ 1, area > 0.003, area > 0.01]]];
Wolfram Language code: mr = DiscretizeRegion[Rectangle[{-1.5, -1.5}, {1.5, 1.5}], MeshRefinementFunction -> f]

See the boundary between large and small triangles:

Wolfram Language code: Show[mr, ContourPlot[x ^ 4 + y ^ 4 == 1, {x, -1.5, 1.5}, {y, -1.5, 1.5}, ColorFunction -> Hue]]

MeshRefinementFunction can be used with TriangulateMesh:

Wolfram Language code: mr = MeshRegion[{{0, 0}, {1, 0}, {1, 1}, {0, 1}}, Polygon[{1, 2, 3, 4}]]

Triangulate it with smaller triangles near the lower-left corner:

Wolfram Language code: TriangulateMesh[mr, MeshRefinementFunction -> Function[{vertices, area}, area > 0.0001(1 + 20Norm[Mean[vertices]])]]

MeshRefinementFunction can be used with DiscretizeRegion:

Wolfram Language code: r = Disk[{0, 0}, {3, 2}]; DiscretizeRegion[r]

Discretize it with smaller triangles further away from the center:

Wolfram Language code: DiscretizeRegion[r, MeshRefinementFunction -> Function[{vertices, area}, area > 0.0025(25 - 8Norm[Mean[vertices]])]]
Wolfram Research (2014), MeshRefinementFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/MeshRefinementFunction.html.

Text

Wolfram Research (2014), MeshRefinementFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/MeshRefinementFunction.html.

CMS

Wolfram Language. 2014. "MeshRefinementFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MeshRefinementFunction.html.

APA

Wolfram Language. (2014). MeshRefinementFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MeshRefinementFunction.html

BibTeX

@misc{reference.wolfram_2026_meshrefinementfunction, author="Wolfram Research", title="{MeshRefinementFunction}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/MeshRefinementFunction.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_meshrefinementfunction, organization={Wolfram Research}, title={MeshRefinementFunction}, year={2014}, url={https://reference.wolfram.com/language/ref/MeshRefinementFunction.html}, note=[Accessed: 12-June-2026]}