XYQuadrilateralMeshFromBoundaryCurve

Overview

We mesh a curved region with quadrilaterals using a chain of generators: a triangulation in the domain defined by the curve that is biased toward right angles, the conversion of that triangulation into quadrilaterals, the snap of the boundary nodes the conversion added back onto the boundary curves, and an optional smoothing pass. XYQuadrilateralMeshFromBoundaryCurve packages that chain behind one block. It creates the generators as sub-generators:

1. XYFrontalDelaunayGenerator triangulates the region enclosed by "boundary", around the "holes", at the element size "desired_area". Its metric and orientation keep their defaults, which place the right isosceles triangles that recombine well. The triangulation splits the boundary segments that are longer than the desired element size unless "refine_boundary" and "refine_holes" forbid it, which keeps the nodes of the input curves so that the mesh can be stitched to a neighbor sharing them.

2. TriToQuadConverter merges pairs of triangles that reach "eta_min". With "all_quad", its default, the triangles that could not be merged are eliminated and the mesh is purely quadrilateral; without it the mesh is quad-dominant, and the triangles of each subdomain are collected in a subdomain of their own named after it with the suffix _tri. The elimination splits every element, which adds a node in the middle of every boundary segment: a mesh that must keep the boundary nodes of the input curves also needs all_quad = false.

3. MoveBoundaryNodesToCurveGenerator snaps each boundary named in "snap_boundaries" onto the curve of the ParsedCurveGenerator it is paired with in "parsed_curve_generators", one snap per pair. Boundaries whose geometry is not a parametric curve need no pair and are left alone.

4. SmoothMeshGenerator runs a variational smoothing pass, unless "smooth" turns it off. The variational algorithm cannot tangle the mesh and only allows node movement that leaves the domain unchanged, so the snapped geometry survives the smoothing. The boundary nodes may still slide along the boundary, so a mesh that must keep its boundary nodes in place also needs smooth = false.

Example Syntax

A gear profile whose bore is the same gear profile scaled down and rotated slightly, meshed with quadrilaterals in one step. Both boundaries are snapped back onto their curves:

[Mesh<<<{"href": "../../syntax/Mesh/index.html"}>>>]
  [outer_curve]
    type = ParsedCurveGenerator<<<{"description": "This ParsedCurveGenerator object is designed to generate a mesh of a curve that consists of EDGE2, EDGE3, or EDGE4 elements.", "href": "ParsedCurveGenerator.html"}>>>
    x_formula<<<{"description": "Function expression of x(t)"}>>> = '(r+a*sin(n*t))*cos(t)'
    y_formula<<<{"description": "Function expression of y(t)"}>>> = '(r+a*sin(n*t))*sin(t)'
    section_bounding_t_values<<<{"description": "The 't' values that bound the sections of the curve. Start and end points must be included. The number of entries in 'nums_segments' should be equal to one less than the number of entries in this parameter."}>>> = '${fparse 0.0} ${fparse pi} ${fparse 2.0*pi}'
    constant_names<<<{"description": "Vector of constants used in the parsed function (use this for kB etc.)"}>>> = 'r a n'
    constant_expressions<<<{"description": "Vector of values for the constants in constant_names (can be an FParser expression)"}>>> = '1.0 0.07 10.0'
    nums_segments<<<{"description": "Numbers of segments (EDGE elements) of each section of the curve to be generated. The number of entries in this parameter should be equal to one less than the number of entries in 'section_bounding_t_values'"}>>> = '60 60'
    is_closed_loop<<<{"description": "Whether the curve is closed or not."}>>> = true
  []
  [inner_curve]
    type = ParsedCurveGenerator<<<{"description": "This ParsedCurveGenerator object is designed to generate a mesh of a curve that consists of EDGE2, EDGE3, or EDGE4 elements.", "href": "ParsedCurveGenerator.html"}>>>
    x_formula<<<{"description": "Function expression of x(t)"}>>> = '(r+a*sin(n*(t-phi)))*cos(t)'
    y_formula<<<{"description": "Function expression of y(t)"}>>> = '(r+a*sin(n*(t-phi)))*sin(t)'
    section_bounding_t_values<<<{"description": "The 't' values that bound the sections of the curve. Start and end points must be included. The number of entries in 'nums_segments' should be equal to one less than the number of entries in this parameter."}>>> = '${fparse 0.0} ${fparse pi} ${fparse 2.0*pi}'
    constant_names<<<{"description": "Vector of constants used in the parsed function (use this for kB etc.)"}>>> = 'r a n phi'
    constant_expressions<<<{"description": "Vector of values for the constants in constant_names (can be an FParser expression)"}>>> = '0.45 0.035 10.0 0.06'
    nums_segments<<<{"description": "Numbers of segments (EDGE elements) of each section of the curve to be generated. The number of entries in this parameter should be equal to one less than the number of entries in 'section_bounding_t_values'"}>>> = '40 40'
    is_closed_loop<<<{"description": "Whether the curve is closed or not."}>>> = true
  []
  [gear]
    type = XYQuadrilateralMeshFromBoundaryCurve<<<{"description": "Meshes a region bounded by input curves with quadrilaterals in one step, by chaining a frontal Delaunay triangulation, the triangle-to-quadrilateral conversion, the snapping of boundary nodes onto parametric curves, and an optional variational smoothing pass.", "href": "XYQuadrilateralMeshFromBoundaryCurve.html"}>>>
    boundary<<<{"description": "The input MeshGenerator defining the outer boundary of the region to mesh with quadrilaterals."}>>> = 'outer_curve'
    holes<<<{"description": "The MeshGenerators that define mesh holes."}>>> = 'inner_curve'
    desired_area<<<{"description": "Desired (maximum) triangle area of the triangulation the quadrilaterals are built from, or 0 to size the elements from the boundary alone."}>>> = 0.004
    output_boundary<<<{"description": "Boundary name to set on the new outer boundary."}>>> = 'rim'
    hole_boundaries<<<{"description": "Boundary names to set on the holes."}>>> = 'bore'
    output_subdomain_name<<<{"description": "Subdomain name to set on the new elements."}>>> = 'gear'
    parsed_curve_generators<<<{"description": "The ParsedCurveGenerators whose curves the boundaries named in 'snap_boundaries' are snapped onto, one generator per entry of that parameter."}>>> = 'outer_curve inner_curve'
    snap_boundaries<<<{"description": "The boundaries whose nodes are snapped onto the curves of 'parsed_curve_generators', one boundary per entry of that parameter. Use the names given in 'output_boundary' and 'hole_boundaries'."}>>> = 'rim bore'
  []
  # The recombination breaks score ties by element id, so keep the ids stable
  allow_renumbering = false
[]

Figure 1: The gear mesh of the input above.

Input Parameters