CircularBoundaryCorrectionGenerator

Overview

The CircularBoundaryCorrectionGenerator object performs radius correction to preserve circular area considering polygonization effect for full or partial circular boundaries in a 2D mesh within the Z=0 plane.

In a 2D mesh, a "circular boundary" consists of sides that connect a series of nodes on the circle, which actually form a polygon boundary. Due to the polygonization effect, the area within a "circular boundary" in a 2D mesh is actually smaller than a real circle with the same radius. Such a discrepancy could cause issues in some simulations that involve physics that are sensitive to volume conservation.

Therefore, a corrected radius can be used to generate the "circle-like" polygon to enforce that the polygon area is the same as the original circle without polygonization. This can be achieved with either of the following approaches.

Moving Radial Nodes

The most straightforward approach for circular correction is to move the nodes on the circular boundary in their respective radial directions (see the left sub-figure of Figure 1). In that case, the azimuthal angle intervals of the boundary sides do not change. Therefore, the algorithm is relatively simple. This is also the default approach of this mesh generator.

For a polygon region used to mesh a full or partial circle, each side (with index ) corresponds to an azimuthal angle interval, , which is defined by the side and the center of the polygon:

(1)

with

(2)

should be for a full circle and between 0 to for a partial circle. For such a full or partial real circle, the area has the following form:

(3)

To enforce that , the radii of the polygon and the circle have the following relation,

(4)

Where is the correction factor used in this object to ensure volume preservation.

Figure 1: A schematic drawing showing the two approaches to correct the polygonization effect for a partial circular boundary

Usage

This object is capable of correcting multiple circular boundaries within a 2D mesh at the same time. Each circular boundary must be provided as a single boundary id/name in "input_mesh_circular_boundaries".

Moving the nodes on a circular boundary may flip the local elements. In order to reduce the probability of element flipping, a transition area is defined for each circular boundary, which is a ring area covering both inner and outer sides of the circular boundary in which the nodes will be moved based on the correction factor. The moving distance fades as the node-to-circle distance increases. The transition area size is defined using a fraction of the boundary radius, which can be customized in "transition_layer_ratios".

Example Syntax

[Mesh]
  [cir1]
    type = ParsedCurveGenerator
    x_formula = 'r*cos(t*2*pi)'
    y_formula = 'r*sin(t*2*pi)'
    section_bounding_t_values = '0 1'
    constant_names = 'pi r'
    constant_expressions = '${fparse pi} 1.0'
    nums_segments = '10'
    is_closed_loop = true
  []
  [cir2]
    type = ParsedCurveGenerator
    x_formula = 'r*cos(t*2*pi)'
    y_formula = 'r*sin(t*2*pi)+0.5'
    section_bounding_t_values = '0 1'
    constant_names = 'pi r'
    constant_expressions = '${fparse pi} 2.0'
    nums_segments = '15'
    is_closed_loop = true
  []
  [xyd1]
    type = XYDelaunayGenerator
    boundary = cir1
    desired_area = 0.5
    refine_boundary = false
    # output_boundary = '10'
    output_subdomain_name = '10'
  []
  [xyd2]
    type = XYDelaunayGenerator
    boundary = cir2
    holes = xyd1
    refine_holes = 'false'
    desired_area = 0.5
    refine_boundary = false
    output_subdomain_name = '20'
    hole_boundaries = '10'
  []
  [ccg]
    type = CircularBoundaryCorrectionGenerator
    input = xyd2
    input_mesh_circular_boundaries = '0 10'
    custom_circular_tolerance = 1e-8
  []
[]

Input Parameters