1
Generation of Multi-Million Element Meshes for
Solid Mesh-Based Geometries The Dicer Algorithm

• AMD-Vol. 220 Trends in Unstructured Mesh
  Generation
• 1997
• Melander, Tautges, and Benzley

2
Goal Mesh Refinement

• Refine existing, coarse, hexahedal mesh
• 2D surface or 3D volume
• Finite Element Modeling (FEM) applications
• Refinement criteria
• Algorithm robustness
• Mesh quality
• Amount of user interaction
• Memory usage
• Execution speed
• Long-range goals
• 100 million element meshes
• parallel mesh generation

source Melander et al.
3
Some Prior Approaches

• For large (many elements) FEM meshes
• all in one
• Generate entire mesh in single code execution
• Memory-intensive
• Has global information to guide meshing
• all in many
• Generate mesh one piece at a time
• Assemble pieces into an overall mesh
• Not as memory-intensive
• Has only local information to guide meshing

source Melander et al.
4
Dicer Algorithm

• Coarse all-hexahedral mesh is generated by the
  user using an existing approach.
• Determine Dicer loops/sheets with conditions
• Coarse element faces that share an edge must also
  share the fine mesh along that edge.
• Coarse hexahedral elements that share a face must
  also share the fine mesh on that face.
• Coarse edges opposite each other on a given
  coarse mesh must receive the same degree of
  refinement in the fine mesh.
• Specify refinement levels.
• Mesh coarse elements with a Trans-Finite
  Interpolation method (TFI see later slide(s)) .

source Melander et al.
5
Dicer Algorithm

• Goal find coarse edges that must have same
  refinement level.
• Initially all coarse edges are unmarked.
• Create a series of dicer sheets.
• To create a dicer sheet
• Travel from edge to opposite edge, as in Figure
  1.

source Melander et al.
6
Dicer Algorithm

• Coarse all-hexahedral mesh is generated by the
  user using an existing approach.
• Determine Dicer loops/sheets.
• Specify refinement levels.
• Done by the user.
• Can be different for each Dicer sheet.
• Mesh coarse elements with a Trans-Finite
  Interpolation method (TFI see later slide(s)) .

source Melander et al.
7
Dicer Algorithm

• Coarse all-hexahedral mesh is generated by the
  user using an existing approach.
• Determine Dicer loops/sheets.
• Specify refinement levels.
• Mesh coarse elements with some Trans-Finite
  Interpolation (TFI) method (e.g. see next several
  slides).
• Coarse edges are meshed first, using interval
  refinement.
• Surface TFI fills interior of a face.
• Volume TFI fills interior of a hexahedral volume
  element.

source Melander et al.
8
Trans-finite Interpolation (TFI)

• Frey George discuss a form of TFI for mapping a
  discretized unit square from logical
  (parametric) space to physical space topological
  distortion of square into physical space.
• fi(x,h), i 1,4 is parameterization of side i of
  physical space
• ai are the 4 corners
• Assume discretizations of opposite sides have
  same number of points.
• Form quadrilateral mesh in parametric space by
  joining matching points on opposite edges
• This creates internal nodes at intersection
  points.
• Lagrange type of TFI follows formula

correction term based on corners
source Mesh Generation by Frey George
9
Trans-finite Interpolation (TFI) (continued)
Very general definition

• The term transfinite interpolation has been
  often used to describe the problem of
  constructing a surface that passes through a
  given collection of curves
• i.e. the surface must interpolate infinitely many
  points.
• In a more general setting, the interpolation
  problem requires constructing a single function
  f(x) that takes on prescribed values and/or
  derivatives on some collection of point sets.
• In this sense, transfinite interpolation is a
  special type of a boundary value problem.
• The sets of points may contain isolated points,
  bounded or unbounded curves, as well as surfaces
  and regions of arbitrary topology.

source web link for Transfinite Interpolation
with Normalized Functions, University of
Wisconsin, Madison. http//sal-cnc.me.wisc.edu
10
Trans-finite Interpolation using Normalized
Functions w1 and w2
Functions w1, w2 define distance to boundaries.
The functions are used for interpolation.
Function w1 defining the boundary.
Function w2 defining the boundary.
It is unclear where A and B are in these 2
pictures
Continuous function interpolating functions
taking on same values at A and B.
Interpolating function with discontinuities at A
and B due to different values at A and B.
source web link for Transfinite Interpolation
with Normalized Functions, University of
Wisconsin, Madison
11
Special Case

• Geometric boundaries need special care to provide
  mesh quality
• Fine nodes generated during refinement are moved
  closer to geometry boundary (as in Figure 2).
• Applies to
• Nodes on edge owned by bounding curve
• Nodes on face owned by geometric surface.

source Melander et al.
12
Example of Surface Dicing(constant refinement
interval 10)
source Melander et al.
13
Example of Surface Dicing (non-constant
refinement intervals)
Refinement interval of 5 propagates around the
surface.
source Melander et al.
14
Example of Volume Dicing (constant refinement
interval 5)
source Melander et al.