“Generation of Multi-Million Element Meshes for Solid Mesh-Based Geometries: The Dicer Algorithm”

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.