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“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

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Page 1: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

“Generation of Multi-Million Element Meshes for Solid Mesh-

Based Geometries: The Dicer Algorithm”

AMD-Vol. 220: Trends in Unstructured Mesh Generation

1997Melander, Tautges, and Benzley

Page 2: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

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.

Page 3: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

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.

Page 4: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

Dicer Algorithm

1. Coarse all-hexahedral mesh is generated by the user using an existing approach.

2. Determine Dicer loops/sheets with conditions:a) Coarse element faces that share an edge must also share the

fine mesh along that edge.b) Coarse hexahedral elements that share a face must also

share the fine mesh on that face.c) Coarse edges opposite each other on a given coarse mesh

must receive the same degree of refinement in the fine mesh.

3. Specify refinement levels.4. Mesh coarse elements with a Trans-Finite Interpolation

method (TFI: see later slide(s)) .

source: Melander et al.

Page 5: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

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.

Page 6: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

Dicer Algorithm

1. Coarse all-hexahedral mesh is generated by the user using an existing approach.

2. Determine Dicer loops/sheets.3. Specify refinement levels.

a) Done by the user.b) Can be different for each Dicer sheet.

4. Mesh coarse elements with a Trans-Finite Interpolation method (TFI: see later slide(s)) .

source: Melander et al.

Page 7: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

Dicer Algorithm

1. Coarse all-hexahedral mesh is generated by the user using an existing approach.

2. Determine Dicer loops/sheets.3. Specify refinement levels.4. Mesh coarse elements with some Trans-Finite

Interpolation (TFI) method (e.g. see next several slides).

a) Coarse edges are meshed first, using = interval refinement.

b) Surface TFI fills interior of a face.c) Volume TFI fills interior of a hexahedral volume

element.

source: Melander et al.

Page 8: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

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. i(,), 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:)()1()()()()1(),( 4321 F

source: Mesh Generation by Frey & George

))1()1()1)(1(( 4321 aaaa

correction term based on corners

Page 9: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

Trans-finite Interpolation (TFI) (continued)

• “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

Very general definition…

Page 10: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

Trans-finite Interpolation using Normalized Functions 1 and 2

source: web link for “Transfinite Interpolation with Normalized Functions”, University of Wisconsin, Madison

Functions 1, 2 define distance to boundaries. The functions are used for interpolation.

Function 1 defining the boundary. Function 2 defining the boundary.

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.

It is unclear where A and B are in these 2 pictures…

Page 11: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

Special Case

source: Melander et al.

• 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.

Page 12: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

Example of Surface Dicing(constant refinement interval = 10)

source: Melander et al.

Page 13: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

Example of Surface Dicing (non-constant refinement intervals)

source: Melander et al.

Refinement interval of 5 propagates around the surface.

Page 14: “Generation of Multi-Million Element Meshes for Solid Mesh- Based Geometries: The Dicer Algorithm” AMD-Vol. 220: Trends in Unstructured Mesh Generation

Example of Volume Dicing (constant refinement interval = 5)

source: Melander et al.