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# Communication-Free Streaming Mesh Refinement

[+] Author and Article Information
Philippe P. Pébay, David Thompson

For clarity, all figures will display edges on front faces with bold lines, edges on back faces with thin lines, and interior edges with dashed lines.

J. Comput. Inf. Sci. Eng 5(4), 309-316 (Mar 22, 2005) (8 pages) doi:10.1115/1.2052806 History: Received September 30, 2004; Revised March 22, 2005

## Abstract

This article presents a technique for the adaptive refinement of tetrahedral meshes. What makes this method new is that no neighbor information is required for the refined mesh to be compatible everywhere. Refinement consists of inserting new vertices at edge midpoints until some tolerance (geometric or otherwise) is met. For a tetrahedron, the six edges present $26=64$ possible subdivision combinations. The challenge is to triangulate the new vertices (i.e., the original vertices plus some subset of the edge midpoints) in a way that neighboring tetrahedra always generate the same triangles on their shared boundary. A geometric solution based on edge lengths was developed previously, but did not account for geometric degeneracies (edges of equal length). This article provides a solution that works in all cases, while remaining entirely communication-free.

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## Figures

Figure 1

Given some initial tessellation of a finite element’s parameter space, we subdivide edges until some application-dependent criterion is met and we are left with a new, refined simplicial complex

Figure 2

Left: unambiguous face subdivision when 0, 1, or 3 edges are split. Right: the topological ambiguity when ne(f)=2(a) is resolved by deciding to connect the longest edge midpoint to the opposite corner (b).

Figure 3

Potentially ambiguous configurations: from left to right, cases 2a, 3a, 3c, 4a, 4b, and 5

Figure 4

The 6 unambiguous geometric configurations of case 3a

Figure 5

Ambiguous face refinement. 2 different subdivisions are possible (left), unambiguous subdivision thanks to a point insertion (center), and point placement in the case where ∣02∣=∣12∣ (right)

Figure 6

Average ζ quality of the decomposition depending on the point placement, for various values of α (left: α⩽π∕6, right: α⩾π∕6)

Figure 7

Ambiguous configurations of cases 2a and 3a

Figure 8

Ambiguous configurations of cases 3c and 4a

Figure 9

Ambiguous configurations α through ζ of case 4b

Figure 10

Ambiguous configurations 4bη, 5α, and 5β. Right: subdivision (all edges shown) of ambiguous case 2a.

Figure 11

Ambiguous case 3aα(∣01∣=∣02∣>∣03∣): all edges shown (a), after removal of 0467 (b), and after removal of 4367 (c). Ambiguous case 3aβ(∣01∣=∣02∣<∣03∣): all edges shown (d), after removal of 0467 (e), and after removal of a267 (f).

Figure 12

Ambiguous case 3aγ(∣01∣=∣02∣=∣03∣): all edges shown (a), after removal of 0467 (b), and after removal of 62d74a (c). Ambiguous case 3cα(∣01∣=∣12∣>∣03∣): all edges shown (d), after removal of 4153 (e), and after removal of a047 and a207 (f).

Figure 13

Ambiguous case 3cβ(∣01∣=∣12∣<∣03∣): all edges shown (a), after removal of 7153 and 7523 (b), and after removal of a047 and a207 (c). Ambiguous case 3cγ(∣01∣=∣12∣=∣03∣): all edges shown (d), after removal of 415b and b153 (e), and after removal of a047 and a207 (f).

Figure 14

Ambiguous case 4aα(∣03∣=∣13∣>∣23∣): all edges shown (a), after removal of 7893, 670b, and 601b (b), and after removal of 6978, 67b8, and 6b18 (c). Ambiguous case 4aβ(∣03∣=∣13∣<∣23∣): all edges shown (d), after removal of 7893, 670b, and 601b (e), and after removal of 6978, 67b8, and 6b18 (f).

Figure 15

Ambiguous case 4aγ(∣03∣=∣13∣=∣23∣): all edges shown (a), after removal of 7893, 670b, and 601b (b), and after removal of 6978, 67b8, and 6b18 (a). Ambiguous case 4bα(∣02∣=∣12∣<∣13∣<∣03∣): all edges shown (d), after removal of 7823 (e), and after removal of a607, a158, a017, and a718 (f).

Figure 16

Ambiguous case 4bβ(∣02∣=∣12∣>∣13∣>∣03∣): all edges shown (a), after removal of 6523 (b), and after removal of a607, a158, a018, and a708 (c). Ambiguous case 4bγ(∣03∣<∣02∣=∣12∣<∣13∣): all edges shown (d), after removal of a607, a158, a018, and a708 (e), and after removal of 67a8, 6a58, and 6378 (f).

Figure 17

Ambiguous case 4bδ(∣02∣=∣12∣<∣03∣=∣13∣): all edges shown (a), after removal of 7823 (b), and after removal of a607, a158, a01b, ab18, a0b7, and a7b8 (c). Ambiguous case 4bε(∣02∣=∣12∣=∣03∣<∣13∣): all edges shown (d), after removal of a607, a158, a018, and a708 (e), and after removal of d625, d378, d238, and d285(f).

Figure 18

Ambiguous case 4bζ(∣02∣=∣12∣=∣03∣>∣13∣): all edges shown (a), after removal of a607, a158, a017, and a718 (b), and after removal of d625, d378, d235, and d385 (c). Ambiguous case 4bη(∣02∣=∣12∣=∣03∣=∣13∣): all edges shown (d), after removal of a607, a158, a01b, ab18, a0b7, and a7b8 (e), and after removal of d625, d378, d23c, d2c5, dc38, and d5c8 (f).

Figure 19

Ambiguous case 5α(∣02∣=∣12∣, ∣03∣>∣13∣): all edges shown (a), after removal of 6529 and 7893 (b), and after removal of a607, a158, a017, and a718, (c). Ambiguous case 5β(∣02∣=∣12∣, ∣03∣=∣13∣): all edges shown (d), after removal of 6529 and 7893 (e), and after removal of a607, a158, a01b, ab18, a0b7, and a7b8 (c).

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