Meshes optimized for discrete exterior calculus (DEC).

Mousley, Sarah C.; Deakin, Michael A. B.; Knupp, Patrick; Mitchell, Scott A.

doi:10.2172/1414659

Cited by 3 publications

(5 citation statements)

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“…We identified some HOT shortcomings previously in a technical report [7]. The HOT mathematical formulas had undesirable features, such as troughs that led to inverted meshes and collapsed elements.…”

Section: Prior Workmentioning

confidence: 99%

“…Mousley et al [7] proposed an optimal-transport inspired theory extrapolation ET. Here ET = EF = ?HOT when the mesh is regular, but modified to treat negative dual lengths as positive:…”

Section: Quality Metricsmentioning

confidence: 99%

“…Clearly, after flipping the edge, both the primal and dual replacements will have non-negative length. The length of the replacement is monotonic in the magnitude of the negative length, so we propose using Mousley et al's [7] extrapolation ET that increases with increasing magnitude, which is eq. ( 3).…”

Section: Extrapolation To Non-regularmentioning

confidence: 99%

“…Thus the edge energy satisfies condition (1) for being a barrier metric. (In contrast, if we attempt to normalize HOT by the square of the triangle area, as we proposed in our technical report [7], in the final case the energy goes to zero and there is no barrier.) Considering condition (2), observe that E 1 BH 4 for any point in D. This can be demonstrated using the following inequality:…”

Section: Barrier and Pseudo-barrier Energiesmentioning

confidence: 99%

“…Figure3: EF HOT contour plots for a triangle with a single free vertex, and fixed vertices (-1,0) and (1,0), courtesy of Mousley et al[7].…”

mentioning

confidence: 99%

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Not so HOT Triangulations

Mitchell

Knupp²,

Mackay

et al. 2023

Computer-Aided Design

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Section: Prior Workmentioning

confidence: 99%

“…Mousley et al [7] proposed an optimal-transport inspired theory extrapolation ET. Here ET = EF = ?HOT when the mesh is regular, but modified to treat negative dual lengths as positive:…”

Section: Quality Metricsmentioning

confidence: 99%

Section: Extrapolation To Non-regularmentioning

confidence: 99%

Section: Barrier and Pseudo-barrier Energiesmentioning

confidence: 99%

“…Figure3: EF HOT contour plots for a triangle with a single free vertex, and fixed vertices (-1,0) and (1,0), courtesy of Mousley et al[7].…”

mentioning

confidence: 99%

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Not so HOT Triangulations

Mitchell

Knupp²,

Mackay

et al. 2023

Computer-Aided Design

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Local Time Stepping for Discrete Exterior Calculus on Space-Time Mesh with Refinements

Räty,

Mönkölä

2023

Preprint

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Not So HOT Triangulations.

Mitchell¹,

Knupp²,

Mackay³

et al. 2021

Proposed for Presentation at the International Meshing Roundtable Held June 21-25, 2021 in Virtual Online.

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We propose primal-dual mesh optimization algorithms that overcome shortcomings of the standard algorithm while retaining some of its desirable features. "Hodge-Optimized Triangulations" defines the "HOT energy" as a bound on the discretization error of the diagonalized Delaunay Hodge star operator. HOT energy is a natural choice for an objective function, but unstable for both mathematical and algorithmic reasons: it has minima for collapsed edges, and its extrapolation to non-regular triangulations is inaccurate and has unbounded minima. We propose a di↵erent extrapolation with a stronger theoretical foundation. We propose new objectives, based on normalizations of the HOT energy, with barriers to edge collapses and other undesirable configurations. We propose mesh improvement algorithms coupling these. When HOT optimization nearly collapses an edge, we actually collapse the edge. Otherwise, we use the barrier objective to update positions and weights. By combining discrete connectivity changes with continuous optimization, we more fully explore the space of possible meshes and obtain higher quality solutions.

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Meshes optimized for discrete exterior calculus (DEC).

Cited by 3 publications

References 0 publications

Not so HOT Triangulations

Not so HOT Triangulations

Local Time Stepping for Discrete Exterior Calculus on Space-Time Mesh with Refinements

Not So HOT Triangulations.

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