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Title: Meshes optimized for discrete exterior calculus (DEC).

Abstract

We study the optimization of an energy function used by the meshing community to measure and improve mesh quality. This energy is non-traditional because it is dependent on both the primal triangulation and its dual Voronoi (power) diagram. The energy is a measure of the mesh's quality for usage in Discrete Exterior Calculus (DEC), a method for numerically solving PDEs. In DEC, the PDE domain is triangulated and this mesh is used to obtain discrete approximations of the continuous operators in the PDE. The energy of a mesh gives an upper bound on the error of the discrete diagonal approximation of the Hodge star operator. In practice, one begins with an initial mesh and then makes adjustments to produce a mesh of lower energy. However, we have discovered several shortcomings in directly optimizing this energy, e.g. its non-convexity, and we show that the search for an optimized mesh may lead to mesh inversion (malformed triangles). We propose a new energy function to address some of these issues.

Authors:
 [1];  [2];  [2];  [2]
  1. Univ. of Illinois, Urbana-Champaign, IL (United States)
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1414659
Report Number(s):
SAND-2017-13575R
659546
DOE Contract Number:  
AC04-94AL85000
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING

Citation Formats

Mousley, Sarah C., Deakin, Michael, Knupp, Patrick, and Mitchell, Scott A.. Meshes optimized for discrete exterior calculus (DEC).. United States: N. p., 2017. Web. doi:10.2172/1414659.
Mousley, Sarah C., Deakin, Michael, Knupp, Patrick, & Mitchell, Scott A.. Meshes optimized for discrete exterior calculus (DEC).. United States. doi:10.2172/1414659.
Mousley, Sarah C., Deakin, Michael, Knupp, Patrick, and Mitchell, Scott A.. Fri . "Meshes optimized for discrete exterior calculus (DEC).". United States. doi:10.2172/1414659. https://www.osti.gov/servlets/purl/1414659.
@article{osti_1414659,
title = {Meshes optimized for discrete exterior calculus (DEC).},
author = {Mousley, Sarah C. and Deakin, Michael and Knupp, Patrick and Mitchell, Scott A.},
abstractNote = {We study the optimization of an energy function used by the meshing community to measure and improve mesh quality. This energy is non-traditional because it is dependent on both the primal triangulation and its dual Voronoi (power) diagram. The energy is a measure of the mesh's quality for usage in Discrete Exterior Calculus (DEC), a method for numerically solving PDEs. In DEC, the PDE domain is triangulated and this mesh is used to obtain discrete approximations of the continuous operators in the PDE. The energy of a mesh gives an upper bound on the error of the discrete diagonal approximation of the Hodge star operator. In practice, one begins with an initial mesh and then makes adjustments to produce a mesh of lower energy. However, we have discovered several shortcomings in directly optimizing this energy, e.g. its non-convexity, and we show that the search for an optimized mesh may lead to mesh inversion (malformed triangles). We propose a new energy function to address some of these issues.},
doi = {10.2172/1414659},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Fri Dec 01 00:00:00 EST 2017},
month = {Fri Dec 01 00:00:00 EST 2017}
}

Technical Report:

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