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Title: Tetrahedral element shape optimization via the Jacobian determinant and condition number.

Abstract

We present a new shape measure for tetrahedral elements that is optimal in the sense that it gives the distance of a tetrahedron from the set of inverted elements. This measure is constructed from the condition number of the linear transformation between a unit equilateral tetrahedron and any tetrahedron with positive volume. We use this shape measure to formulate two optimization objective functions that are differentiated by their goal: the first seeks to improve the average quality of the tetrahedral mesh; the second aims to improve the worst-quality element in the mesh. Because the element condition number is not defined for tetrahedral with negative volume, these objective functions can be used only when the initial mesh is valid. Therefore, we formulate a third objective function using the determinant of the element Jacobian that is suitable for mesh untangling. We review the optimization techniques used with each objective function and present experimental results that demonstrate the effectiveness of the mesh improvement and untangling methods. We show that a combined optimization approach that uses both condition number objective functions obtains the best-quality meshes.

Authors:
;
Publication Date:
Research Org.:
Argonne National Lab., IL (US)
Sponsoring Org.:
US Department of Energy (US)
OSTI Identifier:
11915
Report Number(s):
ANL/MCS/CP-99689
TRN: AH200119%%114
DOE Contract Number:  
W-31109-ENG-38
Resource Type:
Conference
Resource Relation:
Conference: 8th International Meshing Roundtable, South Lake Tahoe, CA (US), 10/10/1999--10/13/1999; Other Information: PBD: 30 Jul 1999
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; OPTIMIZATION; SHAPE; TRANSFORMATIONS; JACOBIAN FUNCTION; MESH GENERATION

Citation Formats

Freitag, L A, and Knupp, P M. Tetrahedral element shape optimization via the Jacobian determinant and condition number.. United States: N. p., 1999. Web.
Freitag, L A, & Knupp, P M. Tetrahedral element shape optimization via the Jacobian determinant and condition number.. United States.
Freitag, L A, and Knupp, P M. 1999. "Tetrahedral element shape optimization via the Jacobian determinant and condition number.". United States. https://www.osti.gov/servlets/purl/11915.
@article{osti_11915,
title = {Tetrahedral element shape optimization via the Jacobian determinant and condition number.},
author = {Freitag, L A and Knupp, P M},
abstractNote = {We present a new shape measure for tetrahedral elements that is optimal in the sense that it gives the distance of a tetrahedron from the set of inverted elements. This measure is constructed from the condition number of the linear transformation between a unit equilateral tetrahedron and any tetrahedron with positive volume. We use this shape measure to formulate two optimization objective functions that are differentiated by their goal: the first seeks to improve the average quality of the tetrahedral mesh; the second aims to improve the worst-quality element in the mesh. Because the element condition number is not defined for tetrahedral with negative volume, these objective functions can be used only when the initial mesh is valid. Therefore, we formulate a third objective function using the determinant of the element Jacobian that is suitable for mesh untangling. We review the optimization techniques used with each objective function and present experimental results that demonstrate the effectiveness of the mesh improvement and untangling methods. We show that a combined optimization approach that uses both condition number objective functions obtains the best-quality meshes.},
doi = {},
url = {https://www.osti.gov/biblio/11915}, journal = {},
number = ,
volume = ,
place = {United States},
year = {Fri Jul 30 00:00:00 EDT 1999},
month = {Fri Jul 30 00:00:00 EDT 1999}
}

Conference:
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