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Engineering with Computers
Erschienen in: Engineering with Computers 4/2012

01.10.2012 | Original Article

Introducing the target-matrix paradigm for mesh optimization via node-movement

verfasst von: Patrick Knupp

Erschienen in: Engineering with Computers | Ausgabe 4/2012

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Abstract

A general-purpose algorithm for mesh optimization via node-movement, known as the Target-Matrix Paradigm, is introduced. The algorithm is general purpose in that it can be applied to a wide variety of mesh and element types, and to various commonly recurring mesh optimization problems such as shape improvement, and to more unusual problems like boundary-layer preservation with sliver removal, high-order mesh improvement, and edge-length equalization. The algorithm can be considered to be a direct optimization method in which weights are automatically constructed to enable definitions of application-specific mesh quality. The high-level concepts of the paradigm have been implemented in the Mesquite mesh improvement library, along with a number of concrete algorithms that address mesh quality issues such as those shown in the examples of the present paper.
Vorheriger Artikel 2D structured grid generation method producing a mesh with prescribed properties near boundary
Nächster Artikel A numerical investigation on the interplay amongst geometry, meshes, and linear algebra in the finite element solution of elliptic PDEs

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Fußnoten
1
Eventually these reports will become either archive journal papers or part of a monograph on mesh optimization.
 
2
Regrettably, the natural acronym for the target-matrix paradigm is TMP, which has the connotation of temporariness, which we hope is not the future of this method.
 
3
There also exist methods of mesh adaptation via mesh modification which use metric tensor weightings (see [21] for a general discussion).
 
4
In our notation, V plays the role of Q in the QR factorization, i.e, it is the orthogonal matrix. The product \(\Uplambda Q \Updelta\) is R in the QR-factorization. A similar explicit factorization can be given for 3 × 3 matrices as well (see [25]).
 
5
\(\parallel \cdot \parallel_F\) is the Frobenius matrix norm.
 
6
All figures in this example courtesy of Jan-Renee Carlson, NASA-Langley.
 
7
The two metrics are linearly combined, both have an ideal value of zero, and both can go to infinity. The combined metric has an ideal value of zero, which would be attained only if A = R W 1 and A = W 2, with R an arbitrary rotation matrix. In turn, this would require W 2 = R W 1. But W 1 corresponds to an equilateral element, while W 2 does not, in general. Therefore, the ideal value of zero is not attained for most, if not all, of the elements in the mesh. The second metric is known to be strictly convex in the vertex coordinates; convexity is not established for the first metric. Hence, convexity is not assured for this combination.
 
8
The constant 0.4394 was determined by requiring that c k  = 0.9 when d k  = 130°.
 
Literatur
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Knupp P (2009) Label-invariant mesh quality metrics. In: Proceedings of the 18th International Meshing Roundtable. Springer, Berlin, pp. 139–155
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Knupp P (2006) Local 2D metrics for mesh optimization in the target-matrix paradigm. SAND2006-7382J, Sandia National Laboratories, Albuquerque
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Knupp P, van der Zee E (2006) Convexity of mesh optimization metrics using a target-matrix paradigm. SAND2006-4975J, Sandia National Laboratories, Albuquerque
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Knupp P (2008) Analysis of 2D rotation-invariant non-barrier metrics in the target-matrix paradigm. SAND2008-8219P, Sandia National Laboratories, Albuquerque
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Luo X, Shephard M, Lee L, Ge L, Ng C (2010) Moving curved mesh adaptatio for higher order finite element simulations. Engr. Cmptrs., published online 27 Feb 2010
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Knupp P, Voshell N, and Kraftcheck J (2009) Quadratic triangle mesh untanglng and optimization via the target-matrix paradigm. In: CSRI Summer Proceedings, SAND2010-3083P, Sandia National Laboratories, Albuquerque, pp. 141–151
Metadaten
Titel
Introducing the target-matrix paradigm for mesh optimization via node-movement
verfasst von
Patrick Knupp
Publikationsdatum
01.10.2012
Verlag
Springer-Verlag
Erschienen in
Engineering with Computers / Ausgabe 4/2012
Print ISSN: 0177-0667
Elektronische ISSN: 1435-5663
DOI
https://doi.org/10.1007/s00366-011-0230-1

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