## 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.

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## Notes

- 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°.

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## Acknowledgments

Many thanks to Jan-Renee Carlson, Jason Kraftcheck, and Nick Voshell for their important contributions to Mesquite and to the numerical examples.

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## Additional information

This work was funded by the Department of Energy’s Advanced Scientific Computing Research Program (SC-21) and was performed at Sandia National Laboratories.

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Knupp, P. Introducing the target-matrix paradigm for mesh optimization via node-movement.
*Engineering with Computers* **28, **419–429 (2012). https://doi.org/10.1007/s00366-011-0230-1

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### Keywords

- Mesh optimization
- Mesh quality
- Target-matrix paradigm