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
Eventually these reports will become either archive journal papers or part of a monograph on mesh optimization.
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.
There also exist methods of mesh adaptation via mesh modification which use metric tensor weightings (see [21] for a general discussion).
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]).
\(\parallel \cdot \parallel_F\) is the Frobenius matrix norm.
All figures in this example courtesy of Jan-Renee Carlson, NASA-Langley.
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.
The constant 0.4394 was determined by requiring that c k = 0.9 when d k = 130°.
References
Knupp P (2001) Algebraic mesh quality measures. SIAM J Sci Comput 23(1):193–218
Freitag L, Knupp P (2002) Tetrahedral mesh improvement via optimization of the element condition number. Intl J Numer Meth Engr 53:1377–1391
Knupp P, Margolin L, Shashkov M (2002) Reference-Jacobian optimization-based rezone strategies for arbitrary Lagrangian Eulerian methods. J Comp Phys 176(1):93–128
Knupp P (2006) Formulation of a target-matrix paradigm for mesh optimization. SAND2006-2730J, Sandia National Laboratories, Albuquerque
Knupp P (2009) Measuring quality within mesh elements. SAND2009-3081J. Sandia National Laboratories, Albuquerque
Knupp P (2009) Label-invariant mesh quality metrics. In: Proceedings of the 18th International Meshing Roundtable. Springer, Berlin, pp. 139–155
Knupp P Tradeoff-coefficient and binary metric construction algorithms within the target-matrix paradigm. manuscript
Knupp P (2008) Updating meshes on deforming domains. Commun Numer Methods Eng 24:467–476
Knupp P (2010) Introducing the target-matrix paradigm for mesh optimization via node-movement. In: Proceedings of the 19th International Meshing Roundtable. Springer, Berlin, pp. 67–83
Brewer M, Diachin L, Knupp P, Melander D (2003) The mesquite mesh quality improvement toolkit. In: Proceedings of the 12th International Meshing Roundtable, Santa Fe NM, pp. 239–250
Castillo JE (1991) A discrete variational grid generation method. SIAM J Sci Stat Comp 12(2):454–468
Tinoco-Ruiz J, Barrera-Sanchez P et al (1998) Area functionals in plane grid generation. In: Cross M (eds) Numerical grid generation in computational field simulations.. Greenwhich, UK, pp 293–302
Kennon S, Dulikravich G (1986) Generation of computational grids using optimization. AIAA J 24(7):1069–1073
Freitag L (1997) On combining Laplacian and optimization-based mesh smoothing techniques. AMD-Vol. 220, Trends in Unstructured Mesh Generation, ASME, pp. 37–43
Zhou T and Shimada K (2000) An angle-based approach to two-dimensional mesh smoothing. Proceedings of the 9th International Meshing Roundtable, pp. 373–384
Brackbill J, Saltzman J (1982) Adaptive zoning for singular problems in two dimensions. J Comp Phys 46:342–368
Steinberg S, Roache P (1986) Variational grid generation. Num Meth PDE 2:71–96
Liseikin V (1992) On a variational method of generating adaptive grids on n-dimensional surfaces. Soviet Math Docl 44(1):149–152
Winslow A (1967) Numerical solution of the quasilinear Poisson equations in a nonuniform triangle mesh. J Comp Phys 2:149–172
Knupp P, Luczak R (1995) Truncation error in grid generation. Numer Method PDE 11:561–571
Frey P, George P (2008) Mesh generation: application to finite elements. Wiley, New York
Dvinsky A (1991) Adaptive grid generation from harmonic maps on Riemannian manifolds. J Comp Phys 95:450–476
Thompson J, Warsi Z, Mastin C (1977) Automatic numerical generation of body-fitted curvilinear coordinate systems. J Comp Phys 24:274–302
Liseikin V (2004) A computational differential geometry approach to grid generation. Springer, Berlin
Knupp P (2009) Target-matrix construction algorithms. SAND2009-7003P, Sandia National Laboratories, Albuquerque
P. Knupp and J. Kraftcheck, Surface mesh optimization in the target-matrix paradigm. manuscript
Knupp P (2001) Hexahedral and tetrahedral mesh untangling. Eng Comput 17(3):261–268
Franks J, Knupp P (2010) A new strategy for untangling 2D meshes via node-movement. In: CSRI Summer Proceedings, SAND2010-8783P, Sandia National Laboratories, Albuquerque, pp. 152–165
Knupp P (2006) Local 2D metrics for mesh optimization in the target-matrix paradigm. SAND2006-7382J, Sandia National Laboratories, Albuquerque
Knupp P, van der Zee E (2006) Convexity of mesh optimization metrics using a target-matrix paradigm. SAND2006-4975J, Sandia National Laboratories, Albuquerque
Knupp P (2008) Analysis of 2D rotation-invariant non-barrier metrics in the target-matrix paradigm. SAND2008-8219P, Sandia National Laboratories, Albuquerque
Pirzadeh S (2003) VGRID unstructured grid generator. http://tetruss.larc.nasa.gov/vgrid/. Accessed 12 May 2010
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
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
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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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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DOI: https://doi.org/10.1007/s00366-011-0230-1