Abstract
We compare inexact Newton and block coordinate descent optimization methods for improving the quality of a mesh by repositioning the vertices, where the overall quality is measured by the harmonic mean of the mean-ratio metric. The effects of problem size, element size heterogeneity, and various vertex displacement schemes on the performance of these algorithms are assessed for a series of tetrahedral meshes.
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Notes
An important alternative to mesh optimization often used by the unstructured mesh community employs a series of local objective functions.
For hybrid meshes, the exact definition of q can change depending on the element type. However, we assume that the quality metric, shape, for example, is the same for every element.
This approach excludes elements such as pyramids but includes triangles, tetrahedra, wedges, quadrilaterals, and hexahedra.
We show in [6] that averaging is unnecessary in the case of triangular or tetrahedral elements.
Recall that (2) minimizes the inverse mean-ratio objective function, so the stated algorithms use minimization terminology. However, the same algorithms can be used for the maximization problem (1).
References
Bank R, Smith B (1997) Mesh smoothing using a posteriori error estimates. SIAM J Num Anal 34:979–997
Canann SA, Stephenson MB, Blacker T (1993) Optismoothing: an optimization-driven approach to mesh smoothing. Fin Elem Anal Des 13:185–190
Parthasarathy VN, Kodiyalam S (1991) A constrained optimization approach to finite element mesh smoothing. Fin Elem Anal Des 9:309–320
Zavattieri P, Dari E, Buscaglia G (1996) Optimization strategies in unstructured mesh generation. Int J Num Methods Eng 39:2055–2071
Castillo J (1991) A discrete variational grid generation method. SIAM J Sci Stat Comp 12:454–468
Freitag L, Knupp P (2002) Tetrahedral mesh improvement via optimization of the element condition number. Int J Numer Methods Eng 53:1377–1391
Anderson D (1990) Grid cell volume control with an adaptive grid generator. Appl Math Comp 35:209–217
Knupp P (2001) Algebraic mesh quality metrics. SIAM J Sci Comp 23:193–218
Knupp P (1999) Matrix norms and the condition number. Proceedings of 8th International Meshing Roundtable, pp 13–22
Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, Belmont
Nocedal J, Wright SJ (1999) Numerical optimization. Springer, Berlin Heidelberg New York
Munson TS (2004) Mesh shape-quality optimization using the inverse mean-ratio metric. Preprint ANL/MCS-P1136-0304, Argonne National Laboratory, Argonne
Munson TS (2004) Mesh shape-quality optimization using the inverse mean-ratio metric: tetrahedral proofs. Technical memorandum ANL/MCS-TM-275, Argonne National Laboratory, Argonne
Armijo L (1966) Minimization of functions having lipschitz-continuous first partial derivatives. Pac J Math 16:1–3
Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. SIAM, Philadelphia
Griewank A (2000) Evaluating derivatives: principles and techniques of algorithmic differentiation. SIAM, Philadelphia
Sandia National Laboratories (2003) Albuquerque, New Mexico. CUBIT 80.1 Mesh Generation Toolkit
Ollivier-Gooch CF (1998–2002) GRUMMP—Generation and refinement of unstructured mixed-element meshes in parallel. http://www.tetra.mech.ubc.ca/GRUMMP
Acknowledgments
The initial version of the analytic gradient for the inverse mean-ratio metric for tetrahedral elements was provided by Paul Hovland (Argonne National Laboratory). The clipped cube mesh image was provided by Carl Ollivier-Gooch (University of British Columbia). The work of the first, second, and third authors was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contracts W-7405-Eng-48 (UCRL-CONF-205150), DE-AC-94AL85000, and W-31-109-Eng-38, respectively. Part of the work of the fourth author was performed while a member of the Center for Applied Mathematics at Cornell University, supported by Sandia National Laboratories, Cornell University, the National Physical Science Consortium, and NSF grant ACI-0085969.
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Freitag Diachin, L., Knupp, P., Munson, T. et al. A comparison of two optimization methods for mesh quality improvement. Engineering with Computers 22, 61–74 (2006). https://doi.org/10.1007/s00366-006-0015-0
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DOI: https://doi.org/10.1007/s00366-006-0015-0