A logbarrier method for mesh quality improvement and untangling
 Shankar P. Sastry,
 Suzanne M. Shontz,
 Stephen A. Vavasis
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The presence of a few inverted or poorquality mesh elements can negatively affect the stability, convergence and efficiency of a finite element solver and the accuracy of the associated partial differential equation solution. We propose a mesh quality improvement and untangling method that untangles a mesh with inverted elements and improves its quality. Worst element mesh quality improvement and untangling can be formulated as a nonsmooth unconstrained optimization problem, which can be reformulated as a smooth constrained optimization problem. Our technique solves the latter problem using a logbarrier interior point method and uses the gradient of the objective function to efficiently converge to a stationary point. The method uses a logarithmic barrier function and performs global mesh quality improvement. We have also developed a smooth quality metric that takes both signed area and the shape of an element into account. This quality metric assigns a negative value to an inverted element. It is used with our algorithm to untangle a mesh by improving the quality of an inverted element to a positive value. Our method usually yields better quality meshes than existing methods for improvement of the worst quality elements, such as the active set, pattern search, and multidirectional search mesh quality improvement methods. Our method is faster and more robust than existing methods for mesh untangling, such as the iterative stiffening method.
 Fried, E (1972) Condition of finite element matrices generated from nonuniform meshes. AIAA J 10: pp. 219221 CrossRef
 Babuska, I, Suri, M (1994) The p and hp versions of the finite element method, basic principles, and properties. SIAM Rev 35: pp. 579632
 Berzins M (1997) Solutionbased mesh quality for triangular and tetrahedral meshes. In: Proceedings of the 6th international meshing roundtable, pp 427–436
 Berzins M (1998) Mesh quality—geometry, error estimates, or both? In: Proceedings of the 7th international meshing roundtable, pp 229–237
 Knupp P (1999) Matrix norms and the condition number: a general framework to improve mesh quality via nodemovement. In: Proceedings of the 8th international meshing roundtable, pp 13–22
 Knupp, P, Freitag, L (2002) Tetrahedral mesh improvement via optimization of the element condition number. Int J Numer Methods Eng 53: pp. 13771391 CrossRef
 Amenta N, Bern M, Eppstein D (1997) Optimal point placement for mesh smoothing. In: Proceedings of the 8th ACMSIAM symposium on discrete algorithms, pp 528–537
 Munson, T (2007) Mesh shapequality optimization using the inverse meanratio metric. Math Program 110: pp. 561590 CrossRef
 Plaza, A, Suárez, J, Padrón, M, Falcón, S, Amieiro, D (2004) Mesh quality improvement and other properties in the fourtriangles longestedge partition. Comput Aided Geom Des 21: pp. 353369 CrossRef
 Shewchuk J (2002) What is a good linear element? Interpolation, conditioning, and quality measures. In: Proceedings of the 11th international meshing roundtable, pp 115–126
 Nocedal, J, Wright, S (2006) Numerical optimization. Springer, New York
 Tang T (2004) Moving mesh methods for computational fluid dynamics. In: Proceedings of the international conference on recent advances in adaptive computation, vol 383, contemporary mathematics
 Shontz, S, Vavasis, S (2010) Analysis of and workarounds for element reversal for a finite elementbased algorithm for warping triangular and tetrahedral meshes. BIT Numer Math 50: pp. 863884 CrossRef
 Shontz, S, Vavasis, S (2012) A robust solution procedure for hyperelastic solids with large boundary deformation. Eng Comput 28: pp. 135147 CrossRef
 Knupp, P (2007) Updating meshes on deforming domains: an application of the targetmatrix paradigm. Commun Num Method Eng 24: pp. 467476 CrossRef
 Kim J, Sastry S, Shontz S (2010) Efficient solution of elliptic partial differential equations via effective combination of mesh quality metrics, preconditioners, and sparse linear solvers. In: Proceedings of the 19th international meshing roundtable, pp 103–120
 Bank, R, Smith, R (1997) Mesh smoothing using a posterior error estimates. SIAM J Numer Anal 34: pp. 979997 CrossRef
 Freitag, L, Plassmann, P (2000) Local optimizationbased simplicial mesh untangling and improvement. Int J Numer Methods Eng 49: pp. 109125 CrossRef
 Park J, Shontz S (2010) Two derivativefree optimization algorithms for mesh quality improvement. In: Proceedings of the 2010 international conference on computational science, vol 1, pp 387–396
 Escobar, J, Rodriguez, E, Montenegro, R, Montero, G, GonzalezYuste, J (2003) Simultaneous untangling and smoothing of tetrahedral meshes. Comput Method Appl Mech Eng 192: pp. 27752787 CrossRef
 Sastry S, Shontz S, Vavasis S (2011) A logbarrier method for mesh quality improvement. In Proceedings of the 20th international meshing roundtable, pp 329–346
 Parthasarathy, V, Graichen, C, Hathaway, A (1994) A comparison of tetrahedron quality measures. Finite Elem Anal Des 15: pp. 255261 CrossRef
 Brewer M, FrietagDiachin L, Knupp P, Laurent T, Melander D (2003) The mesquite mesh quality improvement toolkit. In: Procedings of the 12th international meshing roundtable, pp 239–250
 CUBIT generation and mesh generation toolkit. http://cubit.sandia.gov/
 Si H, TetGen: A quality tetrahedral mesh generator and threedimensional delaunay triangulator. http://tetgen.berlios.de/
 Knupp P (2003) Sandia National Laboratories, personal communication
 Mehrotra, S (1992) On the implementation of a primaldual interior point method. SIAM J Optim 2: pp. 575601 CrossRef
 Title
 A logbarrier method for mesh quality improvement and untangling
 Journal

Engineering with Computers
Volume 30, Issue 3 , pp 315329
 Cover Date
 20140701
 DOI
 10.1007/s0036601202946
 Print ISSN
 01770667
 Online ISSN
 14355663
 Publisher
 Springer London
 Additional Links
 Topics
 Keywords

 Mesh quality improvement
 Mesh optimization
 Mesh untangling
 Interior point method
 Logbarrier
 Industry Sectors
 Authors

 Shankar P. Sastry ^{(1)}
 Suzanne M. Shontz ^{(2)}
 Stephen A. Vavasis ^{(3)}
 Author Affiliations

 1. The Pennsylvania State University, University Park, PA, 16802, USA
 2. Mississippi State University, Mississippi State, MS, 39762, USA
 3. University of Waterloo, Waterloo, ON, N2L3G1, Canada