Abstract
Optimizing surface and volume triangulations is critical for many advanced numerical simulation applications. We present a variational approach for smoothing triangulated surface and volume meshes to improve their overall mesh qualities. Our method seeks to reduce the discrepancies between the actual elements and ideal reference elements by minimizing two energy functions based on conformal and isometric map**s. We derive simple, closed-form formulas for the values, gradients, and Hessians of these energy functions, which reveal important connections of our method with some well-known concepts and methods in mesh generation and surface parameterization. We then introduce a simple and efficient iterative algorithm for minimizing the energy functions, including a novel asynchronous step-size control scheme. We demonstrate the effectiveness of our method experimentally and compare it against Laplacian smoothing and some other mesh smoothing techniques.
Similar content being viewed by others
Notes
We use \(\vert{\user2{A}}\vert\) to denote the determinant of a matrix \({\user2{A}}\) for brevity.
We label vertices of a tetrahedron by 0, 1, 2, and 3 so that the opposite face of vertex 0 has vertices 1, 2, and 3, leading to formulas that are more concise and more consistent with those for triangles.
References
Hirt CW, Amsden AA, Cook JL (1974) An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J Comput Phys 14:227–253. Reprinted in 135:203–216, 1997
Hansbo P (1995) Generalized Laplacian smoothing of unstructured grids. Commun Numer Methods Eng 11:455–464
Field DA (1988) Laplacian smoothing and Delaunay triangulation. Commun Appl Numer Methods 4:709–712
Zhou T, Shimada K (2000) An angle-based approach to two-dimensional mesh smoothing. In: Proceedings of 9th international meshing roundtable, pp 373–384
Parthasarathy VN, Kodiyalam S (1991) A constrained optimization approach to finite element mesh smoothing. Finite Elem Anal Des 9:309–320
Knupp PM, Robidoux N (2000) A framework for variational grid generation: conditioning the Jacobian matrix with matrix norms. SIAM J Sci Comput 21:2029
Branets LV (2005) A variational grid optimization method based on a local cell quality metric. Ph.D. thesis, University of Texas at Austin
Freitag LA (1997) On combining Laplacian and optimization-based mesh smoothing techniques. In: AMD-Vol 220 Trends in unstructured mesh generation, pp 37–43
Garimella RV, Shashkov MJ, Knupp PM (2004) Triangular and quadrilateral surface mesh quality optimization using local parametrization. Comput Methods Appl Mech Eng 193:913–928
Knupp PM (2000) Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. part i: a framework for surface mesh optimization. Int J Numer Methods Eng 48:401–420
Freitag LA, Knupp PM (2002) Tetrahedral mesh improvement via optimization of the element condition number. Int J Numer Methods Eng 53:1377–1391
Freitag Diachin L, Knupp PM, Munson T, Shontz S (2006) A comparison of two optimization methods for mesh quality improvement. Eng Comput 22:61–74
Munson T (2007) Mesh shape-quality optimization using the inverse mean-ratio metric. Math Program Ser A 110:561–590
Parthasarathy VN, Graichen CM, AF AFH (1993) A comparison of tetrahedron quality measures. Finite Elem Anal Des 15:255–261
Amenta N, Bern M, Eppstein D (1999) Optimal point placement for mesh smoothing. J Algorithms 30:302–322
Knupp PM (2001) Algebraic mesh quality metrics. SIAM J Sci Comput 23:193–218
Hansen GA, Douglass RW, Zardecki A (2005) Mesh enhancement. Imperial College Press
Ivanenko SA (1999) Harmonic map**s. In: Thompson JF, Soni BK, Weatherill NP (eds) Handbook of grid generation. CRC
Floater MS, Hormann K (2005) Surface parameterization: a tutorial and survey. In: Advances in multiresolution for geometric modelling. Springer, pp 157–186
Simpson RB (1994) Anisotropic mesh transformations and optimal error control. Appl Numer Math 14:183–198
do Carmo MP (1976) Differential geometry of curves and surfaces. Prentice-Hall
do Carmo MP (1992) Riemannian geometry. Birkhäuser, Boston
Kreyszig E (1991) Differential geometry. Dover
Hormann K, Greiner G (2000) MIPS: an efficient global parametrization method. In: Laurent PJ, Sablonniere P, Schumaker LL (eds) Curve and surface design, Saint-Malo, pp 153–162
Pinkall U, Polthier K (1993) Computing discrete minimal surfaces and their conjugates. Exp Math 2:15–36
Golub GH, Van Loan CF (1996) Matrix computation, 3rd edn. Johns Hopkins University Press
Degener P, Meseth J, Klein R (2003) An adaptable surface parameterization method. In: Proceedings of 12th international meshing roundtable, pp 227–237
Jiao X (2007) Face offsetting: a unified approach for explicit moving interfaces. J Comput Phys 220:612–625
Frey P, Borouchaki H (1998) Geometric surface mesh optimization. Comput Visual Sci 1(3):113–121
Cutler B, Dorsey J, McMillan L (2004) Simplification and improvement of tetrahedral models for simulation. In: Eurographics/ACM SIGGRAPH symposium on geometry processing, pp 93–102. Meshes available at http://people.csail.mit.edu/bmcutler/PROJECTS/SGP04
Freitag LA, Plassmann P (2000) Local optimization-based simplicial mesh untangling and improvement. Int J Numer Methods Eng 49:109–125
Acknowledgments
This work was supported by National Science Foundation under award number DMS-0809285 and also by a subcontract from the Center for Simulation of Advanced Rockets of the University of Illinois at Urbana-Champaign funded by the U.S. Department of Energy through the University of California under subcontract B523819. We thank Dr. Eric Shaffer of UIUC for pointers to some test meshes. We thank Dr. Patrick Knupp for helpful discussions. We thank the anonymous reviewers for their helpful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jiao, X., Wang, D. & Zha, H. Simple and effective variational optimization of surface and volume triangulations. Engineering with Computers 27, 81–94 (2011). https://doi.org/10.1007/s00366-010-0180-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00366-010-0180-z