Abstract
The development of parallel algorithms for mesh generation, untangling, and quality improvement is of high importance due to the need for large meshes with millions to billions of elements and the availability of supercomputers with hundreds to thousands of cores. There have been prior efforts in the development of parallel algorithms for mesh generation and local mesh quality improvement in which only one vertex is moved at a time. But for global mesh untangling and for global mesh quality improvement, where all vertices are simultaneously moved, parallel algorithms have not yet been developed. In our earlier work, we developed a serial global mesh optimization algorithm and used it to perform mesh untangling and mesh quality improvement. Our algorithm moved the vertices simultaneously to optimize a log-barrier objective function that was designed to untangle meshes as well as to improve the quality of the worst quality mesh elements. In this paper, we extend our work and develop a parallel log-barrier mesh untangling and mesh quality improvement algorithm for distributed-memory machines. We have used the algorithm with an edge coloring-based algorithm for synchronizing unstructured communication among the processes executing the log-barrier mesh optimization algorithm. The main contribution of this paper is a generic scheme for global mesh optimization, whereby the gradient of the objective function with respect to the position of some of the vertices is communicated among all processes in every iteration. The algorithm was implemented using the OpenMPI 2.0 parallel programming constructs and shows greater strong scaling efficiency compared to an existing parallel mesh quality improvement technique.






Similar content being viewed by others
References
Tautges T, Jain R (2012) Creating geometry and mesh models for nuclear reactor core geometries using a lattice hierarchy-based approach. Eng Comput 28:319–329
Aliabadi S, Johnson A, Abedi J, Zellars B (2002) High performance computing of fluid-structure interactions in hydrodynamics applications using unstructured meshes with more than one billion elements. In: Proceedings of the 2002 conference on high performance in computing, lecture notes in computer science, vol 2552. pp 519–533
Komatitsch D, Tsuboi S, Ji C, Tromp J (2003) A 14.6 billion degrees of freedom, 5 teraflops, 2.5 terabyte earthquake simulation on the earth simulator. in: Proceedings of the ACM/IEEE SC2003 conference. pp 1–58113-695, 1 March 2003
Chrisochoides N (2006) A survey of parallel mesh generation methods. In: Bruaset A, Tveito A (eds) Numerical solution of partial differential equations on parallel computers. Springer, Berlin
Nave D, Chrisochoides N, Chew L (2004) Guaranteed-quality parallel Delaunay refinement for restricted polyhedral domains. Comput Geom Theor Appl 28:191–215
Galtier J, George P (1997) Prepartioning as a way to mesh subdomains in parallel. in: Proceedings of the ASME/ASCE/SES summer meeting, special symposium on trends in unstructured mesh generation. pp 107–122
Linardakis L, Chrisochoides N (2006) Delaunay decoupling method for parallel guarantee quality planar mesh refinement. SIAM J Sci Comput 27:1394–1423
Chew L, Chrisochoides N, Sukup F (1997) Parallel constrained Delaunay meshing. In: Proceedings of the ASME/ASCE/SES summer meeting, special symposium on trends in unstructured mesh generation. pp 89–96
Chernikov A, Chrisochoides N (2004) Parallel guaranteed quality planar Delaunay mesh generation by concurrent point insertion. In: Proceedings of the 14th fall workshop on computational geometry. pp 55–56
Chernikov A, Chrisochoides N (2004) Practical and efficient point insertion scheduling method for parallel guaranteed quality Delaunay refinement. In: Proceedings of the 18th annual international conference on supercomputing. ACM Press, pp 48–57
Chrisochoides N, Antonopoulos C, Blagojevic F, Chernikov A, Nikolopoulos D (2009) A multigrain Delaunay mesh generation method for multicore SMT-based architectures. J Parallel Distrb Comput
Löhner R, Cebral J (1999) Parallel advancing front grid generation. In: Proceedings of the 8th international meshing roundtable. pp 67–74
Löhner R, Camberos J, Marsha M (1990) Unstructured scientific compuation on scalable multiprocessors. In: Hehrotra P, Saltz J (eds) Parallel unstructured grid generation. MIT Press, Cambridge, pp 31–64
Löhner R (2013) A 2nd generation parallel advancing front grid generator. In: Proceedings of the 21st international meshing roundtable. pp 457–474
De Cougny H, Shephard M (1999) Parallel refinement and corasening of tetrahedral meshes. Int J Meth Eng 46:1101–1125
Castanos J, Savage J (1999) PARED: a framework for the adaptive solution of PDEs. in: Proceedings of the 8th IEEE symposium on high performance, distributed computing
Oliker L, Biswas R, Gabow H (2000) Parallel tetrahedral mesh adaptation with dynamic load balancing. Parallel Comput 26:1583–1608
Rivara M, Pizarro D, Chrisochoides N (2004) Parallel refinement of tetrahedral edges using terminal-edge bisection algorithm. in: Proceedings of the 13th international meshing roundtable
Williams R (1991) Adaptive parallel meshes with complex geometry. In: Numerical grid generation in computational fluid dynamics and related fields
Rivara M, Carlderon C, Pizaro D, Fedorov A, Chrisochoides N (2005) Parallel decoupled terminal-edge bisection algorithm for 3D meshes. Eng Comput
Chrisochoides N, Chernikov A, Fedorov A, Kot A, Linardakis L, Foteinos P (2009) Towards exascale parallel Delaunay mesh generation. In: Proceedings of the 18th international meshing roundtable. pp 319–336
Freitag L, Jones M, Plassmann P (1999) A parallel algorithm for mesh smoothing. SIAM J Sci Comput 20(6):2023–2040
Gorman G, Southern J, Farrell P, Piggott M, Rokos G, Kelly P (2012) Hybrid OpenMP/MPI anisotropic mesh smoothing. In: Proceedings of the 2012 international conference on computational science, ICCS 2012, procedia computer science, vol 9. pp 1513–1522
Benítez D, Rodríguez E, Escobar J, Montenegro R (2013) Performance evaluation of a parallel algorithm for simultaneous untangling and smoothing of tetrahedral meshes. In: Proceedings of the 22nd international meshing roundtable. Springer International Publishing, pp 579–598
Canann S, Stephenson M, Blacker T (1993) Optismoothing: an optimization-driven approach to mesh smoothing. Finite Elem Anal Des 13:185–190
Parthasarathy V, Kodiyalam S (1991) A constrained optimization approach to finite element mesh smoothing. Finite Elem Anal Des 9:309–320
Shephard M, Georges M (1991) Automatic three-dimensional mesh generation by the finite octree technique. Int J Numer Meth Eng 32:709–749
Bank R, Smith R (1997) Mesh smoothing using a posteriori error estimates. SIAM J Numer Anal 34:979–997
Staten M, Canann S, Tristano J (1998) An approach to combined Laplacian and optimization-based mesh smoothing for triangular, quadrilateral, and quad-dominant meshes. in: Proceedings of the 7th international meshing roundtable. Sandia National Laboratories, pp 479–494
Knupp P (1999) Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part 1—a framework for surface mesh optimization, technical report SAND 99–0110J, Sandia National Laboratories
Sastry S, Shontz S, Vavasis S (2011) A log-barrier method for mesh quality improvement. In: Proceedings of the 20th international meshing roundtable. pp 329–346
Sastry S, Shontz S, Vavasis S (2012) A log-barrier method for mesh quality improvement and untangling. Eng Comput 1–15
Xu H, Newman T (2006) An angle-based optimization approach for 2D finite element mesh smoothing. Finite Elem Anal Des 42:1150–1164
Zhou T, Shimada K (2000) An angle-based approach to two-dimensional mesh smoothing. In: Proceedings of the 9th international meshing roundtable. Sandia National Laboratories, pp 373–384
Mezentsev A (2004) A generalized graph-theoretic mesh optimization model. in: Proceedings of the 13th international meshing roundtable. Sandia National Laboratories, pp 255–264
Djidjev H (2000) Force-directed methods for smoothing unstructured triangular and tetrahedral meshes. in: Proceedings of the 9th international meshing roundtable. Sandia National Laboratories, pp 395–406
Freitag L, Plassmann P (2000) Local optimization-based simplicial mesh untangling and improvement. Int J Numer Meth Eng 49:109–125
Park J, Shontz S (2010) Two derivative-free optimization algorithms for mesh quality improvement. In: Proceedings of the 2010 international conference on computational science, vol 1. pp 387–396
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
Knupp P (2001) Hexahedral and tetrahedral mesh untangling. Eng Comput 17:261–268
Freitag L, Plassmann P (2000) Local optimization-based simplicial mesh untangling and improvement. Int J Numer Meth Eng 49:109–125
Freitag L, Plassmann P (2001) Local optimization-based untangling algorithms for quadrilateral meshes. In: Proceedings of the 10th international meshing roundtable. Sandia National Laboratories, pp 397–406
Vachal P, Garimella R, Shashkov M (2004) Untangling of 2D meshes in ALE simulations. J Comput Phys 196:627–644
Agarwal P, Sadri B, Yu H (2008) Untangling triangulations through local explorations. In: Proceedings of the 2008 symposium on computational geometry (SoCG 2008)
Bhowmick S, Shontz S (2010) Towards high-quality, untangled meshes via a force-directed graph embedding approach. In: Proceedings of the 2010 international conference on computational science, procedia computer science, vol 1. pp 357–366
Toulorge T, Geuzaine C, Remacle J, Lambrechts J (2013) Robust untangling of curvilinear meshes. J Comput Phys 254:8–26
Clark B, Ray N, Jiao X (2013) Surface mesh optimization, adaption, and untangling with high-order accuracy. In: Proceedings of the 21st international meshing roundtable. pp 385–402
Wilson T, Sarrate J, Roca X, Montenegro R, Escobar J (2012) Untangling and smoothing of quadrilateral and hexahedral meshes. in: Topping B (ed) Proceedings of the 8th international conference on engineering computational technology
Kim J, Panitanarak T, Shontz S (2013) A multiobjective mesh optimization framework for mesh quality improvement and untangling. Int J Numer Meth Eng 94:20–42
Garanzha V, Kudriavtseva L (2011) Gradient projection based optimization methods for untangling and optimization of 3D meshes in implicit domains. In: Proceedings of the II international conference on optimization and applications (OPTIMA 2011)
Parthasarathy S, Gandhi R (2004) Distributed algorithms for coloring and domination in wireless adhoc networks. In: Lodaya K, Mahajan, M (eds) FSTTCS 2004: foundations of software technology and theoretical computer science, vol 3328 of lecture notes in computer science. Springer, Berlin Heidelberg, pp 447–459
Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, New York
Durand D, Jain R, Tseytlin D (1994) Distributed scheduling algorithms to improve the performance of parallel data transfers. SIGARCH Comput Archit News 22(4):35–40
Brewer M, Freitag Diachin L, Knupp P, Leurent T, Melander D (2003) The Mesquite mesh quality improvement toolkit. In: Proceedings of the twelfth international meshing roundtable. Sandia National Laboratories, pp 239–250
Inria Mesh Database. http://www-roc.inria.fr/gamma/gamma/download/download.php
Si H (2007) TetGen: a quality tetrahedral mesh generator and three-dimensional Delaunay triangulator
Karypis G, Kumar V (2009) MeTis: unstructured graph partitioning and sparse matrix ordering system, version 4.0. http://www.cs.umn.edu/~metis
Freitag L, Jones M, Plassmann P (1995) An efficient parallel algorithm for mesh smoothing. In: Proceedings of the 4th international meshing roundtable. pp 1–18
Xu Y, Chen Y (2008) A framework for parallel nonlinear optimization by partitioning localized constraints. in: Proceedings of the international symposium on parallel architectures, algorithms, and programming (PAAP-08)
Panitanarak T, Shontz S (2014) A parallel log-barrier based mesh warping algorithm for distributed memory machines (in preparation)
Acknowledgments
The authors are indebted to Thap Panitanarak for the use of his partitioned mesh data structure from his MPI implementation of a parallel log-barrier mesh warping algorithm (PLBWARP) in [60]. The work of the first author was supported by the NIH/NIGMS Center for Integrative Biomedical Computing Grant 2P41 RR0112-553-12 and DOE NET DE-EE0004449 Grant. The work of the second author is supported in part by NSF CAREER Grant ACI-1330056 (formerly ACI-1054459). The authors would also like to thank the three anonymous referee for their comments which improved the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sastry, S.P., Shontz, S.M. A parallel log-barrier method for mesh quality improvement and untangling. Engineering with Computers 30, 503–515 (2014). https://doi.org/10.1007/s00366-014-0362-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00366-014-0362-1