Engineering with Computers

, Volume 33, Issue 4, pp 717–726 | Cite as

Efficient parallel optimization of volume meshes on heterogeneous computing systems

  • Zuofu ChengEmail author
  • Eric Shaffer
  • Raine Yeh
  • George Zagaris
  • Luke Olson
Original Article


We describe a parallel algorithmic framework for optimizing the shape of elements in a simplicial volume mesh. Using fine-grained parallelism and asymmetric multiprocessing on multi-core CPU and modern graphics processing unit hardware simultaneously, we achieve speedups of more than tenfold over current state-of-the-art serial methods. In addition, improved mesh quality is obtained by optimizing both the surface and the interior vertex positions in a single pass, using feature preservation to maintain fidelity to the original mesh geometry. The framework is flexible in terms of the core numerical optimization method employed, and we provide performance results for both gradient-based and derivative-free optimization methods.


Mesh optimization Parallel algorithms GPU applications 


  1. 1.
    Boman E, Bozda D, Catalyurek U, Gebremedhin A, Manne F (2005) A scalable parallel graph coloring algorithm for distributed memory computers. In: Cunha J, Medeiros P (eds) Euro-Par 2005 parallel processing, vol 3648., Lecture notes in computer scienceSpringer, Berlin, pp 241–251CrossRefGoogle Scholar
  2. 2.
    DAmato J, Vnere M (2013) A CPUGPU framework for optimizing the quality of large meshes. J Parallel Distrib Comput 73(8):1127–1134Google Scholar
  3. 3.
    Fleurent C, Ferland JA (1996) Genetic and hybrid algorithms for graph coloring. Ann Oper Res 63(3):437–461CrossRefzbMATHGoogle Scholar
  4. 4.
    Freitag L, Jones M, Plassmann P (1999) A parallel algorithm for mesh smoothing. SIAM J Sci Comput 20(6):2023–2040MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Freitag L, Knupp P, Munson T, Shontz S (2004) A comparison of inexact Newton and coordinate descent mesh optimization techniques. In: Proceedings of the 13th international meshing roundtable, Williamsburg, pp 243–254Google Scholar
  6. 6.
    Freitag L, Knupp P, Munson T, Shontz S (2006) A comparison of two optimization methods for mesh quality improvement. Invit Submiss Eng Comput 22(2):61–74CrossRefGoogle Scholar
  7. 7.
    Freitag LA, Plassmann P (2000) Local optimization-based simplicial mesh untangling and improvement. Int J Numer Methods Eng 49(1–2):109–125CrossRefzbMATHGoogle Scholar
  8. 8.
    Garimella RV, Shashkov MJ, Knupp PM (2004) Triangular and quadrilateral surface mesh quality optimization using local parametrization. Comput Methods Appl Mech Eng 193(911):913–928CrossRefzbMATHGoogle Scholar
  9. 9.
    Gorman G, Southern J, Farrell P, Piggott M, Rokos G, Kelly P (2012) Hybrid OpenMP/MPI anisotropic mesh smoothing. Proc Comput Sci 9(0):1513–1522 (proceedings of the international conference on computational science, ICCS 2012)Google Scholar
  10. 10.
    Gyrfs A, Lehel J (1988) On-line and first fit colorings of graphs. J Graph Theory 12(2):217–227MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jiao X, Alexander P (2005) Parallel feature-preserving mesh smoothing. In: Gervasi O, Gavrilova M, Kumar V, Lagan A, Lee H, Mun Y, Taniar D, Tan C (eds) Computational science and its applications ICCSA 2005, vol 3483., Lecture notes in computer scienceSpringer, Berlin, pp 1180–1189CrossRefGoogle Scholar
  12. 12.
    Jiao X, Bayyana NR (2008) Identification of c1 and c2 discontinuities for surface meshes in CAD. Comput Aided Des 40:160–175CrossRefGoogle Scholar
  13. 13.
    Jones MT, Plassmann PE (1993) A parallel graph coloring heuristic. SIAM J Sci Comput 14(3):654–669MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lewis A, Overton M (2009) Nonsmooth optimization via BFGS. SIAM J Optim (submitted)Google Scholar
  15. 15.
    Liu A, Joe B (1994) Relationship between tetrahedron shape measures. BIT Numer Math 34(2):268–287MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    McLaurin D, Shontz S (2014) Automated edge grid generation based on arc-length optimization. In: Sarrate J, Staten M (eds) Proceedings of the 22nd international meshing roundtable, pp 385–403. Springer, New YorkGoogle Scholar
  17. 17.
    Montenegro R, Escobar J, Montero G (2005) Quality improvement of surface triangulations. In: Hanks B (ed) Proceedings of the 14th international meshing roundtable. Springer, Berlin, pp 469–480CrossRefGoogle Scholar
  18. 18.
    Munson T (2005) Optimizing the quality of mesh elements. SIAG/Optim News Views 16:27–34Google Scholar
  19. 19.
    Munson T (2007) Mesh shape-quality optimization using the inverse mean-ratio metric. Math Program 110:561–590MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Park J, Shontz SM (2010) Two derivative-free optimization algorithms for mesh quality improvement. Proc Comput Sci 1(1):387–396 (ICCS 2010)Google Scholar
  21. 21.
    Sastry S, Shontz S, Vavasis S (2014) A log-barrier method for mesh quality improvement and untangling. Eng Comput 30(3):315–329CrossRefGoogle Scholar
  22. 22.
    Shaffer E, Cheng Z, Yeh R, Zagaris G, Olson L (2014) Efficient GPU-based optimization of volume meshes. In: Bader M (ed) Parallel computing: accelerating computational science and engineering (CSE). Advances in parallel computing, vol 25. IOS Press BV, Amsterdam, pp 285–294Google Scholar
  23. 23.
    Shaffer E, Zagaris G (2011) GPU accelerated derivative-free mesh optimization. In: Hwu W-M (ed) GPU computing gems jade edition. Applications of GPU computing series, chap 13, 1st edn. Morgan Kaufmann, Waltham, pp 145–154 Google Scholar
  24. 24.
    Shewchuk J (2002) What is a good linear element? Interpolation, conditioning, and quality measures. In: Proceedings of the 11th international meshing roundtable, pp 115–126Google Scholar
  25. 25.
    Vartziotis D, Wipper J, Schwald B (2009) The geometric element transformation method for tetrahedral mesh smoothing. Comput Methods Appl Mech Eng 199(14):169–182MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Volkov V, Kazian B (2008) Fitting FFT onto the G80 architecture, vol 40. University of California, Berkeley Google Scholar
  27. 27.
    Walton S, Hassan O, Morgan K (2013) Reduced order mesh optimisation using proper orthogonal decomposition and a modified cuckoo search. Int J Numer Methods Eng 93(5):527–550MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag London 2015

Authors and Affiliations

  • Zuofu Cheng
    • 1
    Email author
  • Eric Shaffer
    • 1
  • Raine Yeh
    • 3
  • George Zagaris
    • 2
  • Luke Olson
    • 1
  1. 1.University of IllinoisUrbanaUSA
  2. 2.Kitware Inc., University of IllinoisUrbanaUSA
  3. 3.Purdue UniversityWest LafayetteUSA

Personalised recommendations