Skip to main content
SpringerLink
Log in

A parallel log barrier-based mesh warping algorithm for distributed memory machines

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

Parallel dynamic meshes are essential for computational simulations of large-scale scientific applications involving motion. To address this need, we propose parallel LBWARP, a parallel log barrier-based tetrahedral mesh warping algorithm for distributed memory machines. Our algorithm is a general-purpose, geometric mesh warping algorithm that parallelizes the sequential LBWARP algorithm proposed by Shontz and Vavasis. The first step of the algorithm involves computation of a set of local weights for each interior node which describe the relative distances of the node to each of its neighbors. The weight computation step is the most time consuming in the parallel algorithm. Based on our choice of the mesh partition and the corresponding distribution of data and assignment of tasks to processors, communication among processors is avoided in an embarrassingly parallel computation of the weights. Once this representation of the initial mesh is determined, a target deformation of the boundary is applied, also in an embarrassingly parallel manner. Finally, new coordinates of the interior nodes are obtained by solving a system of linear equations with multiple right-hand sides that is based on the weights and boundary deformation. This linear system can be solved using one of three parallel sparse linear solvers, i.e., the distributed block BiCG, block GMRES, or LU algorithm, all of which support the solution of linear systems with multiple right-hand side vectors. Our numerical results demonstrate good efficiency and strong scalability of parallel LBWARP on up to 64 processors, as the experiments show close to linear speedup in all cases. Weak scalability is also demonstrated. The performance of the parallel sparse linear solvers is dependent on factors such as the mesh size, the amount of available memory, and the number of processors. For example, the distributed LU algorithm gives better performance on small meshes, whereas the distributed block BiCG and distributed block GMRES algorithms yield better performance when the amount of available memory is limited. Finally, we demonstrate the parallel LBWARP performance for a sequence of mesh deformations which can significantly reduce the runtime of the overall algorithm. When applied to k deformations, parallel LBWARP reuses the weight matrix, that was computed during the first deformation, when the distributed LU linear solver is employed. This gives close to k-time performance for sufficiently many deformations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Notes

  1. The factor LU exists only if A is nonsingular. In the event an element on the diagonal of A is zero or nearly zero, partial pivoting is required.

  2. The MR mesh quality metric is given by

    $$\begin{aligned} \eta&= \frac{12(3v)^{2/3}}{\sum _{0 \le i < j \le 3} l_{ij}^2}~[29], \end{aligned}$$
    (5)

    where v and \(l_{ij}\) denote the volume and various edge lengths of the tetrahedron, respectively. Note the ideal mesh quality occurs when \(\eta = 1,\) and 0 denotes a degenerate tetrahedron. The range of the metric is 0 to 1. Higher values denote better quality.

References

  1. Alan G (1973) Nested dissection of a regular finite element mesh. SIAM J Numer Anal 10:345–363

    Article  MathSciNet  MATH  Google Scholar 

  2. Amano A, Kanda K, Shibayama T, Kamei Y, Matsuda T (2007) Model generation interface for simulation of left ventricular motion. Electron Commun Jpn (Part II: Electronics) 90(12):87–98

  3. Antonopoulos C, Ding X, Chernikov A, Blagojevic F, Nikolopoulos D, Chrisochoides N (2005) Multigrain parallel Delaunay mesh generation. In: Proceedings of the 19th annual international conference on supercomputing. ACM Press, New York, pp 367–376

  4. Barrett R, Berry MW, Chan TF, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C, Van der Vorst H (1994) Templates for the solution of linear systems: building blocks for iterative methods, vol 43. SIAM, Philadelphia

  5. Bavier E, Hoemmen M, Rajamanickam S, Thornquist H (2012) Amesos2 and Belos: direct and iterative solvers for large sparse linear systems. Sci Program 20(3):241–255

    Google Scholar 

  6. Benítez D, Rodríguez E, Escobar JM, Montenegro R (2014) Performance evaluation of a parallel algorithm for simultaneous untangling and smoothing of tetrahedral meshes. In: Proceedings of the 22nd international meshing roundtable. Springer, Cham, pp 579–598

  7. Bertsekas DP (1982) Projected Newton methods for optimization problems with simple constraints. SIAM Journal on Control and Optimization 20(2):221–246

    Article  MathSciNet  MATH  Google Scholar 

  8. Botsch M, Bommes D, Kobbelt, L (2005) Efficient linear system solvers for mesh processing. In: Proceedings of the 11th IMA International conference on the mathematics of surfaces, pp 62–83

  9. 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, pp 133–140

  10. Chernikov A, Chrisochoides N (2006) Parallel guaranteed quality Delaunay uniform mesh refinement. SIAM J Sci Comput 28:1907–1926

    Article  MathSciNet  MATH  Google Scholar 

  11. Chrisochoides N (2005) A survey of parallel mesh generation methods. Tech. Rep. SC-2005-09, Brown University

  12. 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

  13. CyberSTAR: a scalable terascale advanced resource for discovery through computing. The Pennsylvania State University. https://ics.psu.edu/advanced-cyberinfrastructure/ics-aci-infrastructure/

  14. De Cougny H, Shephard M (1999) Parallel refinement and coarsening of tetrahedral meshes. Int J Numer Methods Eng 46(7):1101–1125

    Article  MathSciNet  MATH  Google Scholar 

  15. Estruch O, Lehmkuhl O, Borrell R, Segarra CP, Oliva A (2013) A parallel radial basis function interpolation method for unstructured dynamic meshes. Comput Fluids 80:44-54. (Selected contributions of the 23rd international conference on parallel fluid dynamics ParCFD2011)

  16. Fletcher R (1976) Conjugate gradient methods for indefinite systems. In: Numerical analysis. Springer, Berlin, Heidelberg, pp 73–89

  17. Galtier J, George P (1997) Prepartitioning 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

  18. Gerhold T, Neumann J (2008) The parallel mesh deformation of the DLR TAU-code. In: New results in numerical and experimental fluid mechanics VI, notes on numerical fluid mechanics and multidisciplinary design, vol 96. Springer, Berlin, Heidelberg, pp 162–169

  19. Gorman GJ, Rokos G, Southern J, Kelly PH (2015) Thread-parallel anisotropic mesh adaptation. In: New challenges in grid generation and adaptivity for scientific computing. Springer, pp 113–137

  20. GrabCAD. https://grabcad.com

  21. Guennebaud G, Jacob B et al (2010) Eigen v3. http://eigen.tuxfamily.org

  22. Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Natl Bureau Stand 49(6):409–436

  23. Hunter PJ, Pullan AJ, Smaill BH (2003) Modeling total heart function. Annu Rev Biomed Eng 5(1):147–177

    Article  Google Scholar 

  24. Ijiri T, Ashihara T, Umetani N, Igarashi T, Haraguchi R, Yokota H, Nakazawa K (2012) A kinematic approach for efficient and robust simulation of the cardiac beating motion. PLoS One 7(5):e36,706

  25. Interoperable technologies for advanced petascale simulations (ITAPS) center (2010). http://www.scidac.gov/math/ITAPS.html

  26. Karypis G, Kumar V (1999) A fast and highly quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 20:359–392

    Article  MATH  Google Scholar 

  27. Lachat C, Dobrzynski C, Pellegrini F (2014) Parallel mesh adaptation using parallel graph partitioning. In: 5th European conference on computational mechanics, vol 3. CIMNE-International Center for Numerical Methods in Engineering, pp 2612–2623

  28. Li XS, Demmel JW (2003) SuperLU_DIST: a scalable distributed-memory sparse direct solver for unsymmetric linear systems. ACM Trans Math Softw 29(2):110–140

    Article  MATH  Google Scholar 

  29. Liu A, Joe B (1994) Relationship between tetrahedron shape measures. BIT Numer Math 34(2):268–287

    Article  MathSciNet  MATH  Google Scholar 

  30. Löhner, R (2013) A 2nd generation parallel advancing front grid generator. In: Proceedings of the 21st international meshing roundtable, pp. 457–474

  31. Löhner R, Camberos J, Marsha M (1990) Unstructured scientific computation on scalable multiprocessors. In: Hehrotra P, Saltz J (eds) Parallel unstructured grid generation. MIT Press, Cambridge, MA, pp 31–64

  32. Löhner R, Cebral J (1999) Parallel advancing front grid generation. In: Proceedings of the 8th international meshing roundtable, pp 67–74

  33. Lu Q, Shephard MS, Tendulkar S, Beall MW (2014) Parallel mesh adaptation for high-order finite element methods with curved element geometry. Eng Comput 30(2):271–286

    Article  Google Scholar 

  34. Luke E, Collins E, Blades E (2012) A fast mesh deformation method using explicit interpolation. J Comput Phys 231:586–601

    Article  MathSciNet  MATH  Google Scholar 

  35. Nave D, Chrisochoides N, Chew L (2004) Guaranteed-quality parallel Delaunay refinement for restricted polyhedral domains. Comput Geom Theory Appl 28:191–215

    Article  MathSciNet  MATH  Google Scholar 

  36. O’Leary DP (1980) The block conjugate gradient algorithm and related methods. Linear Algebra Appl 29:293–322

    Article  MathSciNet  MATH  Google Scholar 

  37. Oliker L, Biswas R, Gabow H (2000) Parallel tetrahedral mesh adaptation with dynamic load balancing. Parallel Comput J 26:1583–1608

    Article  MATH  Google Scholar 

  38. Park J, Shontz S, Drapaca C (2013) A combined level set/mesh warping algorithm for tracking brain and cerebrospinal fluid evolution in hydrocephalic patients. In: Image-based geometric modeling and mesh generation, Lecture notes in computational vision and biomechanics, vol 3, pp 107–141

  39. Rivara M, Carlderon C, Pizaro D, Fedorov A, Chrisochoides N (2006) Parallel decoupled terminal-edge bisection algorithm for 3D meshes. Eng Comput 22:111–119

  40. 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

  41. Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7(3):856–869

    Article  MathSciNet  MATH  Google Scholar 

  42. Sastry SP, Shontz SM (2014) A parallel log-barrier method for mesh quality improvement and untangling. Eng Comput 30(4):503–515

    Article  Google Scholar 

  43. Selwood P, Berzins M, Dew P (1997) 3D parallel mesh adaptivity: data-structures and algorithms. In: Proceedings of the 8th SIAM conference on parallel processing for scientific computing. SIAM

  44. Selwood P, Verhoeven N, Nash J, Berzins M, Weatherill N, Dew P, Morgan K (1996) Parallel mesh generation and adaptivity: partitioning and analysis. In: Proceedings of 1996 parallel CFD conference

  45. Shephard M, Flaherty J, Bottasso C, de Cougny H, Özturan C, Simone M (1997) Parallel automated adaptive analysis. Parallel Comput 23:1327–1347

    Article  MathSciNet  MATH  Google Scholar 

  46. Shontz S, Vavasis S (2003) A mesh warping algorithm based on weighted Laplacian smoothing. In: Proceedings of the 12th international meshing roundtable, pp 147–158

  47. Shontz S, Vavasis S (2010) Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes. BIT Numer Math 50:863–884

    Article  MathSciNet  MATH  Google Scholar 

  48. Si H (2015) Tetgen: A Delaunay-based quality tetrahedral mesh generator. ACM Trans Math Softw 41:11

  49. Simoncini V (1997) A stabilized QMR version of block BICG. SIAM J Matrix Anal Appl 18(2):419–434

    Article  MathSciNet  MATH  Google Scholar 

  50. Simoncini V, Gallopoulos E (1996) Convergence properties of block GMRES and matrix polynomials. Linear Algebra Appl 247:97–119

    Article  MathSciNet  MATH  Google Scholar 

  51. Trilinos Project. http://trilinos.org/

  52. Tsai H, Wong A, Cai J, Zhu Y, Liu F (2001) Unsteady flow calculations with a parallel multiblock moving mesh algorithm. AIAA J 39:1021–1029

    Article  Google Scholar 

  53. Williams R (1991) Adaptive parallel meshes with complex geometry. In: Numerical grid generation in computational fluid dynamics and related fields

Download references

Acknowledgements

The work of the first author was funded by the Royal Thai Government scholarship. The work of the second author was supported in part by NSF Grants CNS-0720749 and NSF CAREER Award ACI-1500487 (formerly ACI-1330054 and ACI-1054459). This work was also supported in part through instrumentation funded by the National Science Foundation through Grant ACI0821527. The authors wish to thank the two anonymous referees for their careful reading of the paper and for their helpful suggestions which strengthened it.

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania, 16802, USA

    Thap Panitanarak

  2. Department of Electrical Engineering and Computer Science, Bioengineering Graduate Program, Information and Telecommunication Technology Center, University of Kansas, Lawrence, KS, 66045, USA

    Suzanne M. Shontz

Authors
  1. Thap Panitanarak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suzanne M. Shontz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thap Panitanarak.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Panitanarak, T., Shontz, S.M. A parallel log barrier-based mesh warping algorithm for distributed memory machines. Engineering with Computers 34, 59–76 (2018). https://doi.org/10.1007/s00366-017-0521-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-017-0521-2

Keywords

Search

Navigation