Advertisement

International Journal of Material Forming

, Volume 12, Issue 1, pp 45–55 | Cite as

Speeding-up simulation of cogging process by multigrid method

  • Mohamad RamadanEmail author
  • Mahmoud Khaled
  • Lionel Fourment
Original Research
  • 39 Downloads

Abstract

Calculation time of some material forming processes is tremendously expensive which makes reducing computational time one of the most urgent challenges in this domain. Among strategies that have been developed to speed-up calculations, one of the most flexible solutions is to utilize enhanced linear solvers such as Multi-Grid algorithm. It consists in using several levels of meshes of the same domain in order to more efficiently solve the systems of equations derived from the discretized problem. The speed-up results from the efficiency of coarse meshes in computing the low frequencies of the residual while fine meshes are more efficient in reducing the high frequencies of the residual. The method is integrated in the commercial software Forge® and applied to the industrial cogging process. The obtained results show that the speed-up depends on the number of nodes; for an industrial scale mesh of 50,000 nodes, the multigrid technique allows dividing the computational time by a factor of two.

Keywords

Multigrid Iterative solver Meshing Material forming Cogging Process simulation finite element method 

Notes

Acknowledgements

This work was supported by the consortium « club forgeage libre » which gathered the following industries: ArcelorMittal, Cézus (Areva), Sfarsteel (Areva), Aubert & Duval and Manoir industries and the software developer Transvalor.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Dedieu C, Barasinski A, Chinesta F et al (2016) About the origins of residual stresses in in situ consolidated thermoplastic composite rings. Int J Mater Form.  https://doi.org/10.1007/s12289-016-1319-2 Google Scholar
  2. 2.
    Dedieu C, Barasinski A, Chinesta F et al (2016) On the prediction of residual stresses in automated tape placement. Int J Mater Form.  https://doi.org/10.1007/s12289-016-1307-6
  3. 3.
    Lopez E, Abisset-Chavanne E, Lebel F et al (2016) Flow modeling of linear and nonlinear fluids in two and three scale fibrous fabrics. Int J Mater Form 9:215.  https://doi.org/10.1007/s12289-015-1224-0 CrossRefGoogle Scholar
  4. 4.
    Ammar A, Abisset-Chavanne E, Chinesta F et al (2016) Flow modelling of quasi-Newtonian fluids in two-scale fibrous fabrics. Int J Mater Form.  https://doi.org/10.1007/s12289-016-1300-0
  5. 5.
    Leon A, Barasinski A, Chinesta F (2017) Microstructural analysis of pre-impreganted tapes consolidation. Int J Mater Form 10:369.  https://doi.org/10.1007/s12289-016-1285-8 CrossRefGoogle Scholar
  6. 6.
    Lopez E, Abisset-Chavanne E, Lebel F et al (2016) Advanced thermal simulation of processes involving materials exhibiting fine-scale microstructures. Int J Mater Form 9:179.  https://doi.org/10.1007/s12289-015-1222-2 CrossRefGoogle Scholar
  7. 7.
    Cueto E, Chinesta F (2015) Meshless methods for the simulation of material forming. Int J Mater Form 8:25.  https://doi.org/10.1007/s12289-013-1142-y CrossRefGoogle Scholar
  8. 8.
    Khaled M, Ramadan M, Fourment L (2016) Thermal modeling of cogging process using finite element method. AIP Conf Proc 1769:060008CrossRefGoogle Scholar
  9. 9.
    Ramadan M, Khaled M, Fourment L (2016) Application of multi-grid method on the simulation of incremental forging processes. AIP Conf Proc 1769:130004CrossRefGoogle Scholar
  10. 10.
    Ramadan M, Fourment L, Perchat E (2008) A multi-levels method with two meshes for speeding-up incremental processes such as cogging. In: 9th international conference on technology of plasticity, ICTP, pp 1747–175Google Scholar
  11. 11.
    Guo RP, Xu L, Zong BY, Yang R (2016) Preparation and ring rolling processing of large size Ti-6Al-4V powder compact. Mater Des 99:341–348.  https://doi.org/10.1016/j.matdes.2016.02.128 CrossRefGoogle Scholar
  12. 12.
    Wang C, Geijselaers HJM, Omerspahic E, Recina V, van den Boogaard AH (2016) Influence of ring growth rate on damage development in hot ring rolling. J Mater Process Technol 227:268–280.  https://doi.org/10.1016/j.jmatprotec.2015.08.017 CrossRefGoogle Scholar
  13. 13.
    Ramadan M, Fourment L, Digonnet H (2014) Fast resolution of incremental forming processes by the multi-mesh method. Application to cogging. Int J Mater Form 7(2):207–219.  https://doi.org/10.1007/s12289-012-1121-8 CrossRefGoogle Scholar
  14. 14.
    Ramadan M, Fourment L, Digonnet H (2009) A parallel two mesh method for speeding-up processes with localized deformations: application to cogging. Int J Mater Form 2(1):581–584.  https://doi.org/10.1007/s12289-009-0440-x CrossRefGoogle Scholar
  15. 15.
    Khaled M, Ramadan M, Elmarakbi A, Fourment L (2015) Simulation of incremental forming processes using a thermo-mechanical partitioned algorithm. Key Eng Mater 651-653:1331–1336.  https://doi.org/10.4028/www.scientific.net/KEM.651-653.1331 CrossRefGoogle Scholar
  16. 16.
    Wasserman M, Mor-Yossef Y, Greenberg JB (2016) A positivity-preserving, implicit defect-correction multigrid method for turbulent combustion. J Comput Phys 316:303–337.  https://doi.org/10.1016/j.jcp.2016.04.005 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cools S, Vanroose W (2016) A fast and robust computational method for the ionization cross sections of the driven Schrödinger equation using an O(N) multigrid-based scheme. J Comput Phys 308:20–39MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bauer P, Klement V, Oberhuber T, Žabka V (2016) Implementation of the Vanka-type multigrid solver for the finite element approximation of the Navier–stokes equations on GPU. Comput Phys Commun 200:50–56.  https://doi.org/10.1016/j.cpc.2015.10.021 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Augustin CM, Neic A, Liebmann M, Prassl AJ, Niederer SA, Haase G, Plank G (2016) Anatomically accurate high resolution modeling of human whole heart electromechanics: a strongly scalable algebraic multigrid solver method for nonlinear deformation. J Comput Phys 305:622–646, ISSN 0021-9991.  https://doi.org/10.1016/j.jcp.2015.10.045 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bhowmik SK (2015) A multigrid preconditioned numerical scheme for a reaction–diffusion system. Appl Math Comput 254:266–276.  https://doi.org/10.1016/j.amc.2014.12.062 MathSciNetzbMATHGoogle Scholar
  21. 21.
    Bolten M, Huckle TK, Kravvaritis CD (2016) Sparse matrix approximations for multigrid methods. Linear Algebra Appl 502:58–76.  https://doi.org/10.1016/j.laa.2015.11.008 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rauwoens P, Troch P, Vierendeels J (2015) A geometric multigrid solver for the free-surface equation in environmental models featuring irregular coastlines. J Comput Appl Math 289:22–36.  https://doi.org/10.1016/j.cam.2015.03.029 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jiang Y, Xu X (2015) Multigrid methods for space fractional partial differential equations. J Comput Phys 302:374–392.  https://doi.org/10.1016/j.jcp.2015.08.052 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kang KS (2015) Scalable implementation of the parallel multigrid method on massively parallel computers. Comput Math Appl 70(11):2701–2708, ISSN 0898-1221.  https://doi.org/10.1016/j.camwa.2015.07.023 MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sviercoski RF, Popov P, Margenov S (2015) An analytical coarse grid operator applied to a multiscale multigrid method. J Comput Appl Math 287:207–219. ISSN 0377-0427.  https://doi.org/10.1016/j.cam.2015.03.001 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Santiago CD, Marchi CH, Souza LF (2015) Performance of geometric multigrid method for coupled two-dimensional systems in CFD. Appl Math Model 39(9):2602–2616.  https://doi.org/10.1016/j.apm.2014.10.067 MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ma W, Lu Z, Zhang J (2015) GPU parallelization of unstructured/hybrid grid ALE multigrid unsteady solver for moving body problems. Comput Fluids 110:122–135.  https://doi.org/10.1016/j.compfluid.2014.11.012 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gandham R, Esler K, Zhang Y (2014) A GPU accelerated aggregation algebraic multigrid method. Comp Math Appl 68(10):1151–1160.  https://doi.org/10.1016/j.camwa.2014.08.022 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rey B, Mocellin K, Fourment L (2008) A node-nested Galerkin multigrid method for metal forging simulation. Comput Vis Sci 11(1):17–25.  https://doi.org/10.1007/s00791-006-0054-5 MathSciNetCrossRefGoogle Scholar
  30. 30.
    Mocellin K, Fourment L, Coupez T, Chenot J-L (2001) Toward large scale FE computation of hot forging process using iterative solvers, parallel computation and multigrid algorithms. Int J Numer Methods Eng 52(5–6):473–488.  https://doi.org/10.1002/nme.304 CrossRefzbMATHGoogle Scholar
  31. 31.
    Feng YT, Peric D, Owen DRJ (1997) A non-nested Galerkin multi-grid method for solving linear and nonlinear solid mechanics problems. Comput Methods Appl Mech Eng 144(3-4):307–325.  https://doi.org/10.1016/S0045-7825(96)01183-8 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Arnold DN, Brezzi F, Fortin M (1984) A stable finite element for the stokes equations. Calcolo 21(4):337–344.  https://doi.org/10.1007/BF02576171 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Coupez T (1991) Grandes déformations et remaillage automatique. Ph.D. Thesis, Mines Paristech, FranceGoogle Scholar
  34. 34.
    Fourment L, Chenot JL, Mocellin K (1999) Numerical formulations and algorithms for solving contact problems in metal forming simulation. Int J Numer Methods Eng 46(9):1435–1462.  https://doi.org/10.1002/(SICI)1097-0207(19991130)46:9<1435::AID-NME707>3.0.CO;2-9 CrossRefzbMATHGoogle Scholar
  35. 35.
    Sameh AH, Kuck DJ (1977) Parallel direct linear system solvers - a survey. Math Comput Simul 19(4):272–277, ISSN 0378-4754.  https://doi.org/10.1016/0378-4754(77)90044-1 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Jain SK, Ray RK, Bhavsar A (2015) Iterative solvers for image denoising with diffusion models: a comparative study. Comput Math Appl 70(3):191–211, ISSN 0898-1221.  https://doi.org/10.1016/j.camwa.2015.04.009 MathSciNetCrossRefGoogle Scholar
  37. 37.
    Pearson JW (2016) Fast iterative solvers for large matrix systems arising from time-dependent stokes control problems. Appl Numer Math 108:87–101, ISSN 0168-9274.  https://doi.org/10.1016/j.apnum.2016.05.002 MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Benzi M, Golub G, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14:1–137.  https://doi.org/10.1017/S0962492904000212 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Borzacchiello D, Leriche E, Blottière B, Guillet J (2017) Box-relaxation based multigrid solvers for the variable viscosity stokes problem. Comput Fluids 156:515–525.  https://doi.org/10.1016/j.compfluid.2017.08.027 MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Burstedde C, Ghattas O, Stadler G, Tu T, Wilcox LC (2009) Parallel scalable adjoint-based adaptive solution of variable-viscosity stokes flow problems. Comput Methods Appl Mech Eng 198(21–26):1691–1700, ISSN 0045-7825.  https://doi.org/10.1016/j.cma.2008.12.015 CrossRefzbMATHGoogle Scholar
  41. 41.
    Elman H, Howle VE, Shadid J, Shuttleworth R, Tuminaro R (2006) Block preconditioners based on approximate commutators. SIAM J Sci Comput 27(5):1651–1668.  https://doi.org/10.1137/040608817 MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Oosterlee CW, Gaspar FJ (2008) Multigrid relaxation methods for systems of saddle point type. Appl Numer Math 58(12):1933–1950.  https://doi.org/10.1016/j.apnum.2007.11.014 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag France SAS, part of Springer Nature 2018

Authors and Affiliations

  • Mohamad Ramadan
    • 1
    • 2
    Email author
  • Mahmoud Khaled
    • 1
    • 3
  • Lionel Fourment
    • 4
  1. 1.Energy and Thermo-Fluid Group, School of EngineeringInternational University of BeirutBeirutLebanon
  2. 2.FCLAB, CNRS, Univ. Bourgogne Franche-ComtéBelfort cedexFrance
  3. 3.Univ Paris Diderot, Sorbonne Paris Cité, Paris Interdisciplinary Energy Research Institute (PIERI)ParisFrance
  4. 4.MINES ParisTech, PSL Research University, CEMEF, CNRS UMR 7635, CS 10207 rue Claude Daunesse06904 Sophia Antipolis CedexFrance

Personalised recommendations