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A decomposed subspace reduction for fracture mechanics based on the meshfree integrated singular basis function method

  • Qizhi He
  • Jiun-Shyan Chen
  • Camille Marodon
Original Paper
  • 59 Downloads

Abstract

In this work, we propose a new decomposed subspace reduction (DSR) method for reduced-order modeling of fracture mechanics based on the integrated singular basis function method (ISBFM) with reproducing kernel approximation enriched by crack-tip basis functions. It is shown that the standard MOR approach based on modal analysis (ISBFM-MA) with a direct employment of the crack-tip enrichment functions yields an inappropriate scaling effect to the stiffness matrix, and results in the loss of essential crack features and the erroneous representation of inhomogeneous Dirichlet boundary conditions in the reduced subspace. On the other hand, the solution of ISBFM-DSR is not affected by the arbitrary scaling of the enrichment functions, and it properly captures the singularity and discontinuity properties of fracture problems in its low-dimensional reduced-order approximation. It is also shown that the inhomogeneous boundary conditions can be accurately represented in the ISBFM-DSR solution. Validations are given in the numerical examples.

Keywords

Model order reduction Reproducing kernel approximation Meshfree methods Decomposed subspace reduction Fracture mechanics Integrated singular basis function method 

Notes

Acknowledgements

The support of this work by US Army Engineer Research and Development Center under contract W15QKN-12-9-1006 to UCSD is greatly acknowledged.

References

  1. 1.
    Antoulas AC, Sorensen DC (2001) Approximation of large-scale dynamic systems: an overview. Int J Appl Math Comput Sci 11:1093–1121MathSciNetzbMATHGoogle Scholar
  2. 2.
    Schilders WH, van der Vorst HA, Rommes J (2008) Model order reduction: theory, research aspects and applications. Springer, BerlinCrossRefzbMATHGoogle Scholar
  3. 3.
    Sirovich L (1987) Turbulence and the dynamics of coherent structures I: coherent structures. Q Appl Math 45:561–571MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Lieu T, Farhat C, Lesoinne M (2006) Reduced-order fluid/structure modeling of a complete aircraft configuration. Comput Methods Appl Mech Eng 195:5730–5742.  https://doi.org/10.1016/j.cma.2005.08.026 CrossRefzbMATHGoogle Scholar
  5. 5.
    Amsallem D, Farhat C (2008) Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J 46:1803–1813.  https://doi.org/10.2514/1.35374 CrossRefGoogle Scholar
  6. 6.
    Krysl P, Lall S, Marsden JE (2001) Dimensional model reduction in non-linear finite element dynamics of solids and structures. Int J Numer Methods Eng 51:479–504.  https://doi.org/10.1002/nme.167 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lall S, Krysl P, Marsden JE (2003) Structure-preserving model reduction for mechanical systems. Phys D Nonlinear Phenom 184:304–318.  https://doi.org/10.1016/S0167-2789(03)00227-6 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carlberg K, Farhat C (2011) A low-cost, goal-oriented “compact proper orthogonal decomposition” basis for model reduction of static systems. Int J Numer Methods Eng 86:381–402.  https://doi.org/10.1002/nme.3074 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Niroomandi S, Alfaro I, Cueto E, Chinesta F (2010) Model order reduction for hyperelastic materials. Int J Numer Methods Eng 81:1180–1206.  https://doi.org/10.1002/nme.2733 MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ryckelynck D, Benziane DM, Paristech M (2010) Multi-level a priori hyper reduction of mechanical models involving internal variables. Comput Methods Appl Mech Eng 199:1134–1142.  https://doi.org/10.1016/j.cma.2009.12.003 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Millán D, Arroyo M (2013) Nonlinear manifold learning for model reduction in finite elastodynamics. Comput Methods Appl Mech Eng 261:118–132.  https://doi.org/10.1016/j.cma.2013.04.007 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kerfriden P, Gosselet P, Adhikari S, Bordas S (2011) Bridging proper orthogonal decomposition methods and augmented Newton-Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems. Comput Methods Appl Mech Eng 200:850–866.  https://doi.org/10.1016/j.cma.2010.10.009 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Galland F, Gravouil A, Malvesin E, Rochette M (2011) A global model reduction approach for 3D fatigue crack growth with confined plasticity. Comput Methods Appl Mech Eng 200:699–716.  https://doi.org/10.1016/j.cma.2010.08.018 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Everson R, Sirovich L (1995) Karhunen-Loeve procedure for gappy data. J Opt Soc Am A-Optics Image Sci Vis 12:1657–1664CrossRefGoogle Scholar
  15. 15.
    Ryckelynck D (2005) A priori hyperreduction method: an adaptive approach. J Comput Phys 202:346–366.  https://doi.org/10.1016/j.jcp.2004.07.015 CrossRefzbMATHGoogle Scholar
  16. 16.
    Ryckelynck D, Chinesta F, Cueto E, Ammar A (2006) On the a priori model reduction: overview and recent developments. Arch Comput Methods Eng 13:91–128.  https://doi.org/10.1007/BF02905932 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Craig RR (1981) Structural dynamics: an introduction to computer methods. Wiley, New YorkGoogle Scholar
  18. 18.
    Dickens JM, Nakagawa JM, Wittbrodt MJ (1997) A critique of mode acceleration and modal truncation augmentation methods for modal response analysis. Comput Struct 62:985–998CrossRefzbMATHGoogle Scholar
  19. 19.
    Chinesta F, Ammar A, Cueto E (2010) Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models. Arch Comput Methods Eng 17:327–350.  https://doi.org/10.1007/s11831-010-9049-y MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kerfriden P, Goury O, Rabczuk T, Bordas SPA (2013) A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics. Comput Methods Appl Mech Eng 256:169–188.  https://doi.org/10.1016/j.cma.2012.12.004 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rixen DJ (2004) A dual Craig-Bampton method for dynamic substructuring. J Comput Appl Math 168:383–391.  https://doi.org/10.1016/j.cam.2003.12.014 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kim T, James DL (2012) Physics-based character skinning using multidomain subspace deformations. IEEE Trans Vis Comput Graph 18:1228–1240CrossRefGoogle Scholar
  23. 23.
    Barbič J, Zhao Y (2011) Real-time large-deformation substructuring. ACM Trans Graph 30:91.  https://doi.org/10.1145/1964921.1964986 Google Scholar
  24. 24.
    Kerfriden P, Passieux JC, Bordas SPA (2012) Local/global model order reduction strategy for the simulation of quasi-brittle fracture. Int J Numer Methods Eng 89:154–179.  https://doi.org/10.1002/nme.3234 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Niroomandi S, Alfaro I, González D et al (2012) Real-time simulation of surgery by reduced-order modeling and X-FEM techniques. Int J Numer Method Biomed Eng 28:574–588.  https://doi.org/10.1002/cnm.1491 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Chaturantabut S, Sorensen DC (2010) Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput 32:2737–2764.  https://doi.org/10.1137/090766498 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Akbari Rahimabadi A, Kerfriden P, Bordas S (2015) Scale selection in nonlinear fracture mechanics of heterogeneous materials. Philos Mag 95:3328–3347.  https://doi.org/10.1080/14786435.2015.1061716 CrossRefGoogle Scholar
  28. 28.
    Oliver J, Caicedo M, Huespe AE et al (2017) Reduced order modeling strategies for computational multiscale fracture. Comput Methods Appl Mech Eng 313:560–595.  https://doi.org/10.1016/j.cma.2016.09.039 MathSciNetCrossRefGoogle Scholar
  29. 29.
    Olson LG, Georgiou GC, Schultz WW (1991) An efficient finite element method for treating singularities in Laplace’s equation. J Comput Phys 96:391–410.  https://doi.org/10.1016/0021-9991(91)90242-D MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Georgiou GC, Olson L, Smyrlis YS (1996) A singular function boundary integral method for the Laplace equation. Commun Numer Methods Eng 12:127–134CrossRefzbMATHGoogle Scholar
  31. 31.
    Chen JS, Marodon C, Hu HY (2015) Model order reduction for Meshfree solution of Poisson singularity problems. Int J Numer Methods Eng 102:1211–1237.  https://doi.org/10.1002/nme.4743 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nitsche J (1971) Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Math Semin der Univ Hambg 36:9–15.  https://doi.org/10.1007/BF02995904 CrossRefzbMATHGoogle Scholar
  33. 33.
    Fernández-Méndez S, Huerta A (2004) Imposing essential boundary conditions in mesh-free methods. Comput Methods Appl Mech Eng 193:1257–1275.  https://doi.org/10.1016/j.cma.2003.12.019 MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20:1081–1106.  https://doi.org/10.1002/fld.1650200824 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Chen J-S, Pan C, Wu C-T, Liu WK (1996) Reproducing Kernel particle methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 139:195–227.  https://doi.org/10.1016/S0045-7825(96)01083-3 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Belytschko TB, Krongauz Y, Organ D et al (1996) Meshless methods: an overview and recent developments. Comput Method Appl Mech Eng 139:3–47.  https://doi.org/10.1016/S0045-7825(96)01078-X CrossRefzbMATHGoogle Scholar
  37. 37.
    Williams ML (1952) Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J Appl Mech 19:526–528Google Scholar
  38. 38.
    Williams ML (1957) On the stress state at the base of a stationary crack. J Appl Mech 24:109–114.  https://doi.org/10.1115/1.3640470 MathSciNetzbMATHGoogle Scholar
  39. 39.
    Shabana AA (1991) Theory of vibration, vol. 2. I, Mechanical Engineering Series. Springer, New YorkGoogle Scholar
  40. 40.
    Barrault M, Maday Y, Nguyen NC, Patera AT (2004) An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Math 339:667–672.  https://doi.org/10.1016/j.crma.2004.08.006 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    England AH (2003) Complex variable methods in elasticity. Courier CorporationGoogle Scholar
  42. 42.
    Gunzburger MD, Peterson JS, Shadid JN (2007) Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Comput Methods Appl Mech Eng 196:1030–1047.  https://doi.org/10.1016/j.cma.2006.08.004 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Cosimo A, Cardona A, Idelsohn S (2016) General treatment of essential boundary conditions in reduced order models for non-linear problems. Adv Model Simul Eng Sci 3:7CrossRefGoogle Scholar
  44. 44.
    Li Z-C, Mathon R, Sermer P (1987) Boundary methods for solving elliptic problems with singularities and interfaces. SIAM J Numer Anal 24:487–498MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Yau JF, Wang SS, Corten HT (1980) A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. J Appl Mech 47:335.  https://doi.org/10.1115/1.3153665 CrossRefzbMATHGoogle Scholar
  46. 46.
    Shih CF, Asaro RJ (1988) Elastic-plastic analysis of cracks on bimaterial interfaces: Part I—small scale yielding. J Appl Mech 55:299.  https://doi.org/10.1115/1.3173676 CrossRefGoogle Scholar
  47. 47.
    Passieux JC, Rethore J, Gravouil A, Baietto MC (2013) Local/global non-intrusive crack propagation simulation using a multigrid X-FEM solver. Comput Mech 52:1381–1393.  https://doi.org/10.1007/s00466-013-0882-3 CrossRefzbMATHGoogle Scholar
  48. 48.
    Radermacher A, Reese S (2014) Model reduction in elastoplasticity: proper orthogonal decomposition combined with adaptive sub-structuring. Comput Mech 54:677–687MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Structural EngineeringUniversity of California, San DiegoLa JollaUSA
  2. 2.OptimaRH ConsultingParisFrance

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