Abonnement à la biblothèque: Guest
Portail numérique Bibliothèque numérique eBooks Revues Références et comptes rendus Collections
International Journal for Multiscale Computational Engineering
Facteur d'impact: 1.016 Facteur d'impact sur 5 ans: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Imprimer: 1543-1649
ISSN En ligne: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2013005939
pages 253-287


Pierre Kerfriden
Institute of Mechanics and Advanced Materials, Theoretical and Computational Mechanics, Cardiff University, Cardiff, CF24 3AA, United Kingdom
K. M. Schmidt
Cardiff University, School of Mathematics, Senghennydd Road, Cardiff CF24 4AG, Wales, United Kingdom
Timon Rabczuk
Institute of Structural Mechanics, Bauhaus-Universitat Weimar, Marienstr. 15, D-99423 Weimar, Germany
S. P. A. Bordas
Institute of Mechanics and Advanced Materials, Theoretical and Computational Mechanics, Cardiff University, Cardiff, CF24 3AA, United Kingdom


We propose to identify process zones in heterogeneous materials by tailored statistical tools. The process zone is redefined as the part of the structure where the random process cannot be correctly approximated in a low-dimensional deterministic space. Such a low-dimensional space is obtained by a spectral analysis performed on precomputed solution samples. A greedy algorithm is proposed to identify both process zone and low-dimensional representative subspace for the solution in the complementary region. In addition to the novelty of the tools proposed in this paper for the analysis of localized phenomena, we show that the reduced space generated by the method is a valid basis for the construction of a reduced-order model.


  1. Abdi, H. and Williams, L. J., Principal component analysis. DOI: 10.1002/wics.101

  2. Allix, O., Lévêque, D., and Perret, L., Identification and forecast of delamination in composite laminates by an interlaminar interface model. DOI: 10.1016/S0266-3538(97)00144-9

  3. Allix, O., Kerfriden, P., and Gosselet, P., A relocalization technique for the multiscale computation of delamination in composite structures.

  4. Ammar, A., Chinesta, F., and Cueto, E., Coupling finite elements and proper generalized decompositions. DOI: 10.1615/IntJMultCompEng.v9.i1.30

  5. Antoulas, C. and Sorensen, D. C., Approximation of large-scale dynamical systems: An overview.

  6. Astrid, P., Weiland, S., Willcox, K., and Backx, A. C. P. M., Missing point estimation in models described by proper orthogonal decomposition. DOI: 10.1109/CDC.2004.1430301

  7. Barbone, P. E., Givoli, D., and Patlashenko, I., Optimal modal reduction of vibrating substructures. DOI: 10.1002/nme.680

  8. Barrault, M., Maday, Y., Nguyen, N. C., and Patera, A. T., An “empirical interpolation” method: Application to efficient reduced-basis discretization of partial differential equations. DOI: 10.1016/j.crma.2004.08.006

  9. Belytschko, T., Loehnert, S., and Song, J.-H., Multiscale aggregating discontinuities: A method for circumventing loss of material stability. DOI: 10.1002/nme.2156

  10. Bradley, E. and Gong, G., A leisurely look at the bootstrap, the jackknife, and cross-validation. DOI: 10.2307/2685844

  11. Buffoni, M., Telib, H., and Iollo, A., Iterative methods for model reduction by domain decomposition. DOI: 10.1016/j.compfluid.2008.11.008

  12. Cangelosi, R. and Goriely, A., Component retention in principal component analysis with application to cdna microarray data. DOI: 10.1186/1745-6150-2-2

  13. Carlberg, K., Bou-Mosleh, C., and Farhat, C., Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations. DOI: 10.1002/nme.3050

  14. Chaboche, J.-L., Continuum damage mechanics: Part I - General concepts; Part II - Damage growth, crack initiation, and crack growth. DOI: 10.1115/1.3173661

  15. Chevreuil, M. and Nouy, A., Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics. DOI: 10.1002/nme.3249

  16. Coenen, E. W. C., Kouznetsova, V. G., and Geers, M. G. D., Novel boundary conditions for strain localization analyses in microstructural volume elements. DOI: 10.1002/nme.3298

  17. Filzmoser, P., Liebmann, B., and Varmuza, K., Repeated double cross validation. DOI: 10.1002/cem.1225

  18. Grassl, P. and Bazant, Z. P., Random Lattice particle simulation of statistical size effect in quasibrittle structures failing at crack initiation. DOI: 10.1061/(ASCE)0733-9399(2009)135:2(85)

  19. Haryadi, S. G., Kapania, R. K., and Haryadi, S. G., Global/local analysis of composite plates with cracks. DOI: 10.1016/S1359-8368(97)00034-6

  20. Hotelling, H., Analysis of a complex of statistical variables into principal components. DOI: 10.1037/h0071325

  21. Huttenlocher, D. P., Klanderman, G. A., Kl, G. A., and Rucklidge, W. J., Comparing images using the Hausdorff distance. DOI: 10.1109/34.232073

  22. Jackson, D. A., Stopping rules in principal components analysis: A comparison of heuristical and statistical approaches. DOI: 10.2307/1939574

  23. Karihaloo, B. L., Shao, P. F., and Xiao, Q. Z., Lattice modelling of the failure of particle composites. DOI: 10.1016/S0013-7944(03)00004-3

  24. Kerfriden, P., Passieux, J. C., and Bordas, S., Local/global model order reduction strategy for the simulation of quasi-brittle fracture. DOI: 10.1002/nme.3234

  25. Kerfriden, P., Gosselet, P., Adhikari, S., and Bordas, S., Bridging proper orthogonal decomposition methods and augmented Newton-Krylov algorithms: An adaptive model order reduction for highly nonlinear mechanical problems. DOI: 10.1016/j.cma.2010.10.009

  26. Kerfriden, P., Goury, O., Rabczuk, T., and Bordas, S. P. A., A partitioned model order reduction approach to rationalise computational expenses in multiscale fracture mechanics. DOI: 10.1016/j.cma.2012.12.004

  27. Koutsouleris, N., Gaser, C., Bottlender, R., Davatzikos, C., Deckerm, P., Jäger, M., Schmitt, G., Reiser, M., Möller, H. J., and Meisenzahl, E. M., Use of neuroanatomical pattern regression to predict the structural brain dynamics of vulnerability and transition to psychosis. DOI: 10.1016/j.schres.2010.08.032

  28. Krzanowski, W. J., Cross-validation in principal component analysis. DOI: 10.1007/BF02511446

  29. Kunisch, K. and Volkwein, S., Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. DOI: 10.1137/S0036142900382612

  30. Ladevèze, P. and Lubineau, G., An enhanced mesomodel for laminates based on micromechanics. DOI: 10.1016/S0266-3538(01)00145-2

  31. Ladevèze, P., Passieux, J. C., and N&eacuteron, D., The Latin multiscale computational method and the proper generalized decomposition. DOI: 10.1016/j.cma.2009.06.023

  32. LeGresley, P. A. and Alonso, J. J., Dynamic domain decomposition and error correction for reduced order models. DOI: 10.2514/6.2003-250

  33. Liang, Y. C., Lee, H. P., Lim, S. P., Lin, W. Z., Lee, K. H., and Wu, C. G., Proper orthogonal decomposition and its applications - Part I: Theory. DOI: 10.1006/jsvi.2001.4041

  34. Lilliu, G. and van Mier, J. G. M., 3D Lattice type fracture model for concrete. DOI: 10.1016/S0013-7944(02)00158-3

  35. Lorentz, E. and Badel, P., A new path-following constraint for strain-softening finite element simulations. DOI: 10.1002/nme.971

  36. Massart, T. J., Peerlings, R. H. J., and Geers, M. G. D., An enhanced multi-scale approach for masonry wall computations with localization of damage. DOI: 10.1002/nme.1799

  37. Meyer, M. and Matthies, H. G., Efficient model reduction in non-linear dynamics using the Karhunen-Loeve expansion and dual-weighted-residual methods. DOI: 10.1007/s00466-002-0404-1

  38. Nguyen, V. P., Lloberas-Valls, O., Stroeven, M., and Sluys, L. J., Computational homogenization for multiscale crack modeling: Implementational and computational aspects. DOI: 10.1002/nme.3237

  39. Pearson, K., On lines and planes of closest fit to systems of points in space. DOI: 10.1080/14786440109462720

  40. Pinho, S. T., Iannucci, L., and Robinson, P., Physically-based failure models and criteria for laminated fibre-reinforced composites with emphasis on fibre kinking: Part I: Development. DOI: 10.1016/j.compositesa.2005.04.016

  41. Rabczuk, T., Kim, J. Y., Samaniego, E., and Belytschko, T., Homogenization of sandwich structures. DOI: 10.1002/nme.1100

  42. Ravindran, S. S., Reduced-order adaptive controllers for fluid flows using pod. DOI: 10.1023/A:1011184714898

  43. Rewienski, M. and White, J., A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. DOI: 10.1109/TCAD.2002.806601

  44. Rickelt, C. and Reese, S., A simulation strategy for life time calculations of large, partially damaged structures. DOI: 10.1007/1-4020-5370-3_605

  45. Ryckelynck, D., A priori hyperreduction method: An adaptive approach. DOI: 10.1016/j.jcp.2004.07.015

  46. Ryckelynck, D., Hyper-reduction of mechanical models involving internal variables. DOI: 10.1002/nme.2406

  47. Sanchez-Palencia, E., Non-homogeneous media and vibration theory. DOI: 10.1007/3-540-10000-8

  48. Scrivener, K. L., Crumbie, A. K., and Laugesen, P., The interfacial transition zone (ITZ) between cement paste and aggregate in concrete. DOI: 10.1023/B:INTS.0000042339.92990.4c

  49. Sirovich, L., Turbulence and the dynamics of coherent structures. Part I: Coherent structures.

  50. Stone, M., Cross-validatory choice and assessment of statistical predictions. DOI: 10.2307/2984809

  51. Suquet, P., Elements of homogenization for inelastic solid mechanics. DOI: 10.1007/3-540-17616-0_15

  52. van Mier, J. G. M., van Vliet, M. R. A., and Wang, T. K., Fracture mechanisms in particle composites: Statistical aspects in lattice type analysi. DOI: 10.1016/S0167-6636(02)00170-9

  53. Walraven, J. C., Aggregate Interlock: A Theoretical and Experimental Analysis.

  54. Wold, S., Cross-validatory estimation of the number of components in factor and principal components models. DOI: 10.1080/00401706.1978.10489693

Articles with similar content:

International Journal for Uncertainty Quantification, Vol.3, 2013, issue 6
Anas Batou, Christian Soize
International Journal for Multiscale Computational Engineering, Vol.14, 2016, issue 6
Wing Tat Leung, Maria Vasilyeva, Eric T. Chung, Yalchin Efendiev
Mathematical Simulation of the Admixture Transfer by Turbulent Boundary Layer
International Journal of Fluid Mechanics Research, Vol.29, 2002, issue 6
Eugene Alexeyevich Shkvar
Computer Simulation Based on Non-local Model of the Dynamics of Convective Diffusion of Soluble Substances in the Underground Filtration Flow under Mass Exchange Conditions
Journal of Automation and Information Sciences, Vol.51, 2019, issue 5
Vsevolod A. Bohaienko, Vladimir M. Bulavatskiy
Determination of Dimension of Embedding for Chaotic Attractors
Journal of Automation and Information Sciences, Vol.39, 2007, issue 10
Victor G. Gorodetskyi