Tensor-based methods for numerical homogenization from high-resolution images

Computer Methods in Applied Mechanics and Engineering

Volumn 254, Issue , 2013, Pages 154-169

  Giraldi, L ^a   Nouy, A ^a   Legrain, G ^a   Cartraud, P ^a  

^a INSTITUT DE RECHERCHE EN GÉNIE CIVIL ET MÉCANIQUE GEM   (France)

Author keywords

Goal oriented error estimation; Image based computing; Model reduction; Numerical homogenization; Proper generalized decomposition (PGD); Tensor methods

Indexed keywords

GOAL-ORIENTED ERROR ESTIMATION; IMAGE-BASED; MODEL REDUCTION; NUMERICAL HOMOGENIZATION; PROPER GENERALIZED DECOMPOSITIONS; TENSOR METHOD;

NUMERICAL METHODS;

TENSORS;

EID: 84870706109     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2012.10.012     Document Type: Article

References (57)

1. Ainsworth M., Oden J.T. A Posteriori Error Estimation in Finite Element Analysis 2000, Wiley-Interscience. 1st ed.
2. Ammar A., Chinesta F., Diez P., Huerta A. An error estimator for separated representations of highly multidimensional models. Comput. Methods Appl. Mech. Engrg. 2010, 199:1872-1880.
3. Babin P., Valle G., Dendievel R., Lassoued N., Salvo L. Mechanical properties of bread crumbs from tomography based finite element simulations. J. Mater. Sci. 2005, 40:5867-5873.
4. B. Bader, T. Kolda, Matlab tensor toolbox version 2.4, 2011. http://csmr.ca.sandia.gov/tgkolda/TensorToolbox/.
5. Bensoussan A., Lions J., Papanicolaou G. Asymptotic Analysis for Periodic Structures 1978, North-Holland.
6. Camacho D.L.A., Hopper R.H., Lin G.M., Myers B.S. An improved method for finite element mesh generation of geometrically complex structures with application to the skullbase. J. Biomech. 1997, 30:1067-1070.
7. Cances E., Ehrlacher V., Lelievre T. Convergence of a greedy algorithm for high-dimensional convex nonlinear problems. Math. Models Methods Appl. Sci. 2011, 21:2433-2467.
8. Carroll J.D., Chang J. Analysis of individual differences in multidimensional scaling via an n-way generalization of Eckart-Young decomposition. Psychometrika 1970, 35:283-319.
9. Charras G.T., Guldberg R.E. Improving the local solution accuracy of large-scale digital image-based finite element analyses. J. Biomech. 2000, 33:255-259.
10. Chinesta F., Ammar A., Cueto E. Recent advances in the use of the proper generalized decomposition for solving multidimensional models. Arch. Comput. Methods Engrg. 2010, 17:327-350.
11. Chinesta F., Ammar A., Lemarchand F., Beauchene P., Boust F. Alleviating mesh constraints: model reduction, parallel time integration and high resolution homogenization. Comput. Methods Appl. Mech. Engrg. 2008, 197:400-413.
12. Chinesta F., Ladeveze P., Cueto E. A short review on model order reduction based on proper generalized decomposition. Arch. Comput. Methods Engrg. 2011, 18:395-404.
13. Duster A., Parvizian J., Yang Z., Rank E. The finite cell method for three-dimensional problems of solid mechanics. Comput. Methods Appl. Mech. Engrg. 2008, 197:3768-3782.
14. Falcó A., Hackbusch W. On minimal subspaces in tensor representations. Found. Comput. Math. 2012, 12:765-803.
15. Falco A., Nouy A. A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach. J. Math. Anal. Appl. 2011, 376:469-480.
16. Falcó A., Nouy A. Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces. Numer. Math. 2012, 10.1007/s00211-011-0437-5.
17. Golanski D., Terada K., Kikuchi N. Macro and micro scale modeling of thermal residual stresses in metal matrix composite surface layers by the homogenization method. Comput. Mech. 1997, 19:188-202.
18. Hackbusch W. Tensor Spaces and Numerical Tensor Calculus 2012, Springer.
19. Hackbusch W., Kühn S. A new scheme for the tensor representation. J. Fourier Anal. Appl. 2009, 15:706-722.
20. Harshman R.A. Foundations of the PARAFAC procedure: model and conditions for an explanatory multi-mode factor analysis. UCLA Working Papers Phonet. 1970, 16:1-84.
21. Hollister S., Kikuchi N. Homogenization theory and digital imaging: a basis for studying the mechanics and design principles of bone tissue. Biotech. Bioengrg. 1994, 43:586-596.
22. Jiang M., Jasiuk I., Ostoja-Starzewski M. Apparent thermal conductivity of periodic two-dimensional composites. Comput. Mater. Sci. 2002, 25:329-338.
23. Kanit T., Forest S., Galliet I., Mounoury V., Jeulin D. Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 2003, 40:3647-3679.
24. Kanit T., N'Guyen F., Forest S., Jeulin D., Reed M., Singleton S. Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry. Comput. Methods Appl. Mech. Engrg. 2006, 195:3960-3982.
25. Keyak J., Meagher J., Skinner H., Mote C. Automated three-dimensional finite element modelling of bone: a new method. J. Biomed. Engrg. 1990, 12:389-397.
26. Khoromskij B. Tensor-structured preconditioners and approximate inverse of elliptic operators in r̂d. Construct. Approximat. 2009, 30:599-620.
27. Kolda T.G., Bader B.W. Tensor decompositions and applications. SIAM Rev. 2009, 51:455.
28. Ladevèze P., Passieux J., Néron D. The LATIN multiscale computational method and the proper generalized decomposition. Comput. Methods Appl. Mech. Engrg. 2010, 199:1287-1296.
29. Ladevèze P., Chamoin L. On the verification of model reduction methods based on the proper generalized decomposition. Comput. Methods Appl. Mech. Engrg. 2011, 200:2032-2047.
30. Lamari H., Ammar A., Cartraud P., Chinesta F., Jacquemin F., Legrain G. Recent advances in material homogenization. Int. J. Mater. Form. 2010, 3:899-902.
31. Lamari H., Ammar A., Cartraud P., Legrain G., Chinesta F., Jacquemin F. Routes for efficient computational homogenization of nonlinear materials using the proper generalized decompositions. Arch. Comput. Methods Engrg. 2010, 17:373-391.
32. Langville A.N., Stewart W.J. A kronecker product approximate preconditioner for SANs. Numer. Linear Algebr. Appl. 2004, 11:723-752.
33. Laschet G. Homogenization of the thermal properties of transpiration cooled multi-layer plates. Comput. Methods Appl. Mech. Engrg. 2002, 191:4535-4554.
34. Lathauwer L., Moor B., Vandewalle J. A multilinear singular value decomposition. SIAM J. Matrix. Anal. Appl. 2000, 21:1253-1278.
35. Legrain G., Allais R., Cartraud P. On the use of the extended finite element method with quatree/octree meshes. Int. J. Numer. Methods Engrg. 2011, 86:717-743.
36. Legrain G., Cartraud P., Perreard I., Moës N. An x-fem and level set computational approach for image-based modelling: application to homogenization. Int. J. Numer. Methods Engrg. 2011, 86:915-934.
37. Legrain G., Chevaugeon N., Dréau K. High order x-fem and levelsets for complex microstructures: Uncoupling geometry and approximation. Comput. Methods Appl. Mech. Engrg. 2012, 241-244:172-189.
38. Lewis A., Geltmacher A. Image-based modeling of the response of experimental 3d microstructures to mechanical loading. Scripta Mater. 2006, 55:81-85.
39. W. Lian, Contribution à l'Homogénéisation Numérique du Comportement Elastique de Matériaux à Microstructure Complexe Caractérisés par Imagerie. Ph.D. thesis. Ecole Centrale Nantes. Nantes, 2011.
40. W.D. Lian, G. Legrain, P. Cartraud, Image-based computational homogenization and localization: comparison between X-FEM/levelset and voxel-based approaches. Comput. Mech. (2012), http://dx.doi.org/10.1007/s00466-012-0723-9.
41. Madi K., Forest S., Boussuge M., Gailliégue S., Lataste E., Buffiére J.Y., Bernard D., Jeulin D. Finite element simulations of the deformation of fused-cast refractories based on X-ray computed tomography. Comput. Mater. Sci. 2007, 39:224-229.
42. Maire E., Fazekas A., Salvo L., Dendievel R., Youssef S., Cloetens P., Letang J.M. X-ray tomography applied to the characterization of cellular materials. related finite element modeling problems. Compos. Sci. Tech. 2003, 63:2431-2443.
43. Mishnaevsky L.L. Automatic voxel-based generation of 3d microstructural fe models and its application to the damage analysis of composites. Mater. Sci. Engrg. A 2005, 407:11-23.
44. Moulinec H., Suquet P. A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput. Methods Appl. Mech. Engrg. 1998, 157:69-94.
45. Nouy A. A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations. Comput. Methods Appl. Mech. Engrg. 2010, 199:1603-1626.
46. Nouy A. Proper generalized decompositions and separated representations for the numerical solution of high dimensional stochastic problems. Arch. Comput. Methods Engrg. 2010, 17:403-434.
47. Oseledets I.V. Tensor-train decomposition. SIAM J. Sci. Comput. 2011, 33:2295.
48. Ostoja-Starzewski M. Material spatial randomness: from statistical to representative volume element. Probabil. Engrg. Mech. 2006, 21:112-132.
49. Ozdemir I., Brekelmans W., Geers M. Computational homogenization for heat conduction in heterogeneous solids. Int. J. Numer. Methods Engrg. 2008, 73:185-204.
50. Sanchez-Palencia E. Non homogeneous media and vibration theory. Lecture Notes in Physics 1980, 127. Springer-Verlag, Berlin.
51. Sethian J. Level set methods: evolving interfaces in geometry, fluid mechanics, computer vision and materials sciences. Cambridge Monographs on Applied and Computational Mathematics 1999, Cambridge University Press, vol. 3. 2nd edition.
52. de Silva V., Lim L. Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 2008, 30:1084.
53. Takano N., Zako M., Kubo F., Kimura K. Microstructure-based stress analysis and evaluation for porous ceramics by homogenization method with digital image-based modeling. Int. J. Solids Struct. 2003, 40:1225-1242.
54. Terada K., Asai M., Yamagishi M. Finite cover method for linear and non-linear analyses of heterogeneous solids. Int. J. Numer. Methods Engrg. 2003, 58:1321-1346.
55. Touzene A. A tensor sum preconditioner for stochastic automata networks. INFORMS J. Comput. 2008, 20:234-242.
56. Ulrich D., van Rietbergen B., Weinans H., Rüegsegger P. Finite element analysis of trabecular bone structure: a comparison of image-based meshing techniques. J. Biomech. 1998, 31:1187-1192.
57. Young P., Beresford-West T., Coward S., Notarberardino B., Walker B., Abdul-Aziz A. An efficient approach to converting three-dimensional image data into highly accurate computational models. Phil. Trans. R. Soc. A: Math. Phys. Engrg. Sci. 2008, 366:3155-3173.